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continuation on page 723

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

540

Categorical Topology Proceedings of the Conference Held at Mannheim, 21-25 July, 1975

Edited by E. Binz and H. Herrlich

Springer-Verlag Berlin. Heidelberg. New York 1976

Editors Ernst Binz Universit~t Mannheim (WH) Lehrstuhl fQr Mathematik I Schlo6 6800 Mannheim/BRD

Horst Herrlich Universit~t Bremen Fachsektion Mathematik AchterstraBe 2800 Bremen/BRD

AMS Subject Classifications (1970): 18AXX, 18BXX, 18CXX, 18DXX, 54AXX, 54BXY,, 54CXX, 54DXX, 54EXX, 54FXX, 54GXX, 54HXX, 46 EXX, 46 HXX, 57 DXX, 57 EXX, 58C15, 58C20, 58D99.

ISBN 3-540-07859-2 Springer-Verlag Berlin �9 Heidelberg �9 New York ISBN 0-387-07859-2 Springer-Verlag New York �9 Heidelberg �9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re- printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. �9 by Springer-Verlag Berlin �9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.

FOREWORD

This volume consists of the proceedings of the Conference on

Categorical Topology held at Mannheim from the 21st to the 25th of

July, 1975.

Financial support for the conference was provided by the Volks-

wagen-Stiftung, Hannover. The participants would like to express their

thanks to them and to the many others whose assistance was invaluable:

To the Rector of the University of Mannheim, Professor. Dr.E.Gaugler,

for his hospitality, to the University Administration for handling

much administrative work and to Mrs.K.Bischoff for her help in orga-

nizing the conference and for the typing of some of the papers appear-

ing here.

CONTENTS

The role of nearness spaces in topology

by H.L.Bentley .....................................

Un th~or~me d'inversion locale

by F.Berquier ...................................... 23

Charaktergruppen von Gruppen von Si-wertigen stetigen Funktionen

by E.Binz .......................................... 43

Some cartesian closed topological categories of convergence spaces

by G.Bourdaud ...................................... 93

Topological functors and structure functors

by G.C.L.Br~mmer ................................... ~09

An external characterization of topological functors

by G.C.L.Br~mmer and R.-E.Hoffmann ................. ~36

Homotopy and Kan extensions

~52 by A.Calder and J.Siegel ...........................

Tensor products of functors on categories of Banach spaces

by J.Cigler ........................................

Duality of compactological and locally compact groups

by J.B.Cooper and P.Michor ......................... ~8

Products and sums in the category of frames

by C.H.Dowker and Dona Strauss ..................... 208

Categorical methods in dimension theory 22C by R.Dyckhoff ......................................

Envelopes on the category of Kakutani-M-spaces 243 by J.Flachsmeyer ...................................

Vl

Contents

Compactly generated spaces and duality

by A.Fr~licher ................................... 254

Some topological theorems which fail to be true

by H.Herrlich .................................... 265

Topological functors admitting generalized Cauchy-completions

by R~E.Hoffmann .................................. 286

An external characterization of topological functors

by R.-E.Hoffmann and G.C.L.Br~mmer ............... ~36

Category theoretical methods in topological algebra

by K.H.Hofmann ................................... 345

Lattices of reflections and coref~ections in continuous structures

by M.Husek ....................................... 404

Pro-categories and shape theory v..

by S.Mardeslc .................................... 425

A note on the inverse mapping theorem of F.Berquier

by P.Michor ...................................... 435

Duality of compactological and locally compact groups

by P.Michor and J.B.Cooper ....................... 188

Cartesian closed topological categories

by L.D.Nel ....................................... 439

Epireflective categories of Hausdorff spaces

by P.Nyikos ...................................... 452

Categorical problems in minimal spaces

by J.R.Porter ................. ................... 482

Some outstanding problems in topology and the V-process

by M.Rajagopalan ................................. 504

VII

Contents

Nearness and metrization

by H.-Chr.Reichel .................................. 548

Reflective subcategories and closure operators

by S.Salbany ....................................... ~8

Compactness theorems

by M.Schroder ...................................... 566

Differential calculus and cartesian closedness

by U.Seip .......................................... 578

Homotopy and Kan extensions

by J.Siegel and A.Calder ........................... 452

Products and sums in the category of frames

by Dona Strauss and C.H.Dowker ............ ~ ........ 208

Perfect sources

by G.E.Strecker .................................... 605

Espaces fonctionnels et structures syntopog$nes

by D.Tanr~ ......................................... 625

Filters and uniformities in general categories

by S.J.R.Vorster ................................... 635

Categories of topological transformation groups

by J.de Vries ...................................... 65 Z!

On monoidal closed topological categories I

by M.B.Wischnewsky ................................. 676

On topological algebras relative to full and faithful dense functors

by M.B.Wischnewsky ................................. 688

Are there topoi in topology?

by O.Wyler ......................................... 699

Address list of authors and speakers:

H.L.Bentley

F.Berquier

E.Binz

M.G.Bourdaud

G.C.L.Br~mmer

A.Calder

J.Cigler

J.B.Cooper

C.H.Dowker

R.Dyckhoff

J.Flachsmeyer

A.FrSlicher

H.Herrlich

R~E.Hoffmann

The University of Toledo, Dept. of Math. 2801 W.Bancroft Street Toledo, Ohio 43606, USA

B 41 Toison d'Or Centre du G@n@ral de Gaulle 59200 Tourcoing, France

Universit~t Mannheim, Lehrstuhl f.Math.l 68 Mannheim, A5, BRD

Universit@ de Paris VII U.E.R. de Math@matiques Tour 45-55 5 me Etage 2, Place Jussieu 75005 Paris, France

University of Cape Town, Dept. of Math. Private Bag Rondebosch, Rep.of South Africa

University of Missouri,Dept. of Math. Sc. St. Louis, Missouri 63121, USA

Mathematisches Institut der Universit~t Strudlhofgasse 4 1090 Wien, ~sterreich

Mathematisches Institut der Universit~t Linz/Donau, ~sterreich

Birkbeck College, Math.Dept. London WCIE 7HX., England

University of St.Andrews, Math.lnstitute North Haugh St.Andrews, KY16 9SS

Ernst-Moritz-Arndt-Universit~t Sektion Mathematik Ludwig-Jahn-Str.15a 22 Greifswald, DDR

Facult~ des Sciences Section de Math@matiques Universit@ de Gen@ve 2-4, rue du Li@vre 1211 Gen@ve 24, Suisse

Universit~t Bremen, Fachsektion Mathematik 28 Bremen, Achterstrasse, BRD

Universit~t Bremen, Fachsektion Mathematik 28 Bremen, Achterstrasse, BRD

Address list of authors and speakers

K.H,Hofmann

M.Hu~ek

*V,Kannan

*F.E.J.Linton

S.Marde~i~

P.Michor

L.D.Nel

P.Nyikos

J.R.Porter

M.Rajagopalan

H.-Chr.Reichel

*W.A.Robertson

S.Salbany

M.Schroder

U.Seip

J.Siegel

Tulane University, Dept. of Math. New Orleans~ La. 70118, USA

Matematicky Ustav University Karlovy Sokolovsk~ 83 Praha 8 - Karlin, CSSR

Madurai University, Dept.of Math. Madurai, India

Wesleyan University, Dept. of Math. Middletown, Connecticut 06457, USA

University of Zagreb, Inst. of Math. 41001 Zagreb, p.p.187, Yugoslavia

Mathematisches Institut der Universit[t Strudlhofgasse 4 I090 Wien, ~sterreich

Carleton University, Dept. of Math. Ottawa, Ontario KIS 5B6, Canada

University of lllinois at Urbana-Champaign Department of Mathematics Urbana, Iii.61801, USA

The University of Kansas, Dept. of Math. Lawrence, Kansas 66044, USA

Memphis State University, Dept. of Math. Memphis, Tennessee 38152, USA

Mathematisches Institut der Universit~t Strudlhofgasse 4 1090 Wien, ~sterreich

Carleton University, Dept. of Math. Ottawa, KIS 5B6, Canada

University of Cape Town, Dept. of Math. Private Bag Rondebosch, Rep. of South Africa

University of Waikato, Dept. of Math. Hamilton, New Zealand

Instituto de Matematica e Estatistica Universidade de Sao Paulo Cx. Postal 20.570 (Ag.lguameti) Sao Paulo/Brasil

University of Missouri,Dept. of Math. Sc. College of Arts and Sciences 8001 Natural Bridge Road St. Louis, Missouri 63121, USA

XM

Address list of authors and speakers

J.van der Slot

D. Strauss

G.E.Strecker

D.Tanr@

S.J.R.Vorster

J.de Vries

M.B.Wischnewsky

O.Wyler

Schimmelpenninckstraat ~6 Zwijndrecht, Netherlands

University of Hull, Dept. of Math. Hull, England

Kansas State University, Dept. of Math. Manhattan, Kansas 66502, USA

Universit~ de Picardie Th~orie et Applications des Categories Facult~ des Sciences 33, rue Saint-Leu 80 039 Amiens, France

University of South Africa, Dept. of Math. P.O.Box 392 Pretoria, Rep. of South Africa

Mathematisch Centrum

2 e Boerhaavestraat 49 Amsterdam-O., Netherlands

Universit~t Bremen, Fachsektion Mathematik Achterstrasse 33 28 Bremen, BRD

Carnegie-Mellon University, Dept. of Math. Pittsburgh, Pa. 15213, USA

V.Kannan: "Coreflective subcategories in topology" I

F.E.J.Linton: "The Jonnson-Tarskl Topos"

W.A.Robertson: "Cartesian closed categories of nearness structures"

J.van der Slot: "Categories induced by perfect maps"

(These papers will appear elsewhere)

INTRODUCTION

Categorical topology, i.e. the investigation of topological problems-

pure and applied - by categorical methods, is a rather new and ex-

panding field.

Recent investigations have made apparent that a considerable number

of seemingly typical topological problems can best be understood and

analyzed by means of categorical terms and methods, e.g.

(1) Completions, compactifications, realcompactifications etc. have

been classified as solutions of universal problems; topological

modifications such as sequential-, locally connected-, and compactly

generated-refinements as solutions of dual problems

(2) the importance of factorization structures and the close re-

lations between certain classes of spaces and certain classes of

maps have become apparent

(3) the similarities between topological structures such as topolo-

gies, uniformities, and proximities have been traced down to common

properties of the corresponding forgetful functors, and have led to

the important concept of a topological functor

(4) cartesian closedness has been exhibited as one of the crucial

properties not shared by any of the categories Top, Unif, Prox of

topological, uniform, and proximity spaces respectively

(5) a hierarchy of topological categories - some of them cartesian

closed, others not - has been constructed, and it has been demon-

strated that certain classical problems from extension theory, dimen-

sion theory, homology theory, topological algebra, functional analysis

•

Introduction

(espec. duality theory), and differential topology not solvable in

Top are solvable in the realm of certain of these more appropriate

topological categories.

None of the above ideas and results have appeared in book form yet.

The purpose of this conference was to survey the present state of

categori~al topology in order to stimulate and organize further

research in this area.

The papers in this volume may be classified as follows:

I, Internal aspects of topological categories, such as epireflective

and monocoreflective subcategories of Top, Haus, and Unif

(Husek, Nyikos, Salbany)

II. Categorical aspects of extension theory (Bentley, Porter),

dimension theory (Dyckhoff, Herrlich), and perfectness (Strecker)

III. External aspects of topological categories, such as topological

functors (BrCmmer, Hoffmann), and cartesian and monoidal closed

topological categories and topoi (Nel, Wyler, Wischnewsky)

IV. Concrete alternatives to the classical topological categories,

such as nearness structures (Bentley, Herrlich, Reichel) convergence-

and limit-structures (Binz, Bourdaud) syntopogeneous structures

(Tanr$), frames (Strauss-Dowker), and generalized uniform structures

(Vorster).

V. Applications in topological algebra (Hofmann, de Vries, Wischnewsky)

VI. Applications in algebraic topology (Bentley, Calder, Marde~i~)

XV

Introduction

VII. Applications in functional analysis (Binz, Cigler, Cooper-Michor,

Flachsmeyer, FrSlicher, Rajagopalan, Schroder)

VIII. Applications in differential topology (Berquier, Michor, Seip).

Ernst Binz Horst Herrlich

The Role o f Nearness Spaces in Topo logy

by

H. L. B e n t l e y

In 1973, H. Herrlich [25] introduced nearness

spaces and since that time, these spaces have been

used for several d i f fe rent purposes by topologists.

The aim in this paper is to survey some of the

applications of nearness spaces within topology,

namely: un i f icat ion, extensions, homology and

connectedness. Some topics which would have been

included here are dimension theory and function

spaces but these topics have been covered in the

paper by H. H e r r l i c h [ 28 ] wh ich i s be ing p resen ted

at t h i s con fe rence and so they w i l l not be d i scussed

here .

I . U n i f i e d t h e o r i e s o f t o p o l o g y and u n i f o r m i t y .

One o f the i m p o r t a n t r o l e s t h a t nearness spaces can

p lay i n t o p o l o g y i s t h a t o f u n i f i c a t i o n . The idea o f

u n i f y i n g the d i f f e r e n t s t r u c t u r e s wh ich t o p o l o g i s t s s tudy

by f i n d i n g a more e x t e n s i v e s t r u c t u r e wh ich i n c l u d e s them

i s an idea wh ich has he ld wide i n t e r e s t f o r the l a s t

f i f t e e n y e a r s . Th is idea can be made somewhat more p r e c i s e

as f o l l o w s : F ind a c o n c r e t e c a t e g o r y A wh ich c o n t a i n s

both the c a t e g o r y o f t o p o l o g i c a l spaces and c o n t i n u o u s maps

and the c a t e g o r y o f u n i f o r m spaces and u n i f o r m l y c o n t i n u o u s

maps as f u l l s u b c a t e g o r i e s and, moreover , the c a t e g o r y

shou ld share as many mapping p r o p e r t i e s as p o s s i b l e w i t h

the o r i g i n a l two c a t e g o r i e s . Of c o u r s e , such a vague l y

s t a t e d problem can have many s o l u t i o n s , some o f them

t r i v i a l . I t is a we l l known f a c t t h a t the ca tego ry of

un i fo rm spaces a l r e a d y con ta ins a f u l l subca tegory which i s

i somorph ic to the c o m p l e t e l y r e g u l a r spaces (namely , the

f i n e un i fo rm spaces) . However, few t o p o l o g i s t s would submi t

to being f o r b i d d e n to s tudy non c o m p l e t e l y r e g u l a r spaces.

Severa l e l e g a n t s o l u t i o n s o f the u n i f i c a t i o n problem

have been suggested and one or two have been more or less

deve loped. For example, A. Csaszar [ I 0 ] o f f e r e d the

syntopogenous spaces and M. Kate tov [33 ] the mero top ic ~ v

spaces. Others are D. Do ic inov [ 1 2 ] , D. H a r r i s [ 2 1 ] , and

A.K. S t e i n e r and E. F. S t e i n e r [45 ] (For some of t hese ,

u n i f i c a t i o n was a byp roduc t , not the main o b j e c t i v e ) . The

s o l u t i o n which is o f i n t e r e s t here, one due to H. H e r r l i c h [ 2 5 ] ,

i s the ca tegory Near of nearness spaces and nearness pre-

se r v i ng maps. Th is ca tego ry w i l l now be de f i ned and examined

in some d e t a i l .

A nearness s t r u c t u r e on a set X i s a s t r u c t u r e g iven

by a set ~ of c o l l e c t i o n s of subsets o f X ( i . e . ~ C p2X)

hav ing the f o l l o w i n g p r o p e r t i e s ( c a l l e d axioms of nearness

s t r u c t u r e s ) ( ~ a n d ~ d e n o t e subsets of PX):

(NI ) I f ~ . F e ~ and ~ c o r e f i n e s , ~ then me ~ . (~)

(N2) I f ~ }~ ~ then ~I e ~ . (2 )

(N3) ~b e ~ and { @} ~ ~ .

(N4) I f ~v6~z'e ~ then O~le ~ or ~.Fe ~ . (3)

(N5) I f cl~OI e ~ then We ~ .

(c l~A = {x e X I { { x } , A } e ~ } and c l ~ = {c l~A I A e ( ~ } . )

(N6) I f { { x } , { y } } e ~ then x=y.

(1) ~ corefines ,~ i f f for every A e(C~ there exists B e~ with BCA.

(2) The conven t ion C~ ~ @ is adopted.

(3) O l v ~ = {A U B I A e~ I and B e~F } .

The co l lec t ions which are members of ~ are cal led nearness

co l l ec t ions of the nearness s t ruc ture defined by ~ on X.

The phrase " ~ is a nearness c o l l e c t i o n " could be expressed

more suggest ively as "the sets of 0( are near" and, in fac t ,

the short terminology "(~ is X-near" is customari ly used.

A nearness space is a pa i r X = (Sx, ~ X ) cons is t ing of

a set S X and a nearness s t ruc ture ~ X on S x. As is customary

in such s i t u a t i o n s , X is wr i t t en in place of S X. The

operator cl~ appearing in axiom (NS) is usua l ly w r i t t en cl X.

A mapping f : X § Y of a nearness space X in to a nearness

space Y is cal led a nearness preservin9 map (or simply

nearness map) i f f whenever ~)I is X-near then f ~ is Y-near . ( I )

A nearness space X has an underly ing topologica l space

TX whose closure operator is the operator cl X = cl~ which

appears in axiom (N5). (Throughout th is sect ion, topological

space means Tl-space, i . e . f i n i t e subsets are always closed.)

Also, any topologica l space X has an associated nearness space NX

defined by O~ is NX-near i f f ~c l X ~ t @. These correspondences

T: Near + Top and N: Top + Near are f unc to r i a l and, in fac t , N

is an embedding of Top in Near as a b i c o r e f l e c t i v e f u l l sub-

category. Henceforth, Top w i l l be i d e n t i f i e d with i t s image

under N: Top + Near. Thus, a topolog ica l space is a nearness

space X which s a t i s f i e s the condi t ion:

(T) O~ i s X-near i f f N c l x ~ ~ ~ .

A nearness space X has an u n d e r l y i n g un i fo rm space UX

which can be desc r ibed as f o l l o w s : the u n d e r l y i n g se t o f UX

i s the same as t h a t o f X and an X-cover i s a c o l l e c t i o n ~ f o r

which {X-A I A e ( ~ } i s not X-near . A un i fo rm cover in the

s t r u c t u r e of UX i s de f i ned to be those X-covers (~ f o r which

the re e x i s t s an i n f i n i t e sequence (~ = ~ J l ' ~-%2' ~ c 3 ' " "

(~) f O ~ = { f A I A e ( ~ } .

o f X-covers w i t h ~Jn+l s t a r r e f i n i n g ~7- n f o r each n.

V ice ve rsa , any u n i f o r m space X has an a s s o c i a t e d nearness

space NX d e f i n e d by (~I i s X-near i f f {X-A ] A e ~ } i s no t

a u n i f o r m cove r . The co r respondences U: Near § and N : U n i f § Near

are f u n c t o r i a l and, i n f a c t , N i s an embedding o f U n i f i n Near

as a b i r e f l e c t i v e f u l l s u b c a t e g o r y . H e n c e f o r t h , U n i f w i l l be

i d e n t i f i e d w i t h i t s image under N: U n i f ~ Near. Thus , a

u n i f o r m space i s a nearness space X wh ich s a t i s f i e s the

c o n d i t i o n :

(U): I f ~ l i s an X -cove r then t he re e x i s t s an X - c o v e r b w h i c h

s t a r r e f i n e s (~ .

Ano the r i m p o r t a n t c a t e g o r y wh ich can be embedded in Near

i s the c a t e g o r y Cont o f c o n t i g u i t y spaces (V. M. Ivanova and

A. A. I vanov [ 32 ] and W. L. T e r w i l l i g e r [ 4 9 ] ) .

A c o n t i g u i t y s t r u c t u r e on a se t X is a s t r u c t u r e g iven

by a se t ~ o f f i n i t e c o l l e c t i o n s o f subsets o f X hav ing the

f o l l o w i n g p r o p e r t i e s ( ( ~ and ~ d e n o t e f i n i t e c o l l e c t i o n s o f

subse ts o f X):

(C I ) I f ~ ' e ~ and ~ c o r e f i n e s ~ . g then (~(e ~ .

(C2) I f ~ ~ ~ then ~ e ~ .

(C3) ~ e ~ and {~ } ~ ~ .

C4) I f (p~v,~e ~ then ~ e ~ or ~ e

C5) I f c l c ( ~ e ~ then (C~ e ~ .

( c l~A = { x e X I { { x } , A} e ~ } and c l ~ = { c l ~ A I A e ( ~ } . )

(C6) I f { { x } , { y } } e ~ then x=y.

Note t h a t the c o n t i g u i t y axioms are the nearness axioms w i t h

a b l a n k e t assumpt ion t h a t the c o l l e c t i o n s wh ich are members

o f the s t r u c t u r e ~ must be f i n i t e . A c o n t i g u i t y space i s a

se t endowed w i t h a c o n t i g u i t y s t r u c t u r e . I f X i s a c o n t i g u i t y

space w i t h c o n t i g u i t y s t r u c t u r e ~ then a f i n i t e c o l l e c t i o n

is sa id to be X - c o n t i g u a l p r o v i d e d ~){ e ~ . C o n t i g u a l maps

are d e f i n e d in the expec ted way.

A nearness space X has an underlying contiguity space CX

which can be described as follows: the underlying set of CX

is the same as that of X and a col lect ion (~is CX-contigual

i f f (~is f i n i t e and C~is X-near. V ice versa, any contigual

space X has an associated nearness space NX defined by: the

underlying set of NX is the same as that of X and a col lect ion

is NX-near i f f every f i n i t e subset o f O I i s X-contigual. The

correspondences C: Near § Cont and N: Cont § Near are functorial

and, in fact, N is an embedding of Cont in Near as a b i re f l ec t i ve

f u l l subcategory. Henceforth, Cont w i l l be ident i f ied with i t s

image under N: Cont + Near. Thus, a cont igui ty space is

a nearness space X which sat is f ies the condi'tion:

(C) I f every f i n i t e subset of a col lect ion (~ is X-near

then so is ~ I .

Having embedded these three categories (Top, Unif,

and Cont) into Near, a useful new operation is possible:

one can take intersections of these subcategories.

I t turns out that

(l Top f'1 Cont = the category of compact topological spaces.

(2 Top ~ Unif = the category of paracompact

(= f u l l y normal) topological spaces.

= the category of f ine uniform spaces.

(3) Con.t ~ Unif = the category of precompact uniform spaces.

(4) To_Q~_hUnif NCont= the category of compact Hausdorff spaces.

In 1964, O. Frink [17] applied a construction due to

H. Wallman [52] to what he called a normal base ~ of closed

subsets of a completely regular topological space X to obtain

a compactification w(X, ~ ' ) of X and which since has become

known as a Wallman-type compactification. Because of a questCon

which Frink raised and which remains unanswered to this day

(Is every Hausdorff compactification of X of the form w(X,~)

for some ~ ? ) these Wallman-type compactif%catlons have

attracted considerable attention. E. F. Steiner [46] generalized

the construction to allow Tl-compactif ications of Tl-spaces by

replacing the normal bases of Frink by what Steiner called

separating bases. A separating base on a topological space X

is a base ~ f o r the closed subsets of X which is closed

under f i n i t e unions and f i n i t e intersections and which sa t is f ies :

(S) i f x ~ B e ~ then for some E e~.~ , x e E and E~B : 4.

Of course, these ideas can be placed in the nearness space

setting by going through cont igui t ies. However, nearness

structures allow many more poss ib i l i t i e s than do cont igui t ies

and arb i t rary cardinal res t r ic t ions can be made. A separating

base ~fon a topological space X gives r ise to several nearness

Nk(X,~3F) with underlying set X and with k an i n f i n i t e spaces

cardinal number. A col lect ion ~ i s defined to be not NR(X,~IC )-near

i f f for some subset ~ of~CFwith card ~F' < k, ~.~' corefines (~

and ~C~r' = #. The existence of these structures is one reason

for studying nearness spaces because of the fact that the

theory of contiguity spaces is inadequate for treating the

Wallman-type realcompactifications (A.K. Steiner and

E. F. Steiner [44], R. A. A]o and H. L. Shapiro [ l ] , M. S. Gagrat

and S. A. Naimpally [19], H. L. Bentley and S. A. Naimaplly [7] ).

A nearness space X always has a completion X* (see below).

I f ~ i s a separating base on a topological space X and Y denotes

the nearness space N~(X,~) defined above, then the Wallman-type

compactification of X induced b y ' s i s the same as the completion Y*

of the nearness space Y. I f , in addition, ,~r is closed under

countable intersections and Y denotes the nearness space Nj~o(X,~)

defined above, then the Wallman-type ~--realcompacti f icat ion of X

induced by ~ ' i s the same as the completion Y* of the nearness

space Y.

There is another l ine of ideas which can be unif ied

in the setting of nearness spaces and which involve the

extension of continuous maps from dense subspaces.

w i l l be examined at the end of the next sect ion.

These

2. Extensions of topolog ica l spaces.

Nearness spaces are a most natural tool for studying

extensions of topolog ica l spaces; in fac t , i t was in th is

context that nearness spaces o r g i n a l l y arose. This point

of view has been developed in a recent paper by H. L. Bentley

and H. Herr l ich [4 ] .

An extension e: X § Y is a dense embedding of a

topological space X in to a topologica l space Y ( for technical

s i m p l i c i t y , one usual ly assumes that the map e is an inc lus ion) .

Every extension e: X § Y induces various structures on X.

I . e: X +Y induces the nearness structure

: { ~ c P X I ~ C I y ( ~ ( ~ ~ } .

2. e: X § induces the c o n t i g u i t y s t r u c t u r e

c = { ~ P X I (~ l i s f i n i t e and ~ c l y ~ ~ 9}.

3. e: X +Y induces the g e n e r a l i z e d p r o x i m i t y r e l a t i o n

6 = { ( A , B ) e (PX) 2 I ClyA (~)clyB # @ } .

I t has r e c e n t l y been shown t h a t not a l l nearness spaces

are induced by an e x t e n s i o n (H. L. Ben t l ey [ 3 ] , S.A. Na impa l l y

and J. H. M. W h i t f i e l d [ 3 8 ] ) . C o n t i g u i t y spaces were i n t r oduced

and a x i o m a t i z e d by V. M. Ivanova and A. A. Ivanov [32 ] who

proved t h a t every c o n t i g u i t y s t r u c t u r e on X i s induced by

some T I - c o m p a c t i f i c a t i o n . P r o x i m i t y r e l a t i o n s were de f i ned

by V. A. Ef removi~ [14J and then Y. M. Smirmov [42 ] proved

t h a t every p r o x i m i t y r e l a t i o n is induced by a Hausdo r f f com-

p a c t i f i c a t i o n . M. W. Lodato [35 ] a x i o m a t i z e d g e n e r a l i z e d

p r o x i m i t y r e l a t i o n s . W. J. Thron [51 ] (us ing a c h a r a c t e r i z a t i o n

g iven e a r l i e r by M. S. Gagrat and S. A. Na impa l l y [ 1 8 ] ) showed

t h a t eve ry g e n e r a l i z e d p r o x i m i t y r e l a t i o n is induced by some

T l - c o m p a c t i f i c a t i o n . A l l o f these r e s u l t s can be expressed

in terms o f nearness spaces and t h e r e b y become j u s t d i f f e r e n t

aspects o f a s i n g l e genera l theorem.

A f o r m a l i z a t i o n o f the idea o f a nearness space be ing

induced by some e x t e n s i o n is embodied in the f o l l o w i n g d e f i n i t i o n .

D e f i n i t i o n : A nearness space is c a l l e d s u b t o p o l o g i c a l i f f i t

i s a nearness subspace o f some t o p o l o g i c a l nearness space.

D e f i n i t i o n (G. Choquet [ 9 ] ) : ~ is c a l l e d a g r i l l on X

i f f ~ ~O~cPX and

(G) AUB e ~ i f f A e (~ or B eC~ .

D e f i n i t i o n : I f X is a nearness space then (~ is c a l l e d an

X ? g r i l l i f f O~ is a g r i l l on the u n d e r l y i n g set o f X and ( ~ i s

a nearness c o l l e c t i o n on X.

Theorem: A nearness space X is s u b t o p o l o g i c a l i f f each nearness

c o l l e c t i o n on X is a subset o f some X - g r i l l .

Theorem: Any e x t e n s i o n e: X § Y induces a s u b t o p o l o g i c a l

nearness s t r u c t u r e on X. I f Z denotes the nearness space

w i t h the same u n d e r l y i n g set o f p o i n t s as X and w i t h the

nearness s t r u c t u r e induced by the e x t e n s i o n e: X ~ Y then

TZ = X. Vice ve rsa , i f Z is any s u b t o p o l o g i c a l nearness

space then t h e r e e x i s t s an e x t e n s i o n e: X + Y w i t h TZ = X

and where the nearness s t r u c t u r e o f Z is the one induced

by the e x t e n s i o n e: X § Y on X.

A more f r u i t f u l l i n e o f ideas comes from c o n s i d e r i n g

the conc re te nearness spaces.

D e f i n i t i o n : I f X i s a n e a r n e s s space t hen an X - c l u s t e r i s

a n o n - e m p t y max ima l ( w i t h r e s p e c t t o s e t i n c l u s i o n ) n e a r n e s s

c o l l e c t i o n ,

D e f i n i t i o n : A n e a r n e s s space i s c a l l e d c o n c r e t e i f f each

n e a r n e s s c o l l e c t i o n i s a s u b s e t o f some c l u s t e r .

Every c lus ter is a g r i l l so every concrete nearness

space is subtopological . Any extension e: X § Y which

induces a concrete nearness structure on X can be recovered

by a completion process. A co l l ec t i on (~ of subsets of a

topological space X is said to converge (with respect to X)

to a point x of X provided every neighborhood of x contains

some member of 0~. A nearness space X is said to be

complete provided every X-cluster converges (with respect

to the underlying topologica l space TX of X) to some point

of X. Every nearness space X has a completion X*. There

is an embedding e: X § X* of X as a dense nearness subspace

of X* where X* is a complete nearness space.

Theorem: I f X is a nearness space, then X is concrete i f f

i t s completion X* is topo log ica l .

De f in i t i on (M.H. Stone [47] : An extension e: X § Y is cal led

s t r i c t i f f {cIyA I A C LX } is a base for closed sets of Y.

Theorem: Any s t r i c t extension e: X § Y induces a concrete

nearness structure on X. I f Z denotes the nearness space

with the same underlying set of points as X and with the

nearness structure induced by the s t r i c t extension e: X § Y,

then TZ = X and e: X § Z* is equivalent to e: X § Y. Thus,

two s t r i c t extensions e l : X § Yl and e2: X § Y2 are

1o

e q u i v a l e n t i f f t hey i nduce the same nearness s t r u c t u r e on X.

S ince every c o n t i g u i t y space is c o n c r e t e , the s t r u c t u r e

o f a c o n t i g u i t y space X i s a lways induced by some t o p o l o g i c a l

e x t e n s i o n ( e . g . e: TX § X* ) .

The question of how generalize~ proximity structures

f i t in to the nearness space set t ing remains. There is more

than one natural way to embed the category of generalized

proximity spaces in to Near. The de ta i l s are as fo l lows.

A generalized proximity r e l a t i o n on a set X is a

re la t i on a ~(PX) 2 having the fo l lowing proper t ies:

(P0) I f A 6 B then B 6 A.

(Pl) I f A c B and A 6 C then B ~ C.

(P2) I f A ~ B ~ @ then A 6 B.

(P3) I f A 6 B then A ~ 4 �9

(P4) I f A ~ (BUC) then A ~ B or A ~ C.

(P5) I f A 6 B and B~cl~C then A 6 C.

( c I6A = {x e X I { x } ~ A } . )

(P6) I f { x } a { y } then x = y .

A g e n e r a l i z e d p r o x i m i t y space i s a se t endowed w i t h a g e n e r a l i z e d

p r o x i m i t y r e l a t i o n . Prox ima l maps are d e f i n e d i n the expec ted

way and the c a t e g o r y Prox r e s u l t s .

Let U: Cont § Prox denote the obv ious f o r g e t f u l f u n c t o r

wh ich a s s o c i a t e s w i t h any c o n t i g u i t y space X the g e n e r a l i z e d

p r o x i m i t y space UX w i t h the same u n d e r l y i n g se t as X and w i t h

g e n e r a l i z e d p r o x i m i t y r e l a t i o n 6 d e f i n e d by A ~ B i f f {A ,B } i s

X - c o n t i g u a l . The f o l l o w i n g theorem (wh ich was proved in [ 4 ] )

g i ves the bas is f o r d i s c o v e r i n g many i m p o r t a n t f a c t s about

g e n e r a l i z e d p r o x i m i t y spaces.

Theorem: The f o r g e t f u l f u n c t o r U: Cont + Prox i s

t o p o l o g i c a l , i . e . f o r any g e n e r a l i z e d p r o x i m i t y space X any

fami ly (X i ) of con t i gu i t y spaces and any fami ly i e l

( f i : X § UXi) of proximal maps, there ex is ts a con t i gu i t y i e l

s t ruc ture on X, g iv ing r ise to a con t i gu i t y space Y with

UY = X, which is i n i t i a l wi th respect to the given data,

i . e . such that for any con t i gu i t y space Z and any proximal

map g: UZ + X, the fo l low ing condi t ions are equiva lent :

(a) g: Z § Y is a contigual map.

(b) For each i e l , f l og : Z § X i is a contigual map.

Y UY=X

Z . ~ X i UZ ~ UX i f i ~ f iog

The importance in a functor being topological is well

known (e.g. see H. Herr l i ch [24 ] ) . In p a r t i c u l a r , the

f ibres of U form complete l a t t i c e s and so there ex i s t d iscrete

and ind iscre te con t i gu i t y st ructures on each generalized

proximi ty space. Thus, there ex is t at least two ways to

define a functor E: Prox § Cont to be a r i g h t inverse

for U: Cont § Prox, take E to be e i the r the ind isc re te con t i gu i t y

or the discrete con t i gu i t y . Thus, l e t L: Prox + Cont denote

the l e f t ad jo in t , r i gh t inverse of U which associates with

each generalized proximi ty space the corresponding d iscrete

(smallest) con t i gu i t y and R: Prox + Cont the r i gh t ad jo in t ,

r i gh t inverse of U which associates with each general ized

proximi ty space the corresponding ind iscre te ( la rges t )

con t i gu i t y . Using e i t he r of the two functors R or L (which

are embeddings as f u l l subcategories) i t fo l lows tha t , since

every con t i gu i t y s t ruc ture is induced by some extension, then

12

so is every generalized proximity structure also induced

by some extension. Although e i the r R or L can be used in

establ ish ing th is resu l t , R and not L is the most i n te res t ing

of thest two functors. In fac t , i t turns out that i f a

generalized proximity space X with generalized proximity . v

re la t i on 6 sa t i s f i es the add i t iona l axiom of Efremovlc [14]

(E) I f not A 6 B then there exists C:X with

not A 6 C and not B a(X-C),

so that X becomes a proximity space, then the Smirnov

compact i f icat ion of X is the same thing as the completion

of the con t igu i t y space RX and these are usually d i f f e r e n t

from the completion of the con t igu i t y space LX. Thus, R is

to be favored over L.

An in te res t ing l i ne of inqu i ry arises by asking what

nearness theoret ic property of a nearness structure is

equivalent to i t being induced by an extension e: X § Y with Y

having a certa in topologica l property. I t is worth rephrasing

th is statement in category theoret ic terms. Let Ext denote

t;le category of s t r i c t extensions, ~.e. objects of Ext are

s t r i c t extensions e:X § Y and morphisms of Ext are pairs

of continuous maps ( f , g ) : (e:X § § (e ' :X '§ for which

the fo l lowing diagram is commutative:

X e Y

f L X ' > Y' e'

Then there is the "inducing" functor N: ~ t § Near

where N(e:X § Y) is the nearness space whose underlying set

is the same as that of X and whose nearness structure is

the (concrete) one induced by the s t r i c t extension e:X § Y.

13

For a morphism, N(f,g)=f. Thus, the program which presents

i t s e l f in this context is the following one: Given a

topological property P, what nearness theoretic property P'

is such that a nearness space Z has the property P' i f f Z=N(e:X + Y)

for some s t r i c t extension e:X§ where Y has property P. I f

"property" is understood to mean " fu l l subcategory", then this

program can be rephrased as follows. (In the fol lowing,

subcategory always means isomorphism closed, f u l l subcategory).

I f B is a subcategory of Top then give an " in te rna l " description

of the subcategory B' of Near whose objects are nearness spaces

of the form N(e:X + Y) where Y is in B.

There is a semioperational approach to a solution of

this problem. I f B is a subcategory of Top and B' is a

subcategory of Near, say that B' is a pendant of

i f f B' ~Top = B. I f B' is a pendant of B which also sa t is f ies

the three conditions

(1) Each object of B' is a concrete nearness space.

(2) B' is hereditary ( i . e . i f Y is a B' object and Z is a

nearness subspace Of Y then Z is also in B ' ) .

(3) B' is completion closed ( i . e . i f Y is in B' then the

completion Y* is also).

then the objects of B' are precisely those nearness spaces

of the form N(e:X § Y) where Y is in B.

Given B i t is sometimes possible to find a pendant B' of

so that (1), (2), and (3) are sa t is f ied . A solution of this

type of problem with B any of the following topological

properties appears in the paper by H. L. Bentley and H. Herrlich [4]:

(a) Compact.

(b) Hausdorff.

(c) Compact Hausdorff.

(d) Regular.

14

(e) Paracompact.

( f ) Realcompact .

A s o l u t i o n f o r n o r m a l i t y has not y e t been found .

I t was ment ioned at the end o f the l a s t s e c t i o n t h a t

seve ra l theorems about the e x t e n s i o n o f c o n t i n u o u s maps

from dense subspaces can be u n i f i e d in the s e t t i n g o f

nearness spaces. These a re : the e x t e n s i o n theorem o f

Taimanov and seve ra l g e n e r a l i z a t i o n s o f Ta~manov's theorem

due to R. Enge lk ing [ 16 ] and H. H e r r l i c h [ 2 3 ] . Th i s

u n i f i c a t i o n has been deve loped by H. H e r r l i c h in [ 26 ] and

by H. L. B e n t l e y and H. H e r r l i c n i n [ 4 ] . Ano the r t ype o f

g e n e r a l i z a t i o n o f Ta imanov 's theorem has been g i ven by

S. A. Na impa l l y [ 3 8 ] .

The c r u c i a l p r o p e r t y f o r the e x i s t e n c e o f e x t e n s i o n s

o f maps is r e g u l a r i t y o f the range. I f X i s a nearness

space then f o r A, B C X , A <X B means t h a t {A, X-B} i s not

X-near . A nearness space X i s c a l l e d r e g u l a r i f f f o r any

co l l ec t i on (~ , i f the co l l ec t ion {B~X I for some A e(~, A <X B}

is X-near then so is ~ . This concept of r e g u l a r i t y is a

pendant of the topologica l concept of r egu la r i t y . (In fac t ,

i f Re 9 denotes the f u l l subcategory of Near whose objects

are the regular nearness spaces, then Reg is also heredi tary

and completion closed.) H. Her r l i ch 's theorem is the

fo l lowing one [26].

Theorem: I f Y is a complete, regular nearness space, S is

a dense nearness subspace of a nearness space X and f:S § Y

is a funct ion, the fo l lowing condit ions are equivalent :

(1) f:S + Y is a nearness map.

(2) f can be extended to a nearness map g:X § Y.

Her r l i ch 's theorem has many c o r o l l a r i e s , among them

the fo l lowing ones.

15

Corol lary (A. Weil [54]) . I f Y is a complete uniform space,

S is a dense subspace of a uniform space X and f :S § is

a funct ion, the fo l lowing condit ions are equivalent :

(1) f :S§ is uniformly continuous.

(2) f can be extended to a uniformly continuous function g:X § Y.

Corol lary (H. Herr l ich [23] ) . I f Y is a regular topo log ica l

space, ~j is a base for the closed sets of Y, S is a dense

nearness subspace of a topologica l space X, and f:S § Y is

a funct ion, the fo l lowing are equivalent :

(1) f:S § Y is a nearness map.

(2) f can be extended to a continuous map g:X § Y.

(3) I f ~ C ~ w i t h h~ = r then n { c l x f - I A I A e ~ } = @.

Corol lary (R. Engelking [16]) . I f Y is a realcompact topological

space, S is a dense nearness subspace of a topologica l space X,

and f:S § Y is a funct ion, the fo l lowing are equivalent:

(1) f:S § Y is a nearness map.

(2) f can be extended to a continuous map g:X § Y.

(3) For each countable sequence A l , A 2 . . . . of zero sets of Y, o o c o

A i = @ then i=l~ cl x f - l Ai = @. i f

Corol lary (A. D. Ta~manov [48]) . I f Y is a compact

Hausdorff topologica l space, S is a dense nearness subspace

of a topological space X, and f:S § Y is a funct ion, the

fo l lowing are equivalent :

(1) f:S § Y is a nearness map.

(2) f can be extended to a continuous map g:X § Y.

(3) For each f i n i t e sequence A l . . . . . A n of closed sets of Y,

( ~ n IA i i f A i = ~ then ~ c l x f - = r �9 i = l i : l

(4) For each p a i r A,B o f c l osed sets o f Y,

i f A ~ B = ~ then c l x f - I A ( ' ~ c l x f - I B = @ �9

16

3. Nearness spaces and A lgeb ra i c Topology.

v

E. Cech [8 ] de f ined a homology theory f o r general

t o p o l o g i c a l spaces. His theory was developed by severa l

t o p o l o g i s t s , but most no tab ly by C. H. Dowker [13] and

E. H. Spanier [ 43 ] , who showed tha t the ~ech theory

s a t i s f i e s the E i lenberg -S teenrod axioms, and an e x p o s i t i o n

of t h i s theory appears in the famous book by S. E i lenberg

and N. E. Steenrod [15 ] . Let X be a t o p o l o g i c a l space

and l e t u be the set of a l l open covers of X. For each

open cover O{ of X, l e t NO~ be the nerve of ~ and

l e t Hq(N~) be the q-d imensional homology group of the

s i m p l i c i a l complex N~ . The set ~ can be p a r t i a l l y

ordered by ~ ~0~ i f f (~ref ines~..~ and thus ~ becomes

a d i r e c t e d set . I f ~ , ~ . f e ~ and (~I ref ines~.)" , then

there e x i s t " p r o j e c t i o n s " P~,~[}. : N~ ~ N ~ which

s a t i s f y the cond i t i on tha t f o r each A e(~ , POI, ;~ (A) e ~3

and A C P~ ( ,~ f (A). Using the homomorphisms induced by

such p r o j e c t i o n s as bonding maps between groups Hq(N~ },

a d i r e c t spectrum r e s u l t s and the d i r e c t l i m i t group, Hq(X),

is the q-d imensional Cech homology group of X. C. H. Dowker [13]

observed tha t the above d e f i n i t i o n makes sense i f ins tead of

l e t t i n g ~ denote the set of a l l open covers of X, ~ denotes

any set of covers which s a t i s f y the two axioms:

(UI) I f ( ~ e ~ and (~ r e f i n e s a c o v e r ~ then ~ " e ~.

(U2) I f (~e ~ and ~ e ~ then (~^~. ) 'e ~ . ( i )

As H. H e r r l i c h [27] has observed, these two axioms ( t o g e t h e r

w i th the c o n d i t i o n tha t {X} e ~ and @ # ~) g ive r i s e to

s t r u c t u r e s which are e q u i v a l e n t to seminearness s t r u c t u r e s

( i ) ~ I Z ~ : {A r~ B I A e O( and B e ~ r } .

17

( s t r u c t u r e s wh ich s a t i s f y axioms (N I ) - (N4)~ Thus, in t h i s v

t e r m i n o l o g y , Dowker d e f i n e d the Cech homology groups f o r a

seminearness space. No one d id a n y t h i n g about t h i s f o r many

y e a r s , but in 1972, M. Bahaudin and J. Thomas [ 2 ] i n v e s t i g a t e d

these groups f o r u n i f o r m spaces and then D. C z a r c i n s k i [ I I ]

s t u d i e d them f o r nearness spaces. He showed t h a t these groups

s a t i s f y a v a r i a n t o f the E i l e n b e r g - S t e e n r o d axioms f o r a

homology t h e o r y and he a lso showed t h a t a nearness space and

i t s c o m p l e t i o n have the same homology g roups . M. Bahaudin

and J. Thomas asked whe the r a c o m p l e t e l y r e g u l a r t o p o l o g i c a l

space X and the a s s o c i a t e d f i n e u n i f o r m space UX have the

same homology g ropus . Th i s q u e s t i o n remains unanswered;

however , a p a r t i a l s o l u t i o n i s obv ious i n the s e t t i n g o f

nearness spaces. As nearness spaces, a paracompact t o p o l o g i c a l

space X and i t s a s s o c i a t e d f i n e u n i f o r m space UX are the same.

T h e r e f o r e , in t h i s case, they have the same homology g roups .

4. Connectedness.

The homology t h e o r y ment ioned above y i e l d s , o f

cou rse , a t h e o r y o f connec tedness f o r seminearness

spaces as a b y p r o d u c t . I t i s s t i l l i n t e r e s t i n g to

look e x p l i c i t l y at some o f the f e a t u r e s o f t h i s k ind

o f connec tedness .

A seminearness space X is c a l l e d connected

i f f whenever f : X + Y i s a nearness map from X i n t o a

d i s c r e t e nearness space Y, then f i s c o n s t a n t . Th i s

concept has been s t u d i e d by S. G. Mr~wka and W. J . Pe rv i n [ 36 ]

in the s p e c i a l case o f u n i f o r m spaces. Of c o u r s e , i f X

18

is a topological space ( i . e . T l - space) then th is concept

is equivalent to the usual topological connectedness

(proved by B. Knaster and K. Kuratow~ki [34] in 1921).

One of the pr imary values in s tudy ing connectedness of

seminearness spaces, other than the obvious one of u n i f y i n g

the ideas, is tha t r e l a t i n n s h i p s between the var ious kinds

of connectedness is eas ie r to express. Mr~wka and Pervin

proved tha t a uni form space is un i f o rm ly connected i f f i t s

precompact r e f l e c t i o n is un i f o rm ly connected. Th~s theorem

genera l i zes p e r f e c t l y to the s e t t i n g of seminearness spaces

and in f a c t , a qu i te s u r p r i s i n g , a l b e i t s imp le , f u r t h e r

g e n e r a l i z a t i o n is v a l i d : The precompact r e f l e c t i o n can be

replaced by an a r b i t r a r y b i r e f l e c t i o n . Let S-Near denote

the category of seminearness spaces.

Theorem: Let A be any b i r e f l e c t i v e subcategory of S-Near

such that every discrete seminearness space is an A-object

and le t F: S-Near § A denote the associated re f lec to r .

Then a seminearness space X is connected i f f FX is connected.

5. Remarks.

Many topics have been omitted. As was stated

ea r l i e r , dimension theory and function spaces can be

read about in [28] and in general, the paper by

H. Herr l ich [27] is the basic source of information

about nearness spaces.

19

B ib l i og raphy

I . R. A. AI~ and H. L. Shapi ro , Wallman compact and realcompact

spaces, Con t r i bu t i ons to Extension Theory of Topo log ica l

S t ruc tu res (Proc. Symp. B e r l i n , 1967), B e r l i n (1969), 9-14.

2. M. Bahauddin and J. Thomas, The homology of uni form spaces,

Canad. J. Math. 25 (1973) , 449-455.

3. H. L. Ben t l ey , Nearness spaces and ex tens ions of t o p o l o g i c a l

spaces, Studies in Topology, New York (1975), 47-66.

4. H. L. Bent ley and H. H e r r l i c h , Extensions of t o p o l o g i c a l

spaces, p r e p r i n t .

5. H. L. Ben t ley , H. H e r r l i c h and W. A. Robertson, Convenient

ca tegor ies fo r t o p o l o g i s t s , p r e p r i n t .

6. H. L. Bent ley and S. A. Naimpal ly , Wallman T l - c o m p a c t i f i c a t i o n s

as e p i r e f l e c t i o n s , Gen. Topol . Appl . 4 (1974), 29-41.

7. H. L. Bent ley and S. A. Naimpal ly , L - r e a l c o m p a c t i f i c a t i o n s as

e p i r e f l e c t i o n s , Proc. Amer. Math. Soc. 44 (1974), 196-202.

8. E. Cech, Theor ie genera le de l ' homo log ie dans un espace

quelconque, Fund. Math. 19 (1932), 149-183.

9. G. Choquet, Sur les not ions de f i l t r e et de g r i l l e , Comptes

Rendus Acad. Sci . Paris 224 (1947) , 171-173.

I0. A. Cs~s / zar , Foundat ions of General Topology, New York (1963). v

I I . D. C z a r c i n s k i , The Cech homology theory f o r nearness spaces,

Thes is , U n i v e r s i t y of Toledo (1975). V V

12. D. Do ic inov , On a general theory o f t o p o l o g i c a l , p r o x i m i t y

and uni form spaces, Dokl. SSSR 156 (1964), 21-24; Sov ie t

Math. 5 (1964), 595-598.

13. C. H. Dowker, Homology groups of r e l a t i o n s , Ann. of Math.

56 (1952) , 84-95. . V

14. V. A. Efremovlc, I n f i n i t e s i m a l spaces, Dokl. Akad. Nauk

SSSR, 156 (1951) , 341-343.

20

15. S. E i lenberg and N. Steenrod, Foundations of A lgebra i c

Topology, Pr inceton (1952).

16. R. Engelk ing, Remarks on realcompact spaces, Fund. Math.

55 (1964), 303-308.

17. O. F r i nk , Compac t i f i ca t i ons and semi normal spaces,

Amer. J. Math. 86 (1964), 602-607.

18. M. S. Gagrat and S. A. Naimpal ly , Pooximity approach to

ex tens ion problems, Fund. Math. 71 (1971) , 63-76.

19. M. S. Gagrat and S. A. Naimpal ly , Wallman c o m p a c t i f i c a t i o n s

and Wallman r e a l c o m p a c t i f i c a t i o n s , J. A u s t r a l . Math. Soc. 15

(1973), 417 - 427.

20. M. S. Gagrat and W. J. Thron, Nearness s t r u c t u r e s and

p r o x i m i t y ex tens ions , p r e p r i n t .

21. D. Ha r r i s , S t ruc tu res in Topology, Memoirs Amer. Math. Soc.

115 (1971).

22. M. Shayegan Hast ings , E p i r e f l e c t i v e Hul ls in Near, Thes is ,

U n i v e r s i t y of Toledo (1975).

23. H. H e r r l i c h , F o r t s e t z b a r k e i t s t e t i g e r Abbi ldungen und

Kompakthei tsgrad t o p o l o g i s c h e r R~ume, Math. Z e i t . 96(1967) ,64-72 .

24. H. H e r r l i c h , Topo log ica l f u n c t o r s , Gen. Topol . Appl .

(1974), 125-142.

25. H. Herr l ich, A concept of nearness, Gen. Topoi. Appl.

(1974), 191-212. 26. H. Herr l ich, On the ex tend ib i l i t y of continuous functions,

Gen. Topoi. Appl. 5 (1974), 213-215.

27. H. Herrl ich, Topological structures, Math. Centre Tract 5___22

(1974), 59-122.

28. H. Herrl ich, Some topological theorems which fa i l to be

true, preprint.

29. H. Herrlich and G. E. Strecker, Category theory, Boston (1973).

21

30. W. N. Hunsaker and P. L. Sharma, Nearness s t r u c t u r e s

compat ib le w i th a t o p o l o g i c a l space, Arch iv der Math.

25 (1974), 172-178.

31. M. Hu~ek, Categor ia l connect ions between gene ra l i zed p r o x i m i t y

spaces and c o m p a c t i f i c a t i o n s , Con t r i bu t i ons to Extension Theory

of Topo log ica l S t ruc tu res (Proc. Symp. B e r l i n , 1 9 6 7 ) , B e r l i n

(1969) , 127-132.

32. V. M. Ivanova and A. A. Ivanov, C o n t i g u i t y spaces and b i -

compact ex tens ions of t o p o l o g i c a l spaces, Dok. Akad. Nauk

SSSR 127 (1959) , 20-22.

33. M. Kat~tov, On c o n t i n u i t y s t r u c t u r e s and spaces o f mappings,

Comment. Math. Univ. Caro l . 6 (1965), 257-278.

34. B. Knaster and K. Kuratowski , Sur les ensembles connexes,

Fund. Math. 2 (1921) , 206-255.

35. M. W. Lodato, On t o p o l o g i c a l l y induced gene ra l i zed p r o x i m i t y

r e l a t i o n s , Proc. Amer. Math. Soc. 15 (1964) , 417-422.

36. S. G. Mr6wka and W. J. Perv in , On uni form connectedness,

Proc. Amer. Math. Soc. 15 (1964), 446-449.

37. S. A. Naimpal ly , R e f l e c t i v e func to rs v ia nearness, Fund.

Math. 85 (1974), 245-255.

S. A. Naimpal ly and J. H. M. W h i t f i e l d , Not every near

fami l y is conta ined in a c lan , Proc. Amer. Math. Soc. 47

(1975), 237-238.

L. D. Nel , I n i t i a l l y s t r u c t u r e d ca tego r i es and c a r t e s i a n

c losedness, p r e p r i n t . . O

O. Njastad, A p r o x i m i t y w i t hou t a sma l les t compat ib le

nearness, p r e p r i n t .

W. A. Robertson, Convergence as a Nearness Concept, Thes is ,

Car le ton U n i v e r s i t y (1975).

38.

39.

40.

41.

22

42. Y. M. Smirnov, On p r o x i m i t y spaces, Mat. Sb. (NS) 31 (1952) ,

453-574; Engl ish t r a n s l . , Amer. Math. Soc. T rans l . Ser. 2

38 (1964), 5-36.

43. E. H. Span ier , Cohomology theory fo r general spaces,

Ann. of Math. 49 (1948), 407-427.

44. A. K. S te ine r and E. F. S t e i n e r , Nest generated i n t e r s e c t i o n

r ings in Tychonof f spaces, Trans. Amer. Math Soc. 148

(1970), 589-601.

45. A. K. S te ine r and E. F. S t e i n e r , Binding spaces: a u n i f i e d

complet ion and ex tens ion theo ry , Fund. Math. 76 (1972), 43-61.

46. E. F. S t e i n e r , Wallman spaces and c o m p a c t i f i c a t i o n s , Fund.

Math. 61 (1968) , 295-304.

47. M. H. Stone, A p p l i c a t i o n s of the theory of Boolean r ings to

general topo logy , Trans. Amer. Math. Soc. 41 (1937), 375-48] .

48. A. D. Taimanov, On ex tens ion of cont inuous mappings of

t o p o l o g i c a l spaces, Math. Sbornik N. S. 31 (73) (1952) , 459-463.

49. W. L. T e r w i l l i g e r , On C o n t i g u i t y Spaces, Thes is , Washington

State U n i v e r s i t y (1965)

50. W. J. Thron, Prox imi ty s t r u c t u r e s and g r i l l s , Math. Ann.

206 (1973), 35-62.

51. W. J. Thron, On a problem of F. Riesz concern ing p rox im i t y

s t r u c t u r e s , Proc. Amer. Math. Soc. 40 (1973), 323-326.

52. H. Wallman, L a t t i c e s and t o p o l o g i c a l spaces, Ann. of Math.

39 (1938), 112-126.

53. R. H. Warren, P r o x i m i t i e s , Lodato P rox im i t i es and P rox im i t i es

o f Cech, Thes is , U n i v e r s i t y of Colorado (1971).

54. A. Wei l , Sur les Espaces ~ S t r u c t u r e Uniforme et sur la

Topo log ie Generale, Par is (1938).

UN THEOREME D'INVERSION LOCALE

par Frangoise BEROUIER

I. INTRODUCTION.

Soit E un espace de Banach. On ales deux th@or@mes suivants :

a) Soit f une application de E dans E, continBment B-diff@rentiable sur un

voisinage U d'un point x ~ ten d'autres termes, continOment diff@rentiable

au sens de Fr@chet sur U), i.e. la d@riv@e de f par rapport ~ i'ensemble B

des born@s de E existe en tout point x de U et l'application Dr: U§

x~*Df(x), est continue lorsque L(E,E) est muni de la topologie de la conver-

gence uniforme sur les ensembles de B. Supposons que Df(x o) soit un hom@o-

�9 et f(x ) respective- morphisme Alors il existe des voisinages Vet W de x ~ o ment tels que f induise un hom@omorphisme de V sur W ; de plus l'application

f-l: W+V est B-diff@rentiable au point yo=f(Xo ) et (Df-~)(yo)=(Df(xo)) -~.

b) Soit fune application de E dans E induisant une bijection d'un voisina- -I

ge V ~e x ~ sur un voisinage W de f(Xo)=y ~ et telle que l'application f

soit continue au point Yo" Supposons que f soit B-diff@rentiable sur Vet

que Df(x O) soit un hom@omorphisme. Alors l'application f-1 est B-diff@ren-

tiable au point Yo et (Df-1)(yo)=(Df(xo)) -I.

On salt que les th@or@mes a) et b) sont faux en g@n@ral pour des evt

quelconques, comme le montrent les exemples suivants�9

I) D@signons par C(~,~) l'espace des applications continues d'une variable

r@elle ~ valeurs r@elles muni de la topologie de la convergence compacte,

et par fo l'application nulle de C(R,~). Soit e~: C(~,~)44Z:(~,R) l'applica-

tion exponentielle f~e~(f), o~ exp(f)(x)=eo~p(f(x)) pour tout xE~.

Cette application ne v@rifie pas le th@or@me a) : elle est continO-

ment B-diff@rentiable sur C(R,~) et la d@riv@e D(e~p)(f o) est l'application

Identique ; or e~ n'est surjective sur aucun voisinage de fo (3,5)�9

2) Consid@rons l'espace ~N des suites num@riques muni de la topologie pro-

duit. Soit ~: ~elR une application ind@finiment d@rivable ~ support contenu

dans le segment [I/2,3/2] et telle que @(I)=I. Nous noterons simplement

(x n) l'@l@ment (Xn)ne N de ~N.

25

L'application f: ~N § d6finie par f((Xn))=(Xn-#(Xn)) est continGment

B-diff6rentiable sur ~N et la d@riv@e Df(fo) , o~ fo=(O,O,...)~R N, est un , (I) hom@omorphisme �9 pourtant f n'est injective darts aucun voisinage de fo

-x 2) v@rifie les hypo- 3) L'application f: ~N § d@finie par f((Xn))=(x n n

th@ses du th@or@me a) au point fo (avec les notations de l'exemple pr@c@-

dent) ; or elle n'est injective dans aucun voisinage de fo (5).

4) Consid6rons l'espace ~N muni de la topologie pr@c6dente. Pour n=1,2,...,

posons -2 n -2 n-1 -I

x = (2 ,2 .... ,2 ,I,0,0,...), n Yn = (O,0,.~0,I,0,0 .... }.

n

Pour kEN', on pose x k =n (2-~)Xn et y~ = (2-~)y n. En particu

I ~N ~N Yn = Yn" D@finissons une application f: § en posant

f(x2) = x2 -I (n=1,2 .... ; k=2,3 .... ) ;

I ier x = x et n n

f(Xn) = Yn (n=1,2,...) ;

f(yk) = yk+1 (n=1,2,... ; k=],2 .... ) ;

k f(x) = x six ~ {x k} U {yn }.

L'application f v6rifie les hypoth@ses du th@or@me b) au pont fo=(O,O,...); - (I)

pourtant l'application f I n'est pas B-diff@rentiable au point f(fo)=fo

Or on remarque que, si C(R,~) et ~N ne sont plus munis des topologies

pr@c@dentes - qui leur conf@rent une structure d'evt - mais de topologies

plus fines qui font de ces espaces des anneaux topologiques, les applications

consid@r@es plus haut v@rifient un th@or@me d'inversion locale analogue aux

th@or@mes a) et b).

Soit X une vari@t@ r@elle de dimension finie et C(~,X) I'espace des

applications continues sur X ~ valeurs dans ~. Le but de cet article est de

donner un th@or@me d'inversion locale pour des applications

~: C(~,X) § C(~,X)

et, dans ce cadre, d'examiner les exemples @tudi@s plus haut.

26

II. TOPOLOGIE SUR C(~,X).

Soit (Ui)iE J un recouvrement ouvert, d@nombrable et localement fini

de X, form@ d'ensembles connexes relativement compacts qui sont des domai-

nes de d@finition de cartes de X ; soit encore (ai)iej une partition con-

tinue de l'unit@ subordonn@e au recouvrement (Ui)i~ J. D@signons par (Ki)ie J

l'ensemble des supports des applications ai ; chacun des K i est compact et

l~ensemble (Ki)i~ J constitue un recouvrement d@nombrable locaiement fini

de X. Pour tout i~J, on peut supposer qu'il existe xEK i tel que x ~ Kj si

j # i (sinon on retire K. du recouvrement). i

Soit C(~,K i) l'espace des applications continues de K i dans ~ muni

de la topologie de la convergence uniforme.

Si f~C(~,X), nous poserons Ifl = sup If(x)l. xeK.

i

Pour tout i ~ J, soit Pi l'application de restriction

Pi: C(~,X) + C(R,K i)

et soit I la topolog e sur C(R,X) pour laquelle un syst@me fondamental de

~jPT 1 voisinages de l'appl cation nulle fo est form@ des ensembles i (Wi)'

lorsque W i dTcrit un syst@me fondamental de voisinages de l'application

nulle dans C(R, Ki) , es voisinages des autres points de C(~,X) @tant d@fi-

nis par translation.

Ainsi, un vois nage de f dans C(~,X) contiendra un ensemble o

jWKi ' W=i~ '~i o~ cie~ pour tout i@J et oO

WKi,Ei = { fs / Ifli ~ ~i }"

Dans route la suite, V d@signera le filtre des voisinages de fo dans

T, et ~: C(R,X)xC(~,X) § C(IR,X) sera l'application de multiplication. Si F

et F' sont des filtres sur C(~,X), nous noterons FF' le filtre engendr@ par

les ensembles ~' = K(~x~'), o~ ~eF et ~'cF'.

Un filtre F sur C(IR,X) sera dit born# si le filtre VF converge vers

fo dans T ; une partie McC(R,X) sera dite bornde si ie filtre [M] des sur-

ensembles de M est born@.

Notons ~(T) et B(T) respectivement l'ensemble des filtres born@s et

des parties born@es de C(~,X).

Proposition II-I: (C(R,X),T) est un anneau topologique localement born@

(i.e. to pose@de un voisinage born@ dans l).

27

A I) L'application (f,g)~f+g est manifestement continue de TxT vers T.

2) L'application K est continue au point (fo,fo) : si W=i~jWKi,s iest un

voisinage de fo dans T, alors V=i~jWKi,~ i est un voisinage de fo dans T

et W contient K(VxV).

3) Soit f ~C(~,X) ; l'application Kf: C(~,X)+C(~,X) telle que ~f(g)=fg

~jWKi est un voisinage de f dans est continue en fo: en effet, si W= i 'si o

~jWKi,h , avec ~i = si / Ifli si l#li 0, T, alors W contient Kf(V), oO V= i i

ni=1 sinon.

4) L'ensemble M ~ = { f e C(~,X) / suplf(x)l ~ I } est un voisinage de fo xeX

dans T ; de plus M ~ est born@ car, si West un voisinage de fo dans T, W

contient ~(M xW). ? O

Proposition 11-2: f o de C(~,X).

est adh@rent ~ l'ensemble r des @l@ments inversibles

5 Soit W= N W un voisinage de f dans T. Pour tout i~J, posons i~J Ki,~ i o

A ( i ) = { hEJ / K i ~ K ~ # r } .

I I e x i s t e (4) un ensemble ( 6 i ) i e J de r6e ls s t r i c t emen t p o s i t i f s v @ r i f i a n t :

CA(i) ==> 6 x ~ ~i/card A(i).

L'application f=i~J~i~i est inversible et appartient ~ W. V

L'anneau C(R,X) est ordonn@ par la relation

f ~ 9 4==}, f(x) ~ g(x) VxEX. Soit fl l'application constante sur IER. Posons

[fo,fl] = { ~EC(R,X) / O=i~jai~i, aie[O,1]cR ViEJ } .

Proposition II-3: L'ensemble [fo,fl] ales proprr suivantes :

a) Si ~E [fo,fl], alors fo~<~#,<fl et fl- ~ ~[fo,fl].

b) L 'ensemble [fo,fl] ~ r est complet pour l 'ordre induit.

c) Si #,~E[fo,fl] et si ~+~ ~< f1" alors ~+~[fo,fl].

d) Les ensembles $9 = { ~ ~[fo,fl] / 9~<~ }, pour ~ ~[fo,fl] C~F, en#endrent

un filtre S qui converge vers fo dans T.

e) Si ~[fo,fl]F~s et ~]fo,fl] = [fo,f1~-{fo}, il existe Xr=]fo,f1~ mi-

norant ~ et ~ et telle que x+f1-~e[fo,fl]f'Is

28

A a) Soit ~=i~jai~i C [fo,f1~; comme f1-~=i~j(1-ai)ai , il vient que fl- ~

appartient ~ [fo,f1~.

b) Si (i~jaliai)ic A est une pantie non vide de ~fo,f1~nF, alors i~jai~i

est la borne sup@rieure de cette pantie dams Efo,fl] D r, o0 a.=sup all.

c) Si ~=i~jai~i et 9=i~jbi~i appartiennent ~ [fo,fl], et si ai+b i ~ ~ pour

tout iEJ, alors ~+~=i~j(ai+bi)~ie [fo,fl].

d) Si ~=i@jai~i et ~=i~jbi~i appartiennemt ~ [fo,fl]nF, il existe X dams

[fo,fl]~F qui minore # et ~ : il suffit en effet de prendre X=i~jciai, o0

ci=inf(ai,bi)>O pour tout i6J. Ceci momtre que les ensembles S~, ~E[fo,fl]nF,

engendrent un filtre S sun C(R,X). D'apr@s la prop. 11-2, tout voisinage de

fo dams T contient un ensemble S# ; doric le filtre S converge vers fo dams

T.

e) Soit ~=i~jai~i ~ [fo,fl]nr et ~=i~jbi~id ]fo,fl] ; l'application

X=i~jciai, oO ci=inf(ai,b i) pour tout i~J, minore r et 9 �9 appartient

]fo,fl] et x+fl- ~ est inversible. 2

D6finition 11-I: Une pantie MCC(R,X) sera dite convexe si CM+~M C(~+~)M,

pour tous ~,gE[fo,fl] tels que {+~e [fo,fl]ns

II r6sulte de cette d6finition qu'une intersection de parties convexes

est convexe.

Si A est une pantie de C(~,X), l'enveloppe convexe de A, not6e C(A),

est l'intersection de tousles convexes contenant A.

D@finition 11-2: Une pantie M~C(R,X) sera dite absorbante si, pour tout

f~C(~,X), il existe r [fo,fl] ~F tel que l'ensemble S@.f = m($~x{f}) soit

contenu dams M.

Autrement dit, la pantie M est absorbante si elle appartient ~ tousles fil-

tres S.f = K(Sx[f]), pour f{C(R,X).

De la propri@t@ d), prop.ll-3, r6sulte qu'une intersection finie de

parties absorbantes est absorbante.

D@finition 11-3: Une pantie McC(R,X) est dite sym@trique si -M = M.

Proposition 11-4: Dams (C(~,X),T), l'application nulle fo poss~de un syst@-

me fondamental de voisina~es ferm~s, convexes, absorbants et sym@triques.

29

A Lorsque (mi)iEj d@crit l'ensemble de toutes les suites de nombres r@els

N W forment un syst@me fonda- strietement positifs, les ensembles W=ie J Ki,c i

mental de voisinages sym@triques de f . O

Soit W l'un quelconque d'entre eux.

a) West ferm@ : Soit fEC(IR,X) adh6rent ~ W dans T, et soit ~>O. L'ensem-

ble V=j~jWKj,n j , o~ ni=~ et nj=1 si j # i, est un voisinage de fo dans T.

Donc (f+V)r~W # g~, i.e. ii existe hr~C(IR,X) tel que lhlj ~< mj Vj~j,

Ib-fli .< ~ et lh-flj ~< I si j # i. Par suite I+I ~< If-hli + Ibli-< ~+~i ; cette in@galit6 6tant r@alis@e pour tout ~>0, il vient que Ifli ~< mi" En

faisant le raisonnement pour chaque indice i, on obtient que f appartient

W.

b) West convexe : Soit f et g dans W, # et @dans [fo,fl] tels que ~+9

appartienne ~ [fo,fl]f~s L'application h: X-~R d@finie par

(x) f (x) +~ (x) g (x) h ( x ) =

r

est continue sur x et, pour tout xEKi, on a lh(x)l < ~i; donc lhli ~ ~i" En raisonnant pour tousles indices iEJ, on obtient que h appartient ~ W.

Comme Cf+@g = (~+9)h, il en r@sulte que #W+gWC(9+9)Wo

c) West absorbant : Soit fEC(~,X) ; choisissons un ensemble (5i)i6 J de

r@els strictement positifs v6rifiant

XEA(i) ===> 5 X ~ ~i / card A(i) (cf. prop. 11-2) ;

on peut de plus supposer que 5 i ~ I pour tout iEJ. Posons ai=sup !fix 5. X e A ( i )

e t a i = i n f ( 5 i , ~ - i l ) s i n i 7 O, a i = 6 i s i n o n .

L'application ~=i~dai~i est inversible et appartient ~ [fo,fl]

Soit ~ E S@ et x~Ki; comme le(x)f(x)l~e(x)If(x)l=If(x)l ~ az%(x)~ X ~ A ( i )

)axaX(x), on obtient que : tf l ixsA( i 5 X 5 X

I) si fli / 0, alors l*(x)f<x)141fli x{a(i) ~%(x)~Ifli x~i) 7TIT %(x) <

X~A()6X~x(x) ~ Ei" Done l~f I i ~ ~i ;

2) si fl~ = O, alors l~fli = 0. En faisant le raisonnement pour tousles indices iEJ, il vient que @f appar

tient ~ W, pour tout @ de S~. Donc West absorbant. V

30

I I I . Ol FFERENTIABI LITE.

Rappelons que M est l'ensemble { f~_C(~,X) / suplf(x)l ,< ~ }. o xEX

Proposit ion 111-1: Soit r: C(IR,X)ei:;(IR,X) une application tells que r ( fo )= f o. Les assertions suivantes sont dquivalentes :

a) ( VF~T~(T))( VW~V)( _qVEV)(3~sF) / (f~V, he~) ==> r(fh)EfW ;

b) (VBEB(T))(VwEv)(9VEV) / (feV, heB)==> r(fh)efW ;

c) (VWEV)(~VEV) / (feV, h~Mo)== > r(fh)~-fW ;

d) (YheC(R,X))(VWEV)(~VEV) / (fc=V) ~ r(fh)~ fW ;

e) (YWEV)(3VEV) / (fEV) ==y r(f)~_fW.

A II est clair que a)=#b)==~c) et que a)=~d)=~e). Montrons que c)~a).

Soit FE~(T) ; le filtre VF est plus fin que V et M ~ appartient ~ V, doric

i l existe UeV et g~F tels que Mo:/)U g. D'apr@s la prop. 11-2, i l existe une

application y inversible dans U. Soit WeV ; l'ensemble W' = yW est un voi-

sinage de fo clans I. Pour ce voisinage, i l existe d'apr@s c) un voisinage

U'~V tel que

(f~.U', h~M o) ~ r(fh)EfW' (I).

Posons V = yU'~V ; soit fr et h~ ; ii existe k~U' tel que f = yk. Comme

yh~y~cU~cM et que k~U', on obtient, en utilisant (I), o

r(fh) = r(ykh)~ kW' = y-Ifyw = fW,

ce qui @tablit a).

Montrons enfin que e)=~a). Soit F~_~(T) et WEV. II existe U~=V et ~eF tel

que W~U~. Pour le voisinage U, i l existe d'apr@s e) U'~ V tel que

f~U' ~ r(f)e-fU (2).

II existe par ailleurs UIEV et ~I~F tels que U'~UI~ I. Pour f~U Iet h~En~1,

on aura fh~U1~laU' , donc, en utilisant (2), r(fh)~_fhUcfU~dfW, ce qui

@tablit a).

D@finition II I-I: L'application r sera dite petite si elle v@rifie l'une

des conditions @quivalentes pr@c@dentes.

Notons Hom X l'espace des applications l: C(IR,X)+C(IR,X) qui sont des

homomorphismes continus pour la structure de groupe topologique sous-jacen-

te 8 (C(R,X),T) et qui v@rifient de plus

V f,g~C(IR,X), l(fg) = fl(g).

L'espace Hom X sera muni de la topologie de la convergence uniforme

31

sur les born@s de C(~,X).

Soit ~: C(~,X)~(~,X) et ~oEC(~'X)" S'il existe leHom X tel que l'ap-

plication r: C(R,X)~(~,X) d@finie par

r(h) = 9(~o+h)-9(~o)-/(h)

soit petite, nous dirons que ~ est diff@rentiable en ~o ; 'application 1

sera appel@e d@riv@e de ~ en ~o et not@e Dg(~o).

La notion de diff@rentiabilit@ introduite iciest un cas particulier de

la notion 9@n6rale de diff@rentiabilit6 entre modules quasl-topologiques (2)

(C(~,X) @tant consid@r@ comme module topologique sur lui-m@me). C'est pour-

quoi on retrouve avec une telle d@finition les r@sultats habituels de calcul

diff@rentiel. Mentionnons seulement deux r6sultats particuliers qui seront

utilis@s pour d@montrer le th@or@me d'inversion locale.

Proposit.ion 111-I: Si ~ est diff@rentiable en ~oEC(R'X)' alors ~ est con-

tinue en ~o"

A On a ~(~o+h)-~(~o ) = D~(~o).h + r(h), oi rest petite. II suf~it d'@ta-

blir que rest continue en f . Soit W~V ; il existe U~V et VEV tels que o

W~UV. Utilisant la condition e) pour l'application r, on obtient l'exis-

tence d'un voisinage U'~V tel que

fEU' =~r(f) ~fU.

Pour fEU'~V~ on aura r(f)~fUcVU~W, ce qui @tablit la continuit@ de

ten f . V o

Proposition 111-2 (accroissements finis): Soit t: C(~,X)+C(~,X) une appli-

cation diff@rentiable en tout point de l'ensemble

Ef,g] = { ( f l - r / ~ E r }. WKi'qi tel que ~(h)=D~(h).f I appartienne ~ B Supposons qu'il existe B=i~ J

pour tout h ~'gl" Alors

I{(9)-r ~ nilg-fli , pour tout i~J.

& Soit V=i~WKi,s i un voisinage de fo dans T. Pour McC(~,X), d@signons par

~(M) l'enveloppe convexe ferm@e de M. Posons

= { CE~o,fl]~s / ~((f1-~)f+~g) - t(f)~{(g-f)(a+v)} }.

Montrons d'abord que ~ est non vide. Comme { est diff@rentiable en f, on

peut @crire, pour h~C(~,X) :

t((fl-h)f+h9) - {(f) = r - ~(f) = h(g-f)r + r(h(g-f)),

32

o0 r v@rifie (condition e) ):

(VW'~F)( 3V'~ V) / f'EV' ~ r(f')~f'W' (I).

Pour W'=V, il existe V'EF tel que : f'EV'~ r(f')Qf'V.

Comme le filtre ~(Sx~-f]) (notations de la prop. 11-3, d) ) est plus fin

que V , il existe S~S tel que K(S~x{g-f})CV'. Pour un tel ~ ~o,f1~Nr,

on aura ~(g-f)EV', donc r(~(g-f)) E ~(g-f)V ; par suite

~((f1-#)f+~g) - ~(f) E ~(g-f)B + ~(9-f)V = ~(g-f)(B+V)c~ ~{(9-f)(B+V)}.

Donc ~ est non vide. Comme [fo,fl]~r est complet pour l'ordre induit, il

existe m = sup ~E~fo,fl]~r.

Comme l'application l: [fo, f1~F + C(R,X) d@finie par

~(~) = ~-1(~((fl-~)f+~g) - ~(f)),

est continue, et comme ~{(g-f)(B+V)} est ferm@, Iest facile de voir que

m E ~ �9 Supposons m i f1" Comme ~ est diff@rentiable au point (fl-m)f+mg,

on peut @crire, pour h~C(~,X) :

~((fl-m)f+mg+h) - ~((fl-m)f+mg) = h~((fl-m)f+m 9) + r(h),

o~ r v@rifie (I). Pour W'=V, il existe V'E F tel que : f'~ V'~r(f')~f'V.

Comme pr@c@demment, il existe S~ES tel que K(Scx{g-f}) c-V'.

Pour ~S{, on aura 9(g-f)~V', doric r(~(g-f))E~(g-f)V.

D'apr@s la propri@t@ e), prop. ~I-3, comme ~s at fl-m~]fo,f1~,

il existe xE ~fo,f~ minOrant { et fl-m et tel que x+m C ~o,f1~nr. Pour un tel X , on peut @crire :

~((f1-(m+x))f+(m+x)g) - ~(f) = {((fl-m)f+mg+x(g-f)) - {(f) =

= ~((fl-m)f+mg+x(g-f)) - {((fl-m)f+mg) + ~((fl-m)f+mg) - {(f) =

= X(g-f)~((fl-m)f+mg) + r(x(g-f)) + {((fl-m)f+mg) - {(f)

x(g-f)B + x(g-f)V + m~{(g-f)(B+V)} = X(g-f)(B+V) + mG{(g-f)(B+V)} C

X~{(g-f)(B+V)} + m~{(g-f)(B+V)} ~ (x+m)G{(g-f)(B+V)},

en vertu du lemme suivant :

Lemme : Si M est convexe, alors l'adh@rence M de M est convexe.

A Soit ~ et ~ dans ~o,fl] tels que ~+~eEfo,f1~s Montrons que ~M+~M

est contenu dans (~+~)M. Soit ~ et y dans M ; il suffit d'@tablir que ~+9~

est adh@rent ~ (~+~)M, puisque (~+~)~ = (~+~)M (car ~+~ est inversible).

Soit V un voisinage de ~+~ dans T ; il existe des voisinages V~ de ~,

V~ de ~, ~ de ~ et ~de y tels que VDV~ + V~'. Par hypoth@se, il

existe ~ ~ ~M et ~ ~ 9'~M. Le point #~ +#~ appartient d'une part

V~+V@~', donc ~ V, et ~ ~M+~M d'autre part, donc ~ (~+~)M, puisque M est

convexe.

33

Nous venons d'@tablir que x§ appartient ~ ~ ; or x+m > m car x~]fo,fl] ;

ceci contredit l'hypoth@se m = sup ~. Donc m = fl et ~(9)-~(f)~{(g-f)(B+V)}.

Posons ~i = Ig-fli" L'ensemble (g-f)(B+V) @tant contenu dans l'ensemble

i~jWK i,hi(qi+~i)+~i ' qui est convexe et ferm@, il en r@sulte que @[9)-~(f)

appartient & i~jWKi,~i(ni+~i)+~i, donc que le(g)-e(f)li~Ig-fli(ni+~i)+~i

ViE J. Cette in@galit@ @tant v@rifi@e pour tout ~i>O, on en d@duit que

le(~)-~(f)li ~ Ig-flini �9 v

IV. STRICTE DIFFERENTIABILITE.

D@finition IV-I: Une application ~: C(~,X)@C(~,X) sera dite strictement

diff@rentiable au point ~o EC(~'x) si, dans un voisinage de ~o pour T, on

peut mettre 9 sous la forme

~(f+h) = ~(f) + D~(~o).h + R(f,h),

o~ D~(@o)~Hom X, et o~ l'application R: C(R,X)xC(~,X) § C(~,X), d&finie par

R(f,h) = ~(f+h) - ~(f) - D~(~o).h,

v@rifie la condition

(SD) (YB~B(T))(YV~V)(gU~V)(~U'~V) / (f~o+U ', g~U, h~B)~R(f,gh)~gV.

Si ~ est strictement diff@rentiable en #o' ~ est diff@rentiable en #o"

Proposition IV-I: La condition (SD) est @quivalente ~ la condition suivante :

(SD)' (VV~V)(~U~V)(~U'~V) / (f~o+U ', 9~U, h~Mo)~R(f,gh)~gV.

A II est clair que (SD) implique

et V~V ; comme Mo~F, il existe

application y inversible dans U I

T. Pour ce voisinage, il existe,

tels que

(f~o+U ', 9~U, h~M o)

(SD)'. Montrons la r@ciproque. Soit BEB(T)

U 1~V tel que MO~UIB, et il existe une

L'ensemble yV est un voisinage de f dans �9 o

d'apr@s (SD)', des voisinages U~V et U'~V

==~R(f,gh)EgyV.

Pour f~_r ', gEyU~V et hr=B, il existe g'EU tel que g = yg', et -lh~yBd_

CUIB~:Mo, donc R(f,9h) = R(f,9'yh]E 9'~V = 9~-IyV = 9V, ce qui @tablit (SD).q

Proposition IV-2: Soit ~: C(IR,X)+C(IR,X) une application diff@rentiable sur

un voisinage ~ d'un point ~o et strictement diff@rentiable en ~o" L'~pplica-

tion O~" ~ Hom X, f~*D~(f), eet continue en ~o"

A II faut montrer que, pour tout BEB(T) et tout VEV, i l existe U~_V tel

que $o+UC~ et (D$($o+U)-D~($o)).BCV.

34

Fixons Bet V. II existe un voisinage V l~V sym@trique et tel que V~VI+V I.

Pour V Iet B, il existe, d'apr@s (SD), des voisinages U Iet U I darts V tels

que :

(f ~ ~o+Ui, g~U1, h~B)=~R(f,gh)~gV I (I).

L'ensemble (~o+Ul)~ est un voisinage de @o dans T, donc il existe UEV

tel que ~o+UC(~o+U I)~. Pour f~U, g~U Iet h~_B, on aura, d'apr@s (I),

R(~o+f,gh)~ gV I (2).

Mais R(r = ~(~o+f+gh) - ~(r - D~(~o).gh =

= D~(~o+f) .gh + r (gh) - D~(~o).gh =

= (D r + r ( g h ) . Donc (D~(~o+f ) -D~(~o) ) .gh = R(~o+f ,gh) - r ( 9 h ) . Comme { est d i f f @ r e n t i a b l e au p o i n t ~ o * f , i l e x i s t e U 2 E V t e l que :

9eU 2 ==> r ( g h ) E g V 1. Pour gs 1~U2, on aura, en u ? i l i s a n t (2 ) ,

( D r 9V1+gV 1 cgV. En c h o i s i s s a n t dans U I ~ U 2 un @l@ment 9 i n v e r s i b l e , on o b t i e n t que

( D { ( ~ o + f ) - D ~ ( ~ o ) ) . h C V , donc que (Dr162 dV. q

V. THEOREME DU POINT FIXE.

P r o p o s i t i o n V - l : Soit B u n voisinage born@, sym~trique et ferm@ de fo dans

T et ~: B+B une application v@rifiant :

(VieJ)(Bnis [) / le(g)-e(f)li ~ nilg-fli vf,gs Alors ~ admet un point fixe unique ~ ~B.

Posons ~i,n = ~n(f)IK i ~C(~'Ki) ; ~i,~ = flK i' pour une application f~B.

Montrons que la suite (~i,n)n~N est de Cauchy dans C(~,Ki~. Comme Best bor-

.>0 tel que len(f)-fli ~ c i. Pour m>n, n@, il existe pour tout i@J un r@el c

I tm-n(f) I = l~m-n(f)-fl ~ c on a : lr lKi-flKi i i i"

Supposons que l~i,m_p_1-~i,n_p_ll i ~ qy-P-Ici ; alors

l~i,m_p-~i,n_pJi lem-P(f)IKi-en-P(f)IKili ltm-P(f)-r

nil~m-p-1(f)-~n-p-1(f)li ~ ny-Pc i.

Donc la suite (gi,n)n~N est de Cauchy dans C(~,Ki). I existe par cons@quent

une application #i ~C(~,K i) telle que ~i,n n-~ ~ r dans C(E, Ki).

35

Six ~ Ki~Kj, il est facile de voir que ~i(x)=#j(x). Comme Best ferm@

et que X est une vari&t@ de dimension finie, ii existe ~ B telle que

{iKi={i pour tout iEJ. Montrons que t(~)={. Pour cela, @tablissons que

e(#)jKi=r Ki=#i pour tout i~J. Fixons i~J et s>O. I existe NI>O tel que

n>N1~l~i,n-~il i ~< e/2 (I). D'autre part

l~i,n-~(r i lcn(f)IKi-~(~)IKili = l~n(f)-~(~)

=nil~n-1(f)iKi-ClKil i = nil~i,n_l-~il i.

II existe N2>O tel que : n>N2~l#i,n_1-~il i ~< e/2n i

Pour n>sup(Ni,N2), on aura, en utilisant (I) ef (2),

l~i-r i .< lr i + lr162 < ~.

Donc {(~)IK i

I~(~)-~(~) I i ~=~.~

i ~ ni l~n-l(f)-~l =

(2).

= ~i" Si maintenant 9~ Best tel que 9(~) =~, alors Im-eli niIg-~l;, pour tout i~_J, ce qu[ est contradictoire. Donc

VI THEOREME D'INVERSION LOCALE.

Proposition VI-I: Soit 9: C(~,X)e4S(~,X) une application diffdrentiable en

tout point d'un voisinage ~ de ~o ~C(~'X)" Supposons que ~ soit strictement

diff@rentiable en ~o et que D~(@o)~ Hom X soit un hom@omorphisme. Alors

il existe un voisinage V ~ de ~o et un voisinage W o de ~(~o ) tel que ~ soit

un hom@omorphisme de V o sur W o.

On peut supposer que ~o=fo , ~(fo)=fo et que O~(f o) est l'application i-

dentique Id de Hom X.

D@finissons une application ~: ~-~S(R,X) en posant ~(f) = f-~(f).

L'application ~ est diff@rentiable sur ~ et DT(f o) est l'application nulle

de Hom X (i.e. l'application constante sur fo ). D'apr@s la prop. IV-2, l'ap-

�9 Oonc D~ est continue en f plication D~: fz-hLlom X est continue en fo o"

Pour V=~Mo=i~WKi,i/2 , il existe U~V tel que

D~(U).M C V (I). 0

Choisissons U'EV tel que U'+U'+U'C_U et V ~ = i~jWKi,qi tel que VoC ~ et

VoMoCU'. Pour f EV ~ et ~ E [fo,fl], on a ~f~MoVo; donc tf~U' et, d'apr@s

(~), ~(~f)~v. Comme f-~(f) = ~(f)-~(fo ), on obtient, en appliquant le th@or@me des accrois-

36

sements f i n i s ,

If-~(f)li = l~(f)-~(fo)li ~ ~Ifli ~ ı89 Vi ~J.

Soit gCC(R,X) ; on d@finit une application ~g: VoeC(~,X) en posant

r ( f ) = g + f - r g

Pour g~W ~ !V , = 2 o on aura, d'apr@s ce qui pr@c@de,

ltg(f)li < Igli + If-e(f)l i < ai Vi ~J.

Donc, pour gEW o, l'application r applique V ~ dans V o. Soit maintenant f'

. +(V +V )M r et f" darts V ~ et #dans [fo,fl] L'@l@ment f"+~(f'-f") appartient ~ V ~ o o o

@MoVo+MoVo+MoVoCU'+U'+U'cU, donc, d'apr@s (I),

~(f"+~(f'-f"))~ V.

En appliquant le th@or@me des accroissements finis, on obtient :

ltg(f')-~g(f")li l~(f')-T(f")li < ı89 vi e J .

Le th@or@me du p o i n t f i x e m o n t r e q u ' i l e x i s t e f u n i q u e dans V ~ t e l que

~ g ( f ) = f~e=>g = r Donc r une b i j e c t i o n de V ~ su r W o .

Comme ~ e s t d i f f @ r e n t i a b l e sur ~, on s a l t d@j~ ( p r o p . I I 1 - 1 ) que r c o n -

t i n u e su r ~. M o n t r o n s que r e s t c o n t i n u e su r W o. S o i t pou r c e l a h e t k

dans W ~ ; posons f = r e t 9 = r De la r e l a t i o n

~ ( f ) - r = ( f - ~ ( f ) ) - ( 9 - ~ ( 9 ) ) = ( f - g ) - ( ~ ( f ) - ~ ( g ) ) ,

r@sulte que

l(f-9)(x)l < l(~(f)-~(g))(=)l + l(~(f)-r v=~ x .

F i x o n s i ~ J ; pou r x ~ K i , on o b t i e n t

l(f-g)(x)l ~ l{(f)-r + l~(f)-~(g)li ; I

donc If-91i < I~(f)-r + l~(f)-~(g)li 4 I~(f)-r i + 21f-gli; il vient finalement :

Jr-g1; < 21r162

soit encore

l~-1(h)-r < 21h-kli.

Ceci @tant valable pour tout iEJ, on obtient que ~-I est continue sur W o. V

P r o p o s i t i o n V l - 2 : Avec les notations de la proposition pr@c#dente, l'appli-

cation ~-I est strictement diff@rentiable en ~(@o ) et

(D~-I)(r162 = (Dr & Nous prendrons encore ~o=fo , ~(@o)=fo et D~(~o)=ld.

Comme ~ est strictement diff@rentiable en r on peut @crire :

37

~(f+h) = ~(f) + h + R(f,h),

o~ R v@rifie la condition (SD)'. Montrons que ~-I est strictement diff@ren-

tiable en fo" avec D~-1(f o) = Id. Posons

~-1(g+k ) = ~-1(g) + k + S(g,k),

et v@rifions la condition (SD)' pour l'application S. Soit V#_V ; on doit

trouver UEV et U'EV tels que

(gEU', g'~U, kEM o) ~S(g,g'k)Eg'V (I).

I Comme l'ensemble f1+~Mo est born@, i l existe un voisinage sym@trique V' de

fo tel que V contienne (f1+lMo)V ' . Posons V I = V'f] 3 IMo ~v" Pour le voisinage VI, i l existe U IEV et U~V tels que :

(fEUI, f'EUI, h~M o) ~R(f,f'h)~f'V I (2).

Choisissons WI•V tel que WI+W I soit contenu dans UI/qU Iet que W I soit

sym@trique. Comme ~-I est continue sur We, i l existe W I ~V tel que

Comme M ~ est born@, i l existe U2EV tel que U2M ~ C_ W I

~ ~ CW I mosons U = U ' = U20U ~ �9 Soit enfin U 2~V tel que (fl + Mo)U 2

Pour gF_U', g'~U et k~_Mo, on aura g~_U'c_U2ClW ~, done ~-l(g)~WldlU I ;

g'kEUMoCU2MoCW~, done g+g'kEW~ + W~ et t-1(g+g'k)~W I.

Enfin, ~-1(g+g'k) - ~-I(g)~_WI+WICUIF~UI(/U I.

II est facile de voir que S(g,g'k) = -R(~-1(g),@-1(g+g'k)-~-1(g)) ;'

donc, en utilisant (2), on obtient que

S(g,g'k) ~-(~-1(g+g'k)-~-1(g))V I (en prenant h=f I dans (2)).

II existe ~# ~V 1 tel que S(g,g'k) =-~(r ;

or S(g,g'k) = ~-1(g+g'k)-{-1(g)-g'k ;

donc (f1+9)({-l(g+g'k)-{-1(g)) = g'k.

Mais +~=VIC~Vl O ; donc f1+{ est inversible et (fi+{)-I~ f1+~Mo �9

Par suite {-1(g+g'k)-{-1(g) = (f1+{)-Ig'k, et S(g,g'k)=-R(~-1(g),(f1+{)-19 'k).

Pour gEU', g'~U et kEMo, on a (f1+~#)-Ig'~(f1+ Mo)UC(fI+~Mo)U2~-WIC-UI ,

donc S(g,g'k)@:_(f1+~)-Ig'vicg'(f1+ı89 g'V.

Propos it ion V 1-3: Dans les conditions de la prop. V I-I, 1 'application

O~(f)EHom X est inversible pour tout f E V o.

38

A On a D~(f)=Id-~)~(f)~=@ D~(f)=Id-D~(f).

Montrons d'abord que Id-D~(f) est injective pour tout f~V o. Soit h~C(~,X)

te I que

(Id-D~(f)).h = fo ~ h = D~(f).h (I).

ZM Comme [hi est un ultrafiltre born@, il existe B inversible tel que Bh~ 2 o"

Soit kG2M ~ ~ tel que h = ~-Ik. En remplagant dans (I), il vient

l~-Ik = D~(f).B-Ik ~ k = D~(f).k (2).

Posons y = inf{ ~r=[fo,fl]nr / k~(~m o) }. Comme 2o Zm est ferm@, on ob-

tel que k = yk' I M tient que k Ey(Mo). II existe k'E 2 o . En utilisant (2),

I il vient : k = D~(f).k = yDY(f).k' = ~2D~(f).(k'+k')~2(~Mo),

!M car DY(f).MoC 2 o "

Donc y ~< y/2# par suite y = fo et k = fo" Donc Id-DT(f) est injective.

Montrons maintenant que Id-DT(f) est une application ouverte. Soit WET/ ;

montrons qu'il existe V~ST/ tel que (Id-D~(f)).W ~V.

II existe M inversible tel que ~Mo+~JMoCW. Posons V=~Mo~T/. Etant donn@ kEV,

montrons que l'@quation

(Id-D~(f)).h = k~h = k+DY(f).h (3)

poss@de une solution unique hCW. Recherchons h sous la forme h=h'+k. En

utilisant (3), le probl@me se ram@ne ~ trouver h'~V tel que

h' = D~(f).(h'+k) (4).

D@finissons une application F: V-~S(IR,X) en posant F(h') = DY(f).(h'+k).

Pour h' et k dans V, on aura h'+k = ~(hi+k I), pour h~ et k I dans M ~ ;

donc h'+k = 2M(hl/2 + ki/2)C

F(h') = DT(f).(h'+k)

Ceci montre que F est une appl

born@, sym@trique et ferm@ de

2MM o. Par ailleurs

= 2~O~(f).(hl/2 + kl/2)E 2~(ı89 o) = ~M o.

ication de V=MM ~ V, et Vest un voisinage

f dans T. 0

Soit maintenant h' et h" des @l@ments de V ; on a

F(h')-F(h") = DT(f).(h'+k) - D~(f).[h"+k) = D~(f).(h'-h") = (h'-h")~(f).

Pour fEV o, on a vu que~(f)~ I ~M o. Donc, pour tout i EJ, il vient

IF(h')-F(h")Ii ~ ı89 Le th@or@me du point fixe assure l'existence d'un unique h' EV tel que F(h')=h '

Cet h' est la solution cherch@e de (4).

39

Montrons enfin que Id-D~(f) est surjective. Soit k~C(~,X) ; cherchons

h~C(~,X) tel que (Id-D~(f)).h = k. Soit W un voisinage de fo dans T~ On

vient de voir qu'il existe V CV tel que (Id-DT(f)).W~V. Soit v inversi-

ble tel que vk~V. II existe h IEW tel que (Id-D~(f)).h I = ~k. L'applica-

-lhl tion h = ~ ~C(~,X) r@pond & la question.

Nous avons donc @tabli que Id-D~(f) est un hom@omorphisme, pour tout f ~V o.

Cocollaire VI-I: L'application ~-I est diff@rentiable en tout point g EW ~

et (D9-I)(9) = ((D~)(~-I(B))) -I.

Soit gEW ~ et f = ~-1(g). Comme D~(f) est un hom@omorphism%d'apr@s la

proposition pr@c@dente, on peut supposer que g=fo' f=fo et D~(f)=Id.

D@finissons une application s: C(R,X)+C(~,X) en posant s(k) = ~-1(k)-k,

et montrons que s est petite. Soit V~V ; il faut trouver U~V tel que

(f'~U, k~Mo)== > s(f'k)~f'V (I).

L'ensemble f1+~Mo_ @tant born@, il existe V' sym@trique dans V tel que

(fI+ı89 Soit Vl = V'n~Mo~V.

Par hypoth@se, l'application r: C(~,X)-~S(~,X) d@finie par r(h) = ~(h)-h

est petite, donc pour le voisinage V I de fo' il existe U 1~V tel que

(g'~U1, h~Mo)~r(g'h)~g'V I (2).

Comme 9 -I est continue sur Wo, il existe U'~_V tel que @-1(U')~Ui. Soit en-

fin U3~V tel que (fI+~Mo)U3~UI, et U2~V tel que U2MoCU'.

Posons U = U2~U 3. Pour f'EU et k~Mo, on aura f'k~UMoCU2MoCU'

-I (f'k)EU Iet, en utilisant (2) avec h=fl, on obtient que

r(~-1(f'k)) ~ ~-1(f'k)V I .

II exJste #EV I tel que r(~-1(f'k)) = ~-1(f'k).~.

Or s(f'k) = -r(~-1(f'k)) =-~-1(f'k) = ~-1(f'k)-f'k ; d'o~

f'k = (f1+9)~-1(f'k).

Mais ~ V IC~M 3 o ; donc f1+~ est invers

On peut donc @crire : ~-1(f'k) = (f1+~

Pour f'~U et kCMo, on a (f1+~)-If' E

s(f'k) = -r((f1+~)-Ifvk)~(f1+@)-If'v1

d@monstration. V

,donc

ble, et (f1+~)-1~f1+ı89 o.

-I f'k.

f1+~Mo)U I c(fI+~Mo)U3cU I ;donc

c(f1+ı89 , ce qui ach@ve la

40

On peut r@sumer les r@sultats pr@c@dents dans le th@or@me

Th@or@me Vl-1: Soit ~: C(~,X)eC(R,X) une application diff@rentiable en tout

point d'un voisinage ~ d~un point ~o ~C(FR'X)" Supposons que ~ soit stricte-

ment diff@rentiable en @oet que Dr soit un hom@omorphisme. Alors il

existe un voisina@e V ~ de ~o tel que r soit un hom@omorphisme de V ~ sur

~(V o) ; de plus l'application r@ciproque ~I est diff@rentiable sur ~(V o)

et strictement diff@rentiable en ~(@o ) ; pour tout f ~Vo, on a

(D~-r = (D~f)) -I.

Corollaire Vl-2: Si ~ est strictement diff~rentiable en tout point de ~ ,

alors ~-I est strictement diff@rentiable en tout point de @(V ) et -I o

( D ~ - l ) ( ~ ( f ) ) = ( D ~ ( f ) ) pour tout f ~ V o.

Vll. EXEMPLES.

Reprenons les exemples de l'introduction.

I) Cas de l'application exponentielle.

Consid@rons R comme une vari@t@ de dimension I, et consid@rons sur

C(~,~) la topologie T d6finie plus haut.

Proposition Vll-1: L'application ex~p: C(R,~)-K~(~,~) est strictement diff@-

rentiable en tout point de C(~,~).

A Soit @~C(~,~) ; d@finissons une application R: C(~,~)xC(~,~)e4S;(~,R) en

posant R(f,h) = e~(f+h) - ex~(f) - he~(@), et montrons que R v@rifie la

condition (SD)'. Soit f'EC(~,~) et x~R. On a :

R(f,f'h).x = exp[f(x)+f'(x)h(x)] - expel(x)) - f'(x)h(x)exp(r =

e x p ( @ ( x ) ) [ e x p ( ( f ( x ) - ~ ( x ) + f ' ( x ) h ( x ) ) - e x p ( f ( x ) - @ ( x ) ) - f ' ( x ) h ( x ) ] .

LI e s t c l a i r par a i l l e u r s que I V a p p l i c a t i o n R ( f , f V h ) / f T e s t con t i nue sur ~ ,

pour t o u t f v E c ( ~ , ~ ) .

En a p p l i q u a n t le th@or~me des acc ro issements f i n i s sur ~ , on o b t i e n t que

R(f,f'h).x _ h(x)exp(@(x))[ ex~(f(x)-@(x)+e f'(x)h(x)) -I ] = f'(x) x

f' (x)h(x)~exp(O~( f(x)-~(x)~O f'(x)h(x~), = h ( x ) e x p ( @ ( x ) ) E f ( x ) - O ( x ) + O x x

o~ e x et O'x appartiennent ~ ]O,I[cR.

~jWKi ' EV ; on cherche U et U' dans V tels que Soit W= i r

( f-@EU', f'~U, h~M o) =~R(f,f'h)~f'W (I).

41

Soit i~J- alors JR(f'f'h) I

posons n i = leli ; on peut toujours trouver deux hombres hi et Vi stricte-

ment positifs tels que

(~i+~i)ex~o(Zi§ ~ ~iex~(-~i)-

~jWKi ' et sent deux veisinages de fo qui r@pondent Alors U= i hi U'=i~jWKi ,~i

la question. V

2) L'espace ~N = C(~,N) sera muni de la topologie d@finie pr@c@demment.

Ainsi, un voisinage de fo=(0,0,...)E ~N contiendra un ensemble i~NVi , oO

V iest un voisinage de 0 dans ~ pour tout i ~J.

Soit r ~-~IR une application ind@finiment d@rivable ~ support dans

1 3 AN _IR N )) = (Xn_r [~,~] et telle que r Soit f: telle que f((x n .

Proposition Vll-2: L'application fest strictement diff@rentiable sur A N.

Soit Xo=(Xon)~N, et d6finissons une application R: A N x~R N § par

= )+(h )) - f((x )) - (hn-hnDr )). R((Xn)'(hn)) f((Xn n n on

Montrons que R v@rifie (SD)'. Soit (a)~N. On a : n

R((xn),(anhn)) = (-r162162 =

= (anhn).(Dr162 o~ enE]0,1[ ,

= (anhn).(Xon-Xn-enanhn).(D2r ,

o~ @in~]0, i[. Soit W = i~jVi, avec Vi=[-ei,ei]C~. 0n cherche deux voisinages

U=i~jW iet U'=i~jWI, avec Wi=~ni,ni] et Wi=E-nl,nl], et te~s que la relation

(I) de la prop. pr@c@dente soit v@rifi@e.

�9 ' stric- soit M = 0n peut toujours choisir deux nombres n iet n i

tement positifs tels que ni+n I < ~i/M. Ces deux nombres r@pondent & la ques-

tion. V

-x2). 3) Soit f: ~N § telle que f((Xn)) = (x n n

Proposition VII-3: L'application fest strictement diff@rentiable sur ~N.

A Soit (Xon)~ A N ; d@finissons une application R: ~Nx~RN § par

R((x ),(h )) = f((x )+(h )) - f((x )) - (h -2x h ) = n n n n n n on n

= (2h (x -x )-h2). n on n n

42

Soit (a n )~R N ; on obtient R((xn),(anhn)) = (anhn)(2(Xon-~n)-anhn).

Reprenons les notations de la d6monstration de la proposition VII-2 ;

' tels que 2qi+ni ~ ~i on peut toujours choisir deux nembres qi et H i

et ces hombres sont solution du probl@me. V

4) Reprenons les notations de l'exemple 4), et soit W=i~jVi, o~ Vi=]-1,1 [.

Sur ce voisinage, l'app cation f induit ~l'identit@, donc elle est stric-

tement diff@rentiable.

On peut ainsi app quer le th6or@me VI-I aux quatre exemples pr@c@-

dents.

(1)

BIBLIOGRAPHIE

V.l. Averbukh and O.G. Smolianov , The various definition~of the deri-

vative in linear topological spaces, Russian Math. Surveys 23 (1968)~

n ~ 4, 67-113.

(2) F. Berquier , Calcul diff6rentiel dans les modules quasi-topologiques.

Vari6t6s diff@rentiables (~ para?tre).

(3) J. Eells ~ A setting for global analysis, Bull. Amer. M.S. 72 (1966),

751-807.

(4) Vu Xuan Chi, C.R.A.S. ~r~s 27~ (I~73).

(5) S. Yamamuro , Differential Calculus in topological Linear Spaces~

L.N. 374, Springer Verlag 1974.

Th6orie et Applications des Cat6gories

Facult6 des Sciences

33 rue Saint-Leu

80039 AMIENSj FRANCE

Charaktergruppen von Gruppen von Sl-wertigen stetigen Funktionen

E.Binz

Die vorliegende Arbeit erg~nzt den in [Bu] und [Bi] beschriebenen

Versuch, die Pontrjaginsche Dualit~t f~r lokalkompakte Gruppen auf eine

grUYere Klasse von Gruppen auszudehnen~ Dies geschieht im folgenden

allgemeinen Rahmen.

Es ist wohlbekannt, da~ die Kategorie T der topologischen R~ume nicht

kartesisch abgeschlossen ist~ Die Versuche, geeignete kartesisch ab-

geschlossene Kategorien topologischer R~ume zu konstruiezen, sind

mannigfach.

In [Bi,Ke] wurde T als volle Unterkategorie in eine grUYere Kate-

gorie, der Kategorie der Limesr~ume eingebettet. Unter Verwendung der

Limitierung der stetigen Konvergenz auf den Funktionenr~umen, wird

letztexe zu einer kartesisch abgeschlossenen Kategorie.

Im Rahmen dieser Kategorie lassen sich bekannte Dualit~tstheorien, wie

unten kurz skizziert werden soll, ausdehnen. Zu diesem Zweck wieder-

holen wit aus [Bi] und [Bi,Ke] kurz die Elemente der Terminologie

aus der Theorie der Limesr~ume.

Eine Menge X f~r die in jedem Punkt p 6 X ein System A(p) yon

Filtern auf X vorgegeben ist, heist ein Limesraum, falls f~r jedes

p 6 X die folgenden Bedingungen erfOllt sind:

Die Arbeit entstand w~hrend eines Aufenthaltes im WS 1973/74 am

Forschungsinstitut der ETH, Z~rich.

i)

2)

und

3)

44

p, der von {p} c X erzeugte Filter geh~rt

A(p) an.

Mit ~,~ 6 A(p) geh~rt das Infimum �9 A

zu A(p) .

Mit jedem �9 6 A(p) und �9 ~ �9 ist auch

6 A(p)o

Die Filter in A(p) heiBen die gegen p konvergente Filter. Offen-

bar ist Jeder topologische Raum ein Limesraum. Die Stetigkeit von

Abbildungen zwischen topologischen R~umen verallgemeinert sich wie

folgt:

Eine Abbildung f vom Limesraum X in einen Limesraum Y heiBt

stetig, wenn fur jeden Punkt p 6 X und fur jeden gegen p konver-

genten Filter ~, der Filter f (~) gegen f(p) konvergiert.

Auf C(X,Y) , der Menge aller stetigen Abbildungen von X nach Y,

fUhren wir nun die Limitierun~ der steti~en Konver~enz ein. Dabei be-

zeichnen ~ : C(X,Y) • X ) Y die Evaluationsabbildung, die jedes

Paar (f,p) nach f(p) abbildeto In C(X,Y) konvergiert ein Filter e

gegen f 6 C(X,Y) , falls fur jeden Punkt p 6 X und jeden gegen p kon-

vergenten Filter ~ der Filter ~(8 x ~) gegen f(p) konvergiert.

Tr~gt C(X,Y) diese Konvergenzstruktur, so schreiben wir C (X,Y) . c

Wenn X ein lokalkompakter to~olo~ischer Raum ist, und Y uniformi-

sierbar ist, dann ist die Limitierun~ der steti~en Konver~enz mit der

To~olo~ie der kompakten Konver~enz identisch. Die erwahnten Dualit~ts-

theorien, wie sie in [Bi] mehr oder weniger ausfUhrlich beschrieben

wurden, sind die folgenden:

45

a) Dualit~t zwischen Limesr~umen und limitierten Funktionen-

algebren:

FOr jeden Limesraum X ist Cc(X,~) , kurz Cc(X) , eine

Limesalgebra (Operationen sind stetig) o Versehen wit die

Menge Hom C (X) aller stetigen reellwertigen ~-Algebren- c

homomorphismen mit der Limitierung der stetigen Konvergenz,

erhalten wir den Limesraum HOmcCc(X) for den

i x : X > HOmcC(X ) ,

definiert durch ix(P) (f)=f(p) fur alle f 6 Cc(X) und

alle p 6 X , eine stetige Surjektion ist~ Ein Raum X heist

c-einbettbar, wenn i x bistetig iSto Jeder vollst~ndig re-

gulare topologische Raum ist c-einbettbar. FOr jeden Limes-

raum X ist insbesondere Hom C (X) c-einbettbaz. Jede ste- cc

tige Abbildung f zwischen zwei Limesr~umen induziert durch

Kompositionen mit f einen stetigen ~-Algebrenhomomorphismus

und, falls die Raume c-einbettbar sind, wird jeder stetige

~-Algebrenhomomorphismus dutch eine stetige Abbildung indu-

ziert. Zwei c-einbettbare Limesr~ume X und Y sind des-

halb genau dann homSomorph, wenn Cc(X) und Cc(Y) bistetig

isomorph sindo Welter gilt for jeden Limesraum X, daS Cc(X)

und Cc (H~ bistetig isomorph sindo Die eben beschrie-

bene Dualit~t extendiert die Beschreibung kompakter topolo-

gischer R~ume X mit Hilfe der Banachalgebren C (X) . c

46

b) Zur universellen Darstellun@ yon Limesalgebren:

Sei A eine kommutative, assoziative Limesalgebra mit Ein-

selelement (~ber ~) , far die Hom A, die Menge aller

stetigen reellwertigen R-Algebrenhomomorphismen nicht leer

sei. Dieser Raum, versehen mit der Limitierung der stetigen

Konvergenz, heiBt Hom A, c

Der Homomorphismus

und wird Tr~ger yon A genannt.

d : A > Cc( HOmcA) ,

die universelle Darstellung von A, ist definiert durch

d(a) (h)=h(a) far alle a 6 A und h 6 Hom Ao Dieser Homo- c

morphismus ist stetig. Er ist genau dann bistetig, wenn A

die von d initiale Limitierung tr~gt, vollst~ndig ist und

auBerdem A auf Hom A dieselbe schwache Topologie erzeugt

wie sie C(HOmcA) induziert. Die Theorie der universellen Dar-

stellungen wird dutch die Absenz eines gen~gend starken

Stone-Weierstrass'schen Satzes erheblich kompliziert.

Wenn A topologisch ist, ist Hom A in der Kategorie der c

Limesr~ume induktiver Limes von kompakten topologischen

R~umen, also ein lokalkompaktes Objekt. (Der zu Hom A c

assoziierte topologische Raum ist dann ein k-Raum.) Die Li-

mitierung auf C (Hom A) wird in diesem Falle mit der Topo- c c

logie der kompakten Konvergenz identisch.

Diese Dualit~tstheorie extendiert die Darstellung gewisser

topologischer Algebren als Funktionenalgebren, versehen mit

der Topologie der kompakten Konvergenz ~ber einem k-Raum

als Tr~ger.

47

c) Die lineare Dualit~tstheorie:

Sei E ein ~-Limesvektorraum (Limesraum, fur den die

[R-Vektorraumoperationen stetig sind). Der Dualraum E, die

Menge aller reellwertigen stetigen Abbildungen, sei mit

LE bezeichnet. Dieser Raum, versehen mit der Limitierung

der stetigen Konvergenz heiBt der c-Dual von E und wird

mit L E bezeichnet. c

Die nat~rliche Abbildung

i E E > L L E,

c c

die durch iE(e) (1) = l(e) fur alle e 6 E und alle

1 6 L E definiert ist, ist stetig. Ein Raum E heiBt c

c-reflexiv, wenn i E ein bistetiger Isomorphismus ist.

Als Beispiele c-reflexiver Limesvektorr~ume seien Cc(X)

ffir jeden Limesraum X und jeder vollst~ndige lokalkonvexe

topologische Vektorraum genannt.

Wenn E topologisch ist, dann ist L E induktiver Limes c

(in der Kategorie der Limesr~ume) aller abgeschlossenen

gleichstetigen Mengen, versehen mit der schwachen Topologie

(diese Mengen sind kompakte topologische R~ume) . L L E c c

tr~gt in diesem Falle die Topologie der kompakten Konvergenz,

ist also lokalkonveXo Diese Dualit~tstheorie verallgemeinert

mithin die von Grothendieck zur Vervollstandigung lokalkon-

vexer R~ume herangezogene Dualit~t.

48

Die Ausarbeitung all dieser Dualit~tstheorien ist nicht abgeschlossen.

Sie birgt zum Teil erhebliche Schwierigkeiten. Andererseits ~ffnen die

Ausdehnungen ein reiches Feld von Zusammenh~ngen verschiedener Struk-

turen vergleichbar denen, die zwischen funktionalanalytischen Eigenschaf-

ten von C (X) und topologischen Gegebeneheiten in X hezrschen (s" [Bi]) " c

Wit wenden uns nun dem Versuch zu r die klassische Pontrjaginsche

Dualit~t lokalkompakter Gruppen auszudehnen. Dabei beschr~nken wir uns

im Wesentlichen auf den Bereich der Reflexivit~t und der Beschreibung

einiger auftretender Charaktergruppen, lassen abet die Zusammenh~nge

zwischen Gruppen und Charaktergruppen (Untergruppen, Annihilatoren)

der Schwierigkeiten, die bei der Erweiterung yon Charakteren auftreten,

auBer Acht. Herrn Bernd MUller und Herrn Heinz-Peter Butzmann verdanke

ich die Form des Beweises von Lemma 8.

FUr eine beliebige kommutative Limesgruppe G (kommutative Gruppe,

versehen mit einer Limesstruktur fur die die Gruppenoperationen stetig

sind) bezeichnet F G die mit der Limitierung der stetigen Konvergenz c

i versehene Gruppe aller stetigen Gruppenhomomorphismen von G in S

Wit nennen F G die Charaktergruppe von G. Wenn die Limesgruppe G c

eine lokalkompakte topologische Gruppe ist, dann ist die Limitierung

der stetigen Konvergenz auf F G mit der Topologie der kompakten c

Konvergenz identisch. F~r die reichhaltige Theorie der lokalkompakten

topologischen Gruppen sei auf [Po] verwiesen. Der nat~rliche Homo-

morphismus

Js : S > F F G , c c

definiert dutch jG(g) (y) = y(g) f~r jedes g 6 G und jedes y 6 FcG

49

ist f~r jede Limesgruppe G stetig. Falls nun G eine lokalkompakte

topologische Gruppe ist, wird JG' wie aus der Pontrjaginschen

Dualit~tstheorie bekannt ist, ein bistetiger Isomorphismus. Wir nennen

deshalb eine Limesgruppe G f~r die JG ein bistetiger Isomorphis-

mus ist, P - reflexiv. Im Zuge der Ausdehnungsversuche der Pontrja- c

ginschen Dualit~tstheorie ist fur gewisse Typen von nicht-lokalkompak-

ten topologischen Gruppen (ja sogar f~r Limesgruppen) die P -Reflexi- c

vitat festgestellt wordeno So z.B. ist wie in [Bu] gezeigt wurde, f~r

jeden Limesraum X die mit Limitierung der stetigen Konvergenz ver-

sehene R-Algebra Cc(X) aller stetiger reellwertiger Funktionen von X,

aufgefaBt als Limesgruppe, P -reflexiv. Daraus ergibt sich insbesondere, c

dab ein topologischer lokalkonvexer Limesvektorraum, aufgefaBt als

topologische Gruppe, genau dann P -reflexiv ist, wenn er vollst~ndig c

ist (siehe [Bi]) . Jede dieser Typen von P -reflexiven Limesgruppen C

tr~gt aber eine zus~tzliche algebraische Struktur~ die zum Nachweis

der P -Reflexivit~t auch ausgen~tzt wurde. c

I Unter C (X,S ) c verstehen wit die Gruppe aller Sl-wertigen Funktionen

von X, die mit der Limitierung der stetigen Konvergenz versehen s•

Die Gruppenoperationen sind punktweise definiert. In der vorliegenden

nun f~r gewisse Gruppen der Form Cc (X,SI) , wobei X ein Note soll

normaler topologischer Raum ist, der eine (einfach zusammenhangende)

universelle Uberlagerung zul~t, die P -Reflexivit~t nachgewiesen wer- c

den (s.Satz 24). Dabei wird sich am SchluB f~r gewisse CW-Komplexe

eine nat6rliche funktionalanalytische Interpretation der ersten singu-

l~ren Homologiegruppen ergeben.

50

Zum Verst~ndnis des Aufbaus der vorliegenden Arbeit, skizzieren wir

kurz den Nachweis der P -Reflexivit~t von C (X,SI) . C c

Der Charakter

i : ~ > S ,

2Zir der jeder reellen Zahl r die komplexe Zahl e zuordnet, in-

duziert (dutch Komposition) einen stetigen Homomorphismus

• : C (X) > C (X,SI), X c c

der sogar eine Quotientenabbildung auf das Bild, aufgefaBt als Unter-

raum von Cc(X,SI) , ist. Die Gruppe

C(X,Sl)/• x C(X) ,

die sogenannte Bruschlinskische Gruppe [Hu], die mit ~i (X) be-

zeichnet wird, wird durch Einf~hren der Quotientenlimitierung eine

i L i m e s q r u p p e . S i e h e i B t H (X) . D a m i t e r h ~ l t man e i n e t o p o l o g i s c h e c

exakte Folge 1 o >~x c (x) ~ c (x s ) >H i(x) > o C C t C

Um nun die P -Reflexivit~t Yon C (X,S I) nachzuweisen, zeigen wir c c

erst, dab ~X C c ( X ) u n d HI(X)c ( f a l l s v o l l s t a n d i g ) P c - r e f l e x i v s i n d

und verwenden dann, um unser Ziel zu erreichen, eine g~ngige Schlug-

weise Ober das FOnferlemma. Dabei beinhaltet ein groBer Teil der Ar-

beit den Nachweis, dab die Duale und die Biduale der obigen kurzen

exakten Folge, exakt sind. Leider ist uns nicht bekannt, unter welchen

Bedingungen HI (X) vollst~ndig ist. Hinreichend fur die Vollst~ndig- c

keit sind: HI(x) ist diskret (also etwa, wenn HI(X) oder Homl (X,2) c

endlich erzeugt sind)oder HI(X) = Hom(~l (X) ,~) gilt. Letzteres ist

r i c h t i g , f a l l s X e i n C W - K o m p l e x i s t ( [ S p ] p . 4 2 7 ) .

51

Zum Studium der Charaktergruppe yon M x Cc (X) machen wir weitgehend

vonder l i n e a r e n D u a l i t f i t s t h e o r i e v o n C (X) G e b r a u c h . D a b e i w i r d v o n c

den Voraussetzungen an X nur dex wegeweise Zusammenhang ausgen~tzt.

Bei den Untersuchungen von F H I (X) verwenden wir die Theorie der c c

n 1 0berlagerungen um (X) einerseits als Quotient yon Gruppen reell- c

wertiger Funktionen von ~ (der universellen 0berlagerung yon X)

und andererseits als Teilgruppe Hom(r[ 1 (x),~) aller ~-wertigen Homo-

morphismen der Fundamentalgruppe II 1 (X) von X (in natfirlieher Weise)

zu interpretieren. Dabei wird sich herausstellen, dab Ill (X) mit einer c

topologischen Teilgruppe der mit der Topologie der punktweisen Konver-

genz versehenen Gruppe HOms(II 1 (X),2) identifiziert werden kann. Diese

_ H i Tatsache wird hernach beim Nachweis der P Reflexivit~t von (X) c c

wesentlich verwendet.

i I) Beziehungen zwischen C(X) und C(X,S )

Sei X ein Hausdorffscher topologischer Raum. Die Beziehungen zwischen

C ( X ) u n d C ( X , S 1 ) , d i e i m f o l g e n d e n k u r z b e s c h r i e b e n w e r d e n , e r m 6 g -

lichen es uns, die P - Reflexivit~t der Limesgruppen vom Typ C (Y) , e c

wo Y i r g e n d e i n L i m e s r a u m i s t , z u m N a c h w e i s d e r P - R e f l e x i v i t N t c

von C (X,S I) fur gewisse Typen von topologischen R~umen X, heran- c

zuziehen.

i Mit M : R ) S bezeichnen wir die Exponentialabbildung, de-

finiert dutch M(r)= e 2~ir fur jedes r 6 ~. Bez~glich K ist

die universelle 0berlagerung von S I. Offenbar ist M bez~glich der

addiven Gruppenstruktur von ~ ein stetiger Gruppenhomomorphismus.

Dieser induziert den Gruppenhomomorphismus,

i) ~ : C(X) > C(X,S i) , X

52

der jedes f 6 C(X) nach M o f abbildet. Der Kern yon M besteht X

aus all denjenigen Funktionen aus C(X) , die X nach 2 c ~ abbil-

den. Ist X zusammenh~ngend, gilt ker MX=~, wo ~ die Gruppe aller

konstanten Funktionen aus C(X) deren Werte in ~ liegen bezeichnet.

Wit haben also in diesem Fall die exakte Folge

i X 2a) o > _~ ) C(X) > Mx(C(X)) > o.

Dabei bezeichnet i I die Inklusionsabbildung. Offenbar ist in (i)

1 die Abbildung M X i.a. nicht surjektiv, da z.B. die Identit~t auf S

1 nicht als ]4x(g) wo g 6 C(S ) darstellbar ist. Um die technischen

Auswirkungen dieses Umstandes abzumildern, setzen wir f~r alles Weitere

voraus, daS X ein wegeweise, lokalwegeweise und lokal-halbeinfach-

zusammenh~ngender normaler (Hausdorffscher) topologischer Raum ist.

([Sp] (chapter 2). Diese Voraussetzungen garantieren eine einfachzu-

sammenh~ngende universelle Oberlagerung ([Sp] (chapter 2). Diese heiBt

. Die kanonische Projektion von ~ auf X bezeichnen wir mit u.

-I Die Faser u (p) eines jeden Punktes p 6 X kann in nat~rlicher

Weise mit der Fundamentalgruppe HI (X,p) identifiziert werden. In

spateren Ausf~hrungen werden wir oft vonder eindeutigen Wegehoch-

hebungseigenschaft Gebrauch machen. Diese Eigenschaft von ~ besagt,

dab es zu jeder stetigen Abbildung ~ : [O,i] > X nach Festlegung

eines Punktes q 6 u-I (a(o)) eine eindeutig bestimmte stetige Abbildung

a' : [O,i] > X gibt, f~r die o' (o) = q und u o a' = Q gilt.

Diese eindeutige Wegehochhebungseigenschaft von X garantiert einen

f~r das folgende sehr wesentlichen Umstand (siehe [Sp] chapter 2):

ZU jeder stetigen Abbildung s von einem einfach zusammenh~ngenden

wege- und lokal wegezusammenh~ngenden Raum Y nach X gibt es eine

stetige Abbildung s' : Y > ~ mit s=u o s'. Setzt man f~r s'

53

-i ffir einen beliebigen Punkt p 6 X einen Wert aus M (s(u(p))) lest,

dann ist s'

t 6 C(X,S )

2b) o

eindeutig bestimmt. Mithin l~Bt sich jede Abbildung

zu einer stetigen Funktion ft' heben. Also ist

> 2 i ) c(~) > c(x,s I) > o

exakt. Neben diesen Sachverhalten werden wit noch ffir das Bestimmen

der Charaktergruppen yon C (X,S I) das im folgenden aufzustellende c

Diagramm (4) verwenden.

Die Projektion

morphismen:

3a) u : C (x)

3b) ~u : C(X,S I)

X induziert die beiden injektiven Homo-

und

> c(x,s I )

definiert dutch u (f) = f o u und

resp. t 6 C(X,SI).

u(t) = tou f~r jedes f s C(X)

Allgemein bedeuten f~r eine Abbildung a zwischen zwei Mengen A und B

die Abbildungen aS: ~(B,~) >~A,m) und ~a : JG4(B,S I) > ~(A,S I) die

zwischen den Raumen aller ~-wertigen bzw. S I- wertigen Abbildungen

von A und B induzierten Algebren beziehungsweise Gruppenhomomorphis-

men de finiert durch a (f) = f o a und a (t) = t o a f~r j ede Abbil-

dung f : B >~ und t : B ) S resp..

Offenbar ist

4)

c(~) ) c(x,s )

]u, x T *u c(x) ) c(x,s I )

54

kommutativ. Dabei ist M~ eine Surjektion ([Sp] chapter 2). Diesen

Umstand werden wir sparer wesentlich verwenden. Far die weitere tech-

* ,S 1 nische Ausnutzung von (4) sollen die Abbildungen in u(C(X )) ,

u (C(X)) und in Mx(C(X)) gekennzeichnet werden.

FGr jedes t 6 C(X,S I) ist u(t) :

tigen reellwertigen Funktion, sie heiBe wieder

gibt eine Funktion f 6 C(~) , f~r die t

5 )

ft )

i n t i 1 X )- S

i ) S zu einer ste-

ft' hebbar, d.h. es

kommutiert. Offenbar ist ft bis auf eine konstante ~-wertige Abbil-

dung bestimmt. Diese Funktion f bildet far jedes p 6 X die Faser t

-i -i u (p) in die Faser M (t(p)) c ~ ab, d.h. ft ist fasertreu.

Da X bezGglich u die Quotientenlimitierung, d.h. die feinste Li-

mitierung tr~gt, far die u stetig ist, ist for jede fasertreue Abbildung

f 6 c(X,S I)~ die Abbildung M o f im Bild yon u. Also haben wit:

Lemma 1

1 Far eine Abbildun~ t 6 C(X,S ) ist M~(ft )

u enthalten, wenn f fasertreu ist. t

@enau dann im Bild von

Daraber hinaus folgert man weiter:

Lemma 2

Eine Abbildun9 t s C(X,S ) is_~t ~enau dann in u (C(X))

wenn ft auf jeder Faser konstant ist.

enthalten,

55

2) Zu den Beziehungen zwischen C (X), C (~), c c

C ( X , S 1) u n d C ( X , S ) c c

Wit versehen MxC(X) mit der feinsten Limitierung f~r die

MX : Cc(X) ) MxC(X) stetig ist. Die Gruppe, versehen mit dieser

Limitierung, der Quotientenlimitierung bez~glich MX' heiBe MxCc(X) .

Diese ist eine Limesgruppe. Offenbar ist die Inklusion

i : MxCc(X) ~ Cc (X'SI) stetig.

Lemma 3

Es ist MxCc(X) ein Unterraum yon Cc(X,SI) .

Beweis: FOr diesen Beweis fassen wir MxC(X) als Unterraum von

Cc (X'Si) auf. Um zu zeigen, dab MxCc (X) ein Unterraum von Cc (X'Sl)

ist, gen~gt es (da MxC(X) eine Limesgruppe ist) , zu jedem in KxC(X)

gegen die konstante Abbildung ! mit Wert 1 6 S I konvergenten Fil-

ter e einen in C (X) konvergenten Filter e' zu finden, f~r den c

MX(~') ~ 8. Sei nun W c S I eine verm~ge M gleichmaBig Oberlagerte

offene Umgebung von I. Eine Menge U in einem topologischen Raum Y

mit Uberlagerung ~ und Projektion u : ~ > Y heiBt gleichmaBig

-I ~berlagert, wenn u (U) die disjunkte Vereinigung yon Mengen (ge-

nannt Bl~tter) ist, von denen jede verm6ge u hom6omorph auf U ab-

gebildet wird. F~r jeden festen Punkt p 6 X gibt es ein T 6 8

und eine offene Umgebung U von p derart, dab

T(u) c w

Weil T von p,U und W abh~ngt t ersetzen wir T durch T(p,U,W) .

Ohne Verlust der Allgemeinheit k~nnen wir annehmen, dab i 6 T(p,U,W) .

Wit heben nun jedes t 6 T(p,U,W) zu einer Funktion f 6 C(X) hoch,

i i

f~r die f(U) c W, wo W dasjenige Blatt ~ber W ist, das o ent-

56

h~lt. Damit ist die Hochhebung f f~r jedes t eindeutig bestimmt.

Die Gesamtheit all dieser Hochhebungen von Abbildungen aus T(p,U,W)

!

bezeichnen wir mit T' (p,U,W ) . Das Mengensystem e aller Mengen der

! Form T' (p,U,W ) besitzt die endliche Durchschnittseigenschaft, denn

sie enthalten alle die konstanten Funktionen o. (Konstante Funktionen

sollen stets mit dem unterstrichenen Symbol, das f~r den Wert gew~hlt

wird, bezeichnet werden) . Offenbar konvergiert der von 8' in C (X) e

bestimmte Filter in Cc(X) gegen o . Nun bieibt uns noch zu verifi-

n zieren, dab MX(8~') ~ 8. Sei /'~ T' (Pi,Ui,W i) 6 e vorgegeben. Die

i=i

Punkte PI' ..... 'Pn verbinden wir durch einen Weg s. Da die Limi-

tierung der stetigen Konvergenz auf C(X,S I) feiner als die Topolo-

gie der kompakten Konvergenz ist, gibt es eine Menge T 1 6 8, die das

n Bild von s in /'~ W i abbildet. Sei

i=l

n

T o = T 1 N /~ T(Pi,Ui,Wi) i=l

F~r jedes t 6 T ist der Weg f o s eine Hochhebung von t o s, o

falls f eine H o c h h e b u n g v o n t i s t . Da d i e W e g e h o c h h e b u n g e i n d e u t i g

i s t , f a l l s d e r A n f a n g s p u n k t v o r g e g e b e n w i r d , l a s s e n s i c h a l l e t 6 T o n

derart hochheben, dag sie das Bild yon s und damit ganz U u i nach n i=l

~ W i , a b b i l d e n . F o l g l i c h g i l t i = l

n TI o v MX( ~ (Pi,Ui Wi)) D T o ,

i=l

woraus sich MX(8') ~ e ergibt. (Man beachte, dab von X nur der

wegeweise Zusammenhang verwendet wurde.)

Da X bez~glich u die Quotientenlimitierung tr~gt, sind

u : C (X) > C (5) c c

* ( ~ , s 1 und u : Cc(X,S1) > C ) c

57

HomSomorphismen auf Teilr~ume.

Diese Tatsache, mit der Aussage yon Lemma 3 (X ist wegeweise zusammen-

h~ngend!) zusammengefa~t, ergibt:

Lemma 4

~ i) ES ist u : Cc(X,SI) > Cc (X,S

einen Teilraum. Im kommutativen Diagramm

6)

ein Hom6om.ozphis, mus auf

c (~) ~x w I > c (x,s ) c c

NX C c (X) > ~xCc (X)

sind die horizontalen Homomor~hismen Quotientenabbildungen (in der

Kate~orie der Limesr~ume) auf die Bilder und die vertikalen Homo-

mor~hismen HomSomorphismen auf Teilr~ume.

Bemerkung 5 Die Limesgruppe C (X,S I) ist topologisch, falls X c

lokalkompakt ist. Sie tra~t die Topologie de____rr kompakten Konver~enz.

Die Aussa~e in Lemma 4 ~ilt (der Regularit&t yon X we~en)

auch dann, wenn die Limitierun9 de r stetigen Konvergenz Gberall durch

di__~e Topologien der kompakten ~nvergenz ersetzt wird.

Beweis: Wenn Z ein kompakter Raum ist, dann stimmt auf C(Z,S I) die

Topologie der kompakten Konvergenz mit der Limitierung der stetigen

Konvergenz ~berein. Also ist C c (Z,S I) topologisch. Welter tr~gt f~r

1 jeden Raum Y die Limesgruppe Cco (Y,S ) die grSbste Topologie for

die die Restriktionshomomorphismen von Cco (Y,S I) und C c (K,S I) f~r

58

alle kompakten Teilmengen K c y stetig sind. F~r einen lokalkompakten

Raum X ist damit die E valuationsabbildung ~ : C (X,S I) x X ~ S I, co

die jedem P aar (t,P) den Weft t (p) zuordnet, stetig. Mithin ist in

diesem Fall C (X,S I) mit C (X,S I) identisch. Damit ist der erste co c

Teil der Aussage in Bemerkung 5 bewiesen. Als n~chstes zeigen wir, dab

jede kompakte Teilmenge K c X als Bild unter u einer kompakten Menge

K' c ~ dargestellt werden kann, sobald X regul~r ist. Sei also K c X

kompakt. Weil X regul~r ist, kann K mit endlich vielen gleichm~Big

Gberlagerten abgeschlossenen Mengen Ul, ..... ,U n so Gberdeckt werden,

dab fGr jedes i = l,...,n Gber jedem U. ein abgeschlossenes Blatt l

-1 V. liegt. Offenbar sind V O u (K) kompakt fur i=l,,..,n und damit l l

n u_ I K' = ~J V n (K) kompakte Teilmenge in ~ Zudem ist u(K') = K. l

i=l Daraus ergibt sich nun sofort, dab

und

u : Cco (X) ~ Cco

u : C (X,S I) > C (X,S I) co co

Hom6omorphismen auf Teilr~ume sind. Zum Beweis, dab MX und damit auch

M x Quotientenabbildungen auf ihre Bilder aufgefaBt als Teilr~ume von

C (X,S I) und C (X,S sind, verfahre man wie im Beweis von Lemma 4, C O CO

ersetze jedoch die U. durch kompakte Mengen. 1

Es sei noch hervorgehoben, dab Lemma 4 fur jeden wegeweise zusammen-

h~ngenden topologischen Raum gilt. Die erste Aussage in Bemerkung 5

gilt fur jeden topologischen Raum, w~hrend die darauffolgende vonder

Regularit~t und dem wegeweisen Zusammenhang Gebrauch macht.

Charaktergruppe von Cc(X,S I) wird das bis jetzt zu- Das Studium der

sammengestellte Material sowie einige Resultate Uber die Charaktergrup-

pen von C (X) und ~ Cc(X) ben~tigen. c X

59

3) Die Charaktergruppe von C (X) c

Da zur Bestimmung der Charaktergruppe yon Cc(X,S I) diejenige von

Cc(X) herangezogen wird, stellen wir kurz einige Ergebnisse fiber

F C (X) zusammen. Diese gelten auch ohne die 0her X gemachten Voraus- c c

setzungen. Sie gelten in der Tat f~r jeden Limesraum. Mit Hom c (X) c c

sei die mit der Lim• der stetigen Konvergenz versehene Menge

aller unitaren ~-Algebrenhomomorphismen bezeichnet. F~r jeden Punkt

p 6 X geh6rt ix(P) : Cc(X) > ~, der jede Funktion f 6 C(X)

nach f(p) ~berf~hrt, dem Limesraum Hom c (x) an. Die Abbildung c c

i : x > Horn c (x), X c c

die jeden Punkt p 6 X nach ix( p ) schickt, ist eine stetige Sur-

jektion [Bi].

Der Limesvektorraum ~C (X) , der mit der Limitierung der stetigen cc

Konvergenz versehene Vektorraum aller stetigen reellwertigen linearen

Abbildungen, enth~it den yon H0m C (X) erzeugte Vektorraum V(X) als c

dichten Teilraum [Bi] und [Bu] 0 Das bedeutet, dab

7) a : Cc (X) ~ ~c < Cc (X) ,

definiert durch a(f) (i) = l(f) f~r jedes f 6 C(X) und jedes

1 6 <Cc (X), ein bistetiger Isomorphismus ist [Bu] und [Bi].

1 Der Homomorphismus M : gR > S induziert einen bistetigen Iso-

morphismus

S) Xc(x) : ~cCc(X) > FcCc(X)'

der jedem 1 6 ~C (X) den Charakter M o i zuordnet ([Bu] , [Bi]) . C c

F~r einen einfachen Beweis sei auf [Bi] verwiesen. Die Gruppe

60

~C(X) (V(X)) ist demnach dicht in I"cCc (X).

Verwendung von (7) und (8) , dab

Daraus ergibt sich unter

9) JC (X) : Cc(X) ) FcFcCc(X) c

ein bistetiger Isomorphismus ist.

Diese Beziehungen werden wir gleich im n~chsten Paragr~hen ausn~tzen.

4) Die Gruppe P (X) , ihre Charaktergruppe c

und die P - Reflexivit~t von MxCc(X) c

Die Gruppe Pc(X) tritt beim Beschreiben der Charaktere von MxCc (x)

auf. Sie ist das Analogon zu V(X) , eingef6hrt im vorigen Paragraphen.

Wir betrachten for jedes p 6 X den Charakter M o ix(P ) : Cc(X ) 1

>s

Ffir jede reelle Zahl r bezeichne r die konstante Funktion von X,

die den Wert r annimmt. Da f~r

ix(P)(D) = u

f~r jedes n s ~ gilt, annulliert M o ix(P) den Kern ~ von

M x : C (X) > KxCc (X) und l~Bt sich deshalb zu einem Charakter, c

1 er heiBe ix(P) : KxCc(X) > S , faktorisieren. Da MxCc(X)

bez~glich M x die Quotientenlimit• tr~gt, ist jx(p) f~r jedes

p 6 X stetig. Die Abbildung

iO) JX : X > FcKxCc(X) ,

die jedem p 6 X den Charakter jx(p) zugeordnet ist, wie man aus

der universellen Eigenschaft der stetigen Konvergenz [Bi] sofort

61

folgert, stetig. Offenbar gilt

JX = )4C (X) o i x

F~r eine Linearkombination

n I

i = l r i �9 ix(Pi) 6 V(X)

wo Pi 6 X und r i 6 ~ f~r i = l,...,n, laBt sich

n M o [ ri i x (pi) : C c (X) > S 1 �9 , g e n a u d a n n z u e i n e m C h a r a k t e r

i = l

n von )4xCc(X) faktorisieren, wenn ~ r i 6 ~. Wit bezeichnen mit

i=l

n n P(X) die Gruppe aller ~{ o ~ r I ix(Pi), ffir die Z ri 6 ~.

i=l i=l

Diese Teilgruppen von FMxCc(X) versehen wir mit der Limitierung der

s t e t i g e n K o v e r g e n z u n d e r h a l t e n s o d i e L i m e s g r u p p e P (X) , e i n U n t e r - c

raum von FcMxC(X) . Unser n~chstes Ziel ist nun, die Charaktergruppe

von P (X) zu studieren. c

Zu diesem Zweck halten wir fest:

Lemma 6

Der Homomorphismus

MX : FcMxC c (X) > FcCc (X) ,

definiert dutch Mx(y) = y o M x f~r jedes y 6 Fc)%xCc(X) , ist ein

Hom~omorphismus auf einen Teilraum.

Der Beweis ergibt sich unmittelbar aus der Tatsache, dab KxCc(X)

bezOglich MX die Quotientenlimitierung tr~gt.

Dieser Homomorphismus MX bildet Pc (X) hom6omorph auf eine Teil-

62

gruppe von FcCc (X) ab. Diese Teilgruppe heisse --MC (X) p(X). Diesen

Umstand n0tzen wir aus, um eine Beziehung zwischen C (X) und P P (X) c c c

herzustellen. Jede Funktion f 6 C(X) definiert eine stetige lineare

Abbildung f : <Cc(X) ~ rR, gegeben durch f(l) = l(f) f~r

alle 1 6 ~cCc(X). Dutch Komposition mit M erhalten wir einen

Charakter K o f : P C (X) > S l Den schr~nken wir auf c c

~x(P (X)) ein und erhalhen sofort einen Charakter auf P (X) Wir c c "

bezeichnen diesen Charakter mit k(f). Offenbar gilt

n n k(f) (~ o ( ~ r i ix(Pi)) = Z ~ o (r i �9 f(pi ) ) f~r jedes Element

i=1 i=l

n 14 o Y r i ix(Pi) 6 P (X) . Ordnen wir jedem f 6 C(X) den Charakter c i=l

k(f) zu, erhalten wir einen Homomorphismus

ii) k : C (X) > P P (X) . c c c

Dieser ist, wie man wiederum vermittels der universellen Eigenschaft

der Limitierung der stetigen Konvergenz sofort folgert, stetig.

Satz 7: Der Homomor~hismus

k : C (X) > P P (X) , c e c

fGr den

n M o 5- r

l

Kern ist

mu s

n n k(f) (M o ~ r i ix(Pi)) = [ M(r i f(p)) f~r jedes Element

i = l i = l

ix(P i) 6 Pc(X) 9ilt, ist eine steti~e Sur~ektion. Sein

�9 . Welter faktozisiert k zu einem bisteti~en Isomor~his-

: ~xCc(X) > P P (x), c c

dessen Inverser

Jx : PcPc(X) > MxCc(X) '

definiert durch jX(y) = y o JX f0r jedes y 6 PcP (X) , iSto c

63

Beweis: Zum Nachweis der Surjektivit~t von k bilden wir P (X) c

v e r m 6 g e d e r K o m p o s i t i o n v o n

~c (x) X c

Fe~xC e(x) ~ FcC c(x) ~<C c(x)

vergenz versehene Vektorraum V(X).

allen Linearkombinationen der Form

und Pi 6 X variieren, und f~r die

y 6 F P (x) c c

homomorph und hom~omorph auf eine Limesgruppe pl (X) c V (X) ab. e c

Dabei bezeichnet V (X) d e r m i t d e r L i m i t i e r u n g d e r s t e t i g e n K o n - c

i O f f e n b a r b e s t e h t P (X) a u s

c n Z r i �9 i x ( p i ) , wo r i 6 ~

i = J . n Z r. 6 ~. Zu jedem Charakter !

i = l

gibt es einen Charakter 0 : PI (X) > S e

mit

12) --i *

u = o o ~c (x) ~ MX c

Man pr~ft leicht nach, dab ~ bestimmt ist durch die Einschr~nkung

auf M (X), die mit der Limitierung der stetigen Konvergenz versehene c n

Menge M(X) aller Linearkombinationen ~ r i i x (pi) 6 Pcl (X) f~r i=l

n die Z r = i. Jede Strecke, die in V (X) irgend zwei Elemente

l c i=l

aus M (X) v e r b i n d e t , v e r l ~ . u f t g a n z i n M ( X ) , d o h . M (X) i s t w e g e - c c c

weise zusammenh~ngend. Wit wollen nun OlMc(X) : Mc(X) > S I

zu einer reellwertigen Funktion O' heben. Dazu halten wir in Mc(X)

einen Grundpunkt etwa ix( P ) wo p 6 X fest und w~hlen in der Faser

~ber O(ix(P) ) eine reelle Zahl r ~ aUSo Dann verbinden wir jedes

Element e aus M (X) durch eine Strecke s mit ix(P). Den Weg c

1 o s in S heben wit zu einem Weg in ~ mit Anfangspunkt r

o

Diese Hochhebung heiBe (c �9 s) ' ~ Der Wert von O' auf e sei dann

! definiert als der Endpunkt von (a o s) . Nun zeigen wir, dab der Weft

von (O o s) ' unabh~ngig vonder Wahl des Grundpunktes ist. Seien

u,v s Me(X) . Je zwei der Elemente ix(P) , u

derart mit Strecken, dab wir einen in ix(P)

und v verbinden wir

geschlossenen Weg s I

64

in M (X) haben. Wit verifizieren nun, dab s I sich in einem in r 6 c o

geschlossenen Weg hebt. Das yon ix(P) ,u und r in Mc (X) aufgespann-

te Simplex ~ liegt in einem endlich dimensionalen Teilraum yon V (X) c

und tr~gt damit nach [Ku] die nat~rliche Topologieo Offenbar ist s 1

in A nullhomotopo Damit hebt sich s I zu einem r O geschlossenen

Weg in ~~ Daraus folgt sofort, dab O unabh~ngig vonder Wahl des

!

Grundpunktes ist. Zum Beweis der Stetigkeit yon ~ betrachten wir

einen gegen ein beliebiges Element u 6 M (X) konvergenten Filter ~. e

Ohne Verlust der Allgemeinheit k0nnen wir annehmen, da~ �9 die Spur

eines in V (X) gegen u konvergenten Filters ist, der eine Filter- c

basis aus bez~glich u sternfOrmig konvexen Mengen besitzto Also be-

sitzt auch �9 eine Filterbasis von Mengen, die bez~glieh u stern-

f6rmig konvex sind. In jeder Umgebung von O' (u) gibt es eine offene

Umgebung V von ~' (u) , die als Blatt einer gleichm~Big ~berlagerten

offenen Umgebung U von y (u) auftritt. Well y stetig ist, gilt es,

eine (bez~glich u) sternf~rmig konvexe Menge F 6 �9 f~r die y(F) c U.

Wegen der sternf0rmigen Konvexit~t gilt a' (F) c V. Mithin konvergiert

~' (~) gegen ~(u) , d.h. ~' ist stetig in Uo Die Hochhebung

!

: M (X) > ~ ist affin, doh. es gilt c

u u

a' ( Z r i ix(Pi)) = Z r i ' ~' (ix(Pi)) i=l i=l

u Z

f~r jede Linearkombination i=l rl ix(Pi) 6 Mc (X). Um das einzusehen,

verifizieren wir erst, da6 ~i auf jeder Strecke in Mc(X) affin ist.

F~r zwei beliebige Elemente u,v ~ 6 Mc(X) sei s : [O, i] > M (X) ,

eine Strecke, die u mit v verbindet, d.ho s(r) = u + r(v-u) f~r o

jedes r 6 [O,i]. Die auf s([O, i]) c M (X) induziezte Limitierung c

ist die nat~rliche Topologieo Auf s ([0, i]) w~hlen wir eine offene

!

zusammenhangende Umgebung V derart, dab ~ (V) in einem Blatt ~ber

einer gleichm~Big ~berlagerten Umgebung U von y(u) ~liegt. FOr jedes

yon u verschiedene feste Element v I 6 V gilt:

65

G' (u + r (v I - u)) = ~' (u) + z �9 (~' (v) - a' (u))

fur alle r 6 [O,I]. Dies verifiziert man wie folgt: Offenbar ist

1 r 1 - u)=v-u. O' additiv. FOr eine gewisse reelle Zahl r gilt �9 (v I

Wir definieren 1 : s([O,l] > ~ durch

l(v) = o' (u) + n �9 (

1 r

n �9 (~' (v I) - ~' (u)))

1 1 (wobei n die kleinste natOrliche Zahl mit -- �9 r ~ i ist) fur alle

n

r 6 s([O,l]) . Es ist 1 eine stetige Abbildung for die M(I (v)) = O(v)

for jedes r 6 s([O,l]) gilt. Somit ist 1 eine Hochhebung yon

Ols([O, I]) , stimmt mithin mit O' Is([O, I]) Oberein. Folglich haben

wit

O' (u + r �9 (v-u)) = O' (u) + r �9 (O' (v) - ~' (u)) ,

was wit zeigen wollten. Nun betrachten wir

n

r i i X (pi) 6 M c (X) i=l

und schreiben diesen Ausdruck in der Form

li! I 1 r �9 r i 1

i x (pi)) + (l-r) �9 i X (pn)

I n i wobei r,r i 6 ~ und Z r. = 1

l i=l

fur i=l,...,n.

Wir nehmen nun an, dab for m ~ n - I for jede Kombination

n . I.

i=iE r.3 "ix(P')3 wo pj 6 X, und r3 6 IR for j=i,... ,m

mit ~ r = 1 gilt: 3

n . n .

(Y' ( X rj ix(Pj) = Z r.~ �9 (~' (ix(Pj)- j=l j=l

Well ~' auf jeder Strecke aus M (X) affin ist, ergibt sich c

66

n C O' ( [ r. ix(Pi) ) = r �9 ~'

i=l l

n-I 1 [ r. ix (Pi +

i=l l (i-r) �9 a' (ix(Pn))

n-I 1

= ~ r �9 r i �9 ~' (ix(Pi) + (l-r) " O' (ix(Pn) i=l

n

= ~ r i �9 O' (ix(Pi)) i:l

n n

Also gilt O' ( [ r ix(Pi) ) = l i = l i ~ l r i �9 o'(ix(Pi)) )

Ersetzen wir nun die stetige Abbildung ~' o i X : X IR dutch das

Symbol f und verwenden die Tatsache, dab O dutch die Werte auf

M (X) eindeutig bestimmt ist, so erhalten wir c

n n

o( Z r i ix(P)) = [ M(r i f(pi ) ) i=l i=l

f~r alle Kombinationen aus P1 (X) . Offenbar ist f 6 C(X) . c

Damit gilt abet wegen (12) :

n n

k(f) (M [ r i ix (pi)) = Z M(r i f(pi ) ) i=l i=l

f~r alle Kombinationen aus P (X) . Also ist k surjektiv und folg- c

-i lich k bijektiv. Die Stetigkeit yon k ergibt sich nun wie folgt:

Die Abbildung

JX : X > P (X) , c

(siehe (IO)) induziert einen stetigen Gruppenhomomorphismus

JX : Pc Pc(x) > ~xCc(X) ,

der jedes y s Fc Pc (X) in y o Jx ~berfiihrt. Wie man m~helos verifi-

z i e r t , g i l t :

*ix ~ k = idF P (X) c c

67

Mithin ist

wiesen.

-i

k = JX' also stetig. Damit haben wit Satz 7 endl• be-

Lemma 8

Es ist P(X) C FcMxCc(X) dieht.

Beweis:

Wiederum verwenden wit den zu Beginn des vorangegangenen Beweises ein-

gef~hrten Homomorphismus

-i

Cc(X) O ~X : Fc~xC e(x) > ~cCc(X) .

Des Bild dieses Homomorphismus besteht aus allen 1 6 ~cCc(X) mit

i(i) 6 ~. Zu jedem 1 6 <C (X) gibt es, einen gegen i konvergenten c

Filter ~, der eine Basis ~ in V(X) besitzt [Bu] und [Bi]. Es gelte nun

1 ( 1 ) 6 ~ . D a n n k ~ S n n e n w i t o h n e V e r l u s t d e r A l l g e m e i n h e i t a n n e h m e n , d a b n

jedes Element Z r. ix(Pi) aus jeder Menge N 6 ~ der Bedingung 1 i=l

n [ ri~ o

i=l gen~gt. Wir f~hren nun N" definiert als

n ~- ri ix (Pi) n

{ 1 (i) i=l ] X r ix(Pi) 6 N } n i i=l Z r i

i = 1

ein. Diese Menge liegt in ~C(X) o Kx(P(X)) . Der Filter �9 erzeugt -i

dutch { N" I N 6~} konvergiert in c~-Ce(X) gegen 1 . Da ~C(X) o MX

ein Hom6omorphismus auf sein Bild ist, ergibt sich ~, dab P(X) eine

dichte Teilmenge von FcMcCc(X) ist.

68

Satz 9 Die Gruppe MX Cc (X)

Inklusion

i : P (X) c

ist P -reflexiv. Zudem ist der yon der c

Fc~xC c (x)

induzierte Homomorphismus

i : FcFcKxC c (X) >F P (X) , c c

der jeden Charakter y 6 FcFc~xC(X)

ti~er Isomor~hismus.

nach y o i abbildet ein biste-

Beweis:

Aus der Kommutativit[t von

FcFc~xCc(X) > F P (X) c c

j)gxC c (X)

~xCc(X)

folgt unter Verwendung von Lemma 8 und Satz 7 die Richtigkeit der

Behauptung des obigen Satzes.

Korollar Io

F~r ~eden zusammenh~n~enden und lokal wegeweise zusammenh~ngenden to-

p o l o g i s c h e n Raum Y m i t e n d l i c h e r F u n d a m e n t a l g r u p p e g l (Y) i s t

KxCc(Y) = C (Y,SI) . Somit ist C (Y,S I) P -reflexiv. c c c

69

Beweis:

Eine Abbildung f 6 C(Y,S I) ist genau dann zu einer reellwertigen

Abbildung hebbar, wenn der von f induzierte Homomorphismus

i f, : ~{X,p) > II 1 (S ,f{p} )

trivial ist ([Sp] chapter 2) . Weil ~i (S1,f(p)) ~ ~ ist f.

fGr j e d e s f 6 C ( Y , S 1) t r i v i a l . M i t h i n i s t C ( Y , S 1) = K x C c ( X ) c

(beachte Lemma 3). Nach Satz 9 ist damit C (Y,S I) P -reflexiv. e c

Man beachte, dab die S~tze 7, 9 sowie Lemma 8 fGr jeden wegeweise zu-

sammenh~ngenden Raum, also ohne die in w generell gemachten Voraus-

setzungen an X, gelten. Wie in der Einleitung vermerkt, soll zum Be-

weis der P -Reflexivit~t yon C(X,S I) gezeigt werden, dab die Folge c

o >Pc~l(X)c > PcCc(X,S I) > FcMxCc(X) > o

sowie ihre Duale exakt sind. Zum Nachweis der fGr diese Dualisierungs-

operationen notwendigen Eigenschaften von i verwenden wir nun die

universelle 0berlagerung von X. Die gesuchten Eigenschaften (siehe

Korollar 13} werden Gber die Beziehungen zwischen FKxCc(X) und

Pc~Cc(~) hergestellt.

5) Zu den Beziehungen zwischen PcMxCc(X) und FcM~Cc(~)

.J Die Uberlagerungsabbildung u : X

des kommutatives Diagramm

13)

X induziert mit (2) folgen-

i z • o "~ �9 ~ c (~') ~ • c(~') ). o

T i d u

il ~X o ~ _~ ~ c c (x) .~ ~xCc (x) ~ o

70

Dabei bedeuten u den Homomorphismus in (3a) und u die Restriktion

% von u in ( 3 b ) a u f N x C c ( X ) . Wie i n (1) v e r e i n b a r t , s o l l i 1 a l s

die Inklusion bezeichnet werden. Sowohl u als auch u sind Injek-

tionen. Durch "dualisieren" von (2a) erhalten wir

M-~ i X 1

o > rcX~C c(~) > F c (~) ~ r = > o C C C--

14)

l u l u I >~ i X 1

o ~ Fc~xCc(x) ~ r c (x) ~F -~ > o c c c

Dieses Diagramm kommutiert. Der Zusammenhang von FcM~Cc (X) und

FcMxCc (X) soll nun durch die Eigenschaften der in (14) auftretendeh

Abbildungen beschrieben werden. (Wir identifizieren 2 mit ~.)

Proposition Ii Es ist

MX i 1

o > Fc~xCc(X) ~ FcCc(X) > FeZ ~ o

exakt. Dabei ist MX ein Hom~omor~hismus auf einen Teilraum und

i 1 e i n e Q u o t i e n t e n a b b i l d u n ~ .

Beweis: DaB MX ein Hom~omorphismus auf einen Teilraum ist, besagt

Lemma 6. Die Surjektivit~t von i I k~nnen wir etwa so einsehen.

Zum Charakter y : ~ > S I findet sich eine lineare Abbildung

1 : ~ > ~ mit ~ o ll~ = y. Offenbar ist i (r) = r 1 r f~r

ein festes r I 6 ~. Nun identifizieren wir in der offensichtlichen

Weise ~ mit der Menge der konstanten reellwertigen Funktionen von X.

Die Abbildung 1 1 : C (X) ~ C (X) , die jedes f 6 C(X) nach c c

71

r I �9 f abbildet, ist stetig und linear. FOr irgendeinen Punkt p 6 X

ist dann i (p)o 1 eine Ausdehnung yon i . Der Charakter ~ O ix(P) X

wird dutch i I auf y abgebildet. Um zu zeigen, da~ i I eine

Quotientenabbildung ist, verwenden wir das Diagramm

i

T [ i r C (X) i ~ r $ >s I c e c t

wobei i die Inklusion bezeichnet und die unmarkierten Pfeile die 2

nat~rlichen HomSomorphismen bedeuten. Das Diagramm kommutiert~ Die

Komposition der oberen Abbildung • da ~ nur eine separierte Vek-

torraumlimitierung, n~mlich die nat~rliche Topologie, tragen kann [Ku],

eine Quotientenabbildung. Well auch M eine solche ist und ~berdies

-I MC (X) ein Hom~omorphismus ist (siehe w , mu~ auch i I eine Quo-

c tientenabbildung sein.

Durch nochmaliges "Dualisieren" des Diagramms 14 erh~lt man der Pc-

Reflexivit~t der auftretenden Gruppen wegen bis auf eine kanonische

Isomorphie das Diagramm 13 zur~ck.

Der n~chste Satz zeigt, dab FcMxCc(X) und FcCc (X) innerhalb der

K a t e g o r i e a l l e r L i m e s g r u p p e n S f ~ r d i e JS e i n H o m ( 5 o m o r p h i s m u s a u f

sein Bild ist (der offenbar alle im Diagramm 14 auftretenden Gruppen

angeh~ren) Quotienten von Fc~Cc(~) respektive von l"cCc (~ sind.

72

Satz 12 Die im Diagramm (14) auftretenden Homomor~hismen *u und (u )

sind Sur~ektionen mit fol@ender universeller Ei~enschaft: F~r ~eden

Homomorphismus h von F M C (X) respektive yon F C (X) in eine c X c c c ~

Limesgruppe S, ist jS o h ~enau dann steti~, wenn JS" h o

respektive Js o h o (u) steti~ ist.

Beweis: Erst zeigen wir, dab die beiden (stetigen) Homomorphismen

*~r u n d (u ) s u r j e k t i v s i n d . Zum N a c h w e i s d e r S u r j e k t i v i t ~ . t v o n

(U ) halten wit erst fest, dag der Limesvektorraum C (X) und der c

lokalkonvexe topologische Vektorraum C (X) denselben Dualraum haben co

[ Bu ] oder [Bi], also dab ~C (X) =~C (X) gilt.(Hier machen wit yon c co

der Normalit~t nur insoweit Gebrauch, dab jeder normale Raum c-einbett-

bar ist [Bi] und mithin ~C (X) =~C (X) gilt). Nun folgert man co c

leicht unter V e r w e n d u n g d e s b i s t e t i g e n I s o m o r p h i s m u s ~C (X) ( s . ( 8 ) ) u n d c , ~

Bemerkung 5 mit gilfe des Satzes yon Hahn-Banach, dab (u) sur-

jektiv ist. Um zu zeigen, dab ~uu surjektiv ist, sei y 6 Fc~xCc(X).

Dann ist y o 14. X 6 F c C c ( X ) . W e g e n d e r S u r j e k t i v i t a t v o n ( u ) f i n d e t

sich eine "Erweiterung" yl 6 F C (~) f~r die yl = * (u*) (yo~ x) . c c

1 Da y die Gruppe ~ annulliert, gilt dies auch f~r y . Also l~Bt

sich yl y,, zu einem Charakter 6 FcM~C c- (~) faktorisieren. Nun veri-

fiziert man leicht, dab *u(y")=y. zur Verifikation der nicht-trivi-

alen R i c h t u n g d e r u n i v e r s e l l e n E i g e n s c h a f t v o n * u g e b e n w i r u n s e i n e n

Homomorphismus

h : Fc~xC c(x) > S 1

for den jS o h o *u stetig ist und folgern daraus die Stetigkeit von

jS o h. Dann bilden wir das Diagramm

73

X JX

FcX~C c (~)

> Fc~xC c (X)

S

worin h i die Abbildung h o JX bezeichnet. Daraus lesen wit ab, da~

h IO u stetig ist. Weil u eine Quotientenabbildung ist, muB h I

stetig sein.

Nun bilden wit

FcM~C c (~) **J~ 3FcM~C (~) ) FcFcFc~Cc (~) c > Fc~Cc (~)

(h o ~u) h e "1/

** JS (h I ) Fc~xCc(X) ~ PcPcS < s

Darin ist sicherlich die rechte H~ifte kommutativ. Hier bedeutet

~j~ den Homomorphismus induziert yon

�9 (x) *j~ : rcrc~C c<~') > • c ,

wobei ~j~ jedes y nach y o j~ abbildeto Man prOft nun nach, da~

die linke H~ifte des obigen Diagramms ebenfalls kommutiert und ~ber-

zeugt sich weiter davon, dab

74

jFcM~Cc(~ ) O **j~ = •215 c (x%

gilt. Also ergibt sich

** 1 *~ *-- h 0 *u = jS o h o *u .

Weil *u surjektiv ist folgt

**h I = jS o h .

**h I , o h stetig sein. Weil und *M~ Da stetig ist mu~ JS *MX

Hom6omorphismen auf Teilr~ume sind, folgert man nun leicht die analoge

universelle Eigenschaft fur (U ) .

Korollar 13 Der yon der Inklusionsbildun~

i : KxCc(X) > Cc(X,S I)

induzierte Homomor~hismus

*i : FcCc(X,SI) > FcMxCc(X)

ist sur~ektiv und besitzt fol@ende universelle Ei~enschaft:

Wenn fur eine Limesqru~pe S und einen Homomorphismus

h : FcMxCc(X) ~ S die Kom~osition h �9 i steti~ ist, dann ist

auch jS o h stetig.

75

Beweis: Die Surjektivit~t von i und die universelle Eigenschaft

folgert man sofort aus Satz 12 und der Kommutativit~t von

**U rc• (g) > rcCc (X, sl)

i , u { V / , , ~

FcMxCc(X)

Gem~B der in der Einleitung beschriebenen Konzeption zum Nachweis der

P -Reflexivit~t von C (X,SI) , wenden wit uns nun dem Nachweis der c c

P c - R e f l e x i v i t f i t von Cc(X,S1) /~xCc(X) ( f a l l s v o l l s t a n d i g ) zu.

6) Zur Bruschlinski'schen Gruppe

Wie in der Einleitung bemerkt, heiBt die Gruppe

C (X,Sl) /• c (X)

die Bruschlinski'sche Gruppe. Wir ersetzen dieses Symbol durch das

einfachere ~i (X) . Sparer werden wit auf eine andere Beschreibung

nl 1) von (X) zur~ckkommen. Den nat~rlichen Homomorphismus von C(X,S

auf ~I (X) heiBe b. Die Gruppe El(x) versehen wit mit der von b

induzierten Finallimitierung und bezeichnen die resultierende Haus-

dorffsche Limesgruppe mit ~i (X) . c

I Unser erstes Ziel ist es, zu zeigen, dab ~c(X) topologisch ist.

Zur leichteren technischen Handhabung unseres Problems werden wir

die Limesgruppe nl (X) als Quotient von Limesgruppen c

76

yon Funktion aus C (~) beschreiben. Dazu stellen wir zuerst c

~1 (X) a l s Q u o t i e n t v o n G r u p p e n v o n F u n k t i o n e n a u s C ( ~ ) d a r . W i r

fixieren irgend einen Punkt p 6 X und f~hren die Gruppe G aller P

in p verschwindenden Abbildungen aus C(X,S l) ein. Wir setzen weiter

G ~ = MxC(X) D G . Offenbar sind G IG ~ und HI(x) isomorph. P P P P

Nun bilden wir M~I (*U(Gp)) und M~I (~u(G~)) , die wir kurz mit Cp

und C ~ respektive bezeichnen. P

N Nach Lemma 1 besteht C aus allen Funktionen aus C(X) , die auf der

P -i

Faser u (p) ihre Werte in ~ annehmen. Eine Charakterisierung der

Funktionen in C ~ erhalten wir Gber folgende Versch~rfung von Lemma 2: P

Lemma 14

* 1) * Eine Funktion f 6 M~I( u(C(X,S )) 9ehSrt 9enau dann u (C(X)) a__n_n,

-i Wenn sie auf der Faser u (pl) ~ber ir9end einem Punkte Pl 6 X

konstant ist.

Beweis: Nach Lemma 2 ist u (g) f~r jedes g 6 C(X) auf jeder

~i -I Faser konstant. Sei umgekehrt f 6 (*u(C(x,sl))) auf u (pl)

fGr irgend einen Punkt Pl konstant, sagen wir gleich r. Wit w~hlen

p 6 X Zu zeigen ist, dab flu-1(p) konstant ist. Aus Lemma 2 ergibt

sich dann Lemma 14. Wir verbinden Pl mit p dutch einen Weg a.

Diesen Weg heben wir zu einem Weg 01 in ~ hoch, der als Anfangspunkt

ql 6 u-1(pl) besitzt. Der Endpunkt yon g liegt in u-1(p). Sel

* i f = u(g) . Nun heben wir *u(g) o a, der in *u(g) (pl) 6 S beginnh

und in ~u(g)(p) 6 S I endet,zu einem Weg ~" in R mit Anfangspunkt f(ql)

hoch. Der eindeutigen Wegehochhebung in R wegen, ist ~" = f 0 a I.

-i Der Endpunkt von O" ist r. Da ql 6 u (pl) beliebig ausgew~hlt

war und es nur eine Hochhebung von u (f) o C mit Endpunkt r gibt,

ist flu-i (p) konstanto

77

Aus Lemma 14 ergibt sich, dab C ~ gerade aus allen Funktionen in C P P

besteht, die auf u-i (p) konstant sind.

Der Homomorphismus

-i b = b o ~u o M~ : C > II 1 (X) p x p

-1 (man beachte, dab ~u nur auf dem Bild von

sur3ektiv und sein Kern besteht gerade aus C ~ P

zu einem Isomorphismus

u definiert ist) ist

Dieser faktorisiert

: Cp/Cp > HI(X). P

Fixieren wit einen Punkt q 6 u-i (p) und bezeichnen mit C q und M P P

alle Funktionen aus C respektive C ~ die auf q verschwinden, dann P P

faktorisiert b o*u-lo M~Icq P offenbar zu einem Isomorphismus

15) b : C q / M > I~ 1 (X). P P P

Damit kann HI(X) als Quotient von Gruppen von Funktionen aus C(X) ~

aufgefaBt wexden. Um die Limesstruktur auf ~i (X) in Abh~ngigkeit der

-i Faser u (p) beschreiben zu k6nnen (Lemma 15) interpretieren wir diese

Gruppe nochmals um, und zwar als Gruppe von ~-wertigen Funktionen von

-i u (p).

-i Bekanntlich l~Bt sich u (p) mit der Fundamentalgruppe HI (X,p) von

X mit Aufpunkt p identifizieren, und zwar so, dab ein festgew~hlter

Punkt, es sei q, zum Einselelement e 6 ~i (X,p) wird ([Sp] chapter 2) .

-i Statt u (p) schreiben wir nun F o Diese Faser tr~gt also eine

P

Gruppenstruktur mit q 6 F als Einselelement. Jede Abbildung t 6 G P P

induziert dann einen Homomorphismus [Sp]

t, : ~i (X,p)

78

~" I ~ 1 ( S 1 , 1 ) ,

der die Homotopieklasse [O] eines jeden in p geschlossenen Weges O

nach [f o ~] ~berfdhrt. Identifizieren wit ~i (sl,l) mit ~ derart,

dab das Einselelement in der Fundamentalgruppe mit O 6 ~ zusammen-

f~llt, erhalten wir eine Abbildung

t, : F > ~t P

q die eindeutig bestimmte die q nach Null abbildeto Bezeichnet ft

Hochhebung von t, die in q verschwindet, so gilt, wie-man leicht

nachpr~ft,

q I F t, = ft p

Folglich haben wir eine Abbildung

d : C q P P

Horn(F ,Y) , P

die jedem f 6 C q die Einschr~nkung auf P

d ist gerade M . Mithin faktorisiert P P

F zuordneto Der Kern yon P

d zu einem Monomorphismus P

: C q I M ~ Hom(F ,2~) . P P P P

Also haben wir einen nat~rlichen Monomorphismus

-i o b : nl(x) > Hom(F ,Z).

P P P

Wir bezeichnen das Bild von d mit HI(X) . Alle Homomorphismen aus P P

~i (X) lassen sich mithinein zu stetigen reellwertigen Funktionen P

aus C q c C(X). P

79

Nun versehen wir nl (X) mit verschiedenen, ffir die nat~rliche tech-

1 nische Handhabung von H (X) angepaBten Limitierungen und zeigen, dab

sie alle hom6omorph sind. Die G ruppe C q c C(X) fassen wit als Teilxaum P

yon C (X) auf. Damit ist C q eine Limesgruppe. Die Gruppe Hom(Fp,Z) c p

und El(X) versehen wit je mit der Topologie der punktweisen Konver- P

genz und erhalten die topologischen Gruppen HOms(F ,~) und HI (X) . p s p

Die Gruppe HI (X) versehen wir mit der von

b : C q > H 1 (X) P P

induzierten Qdotientenlimitierung. Diese Li~itierung heine HI (X) . c i

Offenbar ist id : H i (X) > HI (x) stetig. c I c

Unser nAchstes Ziel muB es sein, den stetigen Homomorphismus -i

P P

genauer zu studieren.

Zun~chst beschreiben wit eine bequeme Nullumgebungsbasis in HOms (F ,2) . P

Zu irgend endlich viele Elemente el,...,e n gibt es eine Nullumgebung

K c HOms(FpZ ) mit h(e I) = ... = h(e n) = o ffir alle h 6 K. Also

annulliert jeder Homomorphismus h 6 K die von el,..o,e n in Fp

erzeugte Gruppe H Offenbar ist elt...e n �9

{h 6 Horn (F ,E) lh(H ) = o} s p eli...re n

eine Nullumgebung in Homs(Fp,~) . Das System

16) {{h 6 HOrns (Fp,~)lh(H) = o} I H c F endlich erzeugt} P

ist eine Nullumgebungsbasis ~ der topologischen Gruppe Hom (F ,Z) . s p

Diese Basis geschnitten mit HI(x) ergibt eine Nullumgebungsbasis P

n 1 yon (X) s p

80

Lemma 15

Der Isomorphismus

-i

dp p Cl (X) > nl(X)ps

ist ein Hom6omorphismus. Folglich ist

-i o Z : [I 1 (x) > II 1 (x)

p p c s

ein bistetiger Isomorphismus.

-i Beweis: Wir wissen bereits, da~ der Isomorphismus d o b stetig

P P

ist. Zum Nachweis der Stetigkeit der Umkehrabbildung konstruieren wir

erst einen gegen o 6 C q konvergenten Filter ~, der dutch b IC q P P P

auf die Nullumqebungsfilter ~(o) in ~I (X) abgebildet wird. Sei s p

H c F eine endlich erzeugte Gruppe. N bezeichne den kleinsten p H

Normalteiler in F der H enthalt. Wit sagen, da~ N H endlich P

erzeugt sei. FHr jedes h c Hom(Fp,~) gilt offenbar h(NH) =o genau

dann, wenn h(H) =o. Die Gruppe H i umfasse alle Homomorphismen aus

HOms(F ,~) die H annullieren. Der Raum X besitzt eine Uberlagerung P

XNH , (die von ~ ~berlagert wird) deren Decktransformationsgruppe

gerade F IN H ist. Die Projektion von X N auf X heine u N . Die P H H

Projektion von X auf X N sei mit VNH bezeichnet. Dann gilt H

UN~H VNH= u. Die Faser ~ber v N (q) besteht gerade aus N H. Nun sei H

h 6 (NH)l n ~i (X)p. Jede Funktion f 6 C qp mit flFp=h ist auf den

Fasern ~ber X N konstant. Zu p und irgend welchen Punkten Pi 6 X, H

-i wo i=l,.. ,n, wahlen wir neben q 6 u (p) je einen Punkt

l -i Pi 6 u (pi) ~ Weiter wahlen wit um p sowie um Pi je eine abge-

schlossene Umgebung (X ist regular!) U resp. U ,die bez~glich u yon P P•

Bl~ttern V q resp. V ~berlagert werden, die auch Bl~tter bez~glich P Pi

' enthalten Dabei sei N ein endlich er- v N sind und die q bzw. Pi ~

zeugter Normalteiler, f~r den alle f 6 C q mit flF 6 N -L auf P P

81

Up Ui~=l Upi durchfaktorisieren.

I N

, c C q Mit Tp , pL,o..,pn P

sei die Menge aller Funktionen gemeint, deren Restriktion auf jeder

der M e n g e n V ~ , V p l , O . . , V p v e r s c h w i n d e n u n d d e r e n R e s t r i k t i o n e n a u f n

F der Menge N i angeh0ren. P

Solche Funktionen gibt es: Sei f 6 C q mit fJF 6 ~. In C(X) P P

gibt es, der Normalit~t von X wegen, eine Funktion g for die

u Igl u b n v = f <v u v i=l Pi i=l Pi

• * N

Also gehSrt f + u (q) der obigen Menge Tp ,p, '''''Pn an. Das System

der Mengen dieser Form bildet eine Filterbasis ~ : Seien pl. ,...,pn , l l

I ,N l 6 ~ Offen- wo i=l,...,m Punkte aus X. Weiter seien NI,... m

bar gibt es einen endlich erzeugten Normalteiler N c F mit P

N l c l~=im Nil Wir bilden nun analog wie oben die 0berlagerung XN von X.

Wiederum sind die Funktionen aus C q, deren Restriktionen auf F P P

den Normalteiler N annullieren, auf den Fasern Ober X N konstant.

N kann so gro~ gew~hlt werden, dab alle f 6 C q mit f 6 N 1 auf P

U U ~ U durchfaktorisieren. Offenbar gilt P 3,1

P3i I

m Ni Ni /% T i=l P'Pl '''''Pn. ~ T

i 1 P'Pi '''''Pn i m

Also ist ~ eine Filterbasis auf C P. Der von dieser Basis in C p q q

erzeugte Filter heiBe ~. Er konvergiert in C p nach O: Denn seien q

1 U-I P1 6 X und q 6 (pl) und o ein Weg, der Pl mit p verbindet.

1 1 Wit heben diesen Weg zu einem Weg G in X mit Anfangspunkt q

Dez Endpunkt qE yon a I liegt in Fpo Der von qE erzeugte Nor-

malteiler in F heine N. ES ist P

N • c 2

P ' Pl

82

N AuBerdem annullieren alle Funktionen in T das Blatt fiber D~,

P'Pl i

das q enthalt. Folglich konvergiert �9 gegen o s C p . Nach q

Konstruktion konvergiert b (~) und es ist b (~) = ~(o) . Also ist P P

-i o ~ : H 1 (x) > Hl(x)

p p c I s

bistetig. Daraus folgert man mfihelos den Rest des Lemmas. Damit ist

Lemma 15 bewiesen.

Wir bilden die Komposition aus

b : C(X,S I) > HI(x) c

o~ :n i und dp p c (x) > IIl(X)s P c Horns (Fp,~)

und erhalten

1 S I i c Horn (F ,~) : c(x, ) > s(X)p s p

Wir kSnnen Lemma 15 in folgendem festhalten:

Corollar 16

Beide Abbildungen

b I S 1 : C (X, ) c

und

b : Cc(X,sl)

> 51 (x) s

H i (x) c

c Horn s (F ,2) P P

sind Quotientenabbildungen.

Im n~chsten Paragraph soll die

st~ndig) nachgewiesen werden.

83

Pc-Reflexivit~t Yon Ill (X)ps (falls voll-

7) P Reflexivit~t vollst~ndiger Untergruppen von C (Z,~) c s

Die lim• Bruschlinski'sche Gruppe Ec(X) kann nach Lemma 15 als

topologische Untergruppe yon Hom (F,2) aufgefaBt werden. Um nun die s

P -Reflexivit~t dieser topologischen Gruppe nachzuweisen, studieren c

wir kurz die mit der Topologie der punktweisen Konvergenz versehene

Gruppe Cs(Z,~) aller stetigen ~-wertigen Funktionen eines topolo-

gischen Raumes Z. Die Gruppenoperationen sind punktweise definiert.

Lemma 18 Sei Z ein diskreter to~ologischer Raum, G c C (Z,~) eine s

abgeschlossene Untergruppe und h : G >S I ein stetiger Homomorphismus.

Dann kann h zu einem steti~en Homomor~hismus h 1 : Cs(Z,~) > Sl

derart erweitert werden, dab f~r ein g 6 C(Z,2) das nicht G angeh~rt

h I (g) ~ 1 ~ilt.

Beweis: Sei E c Z eine endliche Menge und I(Z) das L-Ideal aller

auf E verschwindenden Funktionen des Ringes C(Z,~) . Die Operationen

seien punktweise definiert. Das System

{ I(E) IE endliche Teilmenge yon Z }

ist eine Nullumgebungsbasis der topologischen Gruppe C (Z,~) . Da s

letztere separiert ist und ~ ~ C (Z,~) abges~blossen, gibt es eine end- s 1

liche Menge E c Z mit g+I(El) D G=~ . Zu jeder gleichmaBig Oberlager-

ten Umgebung V von i 6 S I gibt es eine endliche Menge E c Z mit

h(I(E) N G) c V und I (E) c I (El) . Wenn nun V klein genug ist und

h(I (E) N G) # {i} angenommen wird, gibt es ein gl 6 I(E) N G und

n 6 ~ mit (h(gl))n ~ V. Dem widerspricht abet n-g I 6 I(E) . Also

84

faktorisiert h zu

: G/ICE) N G s

Der stetige Restriktionshomomorphismus r E : Cs(Z,~)

induziert einen Isomorphismus

r : C(Z,~)/I(E) > C(E,~). E

> C (E,~) s

Da nun Cs(E,~) diskret ist, ist die von rEIG auf G/ICE) n G

induzierte Quotientenlimitierung diskret. Zu h gibt es einen

Charakter h auf ~E(G/I(E) N G) derart, da~ h = h o rE. Da

C (E,2) diskret ist, l~t sich bekanntlich h zu s

h I : Cs(E,Z) > S I derart ausdehnen, dab hl (~E(q)) ~ i. Der

Homomorphismus hl~ ~E ist eine stetige Ausdehnung von h, die g

annulliert.

F~r einen diskreten Raum Z, ist Cs(Z,~) das ~-fache kartesische

Produkt von ~. FOr dieses gilt:

Lemma 19 F~r ~eden diskreten to~olo~ischen Raum

P -reflexiVo c

Z ist C (Z,~) s

Beweis: Die Beweisf~hrung ist eine reine Routinesache und soll hier

deshalb nur skizziert werden. Sei ~ das System der endlichen Mengen

von Z. Es ist C (Z,~) der projektive Limes lim Cs(E,Z) der s

diskreten Gruppen Cs (E,~) o Daraus ergibt sich leicht, dab FcCs(Z,~)

der induktive Limes lim F C (E,~) der kompakten Gruppe F C (E,~) ~-~ cs cs

ist. Aus der P -Reflexivit~t von C (E,~) ergibt sich dann die e s

Pc-Pes163 yon Cs(Z,~).

85

Proposition 20 Jede vollst~ndige Untergruppe

~eden topologischen Raum Z P -reflexiv. : c

G ~ C (Z,~) S

ist fur

Beweis: Wir nehmen erst an, dab Z diskret ist. Die Inklusion

v : G > Cs(Z,~) induziert die stetige Surjektion

*v : F C (Z,~) > F G. c s o

(Lemma 18), vonder man unter Verwendung der Darstellung von

F C (Z,2) als lim F C (E,~) mOhelos nachweist, da~ sie eine c s ~ c s

Quotientenabbildung ist. Folglich ist der Monomorphismus

* * v : F F G > FcFcCs(Z,2) c c

ein Hom6omorphismus auf einen Teilraum. Also ist ***v der P c

Reflexivit~t von

tativit~t von

Cs(Z,~) wegen surjektiv (Lemma 18) . Aus der Kommu-

JF G *JG F G c > F F F G > F G c c c c c

\

* (** -I ) /

Ve3Cs (Z,~)

/ P C ( z , ~ ) c s

ergibt sich *'3G~ JFcG = idFcG, also insbesondere, daS JFcG ein

Hom~omorphismus auf sein Bild iSto Weil ***v eine Surjektion ist,

muB * ( * * V ~ jC~(Z,~) ) surjektiv sein. Aus der Surjektivitat von

86

-I *(**v @ JC (Z,~)) folgert man, dab JF G ein bistetiger Isomorphismus

s c

ist. Nun verwendet man wiederum Lemma 18 und zeigt leicht die

P -Reflexivit~t von G. Ist nun z irgend ein topologischer Raum,SO c

ist G eine abgeschlossene topologische Untergruppe von Cs(ZD,~) .Dabei

bedeutet Z den zu Z assoziierten diskreten topologischen Raum. D

Daraus ergibt sich unmittelbar die Aussage in der obigen Proposition.

Damit folgern wir aus Proposition 20 und Lemma 15 endlich:

Satz 21 HOms(Fp,2 ) ist Pc-reflexiv" Die topolo~ische Gruppe

ist dann ( u n d n u r d a n n ) P - r e f l e x l y , w e n n s i e v o l l s t f i n d i g i s t . c

Die folgenden beiden S~tze geben Bedingungen f6r die Kompaktheit

von Fc~(X) an.

H 1 X) c

Satz 22

Wenn F P

k o m ~ a k t .

HOms (Fp,~) ist diskret, wenn F endlich erzeu@t ist. P

endlich erzeugt ist, dann sind F Horn (F ,~') und F H 1 X) c s p c c

Beweis: Mit Hilfe der in (16) dargestellten Umgebungsbasis

folgert man sofort den ersten Teil der Aussage des obigen Satzes.

Der zweite Teil ist eine Konsequenz des ersten zusammen mit der Tat-

sache, dab die Limitierung der stetigen Konvergenz auf FcHOms (Fp,~)

und F [I 1 (X) die Topologie der kompakten Konvergenz ist. FOr die c c

letztere sind aber diese Aussagen richtig [Po].

Satz 23 Ffir ~eden kompakten topoloqis~hen Raum X (der unsere ge-

nerellen Voraussetzungen nicht zu erf~llen braucht) ist HI(X) diskret c

und damit Fc~(X) kom~akt.

87

Beweis: Sei �9 • ein in C (X,S I) gegen das Einselelement kon- c

vergenter Filter~ Zu jedem Punkt p 6 X gibt es zu einer vorgegebenen

1 gleichm~Big Gberlagerten Umgebung V von i 6 S eine Umgebung U(p)

und ein F 6 �9 mit

F (U (p)) c V.

Endlich viele Mengen aus

ein F 1 6 �9 mit

{U(p) [p 6 X} ~berdecken X~ Also gibt es

F 1 (X) c V.

Daraus aber folgt F 1 c MxCc(X) . Mithin hat �9 eine Basis in MxCc(X) -

Daraus schlie~t man, da~ Cc(X,SI) IMxC(X) diskret ist.

8) Zur Pc - Reflexivit~t von Cc(X,S I)

Sei ~l(x) c

vollst~ndig. Die Folge

i b o > XxCc(X) > Cc (x'sl) > H1(X)c > o

ist exakto Lemma 3 besagt, dab i ein Homomorphismus auf das Bild

ist, w~hrend b nach Konstruktion eine Quotientenabbildung iSto

Nun dualisieren wir diese Folge und erhalten:

~b ~i 17) O ~ Fc~Ic(X ) ~ FcCc(X,SI) > Fc~xCc(X) > O

Zum Nachweis der Exaktheit dieser Folge verwendet man einerseits, dab

b eine Quotientenabbildung ist und andererseits, dab nach Korollar 13,

der Homomorphismus ~i eine stetige Surjektion iSto Durch nochmaliges

Dualisieren erhalten wit dann:

88

~i 'S I ~*b El O ~ FcPc~xC (X) > F F C (X ) > P F (X) c c c c c c c O .

Diese Folge ist, wie man unter Verwendung von Korollar 13, der

Kommutativit~t von

~b

FcFcCc(X,sl) > F F HI(x) c c c

lJ Cc (X'SI) I jill (X) c

b Cc (x'sl) > Hie (X)

und der P c

Also ist

- Reflexivit~t yon II 1 (X) (Satz 21) c

O

schlie~t, exakto

O

~i ~b FcFc~xCc(X) > FcF C (X,S I) > F F ~i (X) c c c c c

(x) I Jc c (X S I) I jill (X) t c

b > c (x,s I) > H I (x)

c c

T JMxCc

> ~xCc x)

> o

o

ein kommutatives Diagramm exakter Folgeno Daraus folgt unter Ben~tzung

von Satz 9 und des F~nferlemmas, dab JC (X,S I) ein Isomorphismus ist. c

Die Abbildung

JX : X > FcCc(X,Sl) ,

definiert dutch jx(p) (t) = t(p) f~r alle p 6 X und alle

ist stetig und induziert einen stetigen Homomorphismus

~Jx : FcFcCc (x'SI) ~ Cc (X'SI)'

t 6 C (X,S I) , c

89

der jeden Charakter y 6 F F C (X,S I) nach y o Jx abbildet. Weil nun c c c

~ J x ~ J c ( X , S 1) = i d i s t , h a b e n w i r e n d l • c

Satz 24 Wenn 1"I I (X) vollst~ndig ist (also etwa wenn X kom~akt ist c

oder Hi (X,~) endlich erzeu~t ist oder Ill (X) = Hom(IIl (X),~), was

z.B. dann c t ' i l t , w e n n X e i n C W - K o m p l e x i s t , s . w d a n n i s t C ( X , S 1) , c

P -reflexiv. C

9) Zur Charaktergruppe yon Cc(X,SI)

Die Folge (7) besagt, dab F C (X,S i) eine (uns unbekannte) Erweiterung c C

von I"c II1c (X) und I"c~xCc(X ) ist. Die Gruppe Fc~xCc(X) ist durch

Lemma 8 bestimmt. Wir werden deshalb F rl I (x) unter einschr~nkenden c c

Voraussetzungen noch etwas genauer untersuchen. Dazu studieren wir erst

F Hom (F , ~ ' ) . D i e G r u p p e F m o d u l o d e r K o m m u t a t o r e n g r u p p e i s t i s o - c s p p

morph der ersten singul~ren Homologiegruppe

in ~. Offenbar sind damit HOms(Fp,~) und

isomorph.

HI(X,2) mit Koeffizienten

HOms(Hl (X,~) ,~) bistetig

Wit beginnen mit der Vorbereitung einer spezifischen Beschreibung yon

FcHoms (HI(X'~) ,~) . Dabei setzen wir voraus, dab HI (X,~) endlich er-

zeu~t sei. Demnach gibt es also einen Homomorphismus p von einer

freien, endlich erzeugten, kommutativen Gruppe H auf HI (X,~) . Die

Basis von H heiBe M.

Wenden wit uns nun F c Horn (H,2) zu. Es ist ~ =~ F(SI,s I) . Mithin s

Horn (H,~) mit

F HOrns (H,~) c

Unter

der s t e t i g e n

ist, da die Topologie der punktweisen Konvergenz auf

der Limitierung der stetigen Konvergenz identisch ist,

bistetig isomorph F B (H x Sl,Sl) . (Siehe [Bi,Ke]) . c c

B (H • Sl,S I) verstehen wir die Gruppe B(H x SI,S I) c

Sl-wertigen Bihomomorphismen von H x S I versehen mit der Limitierung

i der stetigen Konvergenz. Dabei tr~gt H • S die Produkttopologie.

Das Tensorprodukt H ~ S I kann in nat~rlicher Weise

90

mit ~ S I identifiziert werden. F6r ~ 6 M bezeichnet S 1 die

Gruppe ~ �9 ~ ~ S I Diese ist nat~rlich isomorph S I. Die Gruppe

@ S 1 versehen wit nun mit der gr6bsten Topologie ffir die die Pro- ~6M

jektion auf S I ffir jedes ~ 6 M stetig iSto Weil M endlich ist, wird

S 1 und damit H ~ S I eine kompakte topologische Gruppeo Wir fiber-

lassen es dem Leser zu verifizieren, da~ der natfirliche Bihomomorphis-

mus ~ von H x S 1 nach H ~ S 1 einen Isomorphismus

*~ : F(H Q S I) > B(H • Sl,S I).

definiert durch *~(y) = ~ o y ffir jedes y E F (H ~ SI), induziert. c

Dieser Homomorphismus ist, wie man aus allqemeinen Resultaten aus

[Bi,Ke] ersehen kann, bistetig, falls beide Gruppen mit der Limi-

tierung der stetigen Konvergenz versehen sind. Mithin ist

**$ : F B (H • SI,s 1 cc

> r r (H G s I) c o

ebenfalls ein bistetiger Isomorphismus= Nach der klassischen Pontx-

j a g i n s c h e n D u a l i t ~ . t s t h e o r i e i s t H ~ ) S 1 b i s t e t i g i s o m o r p h zu

FcFc(H @ S i) und damit auch zu F B (H x SItsI) o c c

Der Homomorphismus

p : H ~ HI(X,~)

induziert eine surjektive Abbildung

p@ id : H~S 1

Wit versehen

durch p~id.

logischen Gruppeo Es ist

*(p ~ id) : Fc(Hl (X,~) ~ S I)

> H:(X,Z) ~ s I.

HI (X,~) ~ S 1 mit der Quotiententopologie definiert

Dadurch wird HI (X,~) ~ S I zu einer kompakten topo-

> r (H 0 S i) c

91

ein Monomorphismus zwischen diskreten Gruppen. Folglich ist, wiederum

nach der klassischen Pontrjaginschen DualitHtstheorie

(p • id) : F F (H O S I) C c > FcF (H i (X,~) ~ S I) c

eine stetige Surjektion, ja sogar eine Quotientenabbildung

(da HI (X,~) ~ S I kompakt ist) . Daraus folgt, dab

JH (X,Z) ~ S 1 : HI (X,~) ~ S 1 I

F F (X,~) O S 1 c c (Hi )

ein bistetiger Isomorphismus ist.

Offenbar sind HOms(Hl (X,~) ,H) , Bc(HI (X,H) • SI,S I) und

F~HI (X,H) ~ S I) bistetig isomorph. Folglich sind auch HI (X,~) O S I und

Fc (HOms (HI (X'2) ,~) bistetig isomorph. ZusammengefaSt haben wir mit-

hin:

Lemma 24 Wenn HI (X,~) endlich erzeu~t ist, dann ist HI (X,H) Q S 1

komgakt und damit P -zeflexiv. Uberdies sind F Hom (HI (X,H) ,~) und c c s

HI (X,2) O S I bistetig isomorph.

Wenn X ein CW-Komplex ist, so ist nach ([Sp] p.427) die Gruppe HI (X)

isomorph zu Horn(Ill (X,p) ,H) und es gilt HI (X,2) = Hom(Hl (X,~)~) =

= Horn(Ill (X,p) ,~). Damit ergibt sich aus Lemma 24:

Satz 25 Wenn der Raum X zusHtzlich ein C_~W-Komplex ist und die Gru_~

H I (X,2) endlich erzeu~t ist, dann sind einerseits [I Ic (X) ,

Fc~(H 1 (X,~) ~ S I) und Fc Homs (H 1 (X,2) ,2) und andererseits Fc IIlc (X)

und H I (X,2) G S I b• isomor~h.

92

LITERATURVERZEICHNIS

[Bi]

[Bi,Ke]

[Bu]

EoBinz: "Continuous convergence on C(X) ". Lecture Notes

in Mathematics 1975, Springer-Verlag Berlin, Heidelberg,

New York.

E.Binz und H.H.Keller: "Funktionenr~ume in der Kategorie

der Limesr~ume". Ann.Acad.Sci.Fenn. Ser. A 1.383 (1966)

1-21.

H.P.Butzmann: "Dualit~ten in C (X)". Dissertation, c

Universit~t Mannheim ( 1 9 7 1 ) .

[Hu]

[Ku]

[Po ]

[sp ]

--."0ber die c-Reflexivit~t von C (X) ". Comment. c

Math. Helv. 47 (1972), 92-101.

S.T.Hu: "Homotopy Theory". Academic Press, New York

and London (1959).

K.Kutzler: "Eine Bemerkung fiber endlichdimendionale,

separierte, limitierte Vektorr~ume". Arch.Math.xx,

Fasc.2 (1969), 165-168.

L.S.Pontrjagin: "Topological Groups". Gordon and Breach,

Science Publishers Inc. New York, London, Paris (1966).

E.H.Spanier: "Algebraic Topology" Mc Graw-Hill Book

Company, New York (1966).

SOME CARTESIAN CLOSED TOPOLOGICAL CATEGORIES OF CONVERGENCE SPACES

by G@rard BOURDAUD

1.CATEGORICAL FOREWORDS.

Let ~ be a cartesian closed topological category (8)

("categorie compl~te ferm~e" in the sense of (1))(*). We

denote IXl the underlying set of an object X of ~ and

Hom the internal hom-functor of C .

D~finition 1.1: A blosed topological subcategory (CTSC) of ~ is

a full subcategory ~' of ~ which satisfies:

(CTSC I) If X and Y are objects of ~' , then Hom(X,Y) is an

object of C'.

(CTSC II) If X is the initial structure defined by a family of

mappings fi: IXI 7 IXil and objects X. of C' then X is an i __ t

object of C'.

The intersection of any family of CTSC of ~ is also a CTSC:

Definition 1.2: If ~ is any subcategory of ~ , let _D be the

intersection of all the CTSC of C which include D . ~ is called

the CTSC of ~ spanned by ~ .

(*) Despite it is now a classical terminology, we think that

"topological category" is far from being satisfactory. It is

rather ambiguous when we speak about categories of "quasi"-topologies

which are...topological. Moreover, the term "topological category"

was already used by EHRES~NN (7) to denote an internal category of

Top .

94

L being a fixed object of C and X any object of C

we have a canonical morphism a X : X >Hom(Hom(X,L),L) , which

maps x to : f > f(x) ; let us denote 1X the initial structure

defined on IXi by this morphism. X is usually finer than 1X; let

~(L) be the full subcategory of ~ whose objects X satisfy to X = 1X .

Proposition I.~: ~(L) is a CTSC of ~ which contains L; moreover, if

X is any object of ~ :

(i) Hom(X,Y) is an object of ~(L) whenever Y is ,

(ii) 1X is a reflexion of X in ~(L) ,

(iii) for all Y~(L) , Hom(1X,Y) = Hom(X,Y) .

Proof: I) Let Y~ be the initial structure defined by

mappings fi: IYi --> i Yii and objects Yi of ~(L) . We have a

commutative diagram:

Y. > Hom(Hom(Yi,L) ,L) 1

fi a i

Y --9 Hom(Hom(Y,L) ,L)

where f is the morphism Hom(Hom(fi,L),L) . If Z is any object of C i

and f a m a p p i n g : IZI > I Y ] s u c h t h a t a o f i s a morph i sm f rom

Z to Hom(Hom(Y,L),L) , then

N a. o f o f = f. o a o f I I I

is a m o r p h i s m f rom Z t o H o m ( H R m ( u ~ But Yit~ ~ ( L ) i m p l i e s t h a t

f. o f is a morphism from Z to Y. . Finally, f is a morphism: Z >Y . 1 1

So C(L) satisfigs to CTSC II .

2) Let YE ~(L) and X~ , U = Hom(X,Y) . We take Z~ ~ and f any

mapping from IZI to IU1, such that a U o f is a morphism from Z

to Hom(Hom(U,L),L) . In order to prove that f is a morphism: Z ----~U ,

95

it suffices to have a morphism:

f : X x Z ~ Y

(x ,z) ~ f (x) . Z

If (x,g)~ IXIX I Hom(Y,L)i , we have a morphism:

[x,g] : U ~ L

h >(g o h)(x),

u : Hom(Hom(U,L),L)

k

hence :

Hom(Xx Hom(Y,L),L)

> ~x~g) )~ k([x,g])!

If we compose u with a U o f , we obtain a morphism:

Z ~ Hom(XxHom(Y,L),L)

z ~ ~x,g) ,g(fz(X))]

thus by adjunction: Xxz ~ Hom(Hom(Y,L),L)

(x ,z) ~ [g , g(fz(X)~

which is precisely ay o ~ . Y being an object of ~(L) , ~ is

a morphism from Xx Z to Y , q.e.d.

3) We prove now that L~(L). Let f be a mapping: IZl ~ ILl such

that a L o f is a morphism : Z ~ Hom(Hom(L,L),L) ; we have morphisms

Ev I : Hom(Hom(L,L),L) ~ L

g(id L) , g

Ev I o a o f : Z ~L

Z Ev1(a(f(z))) = f(z) q.e.d.

4) From (1),(2),(3) we deduce that IXE~(L); on the other hand,

X is clearly finer than 1X . Let f be a morphism from X to Y@~(L).

If we compose a X : IX

Hom(Hom(f,L),L)

we obtain ay o f : 1X

~ Hom(Hom(X,L),L) and

Hom(Hom(X,L),L) ~ Hom(Hom(Y,L),L) ,

> Hom(Hom(Y,L),L) . Y being an object

of ~(L) , it follows that f is a morphism from 1X to Y .

96

5) By adjunction, Hom(IX,Y) and Hom(X,Y) have the same underlying

set; moreover, Hom(1X,Y) is finer than Hom(X,Y) . To prove the

reverse, we have only to see that the evaluation map is a morphism:

Hom(X,Y) x 1X

Ev is a morphism : Hom(X,Y) x X

1 ~om(X,Y) • X]

~(L) being a CTSC , 1 i~ campatible with finite products , which

implies : I[Hom(X,Y)x X] = IHom(X,Y) x IX

= Hom(X,Y) x IX ,

Hom(X,Y) being an object of ~(L) by (2); for the same reason Y = IY

and we obtain the required morphism .

) Y ;

Y , and so , functorially :

7 1Y .

2.CATEGORIES OF CONVERGENCE SPACES.

Let QTop be the cartesian closed topological category of

convergence spaces and continuous mappings (N.B. Convergence spaces

are usually called "espac~ quasi-topologiques" by French authors).

In QTop the structure of Hom(X,Y) is called the continuous convergence.

We denote:

- ~ the topological space on l~l={0,1) whose open sets are :

,{1}, I~1 .

- A the pretopological space on IAi={0,1,2~ whose neighborhood

filters are : ~(0) = { ]AI}, ~(1) ={ IAl}, ~(2) ={ IA1 ,{1,2 }} .

- R the usual space of real numbers .

Let PTop , STop , Top , UTop denote the full subcategories of

QTop whose objects are , respectively :

97

- pseudo-topological (or Choquet) spaces,

�9 t! - pretopological spaces (called "espaces semi-topologlques

in (4),(I0),(3)),

- topological spaces,

- uniformizable topological spaces .

PTop is a CTSC of QTop , but not the three others ; the four

subcategories are reflexive subcategories of qTop . If X~QTop ,

we denote sX , tX , ~X the reflexions of X in ST o~ , Top , UTo~ resp .

~X ~resp sX , tX ) is the initial structure on iXl defined by all

the continuous mappings from X to R (~esp. A,~ ).

If F is a filter on IXI, we write: F X~ x when F converges

to x in the space X . We define ConvxF = {xEIXi/ F x~X} .

Definition 2.1: Let K be any functor from QTop to To~ , compatible with

the forgetful functors . We say that X~QTop is K-closed-domained

if, for each filter F on iXi , COnVxF is KX-closed . A t-closed-

domained space is simply called closed-domained .

Proposition 2.2: The subcategory of QTop whose objects are K-closed-

domained spaces is a CTSC .

Proof: Let Z = Hom(X,Y) , where Y is a K-closed-domained space ;

for x&iXI , ev : Z ~ Y is a continuous mapping , so is : x

ev : KZ > KY . x

By the definition of continuous convergence

C~ = Q r ev~1(C~ Ev(F G)) .

G-~x

ConvyEv(Fx G) being KY-closed , each eVx1(ConvyEv(F xG)) is KZ-closed .

Let Y be the initial convergence space defined by mappings

fi : IYi > IYil and K-closed-domained spaces Yi "

98

ConvyF = igI~ fi I- (C~ �9 1

fi being continuous from Ku to KY i f-1 Convyi i ( f.F) is a KY-closed set, ' i

so is Conv~ .

Remark: The subcategory of closed-domained (resp. ~-closed-domained)

spaces is a CTSC which includes Top ( resp. UTop) .

Definition 2.~: Let K be any functor from QTop to STop, compatible wi~

the forgetful functors . We say that Xg QTop is a K-regular spaoe if ,

for each filter F on iXl and xEiXi , CIKx F ~-~x whenever F x-~X .

A s-regular space is simply called regular .

Here CIKxF denotes the closure of F in the pretopological

space KX ~Recall that STop is isomorphic with the category of closure

spaces . The present notion of K-regularity is somewhat different

from the K-regularity of COCHRAN and TRAIL (6)oBesides s, t, ~, we

define two functors from QTop to STop in the following manner :

Defi~iticn 2.4: Let X be a convergence space . We define the

topological space X" and pretopological space X ~ by their closure

operators :

- Clx~A ={yEIXI /3x~A • ClsXY } ,

- CIx.A :{ yg!X~ /~x~ A ~'~ CltxY }

for any Ac fXl . The correspondances X ~N ~ , X ---~X" define

two functors from QTop to STop , called, respectively, the star-

functor and the point-functor .

99

For details on X ~ and X', see (3),(4) (where "6toile-stable"

stands for "star-regular").

Propositio n ~.~: The subcategorz

spaces is a CTSC .

of QTop whose objects are K-regular

Proof: Let Z = Hom(X,Y) , where Y is K-regular . First, we

prove that, for all AcIZI , BclyI,

(1) Ev(CIKzAX B) C CIKyEV(AX B) .

By the continuity of e~ : KZ > KY , we have , for all x~IXI , x

evx(ClKzA) ~ CIKyeVx(A) .

Hence, for g~CIKz A , x~B , g(x) = eVx(g) ECiKyeVx(A)CC!KyEV( A •

From (I) we deduce, for all filters F on IZI , G on IXI :

(II) Ev( CIKzF • ~CIKzEV(F • G) , therefore :

->

>

--=>

G -~x C!KyEV(F xG) ~f(x) (since Y is K-regular)

G --~x Ev(CIKzF• -~f(x) (from (II))

CIKzF z-~f q.e.d.

Let Y the initial space defined by a family of mappings

fi" I Yl .~" JYi I and K-regular spaces Yi" By the continuity of

fi: KY >KY i , we have, for all AC IYl , fi(CiKyA) CCIKy(fiA) , 1

so, for a filter F on IYI ,

fi(CIKyF) ~CiKYi(fiF) then :

F ~x .-->i i fiF ~ fi(x) l

---> i (fiF) Tkfi(x) CIKYi l

-:-> i fi(clK~z) ~ fi(x) 1

(Yi being K-regular)

100

3.CLOSED TOPOLOGICAL CATEGORIES SPANNED BY Top AND STop.

If ~ is any CTSC of QTop , two types of characterization

can be expected for C :

- a categorical one , e.g. ~ is the CTSC spanned by a well-

known subcategory of QTop

- an internal one : the objects of ~ are exactly those which

satisfy to certain convergBnce properties.

By prop.l.3 , QTo~(A) and QTop(~) are CTSC . The following

theorems give the two expected characterizations for these CTSC.

Lmmma ~.I: For all X,Y objects of QTop , there exists a family of

topological spaces X i and mappings gi: Hom(X,Y) ~ Hom(Xi,Y) ,

such that Hom(X,Y) is the initial space defined by this family .

For the proof see (10) (proof of prop.1.8) .

~heo2em ~.2: QTop(A) is the CTSC spanned by STop . On the other hand,

it is exactly the category of pseudo-topological spaces.

Proof: We remarked that each XE STop is the initial space

defined by all the continuous mappings from X to A ; it follows

that STop ~ QTop(A) . Lemma 3.1 implies that, for any Y~QTop ,

Hom(Y,A) is the initial space defined by Hom(Xi,A) , where X i (and

AX) are pretopological spaces; hence QTop(A)C STop .

PTop being a CTSC which includes STop , we have

QTop(A) C PTop ; for the reverse inclusion, see (4) (Theorem II.4.1).

101

Theorem ~.~: (ANTOINE (1),MACHADO (IO)gBOURDAUD (4)) QTop(~) is the

CTSC spanned by Top . The objects of QTop(~) are called Antoine

spaces . A space X is an Antoine space iff X is pseudo-topological,

star-regular and closed-domained .

Proof: Again by lemma 3.1, we have QTop(~) = To~

topological spaces are pseudo-topological, star-regular, closed-

domained : the same is true for objects of Top , by prop. 2.2, 2.5 �9

For the reverse inclusion, see (4) (Theorem 1.4.4).

4~ CATEGORY OF c-SPACES.

We give now a double characterization of QTop(R). If we

restrict ourselves to Hausdorff spaces, an object of ~To~(R) is a

c-embedded space of BINZ or a c-space (9). Here we call a c-space

any object of QTop(R) (even non Hausdorff ones). First, we need

an analoguous to lemma 3.1 :

Lemma 4.1: If X is any convergence space and Y a T I convergence space,

there exists a family of uniformizable topological spaces X i and

mappings gi: Hom(X,Y) ~ Hom(Xi,Y) such that Hom(X,Y) is the

initial space defined by this family.

Proof: By lemma 3.1 and transitivity of initial structures,

we can suppose that X is topological. Let x be a fixed point of X. The

sets (VXV) UA , where V is a X-neighborhood of x and A the diagonal

of IXl x Ixl, form an entourage bases on IXt - Let X(x) denote the

uniformizable space induced by this uniform structure.

102

X(x) has the following neighborhood filters:

- ~X(x)(y) = Zx(X) i f l x ( y ) ~ Z x ( X ) ,

- l X ( x ) ( y ) = ~ i f ix<y) :V~ Ix(X) . For a mapping f : IXI ~IYI, we have the following lemma:

Lemma: f is continuous from X to Y iff, for all xEIXi , f is

continuous from X(x) to Y .

If f is continous from X(x) to Y , we have :

f(Xx(X)) = f([X(x)(X)) y~f(x) ; thus f is continuous from

X to Y . Let f be continuous from X to Y and xs If ~x(y)~x(X) ,

we have f(y) = f(x) , Y being a T I space , thus :

f(ix(x)(y)) = f(Xx(X)) -y>f(x) = f(y) .

If Zx(y) ~Zx(X) , then f(ZX(x)(y)) = f(#) = f~ -~->f(y~ �9

By the previous lemma IHom(X,Y)I~IHom(X(x),Y) Ifor all x61Xl.

Let us consider the family of canonical injections :

i x : IHom(X,Y)I ----~l Hom(X(x),Y)i �9

1) For each xs I, i x is continuous from Hom(X,Y) to Hom(X(x),Y).

Let F Hom(X,~f . If ~x(y)~x(X) , we have :

Ev(F• = Ev(F• ~--~f(x) = f(y) ;

if lx(y)~Zx(X) , we have :

Ev(F X i X ( x ) ( y ) ) = Ev(Fx #) - - ~ f ( y ) .

~f Thus F Hom(X(x),Y) "

2) Hom(X,Y) is the initial structure defined by the i x .

Let Z be any convergence space and f a mapping : IZI ---~I Hom(X,Y)I,

103

with i x o f continuous from Z to Hom(X(x),Y), for all xEIXJ . We

have the following morphisms :

Z >Hom(X(x),Y) : z I

X(x) > Hom(Z,Y) : y I

Y being TI, so is

mappings : X

Z ~ Hom(X.Y)

~fz

> [z I >fz(y)] .

Hom(Z,Y) ; by the lemma, we have continuous

> Hom(Z,Y) : Yl ~[zJ >fz(y)] , thus

>Hom(X,Y) ~ z] > f q.e.d. z

Theorem 4.2: QTop(R) is the CTSC spanned by UTop .

A uniformizable topological space X is the initial space

defined by all the continuous functions from X to R , therefore Ik

X~QTop(R) , thus UTopC qTop(R) . Conversely, by lemma 4.1, for

any YE QTop , Hom(Y,R) E UTop , which implies that any c-space is an /%

object of UTop .

We shall give now an internal characterization of c-spaces.

Such a characterisation was proposed by ~HRODER ((12) prop.3.5), but

it involves a notion of "strong solid" which is far from being simple.

We introduce first a few notations and lemmas . If X~QTop

and AC JX J , CX will denote the space Hom(X,R) , cX the reflexion

of X in QTop(R) (see prop.l.3) , A ~ the set of functions f ~ JcxJ

such that f(A)~[-q,1] .

Lemma 4.3: If F is a filter on iX J and H a filter on JCX i,

Ev(HxF) R-~0 iff ~ for each r>O, there exists A6F such that

rA~ H .

104

Proof: Ev(H• iff :

V r>0 3A~F 3 B~H : Ev(BXA)C[-r,r]

B~H : Ev(B • [-r,r ] ~>3 B(H

, but :

: B~rA ~ ~ rA~ .

Lemma 4.4: Let F be a filter on IXI and x~ IX l . F c-~X x iff

F ~---~xX and (P) for all filters H on ICXI such that H ~ 0 ,

we have Ev(H • -~->0 .

Proof: Let a be the canonical morphism from X to CCX . We

have : F c-~X x <- ~ aF CC~X a(x)

~---~ V H C~X f Ev(aF x H) ~ f(x) ;

but Ev(aF • = Ev(H E F) , thus :

F -j~>x ~----~V H -g~>f Ev(~• R--~(x) .

If we take H = f , we obtain , if F -~x ,

V f~ICXI fF -~ f(x) , which exactly means that F -~-~ x . On the

other hand, if H is any filter which converges to 0 in CX, we have

Ev(HxF) ~ 0(x) = 0 .

Conversely, let (F,x) satisfy to F ~--~-~x and (P) .

If H ~-~-~f , then H - f C~)~ 0 , thus Ev((H - ;)XF) -~-~0 ;

on the other hand fF -~f(x) . We have :

Ev(HXF)~ Ev((H - f)• F) + fF , which implies :

Ev(H xF) R ~ 0 + f(x) = f(x) q.e.d.

Lemma 4.~: Let X be a 0~-closed-domained space , F a filter on IXI

and x~ X . If F --~x and F o~x~x , then ConvxF = ~ .

105

Proof: Let z be a point of ConvxF ; this set being an

e-closed set which dc~not contain x , there exists f~ICXJ such that

f(x) = 0 and f(z) = 1 . F ~-~-~x ~ ~fF --~-~0 and F -~-2z~-~

fF R~I , which is a contradiction .

Theorem 4.6: A convergence space X is a c-space iff X is pseudo-

topological, ~-regular and 0~-closed-domained .

Proof: R has the three properties of the theorem: it is the

same for any c-space, by prop.2.2, 2.5 �9

Conversely, let X be a space which satisfies to the conditions

of the theorem. In order to prove that X = cX , it suffices to see

that, for each ultrafilter F on JXI , F / / ~ x implies F --/~x . X cX

If F /~ x , then F //~x , cX being finer than ~X . Let ~X cX

us suppose now that F ~---X-~x ; then, by lemma 4.5 , ConvxF ~ ~ o

For any convergent filter G of X, ClaigG is a convergent filter of X,

therefore F ~ Cl~xG and there exists AG~ G such that C~XA G ~ F .

Let H be the filter on ~CX~ spanned by the set~ rAg , where r> 0

and G is a convergent filter of X . It is clear that H C~X 0 �9

Suppose if possible that F c~X x , then, by lemmas 4.3,4.4 ,

there exists an A~F such that A~ H , that is to say : n

A~ ~ rA ~ , with r> 0 and GI,...,G n convergent filters = G i

of X . n

Let K = ~ Clo~xA G . Suppose that zgA - K ; K being an I

0~-closed set, i~ woul~ exist f@JCXl such that f(z) = 2 and f vanishes

on K : thus f/r@A~ for i=1,...,n , fE A ~ and f(z)~ [ -1, ,1] , which 1

is a contradiction.

106

Therefore K~A , which implies that K~F . On the

other hand, F ~ing an ultrafilter, we have X - Cl~xAGEF ,

thus: X - K = ~=~ ~X - C~,xAGi)~ ~ F . This is a contradictioza!

Corollary 4.7: Let X be any Hausdorff convergence space;

then X is a c-space iff X is an u-regular pseudo-topological

space.

This result is due to KENT ((9) Theorem 2.4) and

SCHRODER ((12) Theorem 3.6); they remarked that any

-regular Hausdorff space is w-Hausdorff, and so w-closed-

domained6 In the pretopological case, it was already

established by BUTZMANN and MOLLER (5)~

Remarks: SCHRODER ((12) Ex.4o3.) gave an example of a

compact Hausdorff ~-regular space which is not pseudo-

topologicalo BUTZMANN and MOLLER (5) found a regular

m-Hausdorff topological space which is not ~-regular.

There remains the following problem: find an w-regular

topological space which is not ~-closed-domained (Such

a space must not be Hausdorff !)~

107

5.CATEGORICAL REMARKS.

I. Are the conditions of prop.l.3 characteristic for

categories of ~(L)-type ? More precisely, we ask the following:

let C' be a reflexive CTSC of C which satisfies to (i) and (iii);

does exist L~' such that ~' = ~(L) ? The answer is not clear:

indeed, we ignore if QTop itself is equal to QTop(L), for some

convergence space L .

2. Let -~ be the two-element space with the indiscrete g

topology. In QTop , A~_ appears to be a classifying object for g

regular monomorphisms (or kernels). It is clear that _O_ is an g

object of ~he CTSC of QTop whose objects are, respectively:

- pseudo-topological spaces,

- Antoine spaces,

- c-spaces,

- K-regular spaces, for any K: QTop ~STop ,

- K-closed-domained spaces, for K: QTop ---~To~o with

Thus, Qtop and each of these categories are quasi-topoi,

in PENON's sense (11). This fact seems to us a sufficient reason

to involve non-Hausdorff spaces in our considerations.

108

Bibliography:

(1) ANTOINE P., Etude @l@mentaire des cat@gories d'ensembles

structur@s, Bull. Soc. math�9 Belge, 18, n~ 1966.

(2) ANTOINE P., Notion de compacit@ et quasi-topologie, Cah. Top. l Geom. diff , 14, n~ 1973

(3) BOURDAUD G., Structures d'Antoine associ@es aux semi-topologies

et aux topologies, Comptes rendus Acad. Sc. Paris, 279, s@rie A,

1974, pp 591-594.

(4) BOURDAUD G., Espaces d'Antoine et semi-espaces d'Antoine, Cah. J

Top. Geom. diff., 15 (to appear).

(5) BUTZMANN H.P. and MDLLER B., Topological c-embedded spaces,

Man. F~k. Math. Uni~. Mannheim, nr.31 (1972).

(6) COCHRAN A.C. and TRAIL R.B., Regularity and complete

regularity for convergence spaces, Lecture notes 375,

(7) EHRESMANN C., Cat@gories topologiques, Indag. math., 28, n~

1966.

(8) HERRLiCH H., Cartesian closed topological categories, Math. Coil.

Univ. Capetown, 9, 1974.

(9) KENT D.~ Continuous convergence in C(X), Pacific jour. math.,52,

n~2, 1974.

(10) MACHADO A., Espaces d'Antoine et pseudo-topologies, Cah. Top.

G@om. diff., 14, n~ 1973.

(11) PENON J., Quasi-topos,Cah. Top. G@om. diff., 14, n~ "1973.

(12) SCHRODER M., Solid convergence spaces, Bull. Aust. Math. Sec.,

8, 1973.

0 0 0 0 0 0 0 0 0 0 0 0 0 0

Je remercie particuli@rement Armando MACHADO et Philippe

ANTOINE, dont les id@es sont ~ l'origine des d~veloppements

pr6s~nt@s ici.

TOPOLOGICAL FUNCTORS AND STRUCTURE FUNCTORS

G.C.L. Br~mmer

O. Introduction In this paper we study the construction and

some properties of functors F-- we shall call them structure

functors- which make the triangle

(1) fi ,'/ i T M

commute, where T and M are given faithful functors.

Structure functors arise naturally in the theory of the top

categories of O. Wyler [44], [45]- (also discussed by, among

others, H.-G. Ertel [iO], R.-E. Hoffmann[17], S.H. Kamnitzer [26],

T. Marny [28], [29], W. Tholen [38], M.B. Wischnewsky [41], [42],

[43]) -- and in other situations of greater or lesser generality

treated by M. Hu~ek [22], [23], [24] as well as, for example, by

P. Antoine [i], A.A. Blanchard [3], the author [5], [6], [7],

C.R.A. Gilmour [12], J.W. Gray [13], M.N. Halpin [14], H. Herrlich

[15], [16], R.-E. Hoffmann[20], A. Pultr [31], J.E. Roberts [33],

J. Rosick# [34], S. Salbany [35], [36], [37], and M.B. Wischnewsky

[39], [40].

Research aided by grants to the author and to the Topology Research Group from the South African Council for Scientific and Industrial Research and from the University of Cape Town.

110

This paper is a sequel to the joint paper [8]. The

characterization of topological functors in [8] accounts for the

intimate relationship between topological functors and structure

functors.

In the first section we reconsider the diagram

W

(2) K FI I T I

/ f

�9 > � 9

M

from [8] and drop the requirement that K be full. Then F

as constructed in [8] ceases to be a diagonal in the strict sense,

but is still characterized by an extremal property. (The con-

struction goes back to M. Hu~ek's paper [24], was rediscovered in

[5] and developed in [6].)

In the second and third sections we study the effect of K

and W on the properties of F, in particular the property that

F is right inverse to a given amnestic functor L. We find an

appropriate context for the latter problem to be the case where L

is followed by a functor M such that ML is topological and L

sends ML-initial sources to M-initial sources. The concept of an

M-spanning subcategory of the domain of M (a kind of density condi-

tion) enters into the study of right inverses of L, and therefore

we have a fourth section leading to a characterization of M-spanning

subcategories, overlapping somewhat with work of T. Marny [28],[29].

The results on preservation of initiality in section 2, though

special cases of results of O. Wyler [44], are proved for the sake of

exposition.

For basic properties of topological functors not discussed here

and for more bibliography the reader is referred to H. Herrlich [15],

111

[16, appendix A], R.-E. Hoffmann[17], [19], [20], and further to the

cited works on top categories. In particular [15], [28], [19] and

[7] deal with relatively topological functors. (As our topological

functors are those that are transportable and absolutely topological

in the sense of Herrlich [15], our considerations are independent of

any factorization structures in the base category.)

Many of the results come from my thesis [6], written under the

supervision of Keith A. Hardie, to whom I wish to express my deep

appreciation. The present exposition is however based on the new

point of view offered by the diagonal diagram (2) above.

i. Filling in triangles of faithful functors

section we consider a faithful functor T: k § C.

refer to [8], adding the observation that for objects A,B of A

is consistent with our definitions to write A "iT B if there is a

morphism g:A + B with Tg = ~.

Given a commutative square of faithful functors

W 9 >A

~ --g---~ c

w i t h o u t a s s u m i n g K

a f u n c t o r F : ~ - * - A , a s g i v e n i n [ 8 ] , a s f o l l o w s . F o r X 6 o b ~ ,

s u p p o s i n g t h a t t h e T - i n i t i a l i t y p r o b l e m

(NX, Mf:MX + MKD, ~ I D 6 ob'D, f 6 ~ ( X , KD) )

has a solution, we denote the solution by

(FX, (Mf)' :FX § WD I D 6 obD, f 6 ~(X, KD));

and for g 6 {(X' , X) we define Fg as follows:

Throughout this

For definitions we

it

full, we reconsider Hu~ek's [24] construction of

112

Mf = T (Mf) ' MX ~" MKD = TWD

TFg Mgl ~ T (M (fg)) ' MX'

(Mf) ' FX ~WD

~ ~ ( M ( f g ) ) !

IFX'

Since the left hand triangle commutes, there exists unique

Fg:FX' + FX such that TFg = Mg.

This defines a functor F: ~ § provided each of the relevant

T-initiality problems has a solution and provided, for each X, a

choice is made among isomorphic candidates for FX. For this

functor we shall use the notation

F = <T,M,K:W>.

If T is amnestic, then F is uniquely determined by T,M,K,W.

i.I Proposition Given a commutative square

D w~A

o f f a i t h f u l f u n c t o r s . I f F = < T , M , K : W > e x i s t s , t h e n F i s a

T - c o a r s e s t f u n c t o r ~ § g s a t i s f y i n g t h e c o n d i t i o n s TF = M a n d

FK ~T W. If in addition K is full and T amnestic, then FK = W.

Proof Clearly TF = M. For fixed D O in obD , the source

(FKDo, (Mf)' :FKD ~ § WD) is the solution to the T-initiality problem

, § KD). Taking (MKDo, Mf:MKD ~ § MKD WD) (all D E obD , all f:KD o

in particular f = IDo we define n D = (MIDo)'. As TnDo I for O

arbitrary D O , we have FK ~ T W. If G:~ § is any functor with

TG = M and GK ~ T W, then we have m:GK + W with Tm = (|) and

considering the diagrams

113

f X ~ KD

Mf (Mf)' MX ~ MKD FX > WD

| | PX I mD

TGX TGf~ TKD GX ~ GKD

we obtain Px:GX + FX with TPx = I ; thus G ~T F.

It was shown in [8] that if K is full and T amnestic, then FK = W.

1.2 Proposition Given faithful T: A § and M:K § . Then

every functor F such that TF = M, is of the form F = <T,M,K:W>

for suitable K and W (e.g. K = I x and W = F).

i. 3 Proposition Given a commutative square of faithful functors

W

~ --N--~ c

l e t '0 d b e t h e d i s c r e t e c a t e g o r y w i t h t h e s a m e o b j e c t s a s D ,

K d = KID d , W d = WID d. Then F = <T,M,K:W> if and only if

F = <T,M, d:Wd>

2. Going up a @iven trian@le In this section we consider two

given amnestic functors L: ~ § { , M: K + C . Then ML is amnestic.

We are interested in the functors F: { § ~ for which MLF = M;

these are just the functors of the form F = <ML,M,K:W>.

We shall say that L preserves initialit[ for M if L sends

each ML-initial source to an M-initial source. (We shall briefly

discuss the relation of this concept to the taut lifts of O. Wyler

E41] at the end of this section.)

114

2.1 Proposition Let F = <ML,M,K:W>.

(i) If I~ !M LF , then K !M LW.

(ii) If K !M LW and L preserves initiality for M,

< LF. then I X ~[

Proof (i) From I x % M LF follows K !M LFK and from FK ~ML W (i.i)

follows LFK ~M LW; hence K !M LW.

(ii) We have t:K § LW with Mt = (I). The ML-initiality

problem (MX, Mf:MX § MKD, WD 1D 6 ob9, f:X § KD) has the

solution (FX, (Mf) ':FX + WD I ---). Hence the M-initiality

problem (MX, Mf, LWD I -.-) has the solution (LFX, L(Mf)' I ..-)"

MX Mf ~ ~D LFX ~ LWD

mx' 1 I f

MX X

The definition of "solution" now yields mx:X § LFX with

Mm x = IMX. Thus IX !M LF-

We introduce a simplified form for an important special case

of our notation, as follows. Any s~category U of A determines

an inclusion functor W:~ + A and a functor K = LW:~ § M .

In this case we abbreviate the notation F = <ML,M,K:W> to

F = [L,M:~].

Use of the latter notation will always presuppose that U is a

subcategory of ~. By 1.3, [L,M:U] is unchanged if we make

discrete, so we may regard i simply as a class of objects of

2.2 Lemma If F = [L,M:K], then FLA ~Z~ A for each A 6 ~.

115

Proof As above, we have F = <ML,M,K:W> and by i.i,

FK ~ML W, i.e. FLW !ML W.

2.3 Theorem Consider a functor F: { + ~.

(i) If IX !M LF and FLF = F, then there exists a class

of objects of A such that

F = [L,M:U].

There is even a largest such class, namely

= {A 6 ob~ I FLA !M L A}.

(ii) If F = [L,M:~] for some class N cob ~ and if L

preserves initiality for M, then I~ !M LF and FLF = F.

Proof (i) Kith ~ as above, for each A 6 ~ we have

tA:FLA § A with MLt A = ~. We show that the ML-initiality

problem (MX, Mf:MX § MLA, A [ A 6 U, f:X + LA) has the

solution (FX, tA.Ff:FX § A ] ...). Consider a source

(Y, hf:Y § A I ..-) and u:MLY § MX such that the left-hand

triangle commutes:

Mf t A. F f MX ~ N~A FX ~ A

u h m x l )mY Y

AS I~ 2M LF, we have mx:X § LFX with ~X = I ; and as

FL(FX) = FX we have FX 6 ~, so that m x is one of the indexing

f, giving us a special case of the left-hand triangle which

shows that u = MLh . Hence F = [L,M:K]. If ~ is any m X

class of objects of A such that F = [L,M:U], then by 2.2 A

(ii) Let F = [L,M:U] and let L preserve initiality

for M. Considering the inclusion W: U + ~ and K = LW, we

116

have from 2.1 (ii) that IX !M LF. So there is m:I~ § LF

with Mm = (I). By 2.2 we have tA:FLA § A with MLt A = I.

Now consider that FX and FLFX are given, respectively, by the

ML-initiality problems

(MX, Mf:MX § MLA, A I A 6 ~, f:X + LA)

and

(MLFX, Mg:MLFX + MLA, A I A 6 ~, g:LFX § A).

In these problems we have MLFX = MX and

LtA M(LFX LFf> LFLA LA) = Mf

m x M(X m LFX g > LA) = Mg

so that the two problems are identical. As ML is amnestic, it

follows that FX = FLFX, whence F = FLF.

2.4 Examples Unif, Prox, Creg will denote the categories of

uniform spaces, proximity spaces, completely regular topological

spaces, respectively, without imposition of the To-separation

axiom.

(i) If L:Unif § Creg, M:Cre~ § Ens are the usual forget-

ful functors, L preserves initiality for M. M and ML are

topological.

(ii) If L:Unif § Prox, M:Prox § Ens are the usual forget-

ful functors, L does not preserve initiality for M. Still

M and ML are topological.

(iii) In example (i), if F:Cre~ § Unif is the functor giving

each space the discrete uniformity, then MLF = M but not

I !M LF.

(iv) Let ~ be the category of metric spaces and non-expansive

mappings, ~ the category of metrizable topological spaces and

117

continuous mappings, L = I~, M: ~ § r the functor assigning the

metric topology, F:~ § X given by F(x,d) = F(X,ı89 L and

M are amnestic and L preserves initiality for M, and

I~ ~M LF but FLF ~ F.

The assumption of preservation of initiality in Theorem 2.2

was the best possible in the following sense:

2.5 Proposition Let ~ be topological. If for each class U

of objects of A , putting F = [L,M:U] we have I~ ~M LF and

FLF = F, then L preserves initiality for M.

Proof Given an ML-initial source (A, Fi:A § Ai) I. The

functor F = [L,M:{Ai}I] exists, FLF = F and IX ~M LF, so

we have m:I~ § LF with Mm = (4). By 2.2 we have

ni:FLA i + A i with MLn i = ~. To see that the source

(LA, Lfi:LA § LAi) I is M-initial, consider any source

(X, gi:X § LAi) I and any u:MX + MLA such that the left-hand

triangle commutes :

MLf. Lf 1 1 MLA ~ MLA LA ~ LA. A

1 I/ vj/ u LFX i vl. , ! I

MX X FX Fg i

fi A i

n i

) FLA. 1

As ML(niFg i) = Mgi, there exists v:FX § A such that MLv = u.

Thus M(Lv.m X) = u, and the source (LA, Lfi) I is M-initial.

2.6 Theorem

(i) If there exists a functor F: ~ + A such that I~ <M LF

and FL <ML I&, then L preserves initiality for M.

118

(2) If ML is topological and L preserves initiality for

M, then there exists a functor F: ~ § such that I9~ --<M LF

and FL <ML IA"

(3) In both (1) and (2) , F = [L,M:A].

Proof (3) From I~ <M LF follows F --<ML FLF and from FL <ML I~ A

follows FLF <ML F, whence FLF = F. Thus by 2.3 F = [L,M:U]

where ~ = {A 6 ob~ I FLA <ML A} = obA.

(1) Given any ML-initial source (A, fi:A§ we may

use F = [L,M:ob~] and carry through the remainder of the proof

of 2.5 to show that the source (LA, Lfi:LA § LAi) I is M-initial.

(2) Let F = [L,M:~]. By 2.3 I9( --<M LF, and by i.i, since

F = <ML,M,L:IA> , we have FL <ML I A �9

2.7 Corollary Let L: A§ ~ be an amnestic functor. The follow-

ing two statements are equivalent:

(I) F = [L,I :~];

(2) I~ = LF and FL <L I;s

Moreover they imply:

(3) F is the L-finest right inverse of L;

(4) F is left adjoint right inverse to L;

(5) F is fully faithful;

(6) L preserves initiality for any faithful functor M: ~ § C.

The conditions (4) and (5) form part of R.-E. Hoffmann's

characterization (under side conditions) of topologicity

[17, pp. 84-85], [20].

The condition (3) above does not imply (4):

119

2.8 Example The forgetful functor L:unif § Prox has a unique

right inverse but no left adjoint [24], [6].

A category ~ is said to have trivial centre if there is

just one natural transformation of the identity functor I{ to

itself. This occurs very often: a sufficient condition is that

{ have a generator with just one endomorphism (remark due to

K. A. Hardie, cf. [6].) More on this concept is to be found in

R.-E. Hoffmann [18].

2.9 Proposition Let L: ~+ ~ be amnestic and let ~ have

trivial centre. If L has a left adjoint right inverse F,

then F = [L,I{:A]. Thus L has at most one left adjoint right

inverse.

proof The unit r:I~ § LF and counit s:FL § I~ of the ad-

junction satisfy LsA. rLA = ILA for A s obj. As LF = I{,

r is trivial, hence LSA = |LA' i.e. FL ~L I~, so that by

2.7 F = [L,I~:~].

When L: ~§ K is topological, then F = [L,I~:~] exists.

(The converse is not true: consider the forgetful functor from

To-Spaces, or Tf-spaces, ... to Ens.) By Antoine's theorem

([i], [33], [5], [17], cf. [8] for an external proof), L is

cotopological. Thus the dual of [L,I~:~] - for which we have

no notation: it may be described by the dual of Proposition i.i

or by coinitiality construction - exists, so that by the dual

of 2.7 we have:

2.10 Proposition If L:~ ~ ~ is topological, then L preserves

initiality and coinitiality for any faithful functor M: ~ § C.

In this preposition we may drop the assumption that L

is amnestic.

120

import~t theorem of O. ~ler may in our terminology

be phrased as follows.

2.11 Theorem on "taut lifts" of O. Wyler [44, 45]

Let P:A§ and Q: ~+C be a~estic topological functors

~d ~ ~d R functors such that Q~ = ~.

Then the following two conditions are equivalent:

(i) ~ sends P-initial sources to Q-initial sources

~d S is left adjoint to R;

(2) $ has a left adjoint ~ such that P~ = SQ.

Moreover, ~der either condition it is possible to arrange the

relation between the adjunctions as follows:

Qg = ~Q ~d P~ = eP

with ~ (respectively e ) as unit (respectively counit)

for S ~ R ~d @,6 those for ~ ~ ~.

B < s ~ C

For other elaborations on the theorem, see [45], [29], [38].

For ex~ple, the implication from (i) to (2) does not need

topologicity of Q.

The basic example [44] has P, Q, r , R the usual forgetful

functors between ~ = {topological groups}, B = {groups},

= Top, C = Ens.

~ler [44,45] gives applications to topological algebra; for

generalizations see [iO], [38], [41], [42], [43].

121

Clearly our theorem 2.6 is essentially the case R = I C

of Wyler's theorem.

In proving the implication (i) ~ (2) in Wyler's theorem,

we can construct the functor ~:X+~ as follows. We have the

counit c:SR § I B. For any object X of ~ , we form the

P-initiality problem

(SQX, EpA. SQf , A I A 6 obA , f 6 ~(X,~A)).

The domain of its solution will be ~X.

This exceeds the framework of our construction in section i.

In fact, considering the mentioned example of the topological

groups, one readily sees that it is impossible in general to

express ~ in our notation as a functor of the form ~ = <P,M,K:W>.

3. Right inverses of an amnestic functor Let L: A § ~ be

amnestic. From the following two special cases of the diagonal

construction W

~ > A U ~ >~

// /~ IL L{~ /

/

together with 1.2 one immediately sees that:

3.1 Proposition For F: { § A the following are equivalent :

(i) LF = IX;

(2) There exist functors K and W such that LW = K and

F = <L,IM,K:W>;

(3) There exists a subcategory ~ of ,4. such that

F = [L,I :~].

However, unless L is topological, these characterizations

seem to be of little use in calculating those right inverses

122

that may exist. Also, when X = Ens, the right inverses are so

obvious as to have little interest, and when { is not Ens but

some category of topological type, then it may be quite hard to

verify that L:A § ~ is topological; a nice example is the

forgetful functor from contiguity spaces to Separated Lodato

proximity spaces, shown by Bentley and Herrlich [2, theorem 7.3]

to be topological. The straightforward but interesting situation

often arises that L is not topological but there exists a

functor M: ~ § C such that ML is topological and L preserves

initiality for M, as is the case in the following examples

(taking the usual forgetful functors) :

3.2 (i) A = Unif, ~ = Crew, C = Ens;

(ii) K = Qun (quasi-uniform spaces [5] ), ~ = Top, C = Ens;

(iii) A = Qun, ~ = Pcr9 (pairwise completely regular

bitopological spaces [27], [35], [5] ), C = Ens;

(iv) A = Prox, { = Crew, C = Ens [21];

(v) A = {Lodato proximity spaces}, ~ = {Ro-spaces},

C = Ens [21];

(vi) i = Zero (zero-set spaces), K = Crew, C = Ens [12].

Herrlich [16, pp. 62, 80, 112-113] has argued convincingly

against a simplistic view about certain topological categories

being richer in structure than others; as "forgetful" functors

now go in all directions, there seems to ue a need for tools to

handle them.

For the rest of this section we again consider two amnestic

functors L: K § ~ , M: ~ § C .

123

We shall say that a subcategory V of K is M-spanning

if IK = [I~,M:V]. Again only the objects of V are involved.

3.3 Proposition A subcategory V of ~ is M-spanning if

and only if for each object X of { the source

(X, f:X § V ] V 6 obV , f 6 ~(X,V))

is M-initial.

Two other characterizations (under side assumptions) follow

later. For relations to J.R. Isbell's deeper concept of

adequacy, see [32].

3.4 Examples In the following instances V is M-spanning:

(i) M: ~ § C arbitrary, V = ~;

(2) M:Top § Ens the forgetful functor, V contains as

object a Sierpi~ski space (two points with three open sets) or

any space having a Sierpi~ski subspace;

(3) Write down the obvious proposition of the form "If

is M-spanning, so is every V such that ..." Hint:

Compose M-initial sources [9, p.876];

(4) M:Creg § Ens forgetful, ~ 6 obV ;

(5) M:Pcrg § Ens forgetful, V has as object the real

line with the two topologies

{(-~,x) I -~ s x • ~}, {(x,~) I -~ s x s ~}"

We recall that the notation [L,M:U] presupposes that U

is a subcategory of K .

3.5 Theorem Suppose F = [L,M:~]. Then:

(i) If LF = IK, then L~ is M-spanning.

(2) If Li is M-spanning and L preserves initiality for

M, then LF = I~.

124

Proof (i) We show that the source

(X, f:X + LA ] A 6 U, f 6 X(X,LA)) is M-initial.

source (Y, gf:Y § LA i

triangle commutes:

Consider a

�9 ..) and u:MY § MX such that the first

Mf f MX ) MLA X > LA

, / r ,v l

I I

MY Y

A

fSTtA FX > FLA

t

I !

FY

By 2.2 we have tA:FLA § A with Y~t A = I. Let f' = tA.Ff.

Then t,~f' = Mf, and ML(tA.Fg f) = Mgf. Hence there exists

v:FY § FX with MLv = u. As LF = I, we have Lv:Y § X

with M(Lv) = u, as required.

(2) F = [L,M:U] means that for each X 6 ob~ the source

(FX, (Mf)' 1 A 6 U, f:X § LA) solves the M-initiality problem

(MX, Mf i .--). As L preserves initiality for M, the source

(LFX, L(Mf)' i ..-) is then M-initial; this means that

LF = [IM, M:L~], and by definition of spanning, LF = I x.

3.6 Corollary Every right inverse F of L is of the form

F = [L,M:U] with LU M-spanning.

3.7 Theorem Let V be an M-spanning subcategory of ~ , let

L preserve initiality for M, and let F: X § K be a functor

such that Ix ~M LF. Then F is a right inverse of L if

and only if LFV = V for each V in V .

Proof Let I~ ~M LF and LFV = V for each V in V. By

1.2 and 2.1, F = <~,M,K:W> with K ~M LW. FX is given by

solving the ~-initiality problem

125

(MX, Mf:MX § ~D, WD I D 6 ob9 , f 6 ~(X,KD))

and as L preserves initiality for M, LFX is given by solving

the M-initiality problem

(i) (MX, Mf:MX § ~D, LWD I D 6 obD , f 6 {(X,KD)).

In particular for V 6 V, since LFV = V, we then get V by

solving the M-initiality problem

(2) (MV, Mh:MV § MKD, L~ I D 6 obD , h 6 ~(V,KD)).

As V is M-sufficient, we get X by solving the M-initiality

problem

(3) (MX, Mg:MX + MV, V I V 6 V, g 6 ~(X,V)).

By the well known compositive property of initial sources [9, p.876]

it follows from (2) and (3) that X is given by solving the

M-initiality problem g h

(4) (MX, ~m. Mg, LWD I D 6 obD , V 6 V, X ~ V § KD).

As K ~M LW, we may augment the family of all hg in (4) to the

family of all f:X § KD (since M sends both to the same family).

But then the problem (4) becomes the problem (i) and still has

the solution X. Hence X = LFX.

It is important to observe that in the above theorem the

condition I~ ~M LF cannot be weakened to MLF = M [5, example

3.2]. Thus M-spanning subcategory is not an analogue of

dense subspaceo

3.8 Proposition Suppose F = [L,M:U].

only if F = [L,I~:U].

Proof Routine calculation.

3.9 Proposition

initiality for M,

exists, satisfies

Then LF = IK if and

Suppose ML is topological, L preserves

and L is M-spanning. Then F = [L,M:A]

I~ = LF and FL ~L IA' and is thus left

adjoint right inverse to L.

126

Proof By 3.5, LF = I. Thus by 3.8, F = [L,I :A] and 2.7

can be applied.

We observe that if F and G are right inverses of L, then

F --L < G if and only if F ~ML G

so that we may write simply F ! G. The class of all right inverses

of L will henceforth be equipped with this ordering. Suitable

set theory will safeguard operations in this class.

3.10 Theorem Suppose ~ is topological, L preserves initiality

for M, and LK is M-spanning. Then, in the class of all right

inverses of L, every non-void subclass has an infimum.

Proof Express each F i in the subclass in the form F i = [L,M:~ i]

with largest ~i (2.3), let ~ be the union of the ~i' and

let F = [L,M:~]. By 3.5 each L~ i is M-spanning, hence so is

L~, and LF = I. Clearly F < F.. If G < F. for each i, -- 1 -- l

let G = [L,M:V] with largest V . It follows by 2.3 that U c ~.

Hence G < F.

3.11 Corollary Any one of the following sets of conditions is

sufficient for the class of all right inverses of L to form a

large-complete lattice:

(i) ~ is topological, L preserves initiality for M,

L& is M-spanning, and L has a coarsest right inverse;

(2) ML is topological, L preserves initiality and co-

initiality for M, and L& is M-spanning and M-cospanning;

(3) L is topological.

Call an object X ~ of ~ an L,M-pivot if {X o} is

M-spanning and there exists a unique object A o of ~ with

LA ~ = X o .

127

3.12 Proposition Let L preserve initiality for M and let

have an L,M-pivot X ~ = LA o. If the functor [L,M:{Ao}]

exists, then it is the coarsest right inverse of L.

3.13 Examples of L,M-pivots Let C = Ens, let L and M

be the usual forgetful functors, and let

(i) A = Unif, { = Crew, X ~ = [O,1];

(ii) K = Qun, ~ = Pcr~, x ~ the set [O,1]

equipped with the subspace bitopology from the example in 3.4(5) [35];

(iii) K = Qun, ~ = Top, X ~ the Sierpi~ski space

(cf. 3.4(2)).

3.14 Applications In Example 3.13(i), thinking of embedding in

a uniform product of uniform intervals, one readily sees from

3.12 that the separated-completion reflector 7 in Unif acts

on the coarsest right inverse ~* of L:Unif § Cre@ to produce

v

the Stone-Cech compactification, uniformized; in fact one has

the well-known natural isomorphism y~* = ~*B. There is an

exactly analogous situation in Example 3.13(ii), discovered by

Salbany [35] and further discussed by the author [6], in which

the bitopological analogue of compactness is double compactness,

i.e. the infimum (with respect to !M, which means finerl) of

the two topologies is compact in the usual sense, and the

quasi-uniform analogue of completeness is double completeness,

i.e. the infimum of the quasi-uniformity and its conjugate is a

complete uniformity. For 3.13(iii), see [5]. Returning to L M

Unif § Cre~ § Ens, and taking completions of other right inverses

than the coarsest, one has reflections to generalized compactness

properties. For analogous completions, see Narici, Beckenstein

and Bachman [30].

198

3.15 Example If X ~ = LA ~ is an L,I~-pivot and F = [L,I~:{Ao} ]

exists, then not only is F the coarsest right inverse of L,

but also F gives each object of ~ the L-coarsest A-structure.

Hence, for instance, the forgetful functor Unif § Cre~ is not

topological: not every completely regular space admits a coarsest

uniformity.

We shall call F:~ § & a simple ri@ht inverse of L if there

exists an object A of A such that F = [L,I~:{A}]. Functors

of this form are always right inverses of L (3.1), and in this

form the definition of "simple" has the advantage of being intrinsic:

there is no reference to an extraneous M.

3.16 Proposition (i) If F is a right inverse of L and

F = [L,M:{A}] where A 6 obK , then F = [L,I~:{A}], so that

F is simple.

(2) Let A have small products and let ~ preserve products.

If ~ is a set of objects of ~ such that the functor

F = [L,M:U] is a right inverse of L, then F is simple.

Proof (i) 3.8. (2) It is routine to check that if A is the

product of the objects in the set U , then F = [L,M:{A}].

Then (i) applies.

The author does not know a suitable converse to the above

proposition.

3.17 Examples (i) Consider the usual forgetful functors

L:Qun § Top, M:Top + Ens. For each infinite cardinal number

m let R m be the class of all quasi-uniform spaces of cardinality

not exceeding m, and let F m = [L,M:~m]. As the class ~m

can be skeletized to a set and L~ m is M-spanning, by 3.16 F m

is a simple right inverse of L. It was shown in [5] that if

129

if m < n, then F m is strictly coarser than F n. Thus L has

a proper class of right inverses, forming a large-complete lattice

by 3.11. If G is any simple right inverse of L, say

G = [L,I :{A}], we may take m large enough that A 6 Um, and

= , < G. Hence as by 3.8 F m [L,I~ :U m] it is clear that F m _

the finest right inverse of L is not simple. Halpin [14]

proved that there exists another non-simple right inverse of L:

the finest which gives transitive quasi-uniformities (in the sense

of [ii]) to topological spaces.

(2) Similarly it can be shown that the forgetful functor

Unif § Cre9 has a proper class of right inverses, of which the

finest is non-simple.

4. Bireflectors as structure functors If ~ is a reflective

subcategory of ~ with reflection functor R:{+~ , then we

call the composite x R-+~+{ a reflector in ~. It is a

bireflector if the universal morphisms X + RX are bimorphisms.

Bireflectors given by initiality are of widespread occurrence

(cf., e.g., the talks [4] and [25] in the present conference).

We apply our formalism to one aspect, and refer the reader to

[16, Appendix A] for other.

Throughout this section M:~ § C will be an amnestic functor.

M is called transDortable [17], [16, p.llO] if for each Y 6 obK ,

each C 6 obG and each isomorphism g:C § MY, there exists

X 6 ob K and an isomorphism f:X § Y such that Mf = g. Trans-

portability is self-dual. Each topological functor is transport-

able. (The weaker definition of topologicity in [17] and [15]

does not imply transportability.)

130

We consider the situation

M ~ C

As I~ p r e s e r v e s i n i t i a l i t y f o r M, we h a v e t h e f o l l o w i n g s p e c i a l

case o f T h e o r e m 2 . 3 :

4.1 Proposition For a functor F:~ § �9 the following are

equivalent:

(i) FF = F and I~ ~M F;

(2) There exists a subcategory ~ of ~ such that

F = [I~, M:V].

Moreover the largest subcategory ~ for which condition (2)

holds has object class {X 6 ob~ I FX = X} and this class is

isomorphism-closed.

Proof In 2.3, V is isomorphism-closed by its maximality; also

ob~ = {X I FX ~,I X}, which by I~ s F reduces to {X I FX = X}.

4.2 Proposition If FF = F and I~ ~M F, then F is a

bireflector in ~.

Proof We have rx:X § FX with Mr x = I, so that r x is a bi-

morphism and a reflection to the V of 4.1.

The commonest examples of bireflectors [I~, M:~] are

(i) Top onto Crew, with V = {~ } ~d M:Top § Ens forgetful;

(ii) Unif onto {Totally bounded uniform spaces}, with ~ consist-

ing of the usual uniform space on [O,i] ~Ld M:Unif § Ens

forgetful.

131

4.3 Proposition Let M be transportable and send bimorphisms

to isomorphisms. If F is a bireflector in ~ onto the sub-

category ~, then the functor [I~,M:B] exists and is naturally

isomorphic to F.

Proof We have the universal morphism rx:X + FX such that Mr X

is iso. By the transportability there exists GX and an iso-

morphism nx:GX § FX with ~X = Mrx" For f:X' § X we let

-I Gf = n x .Ff.nx,. This defines a functor G: { § ~ and a natural

-I isomorphism n:G § F. Letting s x = n x rx:X § GX we have Ms X = I,

so that I~ !M G. Hence G ~M GG. The reflection property of

sGx:GX + GGX gives kx:GGX § GX with kxSGx : I, whence ~X = I.

Hence GG = G. Letting ob ~ = {X ] GX = X} we have by 4.1

[I ,M:V], and as ~ is iso-closed, obV = {X I FX = X} = G ob~ t

so that G = [IK,M:B].

4.4 Proposition Let M be transportable and send bimorphisms

to isomorphisms, and let F = [I{,M:~]. Then the bireflective

hull of ~ has object class {X 6 obX I FX = X}.

Proof Let ~ be the full subcategory with objects X = FX.

Then ~ is iso-closed and bireflective. Suppose 8 is any iso-

closed bireflective subcategory between V and ~ . By 4.3 the

bireflector for ~ is G = [I~,M:~]. If X 6 ob~ , then

X = FX = [I ,M:V]X, which means that the source (X, f:X § V I

V 6 V, f 6 ~(X,V)) is M-initial. As ob~ c ob~ , also the

source (X, f:X § B [ B 6 ob~ , f 6 ~(X,B)) is M-initial, which

means that X = GX, so that X s ob~ .

4.5 Theorem Let M:K § ff be topological and send bimorphisms to

isomorphisms. Then a subcategory V of ~ is M-spanning if

and only if the bireflective hull of V is X.

132

Proof If V is M-spanning, then I x = [I~,M:V] whence by

4.4 the bireflective hull of V is ~. Conversely, as M is

topological, F = [I~,M:V] exists. Now the bireflective hull of

V is K, so that by 4.4 ob{ = {X I FX = X}, i.e. F = I{ and

V is M-spanning.

We observe that all proofs in this section go through if we

replace bireflections by M-isoreflections, i.e. reflections whose

~niversal morphisms are sent by M to isomorphisms. Then the

assumption that M sends bimorphisms to isomorphisms becomes

superfluous, and we have, for example:

4.6 Corollary Let M:~ § C be topological. Then a subcategory

U of ~ is M-spanning if and only if the M-isoreflective hull

of U is X.

[1] Antoine, P.

[2]

[3]

[4]

[5]

[6]

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133

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Proximity spaces and Proc. Amer. Math. Soc.

134

[22] Hu~ek, M.

[23] Hu{ek, M.

[24] Hu{ek, M.

[25] Hu{ek, M.

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Full embeddings with a given restriction. Comment. Math. Univ. Carolinae i_~4(1973), 519-540.

Bitopological spaces, compactifications and completions. Thesis, Univ. Cape Town 1970. Math. Monographs Univ. Cape Town No. i, 1974.

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[35] Salbany, S.

[36] Salbany, S.

135

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[38] Tholen, W.

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Relative Bildzerlegungen und algebraische Kategorien. Thesis, Univ. M~nster 1974.

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Harder, A. Topologische Kategorien. Univ. M~nster 1974.

Diplomarbeit,

Topology Research Group University of Cape Town Rondebosch 7700 South Africa

AN EXTERNAL CHARACTERIZATION

OF TOPOLOGICAL FUNCTORS

G.C.L. Br~mmer and R.-E. Hoffmann

0. Introduction

Our concept of topological functor coincides with the

transportable absolutely topological funetors in the sense of Horst

Herrlich [5]. All topological functors are faithful [6], [5]. Our

main result is that a faithful functor T is topological if and only

if whenever K is a fully faithful functor and M and W are

faithful functors such that the outer square in the diagram

W

(1) K F~ I T I

f

M

commutes, there exists a functor F making the diagram commute. As

one implication is only strictly true up to natural isomorphism, we

restrict ourselves to those faithful functors which lift isomorphisms

unique ly.

The first author acknowledges financial aid from the South African Council for Scientific and Industrial Research and from the University of Cape Town to the Topology Research Group. Both authors acknowledge the hospitality of the Mathematisches Forschungsinstitut Oberwolfach.

137

We have to emphasize that our topological functors are not

required to satisfy a fibre-smallness condition, but are assumed to

solve initiality problems over arbitrary - including properly large -

index classes. 0nly when dealing with classes of diagonals of the

diagram (1) do we pass to a higher universe; the main theorem stays

within a single universe.

Amongst the applications we give an external characterization

of Oswald Wyler's [10] taut lifts, and the following result: A full

subcategory S of a topological category A is topological if and

only if there is a retraction of A onto S which respects

underlying structure.

The diagonal F in the diagram (1) is in general not unique.

In Proposition 1.2 we shall construct the T- coarsest diagonal.

The construction was used in [7] (without emphasizing the diagonal

situation) by Miroslav Hu~ek, who has kindly pointed out that the

said proposition can be deduced from [7, Theorem 7, Corollary]. For

expository reasons we again give a full proof.

Further applications of the Hu~ek construction, now explicitly

from the diagonal point of view, are given in [3].

1. Factorizing faithful funetors

We shall mostly work in a single universe U .

1.1 Definitions Let T : A+ C be a functor.

(1) A source in C is a pair (C, (fi)i) where fi : C § C i

138

are C-morphisms indexed over a ~-class I The class I may

be a proper class or a set, and it may be void. The usual notation

for this source will be (C, fi : C+Ci)I

(2) A T-initiality problem consists of an indexed family

(Ai) I of objects in ~ together with a source (C, f• : C§ I

in C . We shall denote this problem by (C, fi : C§ Ai)l "

(3) A solution to the T-initiality problem in (2) is any

source (A, f~ :A+Ai) I in ~ with the following properties:

(i) TA = C and Tf~ = fi for each i in I ;

(ii) given any source (B, gi : B+Ai)I in A and any

morphism u : TB § TA such that fi u = Tg• for each i , there

exists unique v : B § such that f~.v = g• for each i and

Tv = u .

(4) The functor T : A§ C is topological if each T- initiality

problem has a solution (the index classes of the problems may be

proper U- classes and may be void).

(5) As it is known from [6] and [5] that each topological

funetor (even under a weaker definition) is faithful (provided the

domain category has small horn-sets, which we assume), we shall

henceforth consider only faithful functors. This induces a

simplification in (3)(ii) above: the uniqueness of v and the

equations f~v = g• follow from the other conditions.

(6) With Hu~ek [7] we shall call the functor T : A+$ amnestie

if it is faithful and, whenever f is an isomorphism with Tf = 1 ,

then f -- 1 . Equivalently: T is faithful and each T-initiality

problem has at most one solution.

(7) If FI,F 2 : ~§ are functors, we write FI ~T F2 and say

139

F l is T- finer than F 2 , or F 2 is T- coarser than F I , if

there exists a natural transformation n : F I § 2 such that Tn is

the identity natural transformation TF I = TF 2 Observe that if

FI ~T F2 and F2 <T FI and T is amnestie, then F I = F 2 .

1.2 Proposition Let T : A§ C be topological and amnestic. Let

M :~§ , K :9 § and W :D§ be faithful funetors, with K full,

such that TW = MK . Then there exists a functor F : ~§ such

that TF : M and FK = W .

Proof

/ /

K F~ / T /

/ /

M

For X in ob~ we define FX as follows: The

T- initiality problem (MX, Mf : MX§ WD) in which D ranges

through ob D and f ranges through the hom-set ~(X, KD) has a

unique solution which we denote by (FX, (Mf) l : FX+WD) with D and

f ranging as said. For any morphism g : X l § in ~ we define

Fg as follows:

f Mf = T(Mf) l X > KD MX > MKD

X l MX n

(Mf)' FX ~WD

!

Fg I ) ) l I

FX '

Indeed, since the source (FX', (M(fg)) l : FX I +WD) with D and f

ranging as above is such that T(Mf)'.Mg = T(M(fg)) I for each f ,

by 1.1(3)(ii) there exists unique Fg : FX l § with TFg = Mg .

Funetoriality of F is readily verified, and one has TF = M . To

140

prove FK = W , consider a fixed object D o of D , and all

f : KD 0 § KD , D 6 ob D. For each such f , since K is full, there

exists unique T : D o § D with KT = f . The source

(WD0, WT : WD 0 § (indexed over f as before) is then the solution

of the T- initiality problem (MKD0, Mf : MKD 0 § WD) ; indeed,

TW~ = MK~ = Mf and if we are given any source (A, gf : A§ and

any u : TA§ 0 such that TWT.u = Tgf for each f , then

giD ~ : A§ 0 is such that TgID ~ = u . But by definition the source

(FKD0, (Mf) l : MKD 0 § is the solution to the above problem; hence

WD 0 = FKD 0 . This with TW = TFK gives W = FK , since T is

faithful.

By a dia~onal of the equation TW = MK (or of the eommutative

square drawn as above) we shall mean any funetor F such that

TF = M and FK = W .

1.3 Under the assumptions of Proposition 1.2, the functor F : M§

constructed in the above proof is the T- coarsest diagonal of the

given square.

Proof Given any diagonal G : ~ + ~ . For any f : X§ KD the left

hand triangle

Mf (Mf) l MX > MKD FX >WD = GKD

Gf nxi

TGX GX

commutes. Hence there exists n x : GX+ FX with Tn x = 1 . Thus

G ~< T F .

141

1.4 We give some illustrations of the occurrence of the T- coarsest

diagonal.

(1) Non-uniqueness: With T : Top§ Ens the usual forgetful

functor and ~ the empty category, the diagram

/ T /

/ /

/ Ens Ens

has exactly two diagonals Ens § [2, p.56].

(2) Inducing metric topology: Let ~ be the category having one

object, the real line ]R with the usual metric, and morphisms the

non-expansive mappings. Let Mn___~e be the category of metric spaces

and non-expansive mappings, and M : Mne § and T : Top § the

forgetful functors. Let K : ~Mne be the inclusion functor and

W : ~+ Top the functor which gives ]R the usual topology.

W

F1/ / f

/ /

Mne / Ens M

Then the T-coarsest diagonal F I : Mn__~e§ Top gives each metric space

the metric topology, and the ~-finest diagonal F 0 is distinct

from F i "

Proof For a metric space (X, d) , consideration of the mappings

d(a~ -) : X+ ~ shows that the metric topology on X gives the

solution to the T- initiality problem posed by all non-expansive

142

mappings (X, d) § K~ and W~ ; thus FI(X , d) has the metric

topology. On the other hand, F 0 is given by a T - coinitiality

construction (dual to that of FI) , and one readily sees that F 0

gives each totally disconnected metric space the discrete topology.

(3) Comp letin$ upward

Each commutative triangle of faithful functors

A

~( ,> C M can be completed to a square of faithful functors,

W D . . . . ~A

with K full, in which F is the T-coarsest diagonal (e.g. let

~) = ~, K =I , W = F).

(4) Completin~ downward

Each commutative triangle of faithful functors W

9 >A

c a n b e c o m p l e t e d t o a s q u a r e o f f a i t h f u l f u n c t o r s , W 1) > A

V ~ - - ~ - ~ c

with T topological, in which F is the T-coarsest diagonal

( e . g . l e t C : ~L, T = I , M : F ) .

143

1.5 Theorem If T : A§ C is an amnestic functor, then the

following conditions are equivalent:

(1) T is topological;

(2) For arbitrary amnestic functors M, K, W with K full,

such that the outer square in

W

l "l K F~- " t T /

M

commutes, there exists a diagonal F of the above diagram;

(3) Same as (2) but with K a full embedding.

Moreover, if T is merely assumed faithful instead of amnestic,

then still (2) ~ (3) ~ (1).

Proof By 1.2, (1)~(2). Trivially (2) ~(3). To see that

(3) ~ (1), assuming T faithful, consider a T- initiality problem

(C, fi : C~TAi' Ai)l " We construct a category ~[ as follows: We

arrange that the classes ob~, [ C} and I are mutually disjoint

and define

ob~ : obA U{ C} U I .

The hom-sets in ~ are defined as follows (with arbitrary

A s ob~, iE I):

{(A, C) = {A ~ C I u6 C(TA, C)A (V j E I)

IN g E ~(A~ Aj) Tg = fju]} ;

(The notation A u C is a convenient way of writing a 3- tuple

(A, u, C) and serves to make horn-sets disjoint. Observe that

in the definition of ~(A, C) , as T is faithful, g is

uniquely determined by A, u and j , so that we may write

g = gA,u,j) ;

144

f• (c, i) : {c ~ i} ;

~(A, i) = {A g.>i I gE ~(A, Ai) ^ [3 uE{(A, C)

{(A, A) = { i A} ;

]((i, C) = ~9 ;

As for composition in

~ ( C , C) : { 1 c } ;

{ ( C , A ) : @ ;

, we posit that

Tg = fi u] } ;

~(i, i) = {1 i} ;

~[(i, A) :

1 A, i c and i i shall act

as left and right identities at the corresponding objects; then only

one case of composition remains to define:

)i .

We now define a functor M : ~§ e as

MC : C ; Mi : TA i ;

MI c : i c ; MI i : 1TA i ;

fi gA,u,i A u---+C ~i : A

Herewith ~ is a category.

follows:

MA = TA ;

M1A : ITA ;

M(A U-~c) : u ; M(C fi)i) : fi ; M(A-9-+i) : Tg .

Clearly M is a faithful funetor, and as the isomorphisms in 9(

identities, M is amnestic.

We define ])

and we let

atone s tic.

We define a functor W : ])§ ~ as follows:

are

to be the full subcategory of ~ with obD : obA U I ,

K :9 § ~ be the inclusion functor. Thus K is full and

WA : A ; Wi : A i ; W1 A : 1 A ;

Wl i : 1Ai ; W(A l i) = g .

W is a faithful functor, and as the isomorphisms in Then are

identities, W is amnestic. Clearly TW = MK , so that by (3) there

exists a functor F : K+~ with TF = M and FK = W We claim

that the source (FC, F(C fi i))i in A is the solution to the

145

given T - initiality problem (C~ f• : C + TA i , Ai) I �9 Firstly, the

f• eodomain of F(C , i) is Fi : FKi = Wi : A i , and TFC = MC : C ,

and TF(C fi fi ~i) = M(C ~ i) : fi " Secondly, given any source

(A, gi : A§ I in ~ and any u : TA§ C with Tg i : f• for all

i , then A ~> C E~(A, C) so that F(A U-+C)s A(FA, FC) = A(A, FC)

(since FA = FKA = WA = A) , and TF(A U-~C) : M(A U-*c) = u .Q.E.D.

It is intended to present a proof of the following variant of

the main result elsewhere:

1.6

condition is fulfilled, then T is topological:

F o r a r b i t r a r y f a i t h f u l f u n c t o r s M a n d t< , w i t h t(

embedding, such that the outer square in

K F. i T /

I /

M

Theorem Let T be a faithful functor. If the following

commutes, there exists a diagonal F

a full

of the above diagram.

For the sake of the following result we assume that our given

universe ~ is a set in a uniwerse ~ +

1.7 Proposition Let the outer square in

W 9 > A

K F i i T /

I /

/

M > C

146

commute and let T be topological and amnestic, K fully faithful,

W and M faithful. Under the ordering ~T , the class of all

diagonals F forms a complete lattice in ~+

Proof For any C s obC the partial order ~T on the T- fibre

{ A 6 ob~ I TA = C } is such that T- infima of subclasses again belong

to the fibre; indeed, taking the T-initiality problem

(C, i c : C~TAj, Aj)j and calling its solution (A, hj : A+Aj)j , we

have A = infT{Aj I j s J} . Therefore, given diagonals F i (i 6 I)

we may define F = infm{ Fi I i s I} by FX = infT{ FiX I i 6 I} and

the corresponding action on morphisms. By 1.3 the class of diagonals

also has a T- coarsest member; hence the class is a ~T- lattice in

~+

2. Elementary consequences of the main theorem

2.1 Self-duality (Antoine [1]; cf. [9], [2], [6], [5])

If T : ~§ C is topological and amnestic, so is T ~ : A ~ + C ~

Proof

dual.

The equivalent conditions in ~eorem 1.5 are clearly self-

The results in the following proposition are standard

consequences of the diagonalization property in a generalized

factorization system [8] (cf. also [4]).

2.2 Proposition For amnestic functors we have:

(1) The composite of two topological functors is

147

topological.

(2) The full topological functors are precisely the

isomorphisms of categories.

(3 If ST and T are topological, so is S .

(4 If ST is topological and T a retraction, then

S is topological.

(5 If ST is topological and S an embedding, then

T is topological.

(6) In a pullback

s i �9 ~

if T is topological, so is S .

(7) If Ti : ~i § Oi are topological, so is

NT i : NA i + TIC i ,

2.3 Theorem Let T : A§ C be an amnestic topological functor,

S a full subcategory of A , and J :S§ A the inclusion functor.

The following conditions are equivalent:

(1) TJ : ~+A is topological;

(2) There exists a functor E : A+~ such that

EJ = I S AND TJE = T .

Proof (1) ~ (2): J 'l J " TJ

f

I

A. ~'-C T

14B

(2) ~(1):

full such that TJ.W = MK .

D w > s

9. ,/. ~

Consider arbitrary faithful functors W, M, K with K

> C M

As T.JW = MK and T is topological, there exists R : ~+~ with

TR = M and RK : JW . Now TJ.ER = TJE.R = TR = M , and

ER.K : E.RK = E.JW = W , so that ER is a diagonal of the outer

square, and TJ is topological.

3. Taut lifts

We give an external characterization of those fumctors which

carry initial sources from one topological functor to another. These

are essentially 0. Wyler's taut lifts [10].

Given a functor T : A§ , a source in A is called T- initial

if it is a solution to a T-initiality problem. For simplicity of

statement, we agree that all functors in the following theorem are

amnestic.

3.1 Theorem Let T : ~§ C and U : 9§ ~ be topological, and V

and L functors such that UV = LT . The following conditions are

equivalent :

(1) V sends T- initial sources to U- initial sources;

(2) For each commutative outer square

149

Proof

W D > A

l "1 f K ~-" T

/ . /

/

~(' H > C

and for each diagonal G : ~§ B in

D VW >

LM �9 ~

there exists a diagonal F : ~ + ~ in the former square such

that G ~<u VF .

o/:J/vw<

LM

(i) ~ (2): Given the diagonal G between K and U , we let F be

the T- coarsest diagonal between K and T . Condition (1) implies

that the map A~VA of any T- fibre to the corresponding U- fibre

preserves infima. Hence VF is the U- coarsest diagonal between

K and U , and thus G ~u VF .

(2)~(1): Given a T-initial source (A0, h i :A0 § I in A , we

write TA 0 = C , Thi = fi , and we construct ~, ~), K, M, W from the

T- initiality problem (C, fi : C§ Ai)l as in the proof of 1.5.

150

We construct a functor G : [§ ~ as follows: let GA = VA

(A s obK) ; let Gi = VA i (i 6 I) ; let the source

fi (GC, G(C ~ i)) I be the solution to the U -initiality problem

(LC, Lf• : LC+UVAi, VAi) I and accordingly let G(A U-~c) be given

as follows: by the definition of ~(A, C) there exist gi [~A, A i)

with fiu = Tg i ; then Lf i

LC > UVA i

commutes and there exists unique v : VA§ GC such that Uv = Lu ;

we let G(A U-~c) = v . Finally for A-~i in ~ , Gg is given

by the appropriate composite, and immediately G : ~+ B is an

amnestic funetor with GK = VW and UG = LM . By (2) there now

exists a diagonal F between K and T with G ~u VF . From the

proof of 1.5 we know that (FC, F(C fi i))i is the solution to

the T- initiality problem (C, fi, Ai)l ; hence FC = A 0 ,

fi F(C ,i) = h i . We have GC ~u VFC = GA 0 Since Thi = fi ,

1TA0 A 0 > C occurs in ~(A0, C) as a non-identity u , say, and

Gu : GA 0 § is such that UGu = LMu = LITA ~ = 1 ; hence

fi GA0 ~u GC and thus GC = GA 0 . Hence Vh i and G(C ~i) have

the same domain and the same codomain{ the faithful functor U sends

fi both to the same morphism; hence Vh i = G(C ~ i) . Thus V has

sent a given T- initial source to a U- initial source.

The authors acknowledge the benefit of conversations with

Keith Hardie, Horst Herrlich and George Strecker during the

preparation of this paper.

151

References

[1] Antoine, P. Etude 6l~mentaire des cat@gories d'ensembles structur@s. Bull. Soc. Math. Belg. 18(1966), 142-164.

[2] Brim]her, G.C.L.

[3] Br~mmer, G.C.L.

A categorial study of initiality in uniform topology. Thesis, Univ. Cape Town, 1971.

Topological functors and structure functors. Proc. Conf. Mannheim 1976 on Categorical Topology.

[4] Freyd, P. and G.M. Kelly. Categories of continuous functors, I. J. Pure ~?pl. Algebra !(1972), 169-191.

[5] Herrlich, H. Topological functors. Appl. 4(1974), 125-142.

General Topology and

[6] Hoffmann, R.-E. Die kategorielle Auffassung der Initial- und Finaltopologie. Thesis, Univ. Bochum, 1972.

[7] Hu~ek, M.

[8] Ringel, C.M.

Construction of special functors and its applications. Comment. Math. Univ. Carolinae 8(1967), 555-566.

Diagonalisierungspaare, I, II. Math. Zeitschr. 117(1970), 249-266 and 122(1971), 10-32.

[9] Roberts, J.E. A characterization of initial functors. J. Algebra 8(1968), 181-193.

[10]Wyler, 0. On the categories of general topology and topological algebra. Arch. Math. (Basel) 22(1971), 7-17.

G.C.L. Br~mmer Topology Research Group University of Cape Town Rondebosch 7700 South Africa

R.-E. Hoffmann 4 DUsseldorf

Mathematisches Institut der Universit~t D~sseldorf German Federal Republic

HOMOTOPY AND KAN EXTENSIONS

by Allan Calder* and Jerrold Siegel

Let T' C T be a pair of topological categories (i.e. full sub-

cagegories of Top, topological spaces and continuous maps). Let F be

an arbitrary functor on T' having a right Kan extension F K to T. We

will be concerned with questions of the following sort: Suppose F is

a homtopy functor, what relation on the maps of T must F K respect?

The prototype of the sort of result we have in mind is the classical

v theorem that the Cech extension (with respect to finite covers) of

cohomology on polyhedra to normal spaces is a uniform homotopy invariant

but not a "homotopy invariant [3]. It is our intention to provide a

general framework for understanding results of this sort.

Our method is as follows, we first consider general relations R

on the maps of T'. The Kan extension of R is defined to be the

weakest relation RKon the maps of T such that the Kan extension of

every functor that respects R , respects ~. A constructive definition

of ~is then given and some general observations are made. We then proceed

to specific computations. We show, for example, that the Kan extension

of homotopy on Pf to normal spaces, N, is uniform homotopy over ~f

(i.e. maps f,g : X § X' are uniformly homotopic over Pf iff ~f and

~g are uniformly homotopic for all ~ : X ' "§ Q, any map ~ into any

Q in Pf.)

We next show that the ~ech extension (finite covers) [2] of a

functor from Pf to N is naturally isomorphic to the Kan extension.

*Based on a talk given by the first named author at the conference.

153

Hence, in addition to having a different view of the Cech extension, we

have a mild generalization of the classical result mentioned above.

Finally, we address ourselves to the question of when a

particular homotopy functor has a Kan extension that is also a

homotopy functor. The main result we obtain in this direction is

that F = [ ,Y] (homotopy classes), then F K is never a homotopy

functor when Y is a finite complex with non-zero homology.

Notation Again, we will be concerned with TOP, the category of

topological spaces and continuous maps. T, T', T", etc. will

always denote some given full subcategory of Ts ~f (P_c) will

denote the category of finite(locally finite) polyhedra and

continuous maps.

We will assume that functors below defined on T' (F : ~' -+ A_)

admit right Kan extensions [4] ~ : ~ --+A, a some suitable category

usually Ens (sets) or Ens ~

Definition i Given a category T a natural relation (on the maps of T)

is a functor R : T -+ T, where T is a category with Ob(T) = Ob(T)~

R is the identity on objects and R : MorT(X,Y) § Mor~(X,Y) is epi as a

set map.

If R(f) = R(g) we write f ~ g.

Finally, given a pair of categories T' C T and a natural relation

R on T. R will be called the extension of R]~'.

Examples (a) Homotopy induces a natural relation H on any category ~.

We write f ~ g suppressing the subsymbol.

Uniform Homotopy

154

(b) If ~ is a subcategory of the category N of normal spaces,

we say f is uniformly homotopic to g as maps X § X' (f ~ g) if B(f) ~ B(g)

as maps BX § BX' where B is the Stone-Cech compactification. This

particular form of the definition of uniform homotopy is convenient fo~

our purposes and is easily seen to be equivalent to the Eilenberg-Steenrod

definition [3]. We denote the natural relation by SH.

(c) Homotopy over Pf(Pc ) . Two maps gl~ 2 : X § X' are said to be

homotopic over Pf if for any map ~ : X' § Q with Q in ~f we have

~gl ~ ~g2" We write gl ~ g2 (gl ~ g2 )"

Similarly, for normal spaces we have uniform homotopy over ~i (P--c)

written gl ~f g2 (gl ~ g2 )" The induced relations are denoted by

Hf, Hc, ~Hf and ~H c respectively.

These last examples are non-trivial since if X is a single point and

X' = {(x,sin(~) I 0 < x j i} U {(0,y) I 0 j ~I J i} then the maps

g0(Pt ) = (l,sin(1)), gl(Pt) = (0,O) are (uniformly) homotopic over

Pf (P) but not (uniformly) homotopic. Note however, Hf]P@ = H.

Given any relation R : T' § T' there is a natural category theoretic

extension to T. It is this extension that will be of most concern to us

in what follows.

Definition 2 Let T' C T and R : T' § T' be as above. The Kan extension

of R (written R K) is defined by setting f ~K g iff for every functor F of

the form F = FR (F : T --+ A) such that F admits a Kan extension

F K : T --+ A we have FK(f) = FK(g).

The following relationships hold among the various notions mentioned

above.

155

Theorem 3 (a) For the pair Pf C ComP2 (compact hausdorff) the Kan

extension of homotopy is homotopy over Pf.

(b) For the pair Pf C N (normal) the Kan extension of homotopy

is uniform homotopy over Pf.

Before giving a proof of Theorem 3 it is necessary to establish some

elementary facts about natural relations.

Definition 4 Given natural relations R I : ~-+~i and R 2 : ~ --+ 2 we

say R I ~ R 2 if R I factors through R 2 (i.e. R I = QR 2 for Q : T2 + TI )

Note Id : T ~-+ T is a natural relation and for any R < Id. We will

make use of this trivial observation. We also use the following:

Let TiC T and let R : T' § T' be a natural relation. Let

Let F : T' + A

the unique

Theorem 5

R 1 : T § T1 be a natural relation such that R K j R I.

factor through R hence RII~'. We denote by F and F1

factorization (F = FR = FIRII!').

Then ~ = FKR = FIKRI , where = means natural equivalence of functors

and ~K and F1 K are computed over the appropriate categories.

Proof. Since we may choose R 1 = R K it suffices to check F K = FIKRI �9 This

is immediate from the definition of K an extension [4 p. 232] since cones

: c 2_+ F factor uniquely through the cones ~i : c L+ F I.

It is an obvious corollary of Theorem 5 that F K factors through R I.

Again, we will make use of this observation.

We now give a constructive definition of RK. We will base our

computations on this definition.

156

Definition 6 Let T' C T and R : T' § T' be as above. We define a natural

relation ~ on ~ as follows. Given f,g : X § X' we say f ~ g iff for every

map ~ : X' + Q with Q in T' there exists

QI'Q2 ..... Qn in ~' and maps

~i : X ~ Qi' ~i : Qi § Q "~"

(i) ~I~i = ~f, ~n~n = ~g

(2) for each i < 1 < n either

(a) ~i~i = ~i+l~i+l or

(b) ~i = ~i+l and ~i R ~i+l"

Theorem 7 ~ = R K

Proof. To show R K < ~ we must show that for every functor F : T' §

if F factors through R then ~ factors through ~. That is if f ~g then

FK(f) = FK(g) which by the universal characterization of F K is the same

as showing ~(~f) = FK(~g) for all Q in T' and all ~ : X' ~ Q. This

follows at once from the definition of ~ since either

(a) FK(~i~i ) = FK(~i+l~i+l ) since ~i~i = ~i+l~i+l or

(b) ~(*i~i ) = F(~i)FK(~ i) = F(*i+1)PK(~i+l ) = FK(*i+l~i+l ) since

Hi = ~i+l and ~i R ~i+l"

To show ~ < RK we show f ~ g = f ~. g. In particular, we exhibit a R ~

functor F : T' * Ens ~ (op of sets) .~. F factors through R yet FK(f) # FK(g).

If f ~ g, then the~e exists Q in T' and ~ : X' § Q such that there does

not admit a factorization of the type in Definition 6. Let ~ : T § ~. On

the category ~_ consider the Ens ~ valued functor F~ = [_,Q]. One verifies

that ~f ~ ~g ~ [~f] # [~g] in ~,Q] which, in turn, implies

~(~f) ~ ~(~g) ~ ~(f) ~ ~(g)-

157

A A. Let F~ = F~ RIT_'. Since RIT' = R we have trivially that F~

factors through R.

Consider F~ K. From the universal mapping properties of the K an

A~ K extension, there is a natural transformation N : F~ + F~ which is an

equivalence on T', [4 p. 234].

One has F~K(~f)Q(X) = ~(Q)~R(~f)~ ~(Q)F~R(~g)-- FQK(~g)Q(X) (since

n(Q) is an equivalence).

Hence, F~K(~f) ~ F~K(~g) which again in turn implies F~K(f) ~ F~K(g).

Given Theorem 7, it is now possible to complete the proof of Theorem 3.

Proof of 3a. Let H denote the natural relation of homotopy restricted to

the category ~f. For the category ComP2 we must show that homotopy over

H ~ Pf (Hf) is the same as = .

Hf j Trivial since Definition 6 gives a prescription for piecing

together a homotopy ~f ~ zg for any X' ~ Q.

A H j Hf : This follows from the following stronger result which we

prove in an appendix.

Lemma 8 Let X be in Comp 2 and Q' in Pf suppose we are given

h 1% h 2 : X + Q' then we can find Q' in Pf and maps ~ : X + Q,

~i,~ 2 : Q + Q' with

(i) ~i ~ = h I , ~2 ~ = h 2

(2) ~i ~ ~2

A Proof of 3b. Remembering that now H is the Kan extension over N we wish

to s h o w t h a t H = BHf .

A BHf _ < H : L e t h 1 ~ h 2 : X ~ X ' . L e t ~ : X ' ~ Q. We m u s t p r o d u c e a

h o m o t o p y 6 ( ~ h 1) % B ( g h 2 ) . L e t ( { Q i } , { ~ i } , { ~ i } ) b e a f a c t o r i z a t i o n o f ~h 1

158

~h 2 (Definition 6). Noting that any map ~'i : X + Qi extends uniquely to

a map 8(~i) : ~X § Qi' we have ({Qi},{~(~i)},{~i}) is a factorization of

B(~hl) and ~(~h2) , thus B(~hl) ~ ~(vh2) , hence B(~h2) ~ B(~h2).

H J ~ I f : S i n c e BX i s c o m p a c t , we may u s e Lemma 8 t o f a c t o r t h e

homotopy for BX then restrict the factorization to X.

@uestion: Consider the pair P C To F . Let H now be homotopy on P . Does ........ ~ . . . . -'-C

= A HK" H K H ? As before, H < H = However, for the reverse conclusion, we C C --

do not know whether the corresponding version of Lemma 8 holds. One

possible approach is to use Milnors C.W. path bundle over a simplicial

complex [5]. However, it is not known whether the total space is a sim-

plieial complex as would be required by a proof using this approach.

Theorem 3 has several interesting consequences. First, let ComP2, Pf

be the respective homotopy categories (i.e. the quotients of ComP2, Pf by

H). Let F = FH on Pf. We then have

= ~ Theorem 9 F K F~, the homotopy Kan extension of F to ComP2 is naturally

isomorphic to the factorization of the continuous Kan extension to Com~2

through the homotopy category.

Proof.

Hf~H.

~K Denoting HIP f by H, Theorem 3 gives us = Hf but trivially

Now apply Theorem 5 to the relation ~ < H.

Before preceeding to the next result, we need the following

simple observation about Kan extensions from Pf to N.

Theorem i0 On N., FK(Bx) = FK(x).

159

Proof.

ization

For X in N, Q in Pf and~X + Q, we have the unique factor-

BX

X -- ~ Q

Hence, i induces an isomorphism of comma categories (~X + Pf) =~ (X + Pf)

hence an isomorphism FK(Bx) = FK(x) [4].

We now briefly consider the ~ech extension { of the functor F

(finite covers) from ~ to N.

Theorem ii On N, ~ = F K.

Proof. The proof amounts to listing four natural equivalences:

by Theorem i0

by Theorem 9

following Dold [2]

FK(x) = FK(sx)

FK(Bx) = KH(BX)

v v F(SX) = F(X) [3]

The proof of this last equivalence amounts to a selection of a

cofinal family of covers of X that extend in a suitable way to

a cofinal family of covers of 8X.

In light of Theorem ii, the Question raised earlier takes on

added meaning for by [2], we also know that as extensions over P --c

~H = ~ (numerable covers). Hence, F~ = ~K would imply F K =

as extensions over --c P as well as over Pf suggesting a new point of

v view about the nature of Cech extensions.

Finally, we may also use Theorems 9 and i0 to obtain a

geometric description of F K if F is representable. For this, let

F = [ ,Y], homotopy classes of maps into Y.

160

Theorem 12 For the pair Pf C N, consider the functor F_= [_,Y]

on If

Then F K [~ ,Y], homotopy classes of the Stone-Cech compac-

tification of a space into Y.

Proof. Again FK(x) = FK(Bx) = FK(Bx). But, again, following Dold

[2] ~K(X) = [ X,Y] Dold actually gives details for P c Top ' " - - C - '

but as he observes ([2] 3.15) the results hold for other pairs,

in particular ~f c Comp 2.

As a final observation, we use Theorem 12 to study to what

extent Theorem 3 is "best possible" for particular representable

functors. More precisely, if F is a

is F K ever one on N?

homotopy functor on Pf,

This question is studied in [i] for subcategories of N.

There we show that if Y is of finite type with ~i (Y) finite, then

_ = [~_,Y] is, in fact, a homotopy functor on Nf, the category of

finite dimensional normal spaces. However, for N itself the

situation seems quite different as the following negative result

indicates..

Theorem 13 Let F

n > 0, (homology).

= [_,Y], where Y is in Pf and Hn (Y) # 0, some

Then F K = [B ,Y] is not a homotopy functor.

Proof. Consider the standard based path space fibration

~y § pY P§ Y. If [~_,Y] were a homotopy functor ~(p) : BPY § Y

would be homotopic to the constant map. Hence, lifting the reverse

homotopy with initial map constant we could find a map r : BPY § PY

such that pr = B(P). Restricting to the fibre, we have a map

161

s : ~QY + ~Y such that si : ~Y + ~Y (i : ~Y + B~Y) is an isomorphism

in homotopy, (hence homology)and si(~Y) lies in a compact set of

~Y (~QY compact). Thus the homology of Y must be 0 except in a

finite number of dimensions and ~i (Y) must be finite (see remarks

above). But a simple sPectral sequence argument on the fibration

~Y § PY § Y if local coefficients are not present, or if necessary

on (~Y)o = (~) § P~ + Y' Y the simply connected covering complex of

Y, shows that ~Y is never finite dimensional under our assumptions

on Y.

162

APPENDIX

Proof of Lemma 8. Let (D,{V~},%) be an equilocally convex structure

for Q' [6]. That is, A C U c Q' x Q', where A is the diagonal and U

is an open neighborhood of A in Q' x Q'. {V~} is an open cover of

Q' with V B x V B C U for all B. ~ : U x I § Q' is a homotopy of

Pl to P2 : U + Q' where Pl(a,b) = a, P2(a,b) = b, and

~(v~ x v~ x I) = v~.

Since Q' is a finite simplicial complex we may subdivide it

so finely that for every vertex b. of the subdivision (hence- i

f o r t h denoted by Q') we have S t a r ( b i ) ~ V~ f o r some V B.

Let Q ~= u (Star(b.) x Star(hi) ). Q is easily seen to be all h. I

I

a subcomplex of Q' x Q' ( a f t e r s u b d i v i s i o n ) . ~ has ano the r p r o p e r t y

which is essential for our purposes. Let ~ be a lebesgue number

for the open cover {Star(bi)}. Let lho(X) - hl(X) I < ~ for all

x C X~ then we are able to factor ho,h I as required. In particular

A Q, , = ^ = ~. = (ho,h I) : X + Q ~ x Q and ~0 Pll Q' ~i P2] Trivially

~0 ~ = Pl(h0,hl) = h 0 and ~i ~ = h I. Also, %1 ~ x I * Q' is the desired

homotopy.

In general, since X is compact, if H is the homotopy h 0 ~ h 1

we may choose 0 = t O < t I < ...< t = 1 with IHt (x) - H (x) I < n i ti+l

for all x E X.

Let Q = ~, the product of n copies of Q with projections n

0i, 1 ! i j n onto the factors. Define Q c Q by n

Q = {J=l ~ (S.jl, ~2 ) I s.j2 = s(j+l)l 1 _< j _< n - i} . Q may be seen

to be a simplicial complex by considering it as a subcomplex of

163

Q' under the map ((ro,rl),(rl,r2) § n+l .... (rn-l'rn} (ro'rl'''rn)"

Define ~(x) = ~ (H t (x),Ht(x) ) , 0 ~ i j n - 1 and define n i i+l

40 = plPl and 41 = p2Pn . Again trivially, 40~ = h 0

Finally, to check 40 ~ 41 we use the facts that p2p i

plPi ~ p2Pi since, as above, Pl % P2"

and 41~ = h I.

= PlPi+l and

BIBLIOGRAPHY

[1]

[21

[3]

[4]

[5]

[6]

A. Calder and J. Siegel, Homotopy and Uniform homotopy (to appear).

A. Dold, Lectures on Algebraic ToPology , Springer- Verlag (1972).

S. Eilenberg and N. Steenrod, Foundations of Algebraic Topolo$~, Princeton University-Press (1952).

S. MacLane; Categories for the Workin$ Mathematici@n, Springer-Verlag (1971).

J. Milnor, Constructions of universal bundles I, Ann. of Math. (63) 1956, pp. 272-284.

, On spaces having the homotopy type of a C.W. complex, Trans. A.M.S. (90) 1959, pp. 272-280.

Tensor products of functors on categories of

Banach spaces

J. Cigler

I. Sketch of the situation:

In their fundamental paper [11~ B.S. MITYAGIN and A.S. SHVARTS

have laid the foundations for a theory of functors on categories

of Banach spaces. The situation may be roughly described as

follows: The family Ban of all Banach spaces becomes a category

by choosing as morphisms al~ linear contractions, i.e. all

bounded linear mappings ~: X * Y satisfying II~II% I. The set of

all morphisms from X into Y may therefore be identified with the

unit ball of the Banach space H(X,Y) of all bounded linear maps

from X into Y. By a (covariant) functor F: Ban * Ban we mean a

functor in the algebraic sense with the additional property that

the mapping f * F(f) is a linear contraction from H(X,Y) into

H(F(X), F(Y)) for all X,Y. The simplest examples are the functors

E A and H A defined by EA(X)=A ~ X (i.e. the projective tensor

product) and HA(X)=H(A,X).

By a natural transformation ~: F I ~ F 2 we understand a natural

transformation ~ = (~X)X E Ban in the algebraic sense satisfying

II~X!I~ I for all XE Ban. Thus the natural transformations from

F I to F 2 form the unit ball of the Banach space Nat (FI,F 2) of

all natural transformations in the algebraic sense satisfying

!Io11 = sup l!~x'! < ~. X

Denote now by Ban Ban the category whose objects are all fu/Ictors

from Ban into Ban and whose morphisms are all natural transformations.

It is easy to verify that for each AE Ban and each functor F

the equation

165

(I .I) Nat (ZA,F)=H(A,F(1))

holds, where the (isometric) isomorphism is given by e * ~I" (Here

I denotes the one-dimensional Banach space). As a special case we

get

(1.2) Nat (EA,ZB)=H(A,B)

for all A,BE Ban. This may be interpreted intuitively in the

Ban �9 following way: The mapping A ~ E A from Ban into Ban ms an

"isometric embedding", or functors are generalized Banach spaces.

For the functor H A we get the equation

(1.3) Nat (HA,F)=F(A)

given by ~ @ mA (IA) (Yoneda lemma).

As a special case we get

(I .4) Nat (~A,~) = H(B,A)

for all A,BE Ban, which may analogously be interpreted to say,

that Ban op is isometrically contained in Ban Ban

It is now tempting to ask if it is possible to extend the natural

mapping from Ban onto Ban op to a (contravariant) ma~pin@ from

Ban Ban into itself. In other words: Ban Ban Does there exist a contravariant functor D: Ban * Ban

satisfying

1) D ~A =HA for all AE Ban

2) Nat (D FI,D F2) =Nat (F2,FI) for all functors FI,F 2.

If such a D would exist it would be uniquely determined by the

equation

DF(A)--Nat (HA,DF)=Nat (D EA,DF)=Nat (F,ZA).

166

Though it turns out that this functor DF does not satisfy 2) for

all pairs of functors, it nevertheless proved to be of utmost

importance for the theory. It is called the dual functor to F.

MITYAGIN and SHVARTS have begun to compute DF for some concretely

given functors. These computations were rather long and cumbersome.

There was missing some kind of formalism which would be able to

reduce length_ly calculations to simple formulas. The purpose of

this talk is to Show that the concept of tensorproduct for functors

provides us with such a formalism.

2. Functors as generalized Banach modules:

I want to indicate my main ideas by means of a simple analogy,

which I find more illuminating than the corresponding abstract

theory which would be required by contemporary mathematical

standards. Let me state this analogy in the following form ([I~,

[21): "Functors are generalized Banach modules". ThLs has of course

been observed several times before, but nobody seems to have used

this analogy in order to carry over Banach space theory to functors

on categories of Banach spaces by using Banach modules as a sort

of catalyst.

First some definitions: Let A be a Banach algebra. A Banach space V

is called a left A-module if there is a bilinear operation

A• V* V, written (a,v) * av, such that b(av) = (ba)v and

!laVllv~ !fall A llv!l V for a,bE A and vE V. A Banach space W is called

right A-module if w a is defined with similar properties.

167

A Banach space Z will be called A-B- bimodule if it is a left

A-module and a right B-module and if furthermore these module

operations commute:

(a z)b = a(z b).

In order to get a satisfying theory we have to assume that the

Banach algebra A has approximate (left) identities. By this we

mean a net (u) of elements u E A satisfying !lu II~ I and lim u a=a

for all aE A.

The following theorem is well known:

Factoriza~ion theorem (Hewitt-Cohen): Let A be a Banach algebra

with left approximate identity (u) and let V be a left A-module.

Then the following assertions are equivalent for an e~ment vE V:

I) There exist aE A, wE V such that v=aw

2) lim flu v-vll =0.

The set of all such elements forms an A-submodule V of V which e

is called the essential part of V.

Let us now denote by HA(VI,V 2) resp. HA(wI,w2 ) the Banach space

of all left (resp. right) A-module-homomorphisms from V I into V 2

(resp. from W I into W2). Of course ~E HA(VI,V 2) if and only if

~0E H (VI,V 2) and ~(av) =a~(v) for all aE A and vE V I.

In the analogy mentioned above between Banach modules and functors

on categories of Banach spaces the following notions correspond with

each other:

Banach algebra A

left A-module V

right A-module W

~A(vl , v 2 )

HA(wI,W2 )

full subcategory K of Ban

covariant functor F: K* Ban

contravariant functor G: K~ Ban

Nat (FI,F 2 )

Nat (GI,G2).

168

To see this analogy let F: K* Ban, X,YE K, vE F(X), and a: X* Y.

Set av=F(a)v. Then IxV=F(Ix)v=v , llavll =!IF(a)v!l~ llall !Ivl.

If b: Y* Z is a morphism in K then

b(av) =F(b) (F(a)v)=F(ba)v= (ba)v.

This shows in what sense a functor may be considered as a

generalized Banach module.

Let now e: F I * F 2 be a natural transformation. Then

�9 y (F1(a)v)=F2(a) ~x(V)

or without indices ~(av) =a~(v) which may serve as justification

for interpreting natural transformations as generalized module-

homomorphisms.

Once one has recognized this analogy it is easy to give further

notions which correspond with each other.

An important example are Banach algebras with a left approximate

identi~y and full subcategories of A, where A denotes the full

subcategory of Ban consisting of all Banach spaces satisfying the

metric approximation property of Grothendieck. For in this case

there is an approximate identity in the algebra K(X,X) of all

compact operators on X.

In this paper we want to generalize the following assertions

for Banach modules (which may be found in M. RIEFFEL [I 3])to functors

on categories of Banach spaces:

a) For each right A-module W and each left A-module V there is a

Banach space W ~ V, the tensor product of W and V, and a A-bilinear A

mapping w: W x V * W @ V such that the following condition holds: A

For every Banach space Z and each A-bilinear mapping ~:

W • V ~ Z there is a uniquely determined continuous linear

mapping To: W @ V ~ Z such that !IT II =!I~!! and such that the A

169

diagram

Wxv ~ W~ V A Z

commutes. The pair (W @ V,~) is uniquely determined up to an A

isomorphism in Ban.

This tensor product is given by the formula W @ V = (W @ V)/N , A

where N is the closed subspace of W @ V spanned by the elements

of the form wa | v - w @ av.

b) Let W be a right A-module, Z an A- B-bimodule and V a left

B-module. Then

w@ (z@ v) = (w@ z)@ v A B A B

c) Let V be a left A-module, Z an A-B-bimodule and X a left

B-module. Then the socalled exponential law holds:

H A (Z @ X, V) = H B (X, H A (Z,V)). B

This isometry is natural in all variables.

d) If A has a unit element which acts as the identity on the

left module V and the right module W then

A@ V = V and W~ A = W. A A

e) If A has an approximate left identity, then

A @ V = V . A e

170

3. Tensor products of functors:

Let K be a full subcategory of Ban containing I. Let G: K* Ban

be a contravariant and F: K * Ban a covariant functor. Then

(G(Y) X F(X))y, X is a contra-covariant bifunctor on K• K. By a

K-bilinear mapping a from G • F into a Banach space Z we mean a

family (mX)XE K of bilinear mappings

aX: G(X) • F(X) - Z with !fall = sup Ila~l < X

such that

a X (G(m)gy,f X) = ay ( g y , F ( $ ) f X)

(or symbolically a(g~,f) = ~(g,~ f)) for all gyE G(Y), fx E F(X),

and ~ : X * Y.

If we introduce the bifunctor (G(Y) @ F(X))y,x then to a K-bilinear

map a corresponds a dinatural transformationS: G(..) @ F(.) * Z.

Definition: Let F,G,K be as above. By a tensor product G @ F we K m

mean a Banach space together with a dinatural transformation

| G(..)$ F(.)~ G~ F, K

such that for each dinatural transformation r G(..) @ F(.) * Z

into some Banach space Z, there exists a unique continuous linear

mapping T$: G @ F ~such that the diagram K

7,

commutes and IITr = IIr = sup IICXIIX E K" X

171

If such a tensor product G ~ F exists, it is uniquely determined, K

since for Z = I the above condition means that (G ~ F)' coincides K

with the set B(G,F) of all dinatural transformations from

G(..) ~ F(,) into I. It is easy to see that

B(G,F) = Nat(F,G') = Nat ( G , F ' ) .

In order to show the existence of G ~ F, define K

w : G ( . o ) ~ F ( . ) ~ B ( G , F ) '

i.e. a family ~X: G(X ) ~ F(X) ~ B(G,F)'

by Wx(g X| fx)~) = aX(g X| fx ) for aE B(G,F).

Then clearly w is dinatural and

~ h i c h implies IIwtl ~ I.

~et now G ~ F be the closure in B(G,F)' of all finite linear K

combinations of elements of the form w X (gx ~ fx )~ Then G ~ F is

a Banach space. We assert that it has all properties required from

a tensor product.

Let thus r G(..)

Then we must have

X k and

X k

=

11 11~. ' z z X k

s u = 11r l l z ' l ~ Is z x k

I1r sup I~ (z

= !l,tl llz z ~x (g~: X k

F(.) * Z be a dinatural transformation.

mX (gk~ fk)) X k

fXk)ll =

k

kl | f x ) J

172

because Z'o r belongs to B(G,F).

This reasoning implies also that Tr is well defined and linear.

All general theorems on tensor products can be reduced to the

above definition.

It is easy to show (and has been shown in [I]) that

G ~ F = ( Z G(X) ~ FOX))/N K XE K

where E denotes the coproduct in Ban and N is the closed subspace

of this coproduct spanned by all elements of the form

T. G(~) gk | k k

Proposition: Let $ : G I * G 2 and a: F I * F 2 be natural transformations.

Then

B @ a: GI ~K FI~G2~K F 2

is a continuous linear map satisfying 11~ | all ~ II~II llall,

Proof: We define ~ | a by

(S | a ) (gx ~ fx ) = ~ (gx) | ~ (fx)"

Then ~ | a defines a dinatural transformation from the bifunctor

(G I(Y) ~ F I(x))Y,X

into the Banach space G 2 ~ F 2 satisfying !I~ | a!l ~ ll~ll llall.

Therefore by the universal property above it defines also a

continuous linear map

a | ~: GI ~K FI*G2~K F 2

with the same norm.

173

We state now the theorems which we shall use in the sequel and

whose proofs are easy consequences of the above reasonings, but

rather lengthy and shall therefore be omitted. (They may be found

in [31 ).

Theorem I: Let K and L be full subcategories of Ban,

M: K• L * Ban a contra-covariant bifunctor,

G: ~ * Ban contravariant, and

F: K ~ Ban covariant. Then the equation

o@ (M& F) = (G&M)~ F L K L K

holds.

Theorem 2: Let K and ~ be full subcategories of Ban,

M: L • K * Ban a contra-covariant bifunctor, and

FI: L* Ban and

F2: K * Ban be covariant functors

Then the exponential law holds:

Nat (M ~ FI,F 2) = Nat (FI, Nat (M,F2)).

There is also a further generalization of Theorem 2 which turns

out to be very useful.

B Let M: L • K * Ban be replaced by a functor with values in Ban A

the category of all A- B-bimodules and let FI: ~ * Ban B and F2: K ~ Ban A

be functors into the categories of all left B-modules and left

A-modules respectively.

Denote further by ~M ~ FI] B the tensorproduct which is formed from

the bifunctor G(X) @ F(Y) instead of G(X) ~ F(Y); and let NatA(M,F 2) B

denote the space of all natural transformations which are at the

same time A-module-homomorphisms. The we have

174

Theorem ~:

Nat A ([M ~ FI]B,F2) = Nat (F 1,Nat A (M'F2))" K L L B K

This equation is again natural in all variables.

4. Computation of tensor products:

4.1. Let H: K• K* Ban be the restriction of the contra-covariant

bifunctor H(X,Y) to K•

Let G: K * Ban be a contravariant and

F: K * Ban a eovariant functor. Then the equations

H@ F -- F a n d G @ H = G hold.

Proof: Let us prove the first assertion. The second one is

derived in the same way.

It suffices to show that for each A@ K we have

H(.,A) ~ F(-) = F(A). K

By 3. we know that H(.,A) @ F(.) is the closure in K

{B(H(.,A),F(.)')}' F(A)" K

(by the contravariant form of 1.3) of all finite linear combinations

of elements of the form WX(m X| fx) for ~X E H(A,X) and fx E F(X).

Now it is easy to see that this is just the functional on F(A)'

defined by the element F(~x)fxE F(A). Since each fA E F(A) can be

written in this form, e.g. as fA = F(IA)fA' it follows that

H(',A) @ F(-) = F(A), as asserted.

Remark: These equations reduce to the trivial relations

! @ X = X = X ~ I for K = {!} and are the analogs of 2d).

175

The analogs for the equations H(I,X) = X are the Yoneda lemmas

Nat (H,F) = F and Nat (H,G) = G (compare 1.3). K K

4.2. There are some very easy but useful consequences of 4.1.

Let K and L be full subcategories of Ban and let FI: K *

and F2: ~ ~ Ban be two covariant functors. Thsmthe product

F2F I can be written either as tens~product or as space of

natural transformations:

and

F2F I(.) = H(..,F I(.)) ~ F2(..) L

F2FI(.) = Nat(H(FI(.),..),F2(..)). L

This simple observation implies e.g. that left Kan extensions

can be written in the form of a tensor product. This is of course

a well-known fact, but the proof becomes particularly simple:

Nat (FI,F 2 S) = Nat (FI(.) , Nat (H(S(.),..),F2(..))) = K K Ban

= Nat (H(S(.),..) @ FI(.),F2(--)) Ban K

and therefore

Lan S FI('') = H(S(.),'') @ FI(.).

4.3. Another consequence is the equation

Nat (F I,F 2 ~A ) = Nat (FIHA,F2) Ban Ban

for all functors F I ,F2: Ban * Ban.

This follows from the computation:

Nat (FI(.) , F2(A~ .))=

: Nat (FI(-) , Nat (H(A~ .,.-), F2(-.))) =

176

= ~a.t (~(A& .,--) & FI(.),~2(..)) =

= ~2t (H(. ,H(A,.. ) ) & F~ (.) ,F2(-- ) ) =

= Nat (F I(H(A,--)),F2(..)) =

= Nat (F I HA,F2).

The essential point here is of course that ~A is left adjoint

to H A-

Further theorems of this type have been obtained by G. RACHER [12].

4.4. Let G be contravariant, F covariant and let zA be the contra-

variant functor defined by ~A(x) = A ~ X'. Then the equations

and

G ~$ ~A = G(I)& A Ban

~A A ~ _- A& F(1) Ban

hol~. (P. MICHOR [9~).

Proof: G & ~A = ~(') & (" & A) = Ban Ban

= G(.) ~ (H(I,.) ~ A) = Ban {I}

=(G(.) & H(I,')) A A = G(I) ~ A. Ban { I}

The second equation follows in the same way.

4.5. For the next result we need the notion "functor of type ~".

Let F: Ban * Ban be covariant and A~ Ban.

The equation (1.1)

Nat(ZA,F ) = H(A,F(I))

allows the following interpretation. The functor A * Z A from

Ban into Ban Ban is the left adjoint of the forgetful functor

177

F ~ F(I). The coi~nit r EF(I) * F of this adjointness relation

is given by

F CX (~ fi ~ xi) = ~ F(xi)fi for fi E F(I) and xiE X = H(I,X).

The closure Fe(X) of all elements of the form Z F(xi)f i in F(X)

defines a functor Fe, the essential part of F. If F= Fe, then F

is called essential or of type Z.

It is a well-known fact (~f. e.g.V.L. LEVIN [72), that for every X

the restriction of r to the algebraic tensor product F(I) | X

is injective and that F(X) induces on this tensor product a

reasonable norm~ in the sense of A. GROTHENDIECK [5].

For contravariant functors G: Ban* Ban we have the equation

Nat (zA,G) = H(A,G(I)). The counit cG is given by r (Z gi | xi') =~G~i~

and the essential part G e is again defined as the closure in G of

G the image by r

In terms of tensor products of functors F is o• type Z if and only

if H(.,X) ~ F(.) , H(.~) ~ F(') is epi. An analogous relation (I) Ban

holds for contravariant functors.

By dualizing one gets the condition that F is essential if and

only if the mapping F(X)' * H(X,F(I)') defined by

fx ' ~ (x~ F ( x ) ' f x ' )

is injective, etc.

If F I is essential and F 2 arbitrary, then

Nat (FI,F 2) = Nat (F I, ( F 2 )e ). Furthermore

Nat (FI,F 2) * H (FI(I),F2(I))

is injective.

178

Proposition: (P. MICHOR [9]): Let G be contravariant, F covariant

and one of type Z. Then

G $ F = G(I)~ F(I) Ban a

where ~ denotes the completion of the algebraic tensorproduct

with respect to a reasonable tensor norms.

Proof: Suppose without loss of generality that F is essential. Since

(G ~ F)' = Nat (F,G') ~ H(F(I), G(I)') = (G(I) @ F(I))' Ban Ban

is injective, the mapping

G(1) ~ F(1) ~ G

has dense image.

F Ban

Consider the natural transformations G(I) ~ .' ~ G(-) and G(-)*H(-,G(I))

where the first coincides with cG and the second one with

gx ~ (x * G(x)gx). Both have norm S I and induce therefore linear

contractions

G(1) @ F(1) = ~G(1) @ F ~ G @ F ~ H(.,G(1)) @ F = F(G~I)).

Since F(G(I)) * F(I) ~ G(I) has dense image (F is of type Z)

and these mappings act as the identity on G(I) | F(I), the proof

is finished.

4.6. Let XE A. Then there is a net (u) of finite-dimensional - -

operators on X such that flu !I~ I and such that u x ~ x for

all xE X.

Lemma 1: Let F be of type Z and XE A.

equation lim F(u ) f= f holds.

Then for fE Fe(X) the

179

Proof: It is obviously sufficient to consider an f of the form

f=~ F(Xk)f k.

Then

IIF(u:)f-f!l = IIZ(F(U~Xklf k-F(xk)fk)ll ~ Z ~u~x k-xkll llfkll ~ O.

For contravariant functors a similar lemma holds. Let X'E ~. For

every given finite set {x1',...,Xn} r X' and each s> 0 there is a

finite-dimensional u: X' * X' such that II~I~ I and flu xi'-xi'll<r

Let u=E x."l | x-~m'' where (xi") and (~i') are linearly independent.

There are x E X such that lim x. =x." in the topology a(X",X') and

such that !In xia @ xi'!l~ II~I~ I. (cf. D. DEAN [4]). Therefore there

exists a such that v' =E x. | x. is the dual operator to some l~ I

v E H(X,X) and such that !Iv'a x i' -x i'11< r and ilval I_< I.

In other words: If X'E A, there exists a net (v) of finite-dimensional

operators on X such that !Iv 11~ I and such that v'~ x' * x' for all

x'E X'.

Lemma 2 (V. LOSERT): Let X'E A and G a contravariant functor of

type E. Let (v) be a net with the property stated above.

Then lim G(v )g=g for all gE G(X).

Proof: Without loss of generality g=E G(Xk')g k-

Then

1 IG(V )g-gll =!Iz(G(V )G(x k')gk-G(xk')gk]!l =!IZ(G(v 'x k')gk-G(xk')gk)~

~- z11v, 'x k'-x k'll !Igkll ~ 0.

We can now prove the following

Proposition: If XE A then (.' ~ X) ~ F(.) = Fe(X); and if X'E A Ban

then G(.) ~ (X' ~ .) = Ge(X) for each contravariant functor G Ban

and each covariant functor F on Ban.

180

Proof: It suffices to prove the first assertion. The second one

is similar.

Note first that by 4.5. the functor (.' ~ X) ~ F(-) is essential Ban

since .' ~ X is essential.

Since the inclusion F , F is a natural transformation and induces e

therefore a linear contraction

(.' ~ X) ~ Fe(.) into (-' ~ X) ~ F(. and since there is a Ban Ban

linear contraction of the second space into Fe(X)=H(-,X) ~ Fe(-) , Ban

it suffices to show that the canonical map

(.'' ~ X) g Fe(,) ~ H(',X) & Fe(.) Ban Ban

is an isometric isomorphism.

It suffices to show that the adjoint map

Nat (H(-,X),Fe(-)') ~ Nat (.' ~ X, Fe(.)')

is an isometric isomorphism onto. We know that it is injective.

We show first that it is isometric. Let aE Nat (H(.,X),Fe(.)').

Then IIa!I = IIaX(IX)II. It suffices therefore to show that

sup!lax(U ~ )'I = 1!~x(Ix)It.

Bnt

sup 1!~x(U ~)'! = sup sup !<f,ax(U ~)> I : t fE Fe(X)

!If!l_<

= sup sup I < f,F(u t)' aX(Ix)> I = !I fq <_ I

= sup sup ] < F ( u ) f , ~ x ( l x ) > l = f

= sup I ~ f , ~ x ( l x ) > t = I !~x( lx )q . f

Now it remains to show that each ~E Nat ('' ~ X, Fe(.)') is the

181

mmage of some a.

The net ~x(U )g Fe(X )' is bounded. We may therefore assume from

the beginning that it is w -convergent. Let

f'=lim Bx(U )E ~e(X)'

Then to f' corresponds mE Nat (H(-,X), Fe(-)').

We then have for YE Ban, fyE Fe(Y),~yE Y' ~ X:

~fy,By(~y)> = lim <fy,~y(U ~y)> =

= lira <fy,F(gy)' Bx(U ~)> = lim<F(~y)fy,Bx(U ~> =

= <F(~y)fy,f'> = <fy,ay(~y)>; qed.

Remark: This proposition may be regarded as analogon of 2)e).

4.7. If X~ ~ then in general

LF(X) = (.' ~ X) ~ F(') Ban

does not coincide with Fe(X).

The functor LF has been introduced in another way by

C. HERZ and J. WICK-PELLETIER [6] and has been called by them

"computable part" of F.

It has been further studied by P. MICHOR [10].

It would be interesting to study .the following generalization

I~KF of LF in more detail: Let K be a full subcategory of Ban and

let

H~ (A,X) = H(.,X) @ H(A,.). K

Then H ~ is a contra-ccvariant funetor on Ban. For K = {I} we have

A H{!)(A,X) = A ~X, for _K = Ban it coincides with H(A,X) and

for K : Fin, the full subcategory of all finite-dimensional spaces ^

we have H ~ (A,X) : A' • X .

Thus LF(X) could be generalized to

182

A L K- F(X) = H K- ( ' , X ) @ F ( . )

Ban

5. Computation of some dual functors:

Here I want to indicate how tensor products may be used to

compute the dual functor for some concrete functors.

I choose as example functors defined by sequence spaces. The

same method applies also to functors defined by measuz~able

functions but the details are more intricate (and will be given

in another paper).

Let me introduce first of all the notion of a sequence space:

A linear subspace n g 1 co will be called a sequence space if

I) n is an lco-module (as defined in 2.)

2) eiE n and !lei!In = I for e i = (0 ..... 1,0 .... )

3) !lx!l n = sup i l u k X l ln , u k = (1 ,1 . . . . . 1 , 0 , 0 , . . . ) . k

Proposition I: For every sequence space n we have

< llX!In <- llX!ll 11 ~ n~ I co and llx!l co-

Proof: Trivial.

A sequence space n is called minimal if n = c o lco

the subalgebra of 1 oo consisting of all

n, where c o is

null-sequences. An equivalent condition is that the finite sequences

are dense in n.

With the sequence space n we associate the functor n(-) on Ban

defined by

n(X) = {x = (Xk) : XkE X, (llXk! I ) E n} with norm !I (Xk)lln(X) = II (l[Xkll)ll n.

We want to compute the dual functor (D n(.))(X).

This has already been done by MITYAGIN - SHVARTS with a complicated

proof. The use of tensor products reduces it almost to a triviality.

183

~oposition 2: n(X) = l~(X) @ n

Proof: let w: I~(X) x n ~ n(X) be the l~-bilinear map (f,e) * f

We have to show that for each Banach space Z and each 1 ~-bilinear

map ~ : l~(X) x n ~ Z there exists a unique continuous map

T : n(X) ~ Z such that the diagram

1 ~ (X) • , ) n(X)

/ /

/ /

/

Z

w /

/

/ /

/

- %

commutes and lITal ] = lia!l-

NOW w(f I ,e I ) = w(f2,~ 2) implies ~(fl '~ ) = (~(f2'~ because this

means fl ~ = f2 ~2 and therefore

a(f1'el ) = a(fl ~I ~2 I o 1 I +1 ~21 ' 1~11 +1 e21 ) = ~ G 2 I% I +t ~D21 ,fOll +t~2t) =a(f2,e2)-

Every element wE n(X) is the image of an element of I~(X) • n, e.g.

o f ( Y - , I Iw! l ) . qw!]

Now define T

because

by T ( f ~)=e(f,~). Then T

1 % (z fi ~ = l- l fi ~

, z I ~ k l ) :

is well defined and linear

E fi = - - , = , ~ I ~ k l ) = z ~ (fi%) ] %1 i z l %_] i

~ow % (~(f,~)) -_ ~(f,~) ~ q~!l ~ 11%11.

184

On the other hand

! l Tm(w) l l = ! lT a ( ~ " !lwll)ll = fla ( q -~ , I Iw!l) ! l -<t lal ! II IIw!l II n

Corollary: For n minimal we have n(X)= Co(X) @ n. 1 oo

Proof : n = c & o i oo

n

l~(X) @ n=lm(X) ~ (c o @ n)=(l~(X) ~ c o) @ i m i m i ~ i m I m

n--

= Co(X) ~ n.

Proposition 3: Let nX=Hloo (n,1 I). Then n x consists of all

= (pk)E l~176 such that ~ IX k Ukl < oo for all (Xk)E n with

'1 11= qx !11. ilXlln-<1

Furthermore nX(A) = Hl c o (n,1 I(A)) for AE Ban.

Proof: Let ~E Hle o (n,11(A)).

~(X)ek=m(X ek)=m(Xk ek)=Xk ek ~(ek)

Now e k m(e k) = (O,...,ak,0 ,...) =a k ek, akE A.

re(X) = (X1al ' ~2a2 .... )

!Imll= sup S Ikkl Ilak'1=!l(l!ak!l)llnX IIX I1 n--- 1

=~ (a~)~ nX(A).

Proposition 4: For n minimal we have nX(A ' ) = n(A)'

Proof: nX(A )=H l~ (n,11(A')) =

= Hlm (n,co(A)')=H lm (m,H(Co(A),I))=H(Co(A) l~ n, I) = n(A)'

185

Theorem: D(n(.))(A)=nX(A).

Proof: First observe that D 1 cO = 11 because 1 co(X) = Hll (X), and

O (X) (X)=l I $ X=11(X). HIt = Eli

Then (by Theorem 3)

D(n(.))(A)=Nat (n(.), A ~ .) =

=Nat (lco(.) @ n, A~ .) = lco

=H (n, Nat (lco('), A~ .)) = lco

=~ co(n, o ico(.) (A~) = 1

=H co(n, 11(A))=nx(A). 1

Remark: The dual functor satisfies the relation

Nat (FI, D F 2) =Nat (F2, D F I).

It is given by ~ e-~ ~ such that

(~x(fx))Y (fY) = ( ~ (fY))X (fx)"

Therefore also Nat (F, D2F)=Nat (DF, DF).

To the identity on the right side corresponds a natural transformation

j: F * D2F on the left. The functor F is called reflexive if JX

is an isometry of F(X) onto (D2F)(X) for all XE Ban.

The above methods can be used to show that Dn is always reP

flexive. MITYAGIN and SHVARTS have conjectured that every dual

functor is reflexive. It has recently been proved by V.LOSERT

[8] that their conjecture is not true: Consider the funetor A A

A~ �9 �9 Then the dual dunetor coincides with the integral ope-

rators I(A,'). LOSERT has shown that there are Banach spaces

such that I(A,') is not reflexive.

A

186

BIBLIOGRAPHY

[I] J. CIGLER,

[2] J. CIGLER,

[3]_ J. CIGLER,

[4] D.W. DEA~,

Funktoren auf Zategorien von Banachr~umen,

Nonatshefte fttr Mathematik 78 (1974), 15-24

Duality for functors on Banach spaces,

preprint 1973

Funktoren auf Kategorien von Banachr~umen,

Lecture Notes Univ. Wien 1974

The equation L(E,X ~-~) = L(E,X) ~-~ and the principle

of local reflexivity,

PAMS 40 (1973), 146- 148

[5] A. GROTHENDIECK, R6sum6 de la th6orie metrique des produits

tensoriels topologiques,

Bol. Soc.Matem. Sao Paulo 7 (1952), I -79

[6] C. HERZ - J. WICK PELLETIER, Dual functors and integral

operators in the category of Banach spaces,

preprint 1974

[73 V.L. LEVIN, Tensor products and functors in categories of

Banach spaces defined by F~-lineals,

Transl.Moscow Math. Soc. 20 (1969), 41 -77

[8] V. LOSERT, Dissertation Univ. Wien 1975

[9] P. MICHOR,

[ I0] P. MICHOR,

Zum Tensorprodukt yon Funktoren auf Kategorien

yon Banachr~umen,

Monatshefte fur Mathematik 78 (1974), 117-130

Tensor products, operator ideals and functors on

categories of Banach spaces,

Univ. of Warwick 1975

[11] B.S. MITYAGIN - A.S. SHVARTS, Functors on categories of

Banach spaces,

Russian math. Surveys 19 (1964), 65-127

187

[121 G. RACHER, Dissertation Univ. Wien 1974

[13~ M.A. RIEFFEL, Induced Banach representations of Banach

algebras and locally compact groups,

J. Functional Analysis I (1967), 443-491.

DUALITY OF COMPACTOLOGICAL AND LOCALLY

COMPACT GROUPS

J. B. Cooper and P. Michor

We develop a duality for compactological groups, based on

a concrete realization of the dual category of the category of

compactological spaces in terms of a mixed topology on (~(S).

The dual of a compactological group appears as a Hopf-algebra

with mixed topology.

We are able to treat the following notions in terms of the dual:

characterization of locally compact groups, Bohr compactlfication,

almost periodic functions, Pontryagin duality, and some connections

with representation theory.

189

w Preliminaries and Notation

1.1. Compactological spaces: The model far a compactological

space is a topological Nausdorff space together with the

collection of all its compact subsets, disregarding its original

topology. So a c ompactological space S is a set S together with

a collection ~(S) of subsets of 8, each K E %((S) bearing a

compact (Hausdorff) topology ~K such that

(I) ~($) is closed under formation of finite unions and

taking closed subsets.

(2) for each K c L; K, L ~ ~ (S) the inclusion K * L

is continuous.

The category CPTOL of compactological spaces has as morphisms

maps f: 8 * T such that for each K ~ ~(8) there is L c I~(T)

with f(K) c L and flK:K * L is ~K -~L - continuous.

By C~($) we mean the vector space of all bounded complex valued

functions on S whose restrictions to each K E ~<(S) are ~K-continuous.

A compactological space is said to be regular, if C~(S) separates

points on S. We denote the full subcategory of regular

compactological spaces by RCPTOL. We note that compactological

spaces may be reg-~rded as formal inductive limits of systems

of compact spaces. For more information Bee BUC~"JALTER.

190

1.2. The category MIXC : Objects are triples (E, II.II, ~) where

E is a commutative involutive algebra with unit over C, V. II is

a norm on E and �9 is a locally convex topology on E such that:

(I) BII. I I = Ix ~ E: II x II ~< 11 is bounded and complete for

(2) �9 may be defined by a family of seminorms P on E such that

p(xy) = p(x)p(y), p(1) = I and p(x~x) = p(x) 2 holds for all

x , y ~ E an~ p ~ P and II • II : sup I p ( x ) , p ~ P~ f o r a l l x ~ E .

Morphisms are multiplicative linear maps, respecting involution

and unit, contractive for the norm and continuous for the locally

convex topology.

MIXC may be regarded 8s the category of forma:l projective limits

of systems of commutative C~ with unit. For more

information see COOPER 1975.

I .3. The category MIXTOP: Objects are triples (E, II. II, ~) where

E is a vector space (over C), ll. II is a norm on E and �9 is a

locally convex topology on E such that BII. H = Ix ~ E, II x II ~ 11

is ~-bounded. Morphisms are linear maps, contractive for II. II and

N

continuous for ~. (E, II. II, ~) is said to be complete, if BII.I I is

�9 -complete. Then (E, II. II) is a Banach space. The complete objects

in MIXTOP are exactly the formal projective limits of systems

of Banach spaces.

191

1.4. T~e functor C~:

Let S be a compactological space. Let C~(S) be the space

of all bounded complex valued functions on S whose restrictions

to all K ~ ~(S) are TK-continuous. Consider the following

structures on C~(S): II II - the supremum norm

- the topology of uniform convergence

on members of ~(S).

Then (C~(S),II.II ,T) is an object of MIXC . If @: S * T is a

CPTOL-morphis~n, then C~(@): C~(T) ~ C~(S), given by x ~ x o @,

i s a MIXC -morphism.

We have constructed a contravariant functor C~: CPTOL * MIXC

The functor My:

Let (E, II. If, ~) be an object of MIXC .

Denote b y M (E) t h e s e t o f a l l MIXC - m o r p h i s m s E -~. C. We e q u i p

it with the following compactology: members of ~(M (E)) are the

weak -closed subsets of M~(E), whose restriction to BII .If is

* ' ~ - e q u i c o n t i n u o u s , a n d t h e y b e a r t h e r e s t r i c t i o n o f t h e weak - t o p o l o g y .

If ~: E ~ F is a MIXC -morphism, then M (~):M (F) ~ M (E), given

b y f ~ f o ~ , i s a C P T O L - m o r p h i s m . We h a v e c o n s t r u c t e d a

contravariant functor MIXC ~ RCPTOL.

192

.6. Proposition:(C00P~ 1975): The categories MIXC and RCPTROL

are quasi-dual to each other under the functc~s C ~ and % .

The maps 6: s ~ (x ~ x(s)) gives a natural isomorphism S * MyC~~

for each S ~ RCPTOL. We call it the Dirac transformation.

The map : x *(f * f(x)) give a natural isomorphism E * C~v(E )

for each E E MIXC . We call it the Gelfand-Naimark transformation.

These two maps produce the quasi-duality.

I .7: The tensor product in MIXC and MIXTOP:

The category RCPTOL has products (the obvious ones), so MIXC as

the (quasi)-dual category has coproducts. The y-tensorproduct,

which we will n~,; describe, is an explicit construction of the

. coproduct in MIXC .

Yet (E, II. IIE,W E ) and (F, II. II F, XF ) be two objects of MIXC . We

consider the following structures on E @ F, the vector-space

~ensor product of E and F: ll.II~-the inductive tensor product of

the norms If. II E' II. II F (i.e. that induced by the operator norm

via the embedding E @ F * L(E',F).

= WE | ~F -the inductive tensor product of the locally convex

topologies XE' ~F"

Let B denote the closure of [u ~ E @ F, flu II ~ 11 in the completion

of (E | F, ~E | ~F ) and let E ~y F denote the subspace U n B n>O " F, If. II ̂ ,~ ~F ) of this completion, and let I. I be the (E ;y

193

@ is again an object of MIXC . The same construction works for

MIXTOP. A result to be found in COOPER 1975 asserts that

E | F ~ C~(MYY(E) • My(F)) in MIXC , so the Yy-tensor product

is the coproduct in MIXC .

1.8. The strict toDolo~y:

If (E, II.[[ ,~) is an element of MIXC ~, let Yy = Yy [ If. If, T] be the

finest locally convex topology on E which agrees with ~ on

BII. II ; YY is a complete topology. g~ ~w

We note that I"[ If-II 'q:E | ~F ] = 1,[ U. l~,q:E ] @ YY[~-I~,q: F] on E | F.

For further details see COOPER 1975.

The same definition holds of course for objects of MIXTOP.

1.9. It is well known, that locally compact topological spaces

are k-spaces, i.e. their topology is uniquely determined by their

natural compactology. In this spirit we can regard the category

of locally compact topological spaces (we call it LOCCOMP) as a

full subcategory of RCPTOL and we will speak of l__ocall~ compact

comoact ol0gic~l spaces.

I .10. Let (E, If. II ,~) be an object of ~IXC and let P be a defining

family of C -seminorms on E (i.e. a family P satisfying 1.2. (2)).

If p ~ P we denote by Ip the ideal Ix ~ E: p(x) = 01 and by Ap

= o t its annihilator in E, i.e. = IY ~ E: y Ip .

E is said to be perfect (APOSTOL 1971) if the sum Z A is Yy-dense peP P

(1.8) in E.

194

If S is a regular compactological space and if K ~ "~(S) and

PK the associated C . seminorm, then I PK

= Ix ~ c~(s) �9 xlN = ol

and A = IY e C~176 y(s) = 0 for s ~ K} So Z A i s the PK Ke~(s) PK

subspace Cc(S) of functions in C~176 with compact support

( i . e . x ~ Cc(S ) i f f t h e r e i s K ~ ( ( S ) w i t h x ( s ) = 0 f o r s r K) .

1.11. Proposition (COOPER 1975) Let S be a regular compactological

space. S is locally compact if and only if C~176 is perfect.

195

w Compactological grouos and duality

2.1. Definition. A compactological group is a group in the category

CPTOL of compactological spaces, i.e. it is a quadruple (S,m,e,i)

where S is a compactological space and in: S x S * S, e:I *S, i: S * S

are CPTOL-morphisms so that the following diagrams commute:

S x S x S i~ x m S x S

I m x id S [ m

S x S m .-- S

e • id S id S x e

S

d idsx i i X id S d S ~SxS ~ SxS~ ,SxS~ S

e 1 e

I ~ S ~ ,, I

(I is the final object of CPTOL, i.e. the one-point set, d is the

diagonal map). The compactological groups form a category which

we denote by GCPTOL (the morphisms are those CPTOL-morphisms

which respect the maps (m,e,i). A compactological group is said

to be regular if its underlying compactological space is regular,

196

i.e. if C~~ separates S. GRCPTOK denotes the full subcategory

of regular compactological groups.

2.2. Problem: Do there exist compactological groups which are

not regular?

2.3. Definition: CMIXC denotes the category of cogroups in

MIXC . Thus an o b j e c t o f ~IIXC i s a q u a d r u p l e ( E , c , r l , a ) where

E is an object of MIXC , and c: E ~ E | E, ~ E ~ C, a E ~ E

are MIXC - m o r p h i s m s so t h a t the f o l l o w i n g d i a g r a m s commute:

E

e

E| E

e =E| 7 E

I idE| c

C | id E ~ E | E | E

C | | E = E |

C ~ 11 E 11 ~C lc [ (idE,a) E ~ E|

((a,idE)) denotes the canonical morphism from E | ~, the coproduct

in MIXC ( c f 1 . 6 ) i n t o E, d e f i n e d by the m~ps a and idE; C i s t h e

initial object of MIXC~).

197

2.4. If (S,m,e,i) is a compactological group, then clearly

(C~(S),C~176176176 is a cogroup in MIXC . Clearly C ~

lifts to a fLtuctor GCPTOL-~ CMIXC .

If on the other hand (E,c,D,a) is a cogroup in MIXC , then again

(My(E),My(c),My(a)) is a regular compactological group.

to a functor CHIXC * GRCPTOL.

My lifts

Proposition: The functors C ~ and My induce a duality between

CMIXC and GRCPTOL.

2.5. A compactological group (S,m,e,i) is locally compact if and

only if its dual C~(S) is perfect. (cf 1.11).

198

w The Bohr Compactification

3.~. Let (E,c,~,a) be a cogroup in MIXC . If we consider the

C -algebra (E, II.II), then My(E, II. is a compact topological

space, in f a c t , t h e S t o n e - C e c h c o m p a c t i f i c a t i o n o f My(E) . I t

is, however, in general not a topological group, since (E, ll. ll) is

not a c o g r o u p i n t h e c a t e g o r y C o f c o m m u t a t i v e C - a l g e b r a s w i t h ^

unit, since E | E ~ E | E, where E | E, is the C -algebra tensor

p r o d u c t o r t h e i n d u c t i v e t e n s o r p r o d u c t o f B a n a c h s p a c e s .

Let ~ = (id E | a) o c.

3.2. Lemma: There is a largest C -subalgebra ~ of E with the

property that ~ (E) c E | E. ~, with the induced norm and cogroup

structure is a cogroup in C . The assignment E * E is functorial.

Proof: For each ordinal ~, we define a subalgebra E a of E

inductively by

E : =E O

N_ I E~: = c (Ep| E~) (~ = p + I)

Ea: = A IE~, ~ < al (~ is a limit ordinal).

Then t h e f a m i l y I Eal i s e v e n t u a l l y s t a t i o n a r y and we d e n o t e i t s

limit by E.Then ~ is C -subalgebra of E with the desired properties.

3 . 3 . The ~ n c t o r E -~ ~I i s a " f o r g e t f u l ~ n c t o r " f r o m CMIXC i n t o ,

CC . We denote it by U. We c~n now define a functor B: = Myo U o C ~

from GCPTOL into GCOMP, the category of compact groups. If S is

a compactol~gical group, we c811 B(S) the Bo__hr-compactification

199

of S. There is a natural morphism is: S * B(S), Js = My(k), where

X: C~(S) 4 C~(S) is the embedding. Js has dense image, since k is

injective.

3.4. Proposition: B(S) has the following universal property: every

GCPTOL-morphism from S into a compact group factorises over Js"

Proof: If ~: S * T is a GCPTOL-morphism, where T is a compact

group, then C~(~:C(T)+ C~(S) is a CMIXC -morphism. Since

UC(T) = C(T) and U is a functor, acting on morphisms by

restricting them, we conclude that C~(~) = C~(~) maps C(T) into

C~(S), i.e. factors over k: C~176 ~ C~(S).

So ~ = ~ ~ 3s ' ~ = My C~(~).

3.5. If S is a compactological group and x e C~176 then we

define (LaX)(S) = x(as), (RaX)(S) = x(sa) for a, s e S. x e C~(S)

is said to be (left) almost periodic if ILaX , a e Sl is relatively

norm-compact in C~176 We denote by AP(S) the set of left almost

periodic functions on S.

With induced norm AP(S) i s C - s u b a l g e b r a of C'~(S).

3.6 . Le~na: C~ c AP(S).

Proof: If x ~ C~176 there is an x e C(B(S)) so that

x = x o Js by definition of B(S). Then LaX = (Ljs(a) x) o Js and

the result follows from the fact that ~ is almost periodic on

B(S).

200

3.7. Lemma~ Let S be a set, M s finite dimension~l subalgebra

of BF(S), the C -algebra of bounded, complex valued functions

on S. Then there is a finite subset S I of S so that x * x]S~ is

#

a C -isomorphism from M onto C(S 1).

Proof: S 1 is the subset of S on which not all x �9 M vsnish.

result follows then from the uniqueness of C -norms.

^

3 . 8 . Lamina: ~ (~P(S)) c AP(s) ~ AP(S).

The

Proof: Let x �9 AP(S). If s > 0 there is a finite subset M of

~iP(S) so that

ILaX: a e Sl c M + s Bll ]I " By 3.7 we can choose

ISl,...,Snl c S end a basis (Xl,...,Xnl of the linesr span of

~ in C~176 so that xi(s j) = 5ij and

II x II = s u p [ [ x ( s i ) l : i = ~ , , . . . , u l h o l a s f o r a l l x i n t h e l i n e a r

spsn of M.

n Now if a �9 S there is Ikl,...,knl c C so that Ix(as)- Z kixi(s)l~

i=I

for each s �9 S. Hence Ix(asi)-kil < s for i = 1,...,n.

Then

n Ix(as -I) _ z

i=1

n Ix(as -I ) - z

i-I

n T~us II Z(x) - z

i-I

x ( ~ i ) x i (s - I ) I

n

kix i (s - I)1 + II Z i=I

(~i - x(asi))xill ~ 2~

Rsi~X i II 4 28,

so c(x) is in the norm closure of AP(S) | AP(S) in &(S• i.e. in

~P(S) ~ AP(S).

3.9. Propos• C~(S) = AP(S). Proof: 9.6. and 5.8.

201

w The Algebra Mt(~ ) and representations

4.1. Let (E,c,~,a) be a cogroup in MIXC . Equip E with the

strict topology y [ ll. II,~] (I .8) and let Eybe its dual.

Define a multiplication on E 7' in the following way: if f,g ~ ~y ,

then f �9 g be given by x ~ f | g(c(x)).

P.rop.osltion: If E is a cogroup in NIXC , then Ey' is Banach algebra

with identity. It is commutative if E is commutative.

4.2. Let S be a compactologic~l space.

A premeasure on S is a member of the projective limit of the system

[~KI K2: M(KI) * N(K2):K 2 c KI,KI,K 2 ~ ~( (S)I where },,~(K) denotes the

space of all Radon-measures on K.

$

If ~ = I~K]K~ ~ is a premeasure on S and I~KI denotes the outer

measure on K defined by I~KI then we define, for a set C c S

I ~ l * ( c ) = sup I I~KI* (c n ~) : ~ ~ < ( s ) t . A p r e m e a s u r e ~ on S is

said to be tight if for each s > 0 there is a K ~ so that

I~I*(S~K) < s. Equivalent is the existence of an increasing

sequence K n i n ~ ( ( S ) w i t h I l (S n ) * o .

We denote by Mt(S) the space of all tight (pre) measures on S.

If x ~ C~176 and ~ ~ Mt(S) , then the limit lim J l x l K n d~Kn exists n-*-~o

and is independent of the particular choice of the sequence K n.

We write [ x d~ for this limit. J

202

A premeasure ~ = I~K 1 is said to have compact su~L~ort if there

is a K ~ ~(S) so that I~l*(skcx) = o. The space Mo(S) of

premeasures with compact support is identifiable with U M(K). K~S<

Proposition: (COOPER) If S is a regular compactology, then the

dual of [C~(S), TK(S)] is naturally isomorphic to Mo(S) under

the bilinear form (x,~) -* I • d~; and the dual of [C~(S),YK [[.II ,~]]

is naturally isomorphic to Mt(S) under the bilinear form

(x,~) * J• d~.

so c~(s)~ = Mt(S).

4.3. If S is a compactological group, then we can give an

explicit description of the multiplication * (3.1) in Mt(S):

If x E C~(S), ~, v ~ Mt(S), then we have

f• d (~ * v) (~ v) (c(x)) |

= f x ( s . t ) d(~ @ V ) ( s . t ) J

= ffx(st) d (s)dv(t), i.e. we have the ordinary convolution.

~.4. If S is a regular compactological space and E a complete

object of MIXTOP,then define C~(S;E) as the space of all II. ll-bounded

maps f: S * E, equipped with the pointwise linear structure and

the following mixed structure:

If. If - the norm ~f[l =sup [ Ilf(s)I~, s ~ 81.

- the topology of uniform ~E-convergence on members of ~QQ(S).

20S

Proposition: If S is a regular compactological space, than the

embedding 8 : 8 -~ Mt(S ) has the following universal property: for

every complete object E of MIXTOP and every f ~ C~(S;E) there is

a unique T ~ L(Mt(S);E) which extends f via 6.

This gives rise to natural equivalences

C~(S;E} ~ O~(S) | y E ~ L(Mt(S);E ).

4.5. If S is a compactological group and E a complete object in

MIXTOP, then C~(S,E) has a natural map CE: C~(S;E) * C~(SxS;~),

given by OE:X~ ((s,t),* x(st));

i.e. c E = c | idE: C~(S) @I" E * C~(S) | C~(S) My E.

Let (E, If- ~ ,~) be a complete algebra with unit e in MIXTOP, i.e.

there is an associative multiplication m: E • E * E so that (E, II. II )

Ihll.l I is ~ • ~ -continuous. is a Banach algebra and m x BII.I I

Then C~(S;E) has a natural "multiplication".

mE: C~(S;E) • C~176 * C~(S x S;E), given by

mE: (x,y) w ((s,t) w m(x(s),y(t))).

We say that an element x ~ C~(S;E) is orlmitive, if CE(X) = mE(x,x).

4.6. Proposition: Under the identification C~(S;E) ~ L(Mt(S),E)

the primitive elements correspond to the Banach algebra morphlsms.

x c C~~ induces a unit preserving operator if and only if

x(e) = e E.

204

Proof: Suppose that x ~ O~(S;E) is primitive.

in L(Mt(S);E) is defined by ~ ~ l x d~. J S

Then T x (~ * v) = f x d(~ * v) = S

The image T x of x

foE(x) d (~ v) = jx~ xd(~ v) |

SxS SxS

= f x d~ f ydv = Tx(~)Tx(V ). S S

On the other hand, if T ~ L(M~(S);E), then T = T x where x = T o b.

Then CE(X)(s,t ) = x(st) = T(bst )

= T(5 s * 5t) = m(T(Ss),T(Gt))

= m(x(s),x(t)) = mE(x,x)(s,t ).

4.7. Corollary: Let S be a regular compactological group, X a

Banach space. Then there is a one-one correspondence between

(i) the set of st:~ongly continuous representations of S in X.

(ii) the unit preserving Banach algebra morphisms in L(Mt(S);E)

(iii)the primitive elements x of C~(S;E) with x(e) = e E.

(E denotes the object (L(X,X), H. [[ ,~s ) of MIXTOP - ~s is the

strong operator topology#.

2O5

w Pontrya~in duality

5.4. Let (E,c,~,a) be an object of CMIXC . x E E is called

strongly primitive if

(i) c(x) =x~ x

(ii) ~(x) : I

(iii) a(x) : x -1 (in E).

We denote by P(E) the set of strongly primitive elements of E.

5.2. Proposition. P(E), with the topology and the multiplicative

structure induced from (E,1"(l I. II E,~E)), is a topological group.

It is contained in Ix E E: Ixll = 1}.

Proof: P(E) is closed under multiplication:

Let x, y ~ P(E), then

c(xy) = c(x)c(y) : (x | x)(y | y) = xy | xy.

~(xy) : ~(x) ~(y) : 4.4 -- I

a(xy) = a(x)~(y) = x-1 y-1 = (xy)-1

(E is commutative: the C -algebra part).

The constant function I is a unit for P(E).

If x ~ P(E), then a(x) = x -I is an inverse for x. Since

multiplication is T [11.11 E ' ~E) continuous, ( P ( E ) , T [ I1.1~,~ E ] ) is a

topological group. Since II x ~ ~ • II = llx @ x II = I] c(x) ll ~ll • II

and I = I1 1 ~ = I1 y ( x ) II ~ II • II we c ~ c l u d e t h a t II x II = 1 f o r

all x c P(E).

2O6

5~3- Definition: Let S be a commutative compactological group.

A character on S is a GCPTOL-morphism from S into the circle group

^ ^

T. The set S of all characters form a group. S, with the topology

of uniform convergence on members of ~((S), is a topological group,

and it is complete in this uniformity.

5.4. Proposition: Let S be a commutative, compactological group.

A

The S = P(C~(S)) (a t o p o l o g i c a l g r o u p ) .

Proof: If x ~ P(C~(S)), then II x I = I = II • II , so x takes

its values in T.

C(X)(S,t) : x(st) and

(x | x)(s,t) = x(s)x(t) show that the strong primitivity of x is

equivalent to its being a character.

Since Y [ II-II, ~] = ~ on B~. II we see that even the topologies

c oinci de.

3.5. qorollar,y: Let S be a commutative, regular compactological

A

group. The S separates S if and only if the vector space generated

by P(C~(S)) is T-dense in C~~

207

C. APOSTOL

H. BUCHWALTER

REFERENCES

B -algebras and their representations, Jour.

Lond. Math. Soc. (2) 3 (1971) 30-38.

Topologies, bornologies et compactologies,

Lyon

J. B. COOPER The mixed topology and applications.

H. HEYER Dualit~t lokalkompakter Gruppen, Springer

Lecture Notes Nr. 150 (Berlin, 1970).

E. HEWITT, K.A. ROSS The Tannaka-Krein duality theorems,

Jahresber. D.M.V. 71 (1969) 61-83.

Abstract harmonic analysis II (Berlin, 1970). ,

K. H. HOFMANN The duality of compact semigroups and C -

bigebras, Springer Lecture Notes Nr. 129

(Berlin 1970).

P. MICHOR Duality in Groups, unpublished note - 1972.

J.W. NEGREPONTIS (J.W. Pelletier) Duality in analysis from

the point of view of triples, Jour. of

Algebra 19 (1971) 228-253.

N. NOBLE k-groups and duality, Trans. Amer. Math. Soe.

151 (1970) 551-561.

D. W. ROEDER Category theory applied to Pantryagin duality,

Pac. J. Math. 52 (1974) 519-527. ,

S. SANKARAN,S.A.SELESNICK Some remakrs on C -bigebras and duality,

Semigroups Forum 3 (1971) 108-129.

S. A. SELESNICK Watts cohomology for a class of Banach algebras

and the duality of compact abelian groups,

Math. Z. 130 (1973) 313-323.

M. TAKESAKI Duality and von Neumann algebras (in "Lectures

on OPerator Algebras" - Springer Lecture Notes

Nr. 247 - Berlin, 1972).

PRODUCTS AND SUmS IN THE CATEGORY OF FP.A~S

by

C. H. Dowker, Birkbeck College, University of London

and

bona Strauss, Universit~ of Hull

Spaces and frames. The open sets of a topological space ~ form a

complete lattice tX in which the join is the union, while the meet is the

interior of the intersection:

~G~ = ~G~, /\G~ = int ~ G~.

h~ny of the properties of a topological space X, e.g. tne properties of

being compact, regular, normal, etc., are actually properties of the topology

tX. Consequently much of what claims to be topology is actually lattice

theory.

A france

It follows that

L is a complet~ lattice with infinite distributivity

o Va b .

a v(b^c) ~ (avb)^ (arc).

A frame map f : L--~ ~ is a function commuting with finite meets and with

arbitrary joins. Thus the topology t~ of a space X, that is its lattice

of open sets, is a frame. ~ach continuous function f : A--~ Y inauces a

frame map tf : tY--@ tX, where tf(G) ~ f-lG. Thus there is determined

a contravariant functor

t : Top ~ Fr

from the category of topological spaces to the category of frames. The

functor t has a righ~ adjoint

s : ~ --~ To pp,

([I], Theorem 8), where sL is a space whose points are the prime elements

of L or, equivalently, the points are the Hmps of L to the two element

frame 40, i~.

209

The dual of the category of frames.

as a concrete category in the following way.

Given a frame map f : L---~M, let g : ~ - ->L

which

= c L u}. Since f commu~es with joins

Hence g(uj = max ~x : f(x)% u}. If f(x) ~ u then

x _~ g(u), f(x) _~ fg(u) ~ u. Thus

f(x) ~ u iff x ~ g(u).

Since f(a) = A{u ~ ~ : f(a) ~ u},

In lattice theory the relative semicomplement

by

b.c = max ~aE L : a^b_~c).

complete lattice L has semioomplements b~c for all

is a frame. The semicomplement has the property that

where

aAb

~e realize the dual category Fr ~

be the function for

x % g(u). Also, if

b~c in L is defined

b~ c 6 L iff L

aAb ~ c iff a ~ b~c.

The function g determineC by (~) has the following properties :

(ii~ if u # l, ~(u~ W i,

<iii) a~g(u) ~ g(f(a~u),

f is given by (B) in terms of g.

We give the proof only of (iii). Let b = a~g(u). Then

g(u). Hence

that is

f(a)^ f(b) = f(a^b) ~_ u,

that is f(b) ~ f(a)~u. Therefore b z g(f(a)%u),

a~g(u) ~_ g(f(a)* u).

We define a function g : ~- -~ L, where ~l and

be an antimap if it satisfies conditions (i), (ii), (iii)

broken arrows for antimaps.

L are frames, to

above. We use

210

by

Hence

Given an antimap g : M - -> L, the function

(B) is a frame map, that is,

(2) f(1) ~ i,

(3) fCa~b) = f(aj^~Cb).

We give the proof only for (3). Let v =

f : L---~ determined

f(a^b). Then a^b ~ g(v).

f(bJ ~ f~a)@v,

i(a) ̂ f(b) ~ v = f(aa b).

Since the reverse inequality is trivial, f(a^b) = f(a)~f(b).

The correspondence between maps f : L---*~ and anti~ps g : ~ - -> L

gives an isomorphism of the dual category ~op to the category of frames

and antimaps. Identifying under this isomorphism, fop is realized as the

function g : ~ -~L given by (A).

It can easily be verified that f is injective iff

while f is surjective iff fop is injective.

For each frame map f : L ) ~, the image f(L) is a frame, and f

factors uniquely into the surjectivemap fl : L---~f(L) followed by the

inclusion j : f(L) ) M. Then g = fop = f opjop. The injective

antimap fl ~ : f(L) - -> L factors uniquely into an isomorphism

h : f(L)---~gf(L) followec by the inclusion antimap i : gf(L) - -~ L.

The inclusion maps (antimaps) are taken as subobjects in ~? (respectively

Fr__~ The duals of inclusions may be taken as quotients, that is

representative surjections. Thus each map f or antimap g can be factored

uniquely

f = j@ h ~ i ~ g = i~hoj ~

into a quotient followe~ by an isomorphism followed by the inclusion of a

subobject.

fop is surjective,

211

Products. Let (L~) be a family of frames. In }in__~s there is a

product set and projections Let a b in the

product set if a~ ~ b~, i.e. ~a ~ ~b, for all o<. With this

order ~L~ is a frame and the projections ~ are frame maps. Let K be

a frame and let f~ : K ~Lg be frame maps. The unique function

f : K >~ L~, for which ~f ~ f~, is a frame map. Hence the frame ~Lc<

with the projections T[~ is the product in the category of frames.

If (Xo~) is a family of topological spaces and i~ : X~ --~k~ are

the canonical injections into the topological sum, the frame maps

t~ : t~X~---~tX~ induce an isomorphism j : tXXg---~tk~.

tZ~

Although, as appears from the above brief description, products of

frsa~es are trivial, they forI~ the basis of the following construction.

Construction. Let B be the subset of [~ L~ consisting of all b

such that either b~ = 1 for all but a finite number of c< , or b~ ~ 0

for all o<. Then B is a subframe of ~L~. If i : B >~L~ is the

inclusion, the composites ~i : B ~L~ are surjective. ~ subset G of

B is callea decreasi_n_~_ if whenever a 6 G and b ~ B with b ~ a, then

b ~ G. Clearly every intersection of decreasing sets is decreasing and

ever~ union of decreasing sets is decreasing. Thus the set T of all decreasing

sets of B is a topology for B. The function ~: T >B for which

~G = ~G is a surjective frame map. Hence ~i~: T )L~ is surjeotive.

Let ~@ : L~ - -> T be the corresponding injective antimap ; ~ ~ (~i~) ~ �9

B /

T

212

Then the functions ~ have the following universal property. If

f~ �9 L~--9~ are framle maps, there exists a unique frame map g : T---~i

with commutativity fYa ~ f~. Thus T has the universal property for a

coproduct except that the morpnisms ~ are in the wrong category.

Coproducts. That coproducts exist in the category of frames was

announced in [8] and was provea in S. Papert's unpublished dissertation [6].

There is still no publishe~ proof, but a detailea proof is containea in our

paper [3], to appear.

Here it is sufficient to state that the frame T constructed above

has a greatest quotient 8 : T )) L such that the composites h~ = @~

are frame maps. It can be shova% that L with the umps h~ : L@ > L is

the coproduct or sum ~-L~ of the frames L~.

L~>-- -- --> T

L

~f ~) is a f~ily of spaces ~d ~p , 77X~--~X~ are the

projections of the product space, there is induced a surjective fram~ map

u �9 ~tX~---->t~X( which in general is not an isomorphism. h~

t gx~

is however an isomorphism if each X~ is a compact

It is an isomorphism also i• the set of indices ~ is

The map u

iausdorff space.

countable and each ix is a locally co~act Hausaorff space.

Elements of s~s. The injective antimap ~op : B - -> T

b 6 B to the set

~~

m a p s e a c h

213

Such sets form a base for the topology

G =

For u 6 L let H = 8~ =

eH = u. Then

where

T. Thus if

Vb~ G r176

V@G ~u ~" Since

A b ~ -

this meet being finite since for each b,

of b~ are 1 or all b~ are 0. Then

Any frame maps ~ : L~---~K induce a frame map

f{u} =

Properties preserved by sums. ~ frame L

G 6 T,

is surjective,

family (a~) of elements of L SUCh that

such that Va~i = i. subfamily

for each u 6 L, there is a family (a~)

Va~ = u and for each ~ , a~ V u =

either all but a finite n~mber

f : L ~ where

is callea compact if each

Va~ = 1 has a finite

A frame L is called regular if

L such that

Similarly the definitions of other properties of the topologies of spaces

are extended to define the corresponding properties of frames.

Each sum of co~act fr~nes is compact (S. Papert, [7] ). Each sum

of regular frames is regular.

Each sum of paracompact regular frames is paracompact (Isbell, [4]).

Each s~a of Lindel~f regular frames is Lindel~f. Each sum of fully normal

frames is fully non~ml. (by A. H. Stone's theorem, fully normal is the

same as paracompact and normal, which is weaker than paracompact and regular.

The properties of being paracompact regular, LindelSf regular or

fully normal are not preserved in topological proaucts. Thus the points of

a topological space are not only irrevelant to such properties as normal,

Lin~el~f, paracompact, etc., which are properties of the topology of the

space. The presence of the points leads us to wor~ in the wrong category,

and thus get less satisfactory theorems.

of elements of

i, where

max ~x ~ L : a A X =

214

Application to the transfer of algebraic structures. Let i/IT 2 be

the category of locally compact Hausdorff spaces and continuous functions.

Since, for countable products ~d, u : ~ t~i---9 t~X i is an isomorphism,

Opt : LKT 2 ~F_Er ~

preserves countable products, up to equivalence.

Since, for each frame K , t h e m a p v : HK(ZL~. ) )~HK(Lw) is an

isomorphism, the functor

H K Op : Fr ~ ~ Ens

preserves products, up to equivalence. Hence

H~ t : LKT2 >Ens

preserves countable products, and in particular finite products. Therefore

H K t transfers algebraic structures (~8] , Theorem 11.3.4).

Let C(K) = ft a, K], that is C(~) is the set of frame maps of

the topology of the reals to the frame K. In particular, if K = t~ for

some space X, t maps [X, ~] bijectively to It ~, tk] ([2], Theorem 1),

so C(t~) may be regardea as the set of continuous real functions on X.

C(K) is the generalization from topologies to frames.

Since H K t transfers algebraic structures, the algebraic structure

of ~ is taken by H i t to It ~, K] = C(~). Thus C(K) is a ring and

a lattice.

There is an injective morphism from the ring ~ to the ring C(K),

taking real numbers to 'constants'; see below. With multiplication by

constants, C(~) is an algebra and a vector lattice.

Binary operations in C~). We describe more explicitly the binary

operations transferred by H K t from ~ to C(~).

Let p : ~2

may be x + y or

or, if we identify

) ~ be a continuous function. Typical cases: p(x, y)

xy or xvy or x^y. Then tp is a frame map

tp : t ~ )t ~2

t ~2 with t ~ + t ~{ under the isonmrphism

u : t ~ § t~ >t~ 2,

Applying H K we have

or, if we identify

which

215

tp : t ~ )t ~ + tR.

It ~ + t~, K] with It ~, ~x[t ~, K],

HK(tF) : it ~, K]~It R, K] ~ It ~, K].

t~ !

hi tp i h2 tR .... t~'§ t~ t~

fl" f2 ~ I t ~" K], let f : t ~ + t ~ >K be the map for

= fi" Then

HK(tP)(fl , f2) = K~(tp~(f) ~ f otp.

For

fh i

When there is no danger of misunderstanding we ~ay write p for

f1' f2 : t R---~K, we have

P(fl, f2 ) = f~tp : t ~ >K.

= x § y, we write fl + f2 instead of P(fl, f~), etc.

HK(tP). Thus for any maps

If p(x, y)

Thus we have

fl + f2 : t ~. ~K,

flf2 : t ~- ) K,

flvf? : t R ~ ~,

flAf2 : t ~ ~ K.

The lattice operations ~ive an order in C(K). Thus fl ~ f2

Formulas z'or binary operations. We write

r " = r = IxE , >r}

for the members oZ the usual subbase for ~. Then, for ex~ple,

t{~)(r-) = L(x, y) : x + y < r},

t(')Cr') = ~(x, y) : xy < r},

t(v)(r') = {(x, y) : x < r and y < r~,

t(A){r-) = i(x, y) : x < r or y < r}.

§ = fl v f2"

216

For each p : ~2 ) ~ ,

t(p)(o) = V(v" w)^~tip)(u) h~V~ h~

and hence

p(f~, f~)(u) = f(t(p)(u~) = Vv~ct(p~(u ) flv^f~w.

Applying this formula to partioular binary operations p gives, for

example, the following :

(i) (fl * fz)(r-) = Vx + y_~ r f! (x-) ̂ f2(Y-~,

t f + ( f l f~)(r§ = V x . y > r f l (x*)A ~(y ).

(ii) (flv f2)(r-) = f](r') A f2(r-),

(flv f~)(r +) = f1(r*) v f?Cr+),

( i i i ) (f~ A f~ ) ( r ' ) = rl~r-~ v f2 (r - ) ,

(fl ̂ f~)(r+)) " f](r +) Arc(r*),

(iv) f! ~ f2 iff f?(r-) z f~(r-) for all r 6 ~,

fl 9 f? iff f!(r*) ~_ f2(r +) for all r 6 ~.

Constants in C(K). For a 6 R, let k : t ~ ---> K be the function

for which ka(G) = 1 if a & G, k,(G) = 0 if a ~ G. Then k~ is

a frame map. One easily verifies that

k = k~ iff a = b,

ka+ b = k~ ~ k h,

kab = k k b �9

%~,,b = k , ^ 4 ,

kay b = k~. v kb,

Thus k is an embeadin~ of R in C(i~). Identifying a with k~,

we may write a : t ~--+K. Then for f : t ~, ,~K,

f _~ a iff f(r-) = 1 for all r > a,

f ~ a iff f(r*) = 0 for all r > a,

f _> a iff f(r') = 0 for all r < a,

f --> a iff f(r § = 1 for all r < a.

One oan verify ~nat if a > O,

a f ( r - ) = f ( ( ~ - ) - ) , a f ( r +) = f ( ( ~ ) + ) ,

217

while if a < O,

af(r-) = f((~)+), af(r +) = f((~)').

The bounded elements of C(K). We shall say that f e C(K) is bounded

if there exist a, b ~ ~ such that a ~ f ~ b. We write C~(K) for

the set of bounded ele~ents of C(K). It is easy to see that C~(K) is a

subalgebra ana a vector sublattice of C(~).

We introduce a non~ in C~(K) as follows :

~ith this norm C~(k) is a normed algebra. ~e shall sho~ that it is a

Banach algebra.

C~(K) is complete. Indeed, let (fn) be a Cauchy sequence in C~(K).

For each r ( ~, let

~r = VmAn> m fn(r-), Vr = VmAn>m fn(r+)"

We shall verity that fr ~ ~r = O, while if r < s, ~s v ~r = i,

First

If r < s, choose u, v with r < u < v < s, and choose

so that

0 < ~ < rain (u - r, s - v).

Let m be such that if n > m then

"s < flu " fm < g

Then

Vx+y>u (~n(X § A (fro fn)(y+)).

If x �9 y ~_ u and x < r, then y > g and (fro " fn )(y~) ~ O.

Hence

fro(u+)) -~ Vx~r fn(x~) = f~ (r§

Similarly it rosy be shown that, when n > m,

f~Qv') L fn(S-).

218

Hence

Usin~ these properties of ~r and ~r,

proof of [2], Theorem 3, that there exists a fraz~e map

f : t ~ ) K

such that for all r ~ Rj

f(r-) = Vs<:r ~s, f(r*) =

The frame n~p with this property is clearly unique.

fn(r t )

1.

one can show, as in the

Hence

and hence

V m ip> m (fp(X-)) a (-fn)(y-) ~ -g(r-),

f - fn ~ "s

Similarly, using (f - fn)(r+), one c~ show that

f - fn ~ ~ "

The proof of the completeness of C~(K) outlined here uses special

properties of frames. ~e do not know a category theory argument for the

transfer of the Banach space structure from ~ to C~(K).

the sequence (fn) converges to f.

Let s > O. Choose no so that if m, n > no,

-~ < fm " fn < ~ "

For r E ~ and n > no,

(f - fn)(r-) = Vx~yc. r (f(x') % (-fn)(Y'))

= Vx+y~r ( ( V m A p ; ~ fp{x- ) ) a ( - fn ) (y - ) ) .

If x * y ~ r and p > n e,

f~(x-) A (-fn)(Y-) ~ (fp - fn)(r') ~ -E(r').

~>r ~s ~

It is to be shown that

219

3

4

5

REFERENCES

C. H. Dowker and Dona Paper~, Quotient frames and subspaces, Proo. London ~ath. Soc. ib (1866), 275-296.

C. H. Dowker and Dona Papert, On Urysohn's lenmm, General topology and its relations to modern analysis and algebra If, Prague, 1966, lll-ll4.

C. H. Dowker and Dona Strauss, Sums in the category of frames, to appear.

J. k. Isbell, Atomless parts of spaces, l~ath. Scandinav. 31 (1972), 5-32.

Dona Paper~ and S. Papert, Sur les treillis des ouverts et les paratopologies, Sgminaire C. Ehresmann 1957/58, exp. no. i (Paris, 19~ 9 ).

S. Papert, The lattices of logic and topology, C~bridge University Ph.D. dissertation, 1959.

S. Papert, nn abstract theory of topological spaces, Proc. Cambridge Phil. Soc. 60 (19~4), 197-203.

H. Schubert, Categories, Springer-Verlag, Berlin Heidelberg New York, 1972.

CATEGORICAL METHODS IN DIMENSION THEORY

Roy Dyckhoff

St. Andrews University, Fife, Scotland.

The classical conception of dimension theory has been substantially

augmented by cohomological methods, and within the last fifteen years by the

theory of sheaves also. Modern interest in sheaf theory is based on the

categorical properties of sheaves, in that they form an elementary topos, and

on the logical properties, in that the topos provides a language in which

certain sheaves can be described more simply than is classically possible; for

example, the sheaf of germs of (continuous) real-valued maps is internally the

real number object, hence classical but intuitionistically valid theorems about

real numbers can therefore be exploited. The main purpose of this paper is to

ask the question "in what sense is dimension of a space an internal property of

the topos of sheaves over the space", and to give evidence that the question is

not uninteresting.

In investigation of this subject, various approaches closely related to

other work reported at this conference have been fruitful, and we therefore

begin with a discussion of two subjects familiar to categorical topologists:

factorisation theory and projective resolutions. We report on our work linking

the two; the former is convenient in our study of dimension-changing maps, and

the latter has a homological interpretation via sheaf theory.

Section three is a survey of basic dimension theory with sheaves of

coefficients; no novelty is claimed for this, but it is useful background to

the rest as well as an interesting way of fixing notation. We give here the

relationship between projective resolutions and dimension. Section four is a

study of dimenslon-changing closed maps; here the novelty is not so much in

the theorems of dimension theory but rather in the relation of categorical

and logical concepts to the theory and the ensuing simplifications. Specifically,

we illustrate some internal category theory in an analysis of the monadic

resolution of a proper map, by associating to a map a directed sheaf in

221

essentially the same way as to a compact Hausdorff space of dimension zero one

may associate a directed system of finite discrete spaces; we then represent the

monadic resolution of the map as an internal direct limit of ~ech resolutions of

locally finite closed covers, with an application to dimension-raising maps.

No bibliography for such a diffuse subject can be exhaustive, hence we have

chosen to refer mainly to survey articles rather than to original sources, with

a few exceptions to be up to date. The reference [77] refers to the [1977]

publication of the last-mentioned author, or the first in that year, and [77a] to

the next in that year. Certain publication dates are necessari~ conjectural.

w Factorisation theory.

To avoid confusion with Benabou [67] and to avoid saying "bicategory in the

sense of Isbell [63] or Kennison [68] without epi- or mono- assumptions", we use

the term Factorisation category for the following basic concept: a triple

( ~, ~ ,~ ) where ~ and Q are cl~sses of morphisms, in the category ~ , each

closed under composition and containing all isomorphisms, such that

i) Any morphism of ~ has an essentially unique (Q, ~)-factorisation

f:pq, p~e, q~Q.

ii) When pf : pg and fq : gq for p �9 ~ , q �9 Q, then f : g.

As in Herrlich [68] 7.2.3, i) implies that O satisfies the (~,~)-diagonal

condition, and ii) implies uniqueness of such a diagonal. Since ~ determines

(~, we use ~-factorisation for brevity. Some would call ~ a ((~,~)-eategory.

For general theorems about such categories consult e.g. Kennlson [68], Herrlich

[68], [72], Ringel [70], [71], Strecker [74], [76], and Dyckhoff [72], [76].

Apart from onto-factorisatlons and the llke, one of the first significant

examples of this type is Eilenberg's light-factorisation in compact metric

spaces [34], due also to Whyburn simultaneously (see [63] for applications to

topological analysis). The essential content of this was extracted by Michael's

[64] light-factorisatlon in Tl-spaces ; in the same vein are Collins' dissonant-

factorisation [71], [71a] in ] = top. spaces, Strecker's superlight-

factorisation in ~ [74], Henriksen and Isbell's perfect-factorisation [68] in

222

Tychonoff spaces, Herrlich's ~-perfect factorisation in Hausdorff spaces [71a]

(where ~ = compact Hausdorff spaces), and the rather more complex factorisation

theorem in To-spaces with skeletal maps due to Blaszczyk [74], where ~ = maps g

with g-l~ = g-lG for open G, and skeletal maps are the same as Herrlich's

demi-open maps (Harris [71]).

Our aim here is to introduce a few more of these theorems, some being of

particular interest in dimension theory, and others being included to complete

various unfilled pictures. Michael's theorem decomposes the domain of a map into

the space of components of fibres of the map; a similar theorem for Tl-spaces is

valid (Dyckhoff [74a]) for the Tl-reflection of the space of quasi-components of

fibres with a strange topology, where Fox's spreads (Michael [64]) (= decQmposing,

or separating, maps in Russia) play the role of ~ . The Tl-assumption for

these results is more or less vital: it ensures that the induced map from the

middle space to the range is light (resp. a spread). A map f: X --> Y is

proper iff perfect and separated (i.e. f • i Z and X --> X • fX are all closed);

the following result is easy and a good substitute for both these theorems in the

non Tl-case :

Proposition i.i. (~ ,I,~) is a factorisation category, where ~ denotes the

category of spaces and proper maps, ~ (resp. ~ ) the light (resp. monotone) maps

in ~ . []

Many generalisations of perfect maps (cf $t~ecker [76]) are used, some

being based on Herrlich's ~-perfect factorisation. Nevertheless, proper maps,

including perfect maps of Hausdorff spaces, behave rather well:

Proposition 1.2. (Dyekhoff [72], [76], cf also Herrlieh [72]).

( ~ ,~ ,~ ) is a factorisation category, where ~ denotes the category

of top. spaces, ~ the proper maps. D

The maps ~ herein are those satisfying a unique diagonal condition wrt ~ ,

which for our purposes is more useful than any intrinsic characterisation; we

call ~ the improper maps in [76], the term anti~erfect being used in [72] for

improper maps of Hausdorff spaces. For a map f: X --> Y of T2-spaces , the

proper-factorisation has as middle space the T2-reflection of the space of

223

ultrafilters on X with convergent image in Y (Dyckhoff [74a]); extension

outside T3-spaces is unlikely by Wyler [71].

The property, or class, ~ of mappings we call hereditary iff whenever

f: X --> Y has ~ and U c Y is open, then f-iu--> U has ~. Almost all the

above mentioned classes ~ in the factorisation categories are hereditary,

as well as (dense maps), (quotient maps). The main exception is (dense, ~-

extendable maps) in Hausdorff spaces, by a counterexample based on a Tychonoff-

like corkscrew due to Herrlich.

Proposition 1.3. Improper maps form a hereditary class.

Proof. (Dyckhoff [76], [76a]). []

Perhaps this gives some justification to our preference for the proper-

factorisation over the X-perfect factorisation; note that the dense ]<-

extendable maps of Tychonoff spaces form a hereditary class, by the same proof

(Dyckhoff [74a]).

In the study of dimension-changing properties of proper mappings, it is

useful to recall by (1.2) that such a map has a monotone factor and a light

factor; in the classical (Nagami [70]) terminology, the former "lowers" dimension,

the latter "raises" it. Our three propositions imply

Proposition 1.4. (~, ~, ~) is a factorisation category, where ~ is the

class of proper light maps, and ~ is a hereditary class (namely, the dense,

hereditarily 2-extendable maps). []

In our later application of the logic of sheaves to the analysis of a mapping,

that is, the systematic quantification over all open sets, this hereditary

property may appear more natural and desirable.

For convenience we here list the factorisation categories referred to:

Proposition 1.5. The following are factorisation categories:

i) Top. spaces, i-i maps, quotient maps.

ii) Top. spaces, embeddings, onto maps.

iii) Top. spaces, closed embeddings, dense maps.

iv) Top. spaces, dissonant maps, concordant quotients (= 2-extendable quotients).

v) Top. spaces, superlight maps, submonotone quotients.

224

vi)

vii)

viii)

ix)

x)

xi)

xii)

xiii)

Top. spaces, proper maps, improper maps.

Tl-spaces , light maps, monotone quotients.

Tl-Spaces, spreads, hereditarily concordant quasiquotients.

T2-spaces , perfect maps, antiperfect maps.

T2-spaces , ~-perfect maps, dense, Z-extendable maps.

Tychonoff spaces, perfect maps, dense C*-embeddings.

Top. spaces and proper maps, light maps, monotone maps.

T0-spaces and skeletal maps, irreducible r.o. minimal maps, e.d. preserving

maps. O

Categorical methods are available for proving most of the above results,

but it is generally best to proceed directly, Parts vi), ix), and x) are the

odd cases, and our next section is designed to show precisely why the categorical

defining property of, e.g. improper maps, is more useful than any intrinsic

characterisation.

For other types of factorisation theorem connected with dimension, see

Kulpa [70] and Lisitsa [73].

w Projective resolutions.

Gleason's classic [58] on projective topological spaces pointed the way

to a profusion of generalisations of his construction of projective covers in

~(: compact Hausdorff spaces) to wider topological categories, the most

significant being due to Banaschewski [68] whose work simultaneously covers

topological algebras, and to Ponomarev [64], whose theory of absolutes

(regarded in [65] by Aleksandroff as 'one of the most outstanding achievements

of set-theoretic topology in the last decade") drops naturally out of the

spectral methods used to study perfectly n-dimensional spaces. Hager [71] and

Mioduszewski and Rudolf [69] can be used as bases for further reading. For all

these authors, a p rojectiv e cover of a space X means a proper irreducible onto

map RX--> X, where RX is extremally disconnected and thus projective wrt

proper onto maps; such a space RX may be constructed when X is Hausdorff as

a space of convergent maximal open filters on X, e.g. as in Banaschewski [68];

225

note also Gonshor's construction [73] by non-standard analysis.

Henriksen and Jerison [65] noted the non-funotoriality of this notion, and

Herrlich [71] asked essentially both for this to be rectified and for the

construction to be done for non-Hausdorff spaces. Shukla and Srivastava [75]

solve the former problem with the idea of "stable corefleotivity" of the

extremally disconnected spaces in the Hausdorff spaces, there are solutions by

Mioduszewski and Rudolf [89] of the former by restriction to skeletal mappings,

and by Blaszczyk [74], [74a] of the latter in the category of T0-spaces.

Nevertheless, in Dyckhoff [72] and [76] we have proposed the following

alternative solution to these problems:

Proposition 2.1. There is a comonad (P,p,q) on ~ such that

i) each counit map PX: PX -->> X is proper, onto, and

ii) PX is always projective in ~ @ , and thus extremally disconnected.

Proof. Apply (1.2) to the discrete modification DX -->> X, taking its proper

factor PX: PX -->> X. By diagram chasing, P is a functor, p a natural

transformation, and PX projective. The comultiplication P --> p2 comes from

an induced faotorisation theorem in the category of eomonads on~ : ef

Dyckhoff [78] or Diets [73] for details. When X is regular, we may think of

PX as the space of convergent ultrafilters on X with DX embedded therein as the

principal ultrafilters (Dyckhoff [74a]). A point of PX, qua ultrafilter on X,

is also an ultrafilter on DX and thus on a subset of PX; on close analysis, this

is the eomultiplication PX --> PPX at X. []

Rainwater's methods in [59] apply to our construction PX --> X in the

Hausdorff case, reducing it to the irreducible map RX -->> X considered above;

RX is a minimal (by Zorn) closed subspaee of PX mapping onto X.

Corollary 2.2. Every Hausdorff space has a projective cover. []

As noted above, this corollary has been proved for T0-spaces by Blaszczyk

[74a], yet we do not know whether the T0-axiom is necessary. At the price of

irreducibility, our construction is both functorial and valid for all spaces;

both constructions evidently have their value.

The semi-simplicial category A consists of the finite ordinals and

226

order preserving maps; these maps factorise into face maps and degeneracies

(MacLane [71])�9 A functor AoP ___>~ is a semi-simplicial diagram in~ ; we

can represent it as a sequence of spaces and maps

X 3 ~---> X 2 > X > X 0 "'" Xn ........ <-- < i .... <

with relations between the face maps --> and degeneracies <-- . Such diagrams,

or their duals, arise naturally in constructions of (co)homology theories

(MacLane [71], Barr, Beck [69], Duskin [75], for example).

Theorem 2.3. (Dyckhoff [76]). For every space X, there is a semi-simplicial

diagram in ~ :

........ p2X --~ PX > X <

natural in X, all face maps being proper onto, each PIX being projective.

Proof. Iterate the comonad of (2.1) in the standard way (MacLane [71]). []

We refer to this diagram as the projective resolution of X; in sheaf theory,

(Godement [64]), "resolution" denotes exactness of a sequence of sheaves, and our

use of the word seems presumptuous. We shall see in the next section how this

diagram converts a sheaf on X into an exact sequence of sheaves in the most

natural way, and regard this as justification for our usage. Indeed, we must do

some sheaf theory before we can relate the projective resolution of X to dimension.

w Sheaves and dimension.

This section aims to outline very briefly the fundamentals of sheaf theory

and cohomological dimension, both to fix our notation and as a basis for further

reference. For details of the sheaf theory, we refer to Godement [64] and to

Bredon [67], and for the application to dimension, to Kuz'minov [68]. Hurewicz,

Wallman [48], Nagami [70], and Pears [75] are excellent references for

topological dimension theory, and Kodama's appendix to Nagami's book is a good

exposition of sheafless cohomological dimension theory; see also Bokshtein [663.

The application of sheaves to dimension is essentially due to Sklyarenko [62],

[84]. We refer to Freyd [72], Kock and Wraith [71], Wraith [75], and Fourman

[78] for the theory of topoi.

228

exact left adjoint. []

The Grothendieck cohomology of X is the sequence H* of right derived

functors of the global section funetor H0: Ash(X) --> Ash(pt) = Ab. grps.,

A~--> A[X!. Interestingly, this factors through Apsh(X). Let U C X be open;

there is an exact functor -U: Ash(X) --> Ash(X) which concentrates a sheaf on U;

formally, if i: X\U --> X is the embedding, then A U = ker (A --> i,i*A).w We say

an abelian sheaf A is soft iff restriction A[X] --> A[F] is onto for all closed

F C X. An exact sequence of abelian sheaves 0 ~ A --> A 0 ---> A 1 --> .... is

said to be a resolution of A.

Theorem 3.5. (Zarelua [69]). Let A be an abelian sheaf on the paracompact space

X, and n ~ 0. The following are equivalent, and if they hold we say dim(X,A) ~ n:

i i) H (A U) : 0 V i > n, V U open in X,

... ~ 0 of A, it) For some, or any, resolution 0 --> A ---> A 0 ~ --> A n

with A. soft for all i < n, then A is soft. [] 1 n

In particular, A is soft iff Hi(A U) : 0 for all i > 0, for all open U C X,

and then dim(X,A) : 0. The theorem is fundamental both in the reduction of

problems in higher dimensions to problems in dimension zero (analogous to the

role of decomposition theorems in the dimension of metric spaces (Nagami [70])),

and in its conversion of the algebraic problem of computation of cohomology

groups to the topological problem of extending sections over closed sets. For

example, it thus suffices to prove the next two theorems for dimension zero, by

gradual extension of sections via Zorn:

Proposition 8.6. Let A be an abelian sheaf on paracompact X, F C X, and (F)

a locally countable closed cover of X. Then

i) dim(F,A) 5 dim(X,A) provided F is closed in X or X is totally normal,

it) dim(X,A) f sup. dim(F ,A).

Proof. See Kuz'minov [68]. We make little distinction between a sheaf on X

and its restriction to a subspace F or F of X. []

Since every open cover of a paraeompact space has a locally finite closed

refinement, there is an obvious corollary.

Theorem 3.7. (Kuz'minov and Liseikin [71]). Let (AI) be a direct system of

227

A sheaf on X is a local homeomorphism p: S --> X; for B C X, the sections

of p over B form a set S[B], with a restriction map S[B] --> S[C] for B D C.

A presheaf on X is a functor to Sets from the category ~(X) ~ of open sets of

X with a morphism U --> V iff U D V. The canonical presheaf of the sheaf S is

the presheaf S: U }---> S[U]. Certain presheaves may be thought of as sheaves by ~

Proposition 3.1. : Sh(x) ---> Psh(X), S ~--> S, has an exact left adjoint. []

Proposition 3.2. Let f: X --> Y be a map and f,: Sh(X) --> Sh(Y) the (direct

image) functor with f,S the sheaf on Y associated by (3.1) to the presheaf

U ~--> S[f-Iu]. Then f, has an exact left adjoint f*.

Proof. (see Grothendieck [72] IV.4.1). For a sheaf T --> Y, f*T --> X is just

the pullback by f. []

Proposition 3.3. (Grothendieck [72]). Sh(X) and Psh(X) are elementary topoi.

Proof. Let p: S --> X be a sheaf on X; then p,p* is right adjoint to - • so

Sh(X) is cartesian closed. As subobject classifier, take X --> ~, where S is

the sheaf on X associated by (3.1) to the presheaf U ~--> ~(U). Similarly for

presheaves. []

In any category with pullbacks, (abelian) groups, rings, modules, partially

ordered sets, categories, etc. are all definable diagrammatically; in any topos,

they are also models for the appropriate axioms written in the internal language

of the topos (Mulvey [74]). The above propositions permit the application of

topos theory in the study of a topological space - they assert the existence of

morphisms between the topoi Sh(X), Psh(X), Sh(Y). Now it is clear what an

abelian group object in Sh(X) must be, we call it an abelian sheaf, denoting the

category of abelian sheaves on X by Ash(X), (and Apsh(X) for abelian presheaves);

it is not only an abelian category but also an abelian category object in Sh(X)

(if we ignore the foundational difficulty). Injectives ~re definable from the

external (or internal) hom-functors Hom (or ~om) in Ash(X) - fortunately, if

Hom(-,l) is exact then so is ~om(-,l); in either sense, we have

Proposition 3.4. (Grothendieck). There are enough in~ective abelian sheaves on X.

Proof. (see Godement [64]). True for singleton X (MacLane [63]), hence true for

discrete X. Now use (3.2) and preservation of injectives by any functor with an

229

abelian sheaves on paracompact X. Then dim(X, lim Al) 5 sup.dim(X,Al). [] _-.-->

For compact Hausdorff spaces and hereditarily compact spaces this evidently

follows from Godement [64]; in fact, for such spaces cohomology H* commutes with

direct limits of sheaves, a result typical of many in algebraic geometry

(Grothendieck [72] VI). According to Bredon [67], H* does not commute with

direct limits of sheaves even on a space as nice as ~ ; it is essentially the

characterisation (3.5) of dimension in terms of soft sheaves which permits a

result such as 3.7.

The relationship between cohomologlcal dimension and covering dimension is

due to Aleksandroff and Dowker:

Theorem 3.8. Let X be a paracompact space of finite covering dimension.

Then dim X = dim(X, ~ ).

Proof. ~ is easy (Godement [64]), once the ~ech cohomology theory has been

shown identical with Grothendieck's, since it is based on coverings. For both

parts, see Kodama 36.15 (Nagami [70]) or Kuz'minov [68] II w Bokshtein [56]

has a bypass to the Hopf extension theorem. []

Clearly there are decent sheaves A for which dim(X,A) # dim X < 0o: for

example, the sheaf~ of Dedekind real numbers on a paracompact space is soft

(Huber [61]), and a converse result holds (Zakharov [74]), that for locally

compact X, softness of this sheaf implies paracompactness, if we reinterpret

softness as zero-dimensionality. See also Mulvey [76], for a proof that X is

paracompact iff ~ is a projective~-module. The whole point of using sheaves

as coefficient groups is not just that lots of different dimension functions

thereby arise, but, for example, that a map f: X --> Y can turn the trivial

sheaf ~ on X into the non-trivial sheaf f,~ on Y. It is vital for us to

relate dim(X,A) to dim(X,Z ) and thus to dimX; we state the relationship in the

next result, proved for example in Kuz'minov [68], the essentials being a lemma

of Grothendieck and (3.7); the idea is to use (3.7) and Zorn to find a maximal

subsheaf A' of A for which dim(X,A') 5 dim(X,~ ) and show A' = A. Of course,

the theorem is unnecessary for finite dimensional X if we use the proof in (3.8)

that dim(X,A) S dim X and the equality in (3.8).

230

Theorem 3.9. Let A be an abelian sheaf on paracompact X. Then

dim(X,A) S dim(X,~ ). []

For the representation of cohomology with coefficients in a group as the

homotopy classes of maps into Eilenberg-MacLane spaces, see Huber [61], Bartik

[68] or Goto [67].

Fundamental to the above discussion is the existence, for an abelian sheaf

on X, of a soft resolution 0 --> A--> A 0 --> A I ---> ..., finite or infinite.

For paracompact X, there is a variety of constructions of such resolutions:

injective, canonical flabby, semi-simplieial flabby (Godement [64]), all of which

are soft since on paracompact X injective and flabby imply soft, or by the easy

Lemma 3.10. Let,f: X --> Y be continuous, Y paracompact.

Then f,A is soft when A is soft. []

Let us reconsider the "projective resolution" of a space X defined in w

an abelian sheaf A on X can be pulled back to a sheaf Pi A on PIX, and the

direct image pi~Pi:'A formed on X; the face maps can be transferred too, and

we have a (co)semi-simplieial diagram in Ash(X)

> . >

--~-- > *A '> PI*PI*A < " < Pn*Pn ~ ........ 0 > A >

Let us remove the degeneracies <-- and simplify the face maps ~> by taking

alternating sums:

Theorem 3.11. The sequence 0 --> A --> .... --> pn,Pn*A --> ..... is a

resolution of A; all the sheaves pn,Pn*A are soft when X is paracompact, and

thus the cohomology H*(A) is the global cohomology of the semi-simplicial

complex (2.3) with coefficients A.

Proof. Dyckhoff [76]: by (3.10) and softness of all sheaves on paracompact

extremally disconnected spaces. []

Our [76] discusses also the interesting analogy with cohomology of groups,

which is definable both from projectives and injeetlves; so is the cohomology

of sheaves by (3.11), bearing in mind that since Ash(X) lacks projectives we

construct them in the category of spaces and proper maps, over X, instead.

Corollary 3.12. Let X be finite dimensional paracompact.

231

Then dim X ~ n iff ker(Pn+l,Pn+l~'~ --> Pn+2,Pn+ 2 ) is soft. []

The theorem and its corollary give some point to Gleason's suggestion [58]

that his projective spaces have homologieal significance; but we do not claim,

for example, that they provide any method for the calculation of cohomology or

the estimation of dimension: for that the ~ech theory and Huber's representation

theorem are more appropriate. There may be a link between our theory restricted

to compact Hausdorff spaces and the injective C*-algebras (cf Gleason [58]).

Our next section will discuss finite-to-one closed mappings and certain

resolutions associated thereto; meanwhile, there is the following amusing and

easy observation concerning projective covers and a "dimension" function:

let edimX J n iff every locally finite closed cover of X has a locally finite

closed refinement of order S n+l.

Proposition 3.13. (Dyckhoff [74a]) edimX S n iff the projective cover RX --->> X

has order S n+l; but edim]R is infinite. []

Problem 3.14. What significance have the degeneracies in the projective

resolution of a space?

w Dimension-changing closed maps.

Proper maps preserve many topological properties, particularly those

concerning coverings, but not of course dimension: the classical theorems of

Hurewicz on the relationship between the dimensions of two spaces related by a

proper map are well known, and of importance both in the theory of product

spaces and in attempts (e.g. Nagami [62]) to base dimension theory on the

minimum order AX of a proper map onto a given space X with a domain of dimension

zero. See Nagami [70] for the classical theory, modernised, Filippov [72] for

recent contributions based on reduction to the metric case, and Pears and Mack

[74] for a study of AX among other dimension functions. The most precise forms

of these theorems were originally obtained by the theory of sheaves, in

Sklyarenko [62] and Zarelua [69]; they are very fine illustrations of the power

of cohomological methods. Our aim in this section is to make their proofs more

accessible to the category theorist, and in particular to illustrate the advantage

232

of using the concept of a topos.

Let F: ~ --> ~ denote the restriction to abelian objects of a geometric

morphism of topoi, where ~ has enough injectives; let A e ~, n > 0. We say

F is n-exact at A iff the right derived functors RPF vanish at A for all p > n.

Lemma 4.1. Let F: ~ --> ~ , G: ~ --> ~ be two such morphisms, where F is n-exact

at A and G is m-exact everywhere. Then GF is (m+n)-exact at A.

Proof. See Grothendieck's SGA4 V 0.3 [72a]. Apply F to an injective resolution

of A and apply G to the image of that from the n th kernel onwards. []

Theorem 4.2. (Hurewicz, Sklyarenko). Let f: X --> Y be closed, X paracompact,

A an abelian sheaf on X. Then dim(X,A) J sup(dlm(f-ly,A): y e Y) + dim(Y,~).

Proof. Apply to F = f,, G = t,, where t is the terminal map Y ~ point. []

We have deliberately removed the Leray spectral sequence implicit in (4.1) to

stress the simplicity of the argument. By (3.8) and (3.9) and elementary

arguments, all dimension functions here can be replaced by "dim". Then, the

theorem is true (Pasynkov [65]) for X normal but Y paracompact, by consideration

of the ~-perfect factor of f, which has the same dimension as f for X normal.

Filippov [72] shows that normality of Y cannot replace paracompactness.

Corollary 4.3. Let X be compact, Y paracompact. Then dim(X• J dimX + dimY. []

Morita's work [73] suggests that 4.3 is true for all Tychonoff spaces; in any

case, proofs of 4.3 for more general spaces generally involve reduction to easy

special cases such as compact • paracompact.

For an improper map f: X--> Y (of paracompacta), dim X = dim Y, since the

map Bf between the @ech extensions is an isomorphism. Let f: X --> Y be a

proper map, f = gh its light-factorisation by (i.i). By (4.2), dim Z ~ dim Y:

one says that g: Z --> Y raises dimension.

Problem 4.4. What conditions on a proper map guarantee dimY ~ dimX, and is there

a factorisation category involving the maps which raise (resp. lower) dimension?

The fundamental theorem on dimension-raising maps is

Theorem 4.5. (Hurewicz, Zarelua). Let f: X --->> Y be closed, onto, where X is

paracompact, A an abelian sheaf on Y. Let order(f) = sup(If-lyl: y e Y) = k + i.

233

Then

dim(Y,A) ~ k + dim(X,f*A). []

Corollary 4.6. Let f: X -->> Y be closed onto, where X is Tychonoff, Y normal.

Then dim Y < k + dim X, where order (f) = k+l.

Proof. Suppose f has order < k+l; by lemma 4.1 of Pears and Mack [74], so does

the Cech extension Bf, to which we apply (4.5). []

We sketch the proof of (4.5) in categorical terms to suggest that there

is a natural way to look at the problem. Let f: X ---> Y be any map; there is a

monad (f,f*,u,m) on Sh(Y), or on Ash(Y), induced by the change of base adjunction

(3.2). When f is onto, u: I ---> f,f* is a monomorphism and when f is a montone

quotient, u is an isomorphism. Indeed if we form the semi-simplicial functor

from the monad in the usual way (see Barr, Beck [69], or MacLane [71]), and in

Ash(Y) convert by alternate summation to a complex, we have

Lemma 4.7. When f is onto, then the sequence

0 --> I --> f,f* --> ... --> (f,f,)n --> ...

is exact; when f is a monotone quotient, the arrows of the sequence are

alternately isomorphisms and zero.

Proof. To prove exactness of the sequence of the stalks over y E Y of images of

a sheaf A, pick any x 6 f-ly and use evaluation at x to contract the sequence:

cf Godement [64] Appx. The last part depends on monotone quotients forming a

hereditar~y class (w and lightness of structure maps of sheaves. []

Thus for an onto map f: X -->> Y there is a monadic resolution functor Mf

on Ash(Y), taking the sheaf A to the resolution

0 --> A --> f,feA ~> ... --> (f,f*)nA--> ....

Godement's semi-simplicial flabby resolution ([64], Appx.) is an example, and

the original inspiration for monads, being based on the map DX --> X. For two

other examples, consider the ~ech resolution (Godement [64]) deter~nined by an

open, or locally finite closed cover - bearing in mind that a cover ~ of X

induces an onto map @~ -->> X, open or proper respectively. We shall be

concerned with such resolutions determined bY proper maps; meanwhile, we note

that the discrete modification map DX --> X and its proper factor PX --> X

234

induce distinct resolution functors on Ash(X), both of which for X paracompact

have the same global cohomology, by proof similar to (3.11).

Problem 4.8. If gf and g are onto and f is improper, does f induce an equivalence

between the cohomology sheaves of the resolutions for gf and g?

There are two other important descriptions of the monadic resolution of

a proper map; the first gives us a geometrical picture, the second is an aid to

calculation.

Theorem 4.g. (from Grothendieck's SGA4 p. 141 [72a]). Let f: X -->> Y be a

proper map. Then the following two resolution functors are isomorphic:

i) 0--> I--> f,.f*--> __> (f~f,)n __>

ii) 0--> I--> f,f*--> --> fn fn* -->

where f: X • fX • f .... • fX (n factors) --> Y and the maps in ii) are

alternating sums of face maps derived from the 'resolution of Y":

< > ----->

..... - > X • fX • fX -->> X • fX -----><__ X > Y. [] <

Corollary 4.10. The resolution (4.9.i) has a simplicial structure.

Proof. ii) has, from the symmetric group acting on each n-fold product

X • f ... • fX. •

Corollar~f 4.11. The monadic resolution of a proper map f has an "alternatin$

subresolution" Mr. ~ n

Proof. For a sheaf A on Y, (f,f) A has stalk at y �9 Y the abelian group

A (f-ly)n, which has a sUbgroup consisting of all the alternating elements; over Y

y, the sequence of such subgroups is exact by standard homological methods

(Godement [64]); since any such (alternating) element has a local representation

over a nbd of y, which can be made alternating when the nbd is taken small

.,. n enough, the "alternating subgroups" form a subsheaf of (f,f") A. []

For example, the alternating ~ech resolution determined by a locally finite

closed cover; a similar argument works for open covers. Macdonald [68] has a

useful description of these.

Corollar]f 4.12. Let f: X -->> Y be a proper map of order < k+l. Then the

alternating resolution by f of any abelian sheaf A on Y vanishes after the term

235

(f~f*)k+iA. []

We now come to the representation of the monadic resolution of a proper

map in calculable terms; the following procedure applied to any map gives a

result dependent only on the proper light factor. Let f: X--> Y be a map,

which, by (4.7) we shall assume to be onto. A pa~tition of f over open U C Y

is a locally finite closed cover ~ of U and a factorisation f-iu ~> ~-->> U,

the first factor being dense. Since we consider many partitions, the index

will denote the partition f-iUx ---> ~ ~k ~ Ul. We say of two partitions X,

over U that k ~ ~ iff there is a factorisation f-iu -> @~

@{F~ >> U

Note that @~U ~ @ ~ is onto and is the unique map, if any, making the diagram

commute; hence J is a partial order on ~ f(U), the set of partitions of f over

U. Let (U) be an open cover of open U C Y, and let ~ e ~f(U ) be given for

each s in such a way that k s and ~8 always agree over U s hUB. Then the

partitions patch together, forming uniquely a partition X of f over U. Hence A f

is a sheaf of partially ordered sets on Y.

In the internal language of the topos of sheaves over X, due to Benabou,

but see Mulvey [74] for a convenient account, having the open sets of X as

truth values in the intended interpretation, and therefore intuitionistic, we

can say that ~f is a partially ordered object of Sh(Y). To say that Af is

directed would be to say that given ~, U in Af(u), there is a cover (U s) of U

and 9~ in ~f(U s) with vs -> both k and ~ on U~: formally,

V ~V~: (~_<~)^(~<~). By using pullbacks it is easy to see that this is true even with U = U and

s

all 9 equal: see Dyckhoff [74] for the details in different terminology. s

Thus Af is "directed", and thus a "filtered category": see Gray [75] for

internal category theory.

In the language of set theory, a natural resolution on ~ is a functor

from an abelian category ~ to the category of exact sequences in ~ , taking A

236

to the sequence 0 --> A --> A 0 --> A 1 ---> ...... , and f: A --> A' to a

morphism of exact sequences over f. Now we consider the topos Sh(Y); let

denote the internal category, for which ~ (U) denotes the set of all natural

resolutions on Ash(U): this is a presheaf, and actually a sheaf, and thus an

object of Sh(Y); it is also a category object, or internal category. We have

described above an (internal) functor M: Af --> ~, taking, over open U, the

partition k to the monadic resolution M~ determined by @~ --->> U k = U.

The simplest example of a topos is that of finite sets; this is neither

complete nor cocomplete, except internally, - it has finite limits and colimits.

More generally, every topos is internally complete and cocomplete, and externally

complete and cocomplete iff it is a Sets-topos (i.e. there is a morphism to Sets,

i.e. ordinary sets can be pictured inside the topos, by pulling back along the

morphism). Thus, in topos theory, the fundamental notion of, e.g. cocompleteness,

is internal; it is accidental that Sh(Y) is also externally cocomplete. Moreover,

colimits over "filtered categories" commute with finite limits, in particular,

direct limits inside Sh(Y) commute with finite limits (cf Johnstone [74]).

Once these rather complex ideas are absorbed, the following result is

trivial, both parts of (1.4) being borne in mind:

Theorem 4.13. (Dyckhoff [74], [74a]). Let f: X ~> Y be an onto map. Then the

diagram M: A f --> ~ has a colimit, the monadic resolution of the proper light

factor of f. []

Corollary 4.14. Let f: X -->> Y be proper, light, onto. Then its monadic

resolution is representable as an internal direct limit of ~ech resolutions

determined by partitions of f. []

Our [74] does this as a representation theorem for f as a partial inverse limit

of simple maps, i.e. maps looking like finite closed covers. We have replaced

"finite" by "locally finite" to ensure that our ~f is actually a sheaf. Partial

inverse limits are to inverse limits as partial products (Pasynkov [65]) are to

ordinary products.

Now recall 3.7: simply, that over a paracompact space, a direct limit of

soft sheaves is soft. This result was first proved by Zarelua [69] with a mild

237

restriction on the bonding maps, but in a more general form, which can be stated

as a theorem about internal direct limits. We conjecture that there is a formula

in the language of sheaves over X, a paracompact space, asserting of an abelian

sheaf over X that it is soft. Martin Hyland (private communication) has given

an intuitionistically valid proof of the lemma (15.10 in Bredon [67]) of

Grothendieck referred to as the key ingredient in Theorem 3.9. We note also that

Deligne, in Grothendieck's SGA4 V Appx. [72a], studies internal filtered colimits

and extends to topoi a theorem of Lazard, that every flat module is a direct

limit of free modules of finite type. Mulvey's work [74] on internal descriptions

of rings shows that such formulae should be intuitionistically provable in the

topos language; this seems feasible for the Deligne-Lazard theorem and both

ambitious and fascinating for the dimension theory. Thus the problem is this:

give an internal proof of

Theorem 4.15. (Zarelua [69]) An internal direct limit of soft sheaves, over a

paracompact space, is soft, provided the system is regular (bonding maps are

monos, and the cover (U) of U in the condition for directness is just (U). []

A subsidiary problem is to remove the regularity.

Corollary. 4.14 has an alternating version: this and 4.15 are the main

ingredients of the proof of (4.5). To estimate dim(Y,A), resolve A by the

alternating monadic resolution, calculate the dimension of Y with coefficients

in the terms of the resolution (which are soon zero), and use long exact

cohomology sequences: see our [74] for details, where we regret that it was not

made clear that all resolutions considered are alternating.

Thus the monadic resolution of a proper map is a natural tool for studying

the dimension-raising properties, both for geometrical reasons (4.6), and because

internally it is like a ~ech resolution. The resolution, and its alternating

form, determine spectral sequences; in the case of a finite sheeted regular

covering map between locally contractible paracompaeta, we obtain the Caftan

spectral sequence of the action of a finite group on a space; see Skordev [70],

[71]. One wonders about profinite groups in this context: see Grothendieck [72a]

VIII. For the map induced by a locally finite closed cover, the Leray spectral

238

sequence is obtained (to be distinguished from the Leray s.s. of a proper map):

thus the Zarelua spectral sequence is a generalisation of one of Leray's, as

Grothendieck implicitly suggested in [72a] VIII 8.1.

We conclude by repeating the question: in what sense is the dimension of X

expressible in the language of sheaves on X? We can say, for example, that X is

of dimension zero iff the sheaf ~ of integers is soft; is that so expressible,

and what about higher dimensions? There is indeed the characterisation (3.5) in

terms of exact sequences; is there anything more explicit? What, for example,

is the sheaf analogue of Katetov dimension (see Gillman and Jerison [60] or

Pears [75]) - the analytic dimension defined in terms of generators of certain

subrings of C(X)? Some attempt has been made on this problem by Fourman [75],

but the results are not yet quite well enough related to dimension even on

standard spaces.

239

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Envelopes in the cate$ory of Kakutani-M-spaces

by

J~rgen Flachsmeyer

Introduction

Every compact Hausdorff space X is uniquely determined (up to a

homeomorphism) by its system of all real valued continuous functions

on X regarded as an algebraj resp. a vector lattice, resp. a lattice,

resp. a Banach space etc. If these corresponding algebraic structures

are abstractly characterizable as special structures I) then this gives

a dual equivalence of the category COMP of all compact Hausdorff

spaces to the category of such special algebraic structures. Here we

are interested in the category KAKUMI of all Kakutani-M-spaces with

unit. For the subcategory COMp o of COMP consisting of all compact

Boolean spaces (zero-dimensional spaces) there is a dual equivalence

to the category BOODALG of all Boolean algebras. It may be asked for

concepts and theorems concerning Boolean algebras which can be gene-

ralized for Ml-spaces (the objects in KAKUMI).

Our paper is a contribution to such a program. In 1950 Sikorski

[17] has studied dense subalgebras A of Boolean algebras B From

a categorical point of view dense embeddings in BooLALG are enve-

lopes. A subalgebra A of a Boolean algebra B is called dense iff

Under each non-zero element of B lies a non-zero-element of A.

Sikorski has shown that dense embeddings preserve all existing extre-

mas and every element of B is the supremum (resp, infimum) of ele-

i) For example: In the case of algebras these are the self-adjoint

commutative Banach algebras with unit.

In the case of normed vector lattices these are the Kakutani-M-spaces

with unit.

244

ments from X. Our theorem i, (i),(2), generalizes this fact. With

the help of theorem 2 we conclude that for every embedding of a MI-

space into another exist maximal envelopes. The theorem 3 shows that

the injective envelopes in KAKUM~ are characterized in the same

manner as the injective envelopes in B00LALG.

I. Preliminaries

A vector lattice M over the real field R is called a Banach-lattice

iff M is endowed with a complete norm Jl'II and this norm is mono-

tone in the following sense:

Ixl ~ lyl ~ IIxll S IIylJ for all x,y E M .

A Banach-lattice M is called a Kakutani-M-space iff there holds the

following M-condition:

x ~ O, y ~ 0 ~ lJ x v y lJ = ILxJlvllyll for all x,y s M .

(whereby v means the supremum in M resp. in R ).

A Kakutani-M-space M is said to have a strong order unit u iff

there is a greatest element u in the unit ball (x I 11xll ~ I).

For M ~ (0} this order unit satisfies IIuJl = i and it is unique.

These M-spaces with unit - abbreviated as Ml-spaces - are abstract

descriptions of the concrete Banach-lattices C(X,R) of all continuous

realvalued functions on compact Hausdorff spaces X. Namely, by a

wellknown representation theorem of Xakutani [~2] (see also [15J)

to every Ml-space M corresponds a topological unique representation

space X (compact T 2) such that M is linear lattice isomorphic iso-

metric to C(X,R) and the unit goes to the unit. Let us call this

correspondence M ~ X that sends each Ml-space to the represen-

tation space the Kakutani-functor~ By a Ml-homomorphism is meant a

linear continuous lattice homomorphism of one Ml-space into another

which preserves the unit. Then they must have norm i. KAKUMI denotes

the category of all Ml-spaces as objects and the Ml-homomorphism as

~5

mosphisms. Via the Kakutani-functor the category KAKUMI is dual equi-

valent to the category COMP of all compact Hausdorff-spaces with

continuous maps.

Now let us note the meaning of some categorical notions in KAKUM~.

(For category theory we refer to [11], [14] and [15]).

Isomorphism linear isometric lattice isomorphism preser- ving the unit

monomorphism linear isometric lattice injection preserving the unit (embedding)

epimorphism linear lattice surjection of norm I preser- ving the unit

injective object M each Ml-homomorphism h : A ~ M of a Ml-subspace A of a Ml-space B can be ex- tended to a Ml-homomorphism of B into M.

The internal characterization of the injective Ml-spaces is as follows.

A MI-space M is injective iff M is an order complete MI-spaces, e.g.

every bound set in M has a supremum and an infimum. This follows

from the Gleason characterization [91 of the projective objects in

COMP as the extremally disconnected spaces and the Stone-Nakano

theorem [13], [19], which says that the Ml-space C(X,R) is order

complete iff X is extremally disconnected.

Remark: We remember that the first theorem with respect to injectivi-

ty in some category is Sikorski's theorem on extensions of homomor-

phisms in the category of Boolean algebras [16]. This theorem stated

that every complete Boolean algebra is injective.

2. Envelopes

The categorical concept of an envelope (Semadeni [15]) gives for

KAKUMI the following interpretation: By an envelope of a MI-space A

is meant an embedding A �9 B such that for every homomorphism

B ~ C for which h o E is an embedding h itself is an embedding.

If we regard A as a subspace of B then the inclusion A C~ B

246

is an envelope iff each extension of any Ml-isomorphism A ~ C

to a Ml-homomorphism of B into C is itself a Ml-isomorphism.

To have internal characterizations of envelopes we prove the following:

Proposition i Let be A a MI-subspace of the MI-space B. Then the

following conditions are equivalent:

(1) Every element of B is the supremum of elements of A.

(2) Every element of B is the infimum of elements of A.

(3) Every open internal ] O,x [ in B, x > O, contains elements

of A.

Proof: In vector lattices it holds for subsets S :

x = sup S ~ - x = inf(-S). Thus (I) and (2) are equivalent.

(I) ~ (3). If x > 0 and x = sup{yly E A, 0 ~ y K x}, then there

must be one Yo E A : 0 < Yo S x, otherwise x : O. For such Yo is

0 < ! Yo < x. 2

(3) ~ (1) Let be x E B we may assume x > O. The general case

follows by translation with a suitable X.1(1 the unit in B).

S : = (yty E A, 0 S Y S x}. Let be z an upper bound of S. We will

show that z S x implies z=x. Assume, z < x. Then for x-z > 0

is an element v E A: 0 < v < x-z. Then v+y S z for all y E S and

we get 2v < x-y for all y E S, e.g. 2v + S c S. Then 2v + y S z

for all y E S and we get 3v < x-y for all y E S, e.g. 3v + S c S.

By induction it follows that (*) kv+S c S for every natural number k.

Now a vector lattice is archimedean but this gives a contradiction

to (*)

Theorem i Let be A a MI-subspace of the MI-space B. Then the

following are equivalent:

(I) A c_~ B is an envelope

(2) A is order dense in B in the sense of proposition 1.

(3) Every norm closed non-zero lattice ideal I in B contains

non-zero elements of A.

247

(4) If B = C(X,R) and Y is a proper closed subset of X then A

contains a non-zero funation which vanishes on Y.

s (i) ~ (2) The equivalence of (1) and (2) follows by a

dual argument and a theorem proved in [8]. A ~B gives by the Kaku-

tani-functor a surjection of the representation space of B onto the

representation space of A. Now envelopes in KAKUMI means coenve-

lopes in COMP Coenvelopes are described by irreducible surjections

(see [15, p.446]). But the irreducibility of a surjection in COMP

is equivalent to the order density of the corresponding embedding of

the associated function lattices (see[8]).

(S) ~ (3) The norm closed lattice ideals I with i E I are the

cernels of Ml-homomorphisms

Now I N A = {0} ~=~ ~ i A is a monomorphism.

Thus (i) ~ (3)

(3) ~=~ (4) The proper closed subsets Y of X are in one-one corres-

pondence to the proper norm closed lattice ideals I of C(Z,R).

I = {flf [ C(X,R), f ~ 0 on Y}.

Remarks: i. The notion of an envelope is firstly due to Eckmann and

Schopf [61 for modules. The envelopes are there called essential

extensions and they are defined through the corresponding property (3)

of the preceding theorem.

C ~ 2. By the equivalence of the categories KAKUMI and - ALG

of C - algebras with unit and their homomorphisms the equivalence of

(i),(3) and (4) is only a translation of the corresponding properties

formulated by Gonshor [i0].

In the next statement we give a test for beeing an envelope by further

reduction to norm dense subspaces.

248

Theorem 2. Let be A a MI-subspace of the MI-space B with a given

vector sublattice C of B containing A. The norm closure ~ of C

in B is an envelope of A iff A is dense in C in the sense that

for all x C C

SUPc(]*-- , x] N A) : x : infc([X ,--~ [ N A)

Proof: We have only to prove that from the density of A in C

follows the density with respect to ~. First we show that for every

x E C: sup~ (] ~--, x] N A) = x and then we show it for arbitrary

x C ~. We interprete the situation for B = C(X,R).

i. Let be f E C and h an upper bound in ~ of U(f):=] ~--, f] N A,

with h S f. It should be proved h = f. We have a sequence

h n E C with h n > h uniformly. For a suitable subsequence hnv

+ ~ Now h + i > h and h + i E C Observe holds h S h n . n ~ n

hn^ f--~h ^ f.Therefore we may start with a sequence h n E C such that

h s h n S f and h n > h. Every h n is an upper bound for U(f).

This gives f S h n and therefore h = f.

2. Now consider f s 3. Let be h an upper bound of U(f). We have

a sequence f E C with f ~ f uniformly. It must be shown n

f S h. Assume, there is a point x E X: f(x) > h(x). Take an s > 0

with (*) f(x) - c > h(x). Now for almost all n holds

- ~ belong to C, there- (**) f-s < fn [ $ f" The functions fn

e _ e : sup@ U(f n- 7) Then from fore from the first part follows fn [

(**) we get h ~ fn- 7 for every n. This contradicts (*).

From the preceding theorem we conclude the

Corollary Let be A a MI-subspace of the MI-space B and let be

A c A a c B an increasing family of MI-subspaces of B all envelopes

of A. Then the norm closure U A a is an envelope of A.

Proof: C:= U A a is a vector sublattice of B. A being envelopes

of A implies that for every x E C holds sup U(x) = x = inf O(x),

249

where U(x) = ] +--, x ] Q A and O(x) = [ x~ --~ [ N A.

By theorem 2 C is an envelope of A,

Corollary If A is a MI-subspace of the MI-space B, then a maximal

envelope of A in B exists.

Proof: The preceding corollary with the Zorn lemma gives this result.

Remark In the paper [iO]

C ALG by another approach.

of Gonshor this was shown in the category

3. Injective envelopes

For every object in KAKUMI exists an injective envelope and this is

unique up to an isomorphism. Therefore it is also named the injective

hull. The existence and the unity follows from the corresponding facts

in the category COMP �9 Every object in C0MP has a projective coen-

velope and this is unique (Gleason [9]), It can be constructed for

X C COMP as the natural projection p : pX ) X of the Stone

representation space pX of the Boolean algebra Ro(X) of all regu-

lar open (closed) sets. The injective envelope of M E KAKUMI is then

the Dedekind-MacNeille completion of M by cuts. Analog to the charac-

terization of the injective hull of a Boolean algebra in BooLALG

holds the following:

Theorem 3. Let be A a MI-subspace of the MI-space B, Then the

following are equivalent:

(1) A ~B is an injective envelope

(2) a) B is order complete

b) The injection A ~-~ B is an order complete isomorphism, e,g.

for every family F ~ A for which A-sup F resp. A-inf F

exist holds A-sup F=B - sup F resp.

A-infF = B-infF.

a) There is no proper complete MI-subspace of B containing A.

250

Proof: From the category ~0MP follows by duallzing that every en-

velope A ~-~ B can be realized in the injective hull. But the Dede-

kind-MacNeille completion of arbitrary ordered sets preserves all

existing extrema (see [i]).

Therefore every envelope A ~ B preserves all existing extrema.

Thus from (i) follows 2)a) and b).

c) No proper complete Ml-space C of B contains A, because the

injective hull is the smallest injective extension. Now let be holds

(2). We take a maximal envelope A c C c B from A in B. This

G fulfills a) and b). Then by c) it must be C = B.

Remark: For the category BOOLALG is known the notion of the

m-completion given by Sikorski [18] (m-a given infinite cardinal).

In our paper [8] we have shown that this notion can be generalized

to the category KAKUMI Thus for every object in KAKUMI is

(unique) m-injective envelope. This gives for every object in COMP

a (unique) m-extremally disconnected irreducible preimage.

Our next theorem looks for the special Ml-space C(X,R) over a hyper-

stonian space X. This gives a generalization of a theorem of Gon-

shor ([iO, Theorem 7]) if it will be translated in the category

C -ALG,

Theorem 4 Let be A

a hyperstonian space

(i) The injection A ~--~B is an injective envelope

(2) For every closed set Y c X for which not all hyperdiffuse

measures on X vanish exists a non-zero functions f E A

which vanishes outside from Y.

a Ml-subspace of the Ml-space B = C(x,R) over

X. Then the following are equivalent:

Proof: A Ml-space B is isomorphic to C(X,R) over a hyperstonian

X iff B is the second dual of a Ml-space C (see for hyperstonian

spaces Bixmier [5]). Then B is order complete. For every nonvoid

251

set G in X exists a nontrivial hyperdiffuse measure ~ on X

with supp ~ c G. Therefore a closed set y c X for which not all

hyperdiffuse measures vanish has a nonvoid interior (equivalently:

contains a nonvoid clopen set).

Now under every indicator function of a nonvoid clopen set lies a

positive non-zero function of the subspace A iff A e_~ B is an

envelope (theorem 1,(2). Thus we have (I) ~ (2).

References

[ i ] B.Banaschewski:

[ 2 ] B.Banaschewski: G.Bruns:

[ 3 ] G.Birkhoff:

[ 4 ] R.P.Dilworth:

[ 5 ] J.Dixmier:

[ 6 ] B.Eckmann: A.Schopf:

[ 7 ] C.Faith:

[ 8 ] J.Flachsmeyer:

[ 9 ] A.M.Gleason:

[ 10 ] H.GQnshor:

252

H~llensysteme und Erweiterungen von Quasi-

Ordnungen. Z. math. Logik Grundl. Math. 2,

(1956), 35-46.

Categorical characterization of the

MacNeille completion. Archiv Math. 18,

(1967), 369-377.

Lattice theory 3. ed.Amer.Math. Soc.Colloq.

Publ.(1967) .

The normal completion of the lattice of

continuous functions. Amer. Math. Soc. 68,

(1950), 427-438.

Sur certains espaces consid@r@s par

M.H.Stone. Summa Brasil. Math. 2, (1951),

151-182.

Ober injektive Moduln. Arch. Math. 4,

(1953), 75-78.

Lectures on injective modules and quotient

rings. Lecture Notes in Math., Springer,

(1967).

Dedekind MacNeille extensions of Boolean

algebras and of vector lattices of con-

tinuous functions and their structure

spaces. General Topology and its Appl.

(to appear).

Projective topological spaces, lll.J.Math.

2, (1958), 482-489.

Injective hulls of c algebras. Trans.Amer.

Math. Soc. 131, (1968), 315-322.

[ 11 ] H.Herrlioh: Category theory. Allyn and Bacon Inc. G.E.Strecker: Boston (1973).

[ 1 2 ]

[13]

[14]

[15]

[16]

[17]

I t 8 ]

[19]

S.Kakutani:

H.Nakano:

Z Semadeni:

Z Semadeni:

R Sikorski:

R Sikorski:

R Sikorski:

M H.Stone:

253

References

Concrete representation of abstract (M)-

spaces. Ann. of Math. 42, (1941), 994-1024.

Dber das System aller stetigen Funktionen

auf einem topologischen Raum. Proc. Imp.

Acad. Tokyo~ 17, (1941), 308-310.

Projectivity, injectivity and duality.

Dissertationes Math.35, (1963), 1-47.

Banach spaces of continuous functions I.

PWN Warszawa (1971).

A theorem on extension of homomorphisms.

Ann. Soc.Pol.Math. 21, (1948), 332-335.

Cartesian products of Boolean algebras.

Fund. Math. 37, (1950), 125-136.

Boolean algebras, Berlin-GSttingen-Heidel-

berg.

Boundedness properties in function lattices.

Canad. J. Math. 1, (1949), 176-186.

COMPACTLY GENERATED SPACES AND DUALITY

by Alfred Fr~licher

1. Introduction

Duality theory within the classical topological frame-work (e.g. for topo-

logical vector spaces E) does not give very satisfactoy results. A main reason

for this is the fact that there is no good function-space topology available

(e.g. such that the canonical map of E into its bidual E** is always continuous).

It therefore seams advantageous to use a cartesian closed category instead of

the category of topological spaces. This has been done very successfully by

E. Binz who used the cartesian closed category of limit-spaces and so obtained

many interesting and useful results on duality [ i ] . General considerations

with an arbitrary cartesian closed category have been made by D. Franke for the

case of algebras [4 ] . We shall use the cartesian closed category K of com-

pactly generated spaces.

Only in special cases (e.g. for vector spaces), the dual of an object X of

a category A is an object of the same category. In general, one has contravariant

functors

~-: A ~ B and ~ : B ~ A

and~X is called the dual,~-X the bidual of the object X.

We shall work with categories A and B whose objects are sets with an al-

gebraic structure of some type and a compatible compactly generated topology.

It is essential, that "compatible" means the continuity of the algebraic ope-

rations with respect to the categorical, i.e. the compactly generated product

and not with respec~ to the product topology. Using the cartesian closedness

of K we then get in each case morphisms X ~-X forming a natural transfor-

mation of the identity functor I A of A into ~-.

The categories we are going to examine are examples of so-called enriched

categories, consisting of a category A together with a faithful functor into

a cartesian closed category. We omit a general categorical outline in this

direction, and shall directly examine the following categories A.

255

- the category of real (or complex) compactly generated vector spaces ;

- the category of compactly generated spaces (without additional alge-

braic structure) ;

- the category of compactly generated *-algebras.

Other categories shall be investigated later in the same way ; in particular

the category of compactly generated abelian groups and that of compactly gene-

rated groups.

The particular problems which shall be discussed here can be summarized as

follows. A duality A ~ ~ ~ with a natural transformation IA~ ~o~- being

established one asks for further information on the morphisms X~-X. We

shall give necessary and sufficiant conditions on X in order that X-~X shall

be

(a) a monomorphism (an injective map) ;

(b) an extreme monomorphism (X has the compactly generated topology induced

by the injection into the bidual ~-X) ;

(c) an isomorphism (bijective and bicontinuous).

In case (b), X is called imbeddable ; in case (c) it is called reflexive.

Once the imbeddable resp. reflexive objects are determined, one will ask

for properties of the full subcategories of A formed by these. Some results and

some problems of this sort shall be mentioned.

Similar duality problems have been examined from a slightly different

point of view by H. Buchwalter[ 2 ] : he used in one of the involved categories

A, B a topological, in the other a compactological structure. In this way he

obtained excellent results, and many of the methods he developed in his proofs

have been crucial in order to obtain certain results presented in the following.

256

2. Generalities on compartly generated spaces.

Compactly generated spaces, also called K-spaces, were introduced by

Kelley[ 9 ] and have been studied and used in many articles as for example in

[ 7][10] [11 ] .

If X is an object of the category T2of Hausdorff spaces, we can form

a new Hausdorff space kX by putting on the underlying set of X the topology

induced by the inclusions of the compact subspaces of X. It is easily verified

that X and kX have the same compact subspaces. Therefore kkX = kX ; and for

f : X ~ Y continuous, f : kX ~ kY is also continuous. Hence one has a functor

k : T 2 ~ T 2 satisfying k 2 = k. A compactly generated space is an o~ject X of

T 2 for which kX = X ; the full subcategory of T 2 formed by these will be called

Ko The functor k yields a functor k : T 2 ~ K which is a retraqtion and an ad -

joint of the inclusion functor i : K ~ T 2. It then follows that K is a complete

and cocomplete category. As an adjoint, k commutes with limits ; in particular

the categorical product X~Y of two objects of K is given by X ~ Y = k (X •

where X • Y denotes the topological product.

Most T2-spaces , in particular all sequential and hence all metrizable

spaces, are in K ; a simple counter-example is the weak topology of ordinary

Hilbert space.

We write Cco(X,Y ) for the space of continuous maps X ~ Y with the

compact-open topology andC(X,Y) = k Cco(X,Y). Then we get a bifunctor

e: E~215 E ~E

and for X,Y,Z in K one has the following universal property : a map

f : X ~ ~ (Y,Z) is continuous if and only if the map ~ : X ~ Y ~ Z,

defined by~(x,y) = f(x)(y), is continuous. Hence for each Y, the functor

C(Y,-) is adjoint to the functor -~Y, and this caracterizes C up to an

isomorphism. The existence of a functor ~ with this property is expressed

by saying that K is a certain closed category,

257

3. Duality for compactly generated vector spaces.

This duality was examined in [ 6 ] ; more results and all proofs can be

found there. We consider the category KV of compactly generated vector spaces ;

the objects are vector spaces E with a compactly generated topology such that +

E ~ E --~ E and BE ~ E are continuous, the morphisms are the continuous linear

maps. E could be replaced by C . The dual E* of E is again in KV : it is the

space of continuous linear functions E ~ ~ with the universal compactly gene-

rated function space topology. So we have for this case A = B = 107 and

~-E =~E = E*. Using the universal property of the function space topology one

easily shows that the canonical map e E : E ~ E** is continuous. These morphisms

form a natural transformation of the functor IKV into the functor "bidual".

If E is in KV, the convex neighborhoods of zero in E form a basis for

the filter of zero-neighborhoods of a locally convex topology on the under-

lying vector space ; the so obtained locally convex space is denoted by cE.

For a continuous linear map f : E l ~ E2, also f : cE I ~ cE 2 is continuous. By

means of the theorem of Hahn-Banach one gets immediately :

(3.1) e E : E ~ E** injective <==> cE separated

We restrict now to objects E satisfying cE separated and we denote by

KVs the full subcategory of KV formed by these, With LCV denoting the category

of separated locally convex spaces we have functors

c KVs ~ LCV

k

and one easily shows the following : k is adjoint to c ; ckc = c ; kck = k.

We can state now the main results :

(3.2) e E : E ~ E** a subspace ~ kcE = E

(3.3) e E : E ~ E** a homeomorphism~--> kcE = E and cE complete

We remark that by "a subspace" we mean that E has the compactly genera-

ted topology induced by the injection e E ; this topology has the universal

subspace property within compactly generated spaces and can be obtained by

applying k to the subspace topology.

258

Since E* = k L (E ;~ ) and since kck = k, any dual and in particular �9 CO .

any bidual is kc-lnvarlant. But this property goes over to subspaces. Hence

the condition in (3.2) is necessary. In order to show that it is sufficient,

one makes use of the theorem of bipolars. The proof of (3.3) uses Grothendiecks

caracterization of the completeness of a locally convex space.

By (3.2), the category of imbeddable compactly generated vector spaces

is the full subcategory of KVs formed by the objects invariant under kc. Using

the above functors KVs ~ LCV _k KVs and their properties one sees that this

full subcategory of KVs is isomorphic to the full subcategory LCV = of LCV formed

by the objects invariant under ek, and also that ck yields a retraction and ad-

joint to the inclusion functor LCV ~ ~ LCV. Therefore the completeness and co-

completeness of the category LCV yields the same properties for LCV~ and one has

the first of the following results :

(3.4) a) The imbeddable compactly generated vector spaces form a complete and

cocomplete category ;

b) If E l .... ,En, F are imbeddable, the space ~(EI,...,E n ; F) of multilinear

maps with its universal compactly generated topology is also imbeddable ;

c) One has a bifunctor ~) satisfying

~(E 1 ~ E 2 ; E 3) ~ ~(El ; ~(E 2 ; E3)) ;

d) Products " ~" and coproduets " ~ " satisfy

=" i Z 1 \ i~I i : i(I 1 and I i(I i(I '\

e) If E,F are imbeddable, then also~(E,F).

From the caracterization (3.3) it follows that all Fr~chet spaces and in

particular all Banach spaces are reflexive compactly generated vector spaces.

For a reflexive E, C(E, ~) can be shown to be also reflexive. It is not known

however, whether reflexivity carries over from E, F to ~ (E ; F) ; if it would,

then the same would follow easily for C (E ; F). As to the question of com-

pletness and cocompleteness of the category of reflexive compactly generated

vector spaces, there is no problem with products and coproducts : (3.4 d)

shows that they exist and are the same as in the category of imbeddable spaces.

But it is not known either, whether kernels (and hence equalizers) exist. To

summarize : one has not obtained for the reflexive objects as good categorical

properties as for the imbeddable ones.

259

4. Reflexive vector spaces and calculus.

Compactly generated vector spaces have been very successfully used by

U. Seip for calculus [I0 ] . The obtained theory is not only more general than

classical calculus for Banach spaces, but gives much better results, in parti-

cular with respect to functions spaces. In fact, for admissible spaces E,F,OCE

open and o ~ k ~ ~ ~ the function space ck(o , F) formed by the maps O ~ F of k

class C k and equiped with a natural compactly generated C -topology, is again

admissible. It turned out that a convenient notion of "admissible" is the fol-

lowing : E = kcE and cE sequentially complete. Therefore the admissible spaces

are all imbeddable, and in fact they are very close to the reflexive.ones. The

reason for imposing on the imbeddable spaces E the additional condition "cE

sequentially complete" is the fact that by this condition one gets a full sub-

category having the same excellent properties as the category of imbeddable

ones (cf. (3.4)). The crucial difference is that for a sequential~ complete

locally convex space L, the space ckL is also sequentially complete, while L

complete does not imply ckL complete. Otherwise one would have worked with the

reflexive spaces as the admissible ones.

One used to say that for a differentiation theory within a given class of

topological (or similar) vector spaces, two somehow artificial choices must be

made : the remainder condition and the topology (or respective structure) On

the spaces L(E ;F), From a new point of view, both become, as we shall indicate~

very natural within Seip's calculus. Usually the central role in calculus is

attributed to the operator f ~ f', where f : E ~ F and f' : E ~ L(E ;F).

However one could as well work with the operator f ~ Tf, where Tf : TE ~ TF

is defined by TE = E ~ E and Tf(x,h) = (f(x), f'(x)(h)). The operator T has at

least the big advantage of being functorial ; and instead of involving L(E;F)

which later requires an additional structure (topology), it only brings in the

products E~E, F~F which have their natural categorically determined structure.

It therefore seems more natural to define, for a differentiable f, "continuous-

ly differentiable" resp. "twice diff~rentiable" by imposing "Tf continuous"

resp. "Tf differentiable". In Seip's theory however, these conditions become

equivalent to "f' continuous" resp. " f' differentiable", provided one uses

on L(E ;F) the universal compactly generated topology. Furthermore, in analogy

with results of H. Keller [ 8 ~ one can show : in order that f : O ~ F

260

(where 0C E open) is C 1 (i.e. continuously differentiable) it is sufficient

(and of course also necessary) that there exists a continuous map f' : O~C(E;F)

such that the "weak" G~teau condition

lim f(a + Ix) - f(a) = f'(a)(x)

k ~ o

is satisfied for a ~ O, h ~ E ; the "weak" here shall indicate that the limit is

taken with respect to the weak topology of F. The proof uses the modern form of the

mean value theorem. The result shows, that for the definition of "continuously

differentiable on an open set" everything becomes natural : the structure on

L(E ; F)as already indicated, and the remainder condition also, since the above

G~teau condition is the weakest reasonable one.

If F is reflexive, continuous differentiability of a map can be caracterized

by means of its behaviour with respect to the cl-functions ; using the preceding

result one can show :

(4,1) f :O -~ F is C 1 if and only if for each ~ ~ ~(F, ~ ), f* @~CI(O ,R)

and f* : CI(F, R) -~ i(~,~) is continuous.

1 We recall that ~ (O,R) has the natural compactly generated topology which

takes care of the functions and their derivatives ; it is the coarsest making

continuous the map ~I(O,R) ~ TO-~ TF defined by (f,x,h,) ~-~ Tf (x,h). An

analogous caracterization of ck-maps is obtained by induction. The interesting

part of (4.1) is the sufficiency of the condition. The only elements ~in

~I(F, ~) which are available for the proof are the elements of F*. The remark

that in the G~teau condition the weak topology can be used becomes crucial.

Since reflexivity would also have other great advantages in analysis, it

seems very desirable to find a category of reflexive vector spaces which is big

enough and has good categorical properties (possibly those of (3.4)). There

might be a chance to get such a category if one starts with an other cartesian

closed category of topological spaces instead of K(cf. "12 ).

261

5. Duality for compactly generated spaces.

For details and proofs we refer to [ 5 ] . We take now for A the category

of compactly generated spaces without additional algebraic structure, and for

the category KA of compactly generated unitary real algebras (analogous re-

sults hold if one replaces R by C ). One has contravariant functors

~-: K -~ KA and ~: KA -+ K

where, for X in K and A in KA,~-X is the algebra e (X,R) of continuous functions

X ~ R and ~A is the set of continuous unitary algebra homomorphisms A ~ R,

~-X and ~A are equipped witktheir universal compactly generated function space

topologies. The canonical map e X : X ~JC~-X, also called the Dirac transforma-

tion of X, is always continuous. In order to study further properties of ex,

some considerations on uniform and on completely regular spaces are useful.

The classical functor t : Unif ~ Top which associates to a uniform space

the underlying topological space has an coadjoint u : Top ~ Unif ; for T in

Top, uT is the underlying set of T with the finest uniforme structure such that

each uniform neighborhood of the diagonal ~ T is a neighborhood of AT, and uf = f

for., a morphism f, since f : T I ~ T 2 continuous implies f : uT I ~ uT 2 uniformly

continuous. A topological space T is called topologically complete if uT is

complete. It is known (see e.g. Problem L( d ) Chap. 6 in [ 9 ]) that all para-

compact and in particular all metrizable spaces are topologically complete. One

easily verifies utu = u and tut = t. The identity map T ~ ruT is always conti-

nuous, but the topology of tuT can be strictly coarserthan that of T.

We denote by Ks the full subcategory of K whose objects X satisfy tuX

separated. Putting tuX = rX and remarking that rX is completely regular one has

functors r

Ks > CR

k where CR denotes the category of separated completely regular spaces. For

these functors one has : k is adjoint to r ; rkr = r ; krk = k.

The following results are formally completely analogous to those for

compactly generated vector spaces (cf. (3.1) to (3.3)) ; the proofs however

are quite different.

262

(5.1) e X : X ~ ~-X injective ~=> rX separated.

Supposing in the following this condition to be satisfied, one has furthermore :

(5.2) e X : X ~ JC~X a subspace <==> krX = X ;

(5.3) e X : ~ ~ ~-X a homeomorphism~ krX = X and X topologically complete.

Of course, since utu = u, the condition "X topologically complete" is

equivalent to the condition that the associated completely regular space rX is

topologically complete.

For the categorical properties of the imbeddable objects X one shows, as

in the analogous situation for vector spaces, that they form a category iso-

morphic to that of the completely regular spaces T satisfying rkT = T, and this

last category is a reflective subcategory of the complete and cocomplete category

C__RR. This yields the first of the following results :

(5.4) a) The category of imbeddable compactly generated spaces is complete and

cocomplete.

b) For Y imbeddable, C(X,Y) is also imbeddable, and hence the category

of imbeddahle compactly generated spaces is cartesian closed by means

of the restriction of the bifunctor C.

(5.3) shows, that the conditions for being reflexive are not very restrictive ;

e.g. all compactly generated paracompact spaces, in particular all metri-

zable spaces, are reflexive. It is not known however whether the category

of reflexive spaces has as good categorical properties as that of the

imbeddable ones.

263

6. Du__ality for compactly generated *-algebra ~.

Duality for this case is being studied by D. Favrot [ 3 ] . We give a sum-

mary of the results be obtained so far. The category A is now the category KA*

of compactly generated *~algebras, whose objects are the unitary complex alge-

bras with a compatible compactly generated topology and a continuous involution

x ~_~ x*. To such an algebra A one associates a locally multiplicatively convex

*-algebra cA by putting on the underlying algebra the topology determined by

all continuous seml-norms p : A ~ R satisfying p(x.y) < p(x).p(y) and

p(x.x*) = p2(x) (and hence p(x*) = p(x)). Conversely, to a locally multiplica-

tively convex separated *-algebras B one associates, by refining its topology

by means of the functor k, a compactly generated *-algebra kB. One has contra-

variant functors

: KA* ~ K and ~- : K ~ KA*

where, for A in KA__* and X in K, ~A is the space of all continuous algebra-

homomorphisms h : A ~ C satisfying h(a*) = h(a), and~-X is the *-algebra of

all continuous functions X ~C ,~A and ~X being equipped with their universal

compactly generated function space topologies.

For all A in KA* the canonical map e A : A ~ ~-~A, also called the Gelfand

transformation of A, is continuous.

(6.1) e A : A ~ ~-~A injective ~ A separated.

Supposing this condition satisfied, we have furthermore :

(6.2) e~ A ~-~A an extreme monomorphism ~===>kcA = A

(6.3) e~ A -~-~A an isomorphism<==> kcA = A and cA complete,

This result considerably improves the classical theorem of Gelfand-

Nalmark concerning Banach*-algebras. The proof uses the theory of Gelfand and

the methods used by Buchwalter [2 } in his investigations of the Gelfand trans-

formation by means of compactologies.

264

BIBLIOGRAPHY

[i] Binz E.- Continuous Convergence on C(X). - Lecture Notes in Mathematics 469 - Springer, Berlin-Heidelberg - New York 1975.

[2] Buchwalter H.- Topologie et compactologies - Publ. Dept. Math. Lyon - t. 6-2 - 1-74 (1969).

[3] Favrot D.- Thesis - University of Geneva (in preparation).

[ 4 ] Franke D.- Funktionenealgebren in kartesisch abgeschlossen Kategorien - Dissertation - Freie Universit~t - Berlin (1975).

[5] Fr~licher A.- Sur la transformation de Dirac d'un espace ~ g~n~ration compacte - Publ. Dept. Math. Lyon t. i0-2~79-iOO (1973).

[6] Fr~licher A.- Jarchow W.- Zur Dualit~tstheorie kompakt erzeugter und lokalkonvexer Vektorra~me - Comm Math. Helv. Vol. 47 - 289-310 (1972).

[?]

[8]

Gabriel ~ and Zisman M.- Calculus of fractions and homotopy theory - Ergebn. der Math 35 - Springer~ Berlin-Heidelberg - New York 1967.

Keller HH.- Differential Calculus in Locally Convex Spaces - Lecture Nores in Mathematics 417 - Springer - Berlin-Heidelberg New York 1974.

[9] Kelley J

[iO] Seip U.-

[ Ii ] Steenrod

12

.L.- General Topology - Van Nostrand - New York 1955.

Kompakt erzeugte Vektorra~me und Analysis - Lecture Notes in Mathematics 273 - Springer - Berlin-Heidelberg-New York 1972.

N. - A convenient category of topological spaces - Mich. Math. Journ. 14 - 133-152 - (1967).

Wyler O.- Convenient categories for topology.

General topology Vol. 3, 225-242 (1973).

Some Topological Theorems which Fail to be True

by

Horst Herrlich

Consider the following statements:

(I) Products of paracompact topological spaces are paracompact.

(2) Products of compaot Hausdorff spaces with normal topological

spaces are normal.

(3) Subspaces of paracompact (normal) topological spaces are para-

compact (normal).

(4) dim (X • Y) < dim X + dim Y for non-empty paracompact topological m

spaces X and Y .

(5) dim X = dim Y for dense subspaces X of regular topological spaces

Y .

(6) dim X < dim Y for subspaces X of topological spaces Y . m

(7) Continuous maps from dense subspaces of topological spaces into

regular topological spaces have continuous extensions to the whole

space,

(8) X Y• Z (X Y) for topological spaces X,Y and Z.

(9) Products of quotient maps between topological spaces are quotiene

maps.

Although we would like the above statements to be true, we know that

none of them are--provided such operations as the formation of products,

subspaces and function spaces are performed, as usual, in the category

T• of topological spaces and continuous maps. However, there exist

settings--more appropriate it would seem--in which the above state-

ments are valid. The category Top can be decently embedded in larger,

more convenient categories such that, when the mentioned operations

are performed in the larger category, the above statements are not

only true but, in fact, special cases of more general theorems.

266

Especially simple and convenient settings for theorems such as the

above are the category S-Near of Semi-nearness spaces and nearness

preserving maps, introduced under the name Q-Near in [27] , resp. the

isomorphic category of merotopic spaces, introduced by M. Kat~tov [331

in a very important but hardly noticed paper already ten years ago,

and various full subcategories of S-Near. Especially, statements (I)-

(7) are true in the bireflective full subcategory Near of S-Near,

whose objects are all nearness spaces, and statements (8)-(9) are true

in the bicoreflective full subcategory Grill of S-Near, whose objects

are all grill-determined semi-nearness spaces.

Some of our results are new, others are just reinterpretations of

known facts.

I. Structures Induced by Topologies

With any topological space X = (X,cl) there can be associated

various structures, e.g.

(I) a nearness structure, consisting of all collections A of subsets

of X with ~cl~ ~

(2) a covering structure, consisting of all covers of X, which are

refined by some open cover of X

(3) a convergence structure, consisting of all convergent filters in X.

In case, X is a topological R -space, i.e. if x e cl{y} <=> y ecl{x} -- o

each of the above structures determines the topology of X and hence all

the other structures. From now on, topological space means topological

Ro-Space ~ and Top denotes the category of topological (Ro-)Spaces and

continuous maps. (For a slightly more complicated setting covering the

non-R -case see D. Harris [24] and K. Morita [41].) o

There are natural constructions, which--when applied to topological

spaces--automatically yield r~ore general types of structures, e.g.

(I) if ~ is a topological space, and S is a subset of Xo then the

267

(2)

(3)

nearness structure, consisting of all collections ~ of subsets of

S with ~Clx~ # ~, is in general not topological

if.~l and ~2 are topological spaces, then the covering structure

on x I x X2 , consisting of all covers, which can be refined by some

cover of the form {A I x A2 I A i e ~i } , where the ~'l are open

covers of ~i' is in general not topological

if ~ and ~ are topological spaces, C(~,~) is the set of all con-

tinuous maps from ~ into ~, and e: X x C(~,~)~ Y is the evalua-

tion map, defined by e(x, f) = f(x), then the convergence struc-

ture on C(~, ~), consisting of those filters Fwhich have the

property that for every convergent filter ~in X the filter gene-

rated by e(G,~) = {e(G • F) IG e():, F e ~converges in ~, is in

general not topological.

In order to find supercategories of Top which are closed under

the above constructions, we need to concentrate only on one of the

three types of structures described above: nearness structures, cover-

ing structures and convergence structures resp., since--as has been

shown in [27J--they are all equivalent, i.e. just different facets of

the same type of structure.

2. (Se_mi-) Nearness Spaces

For any set X, denote by PX the set of all subsets of X. A

semi-nearness structure on X is a collection ~ of subsets of PX, satis-

fying the following axioms:

(N1) If ~CPX, ~ corefines~Z~, and Z~e~ then ~ e

(N2) If /~CPX and /~ ~ @ then /~ e

(N3) If /~CPX, ~CPX, and {AU BIA s ~ , B s ~ } s [ then R e ~ or ~

(N4) ~ e ~ and {~} ~ ~.

A nearness structure on X is a semi-nearness structure ~ on X,

i~ corefines /~iff for each A~ there exists B a Z3 with B ~ A0

268

satisfying the additional axioml

(N5) If ~ C PX and {cI~AIA e ~ } e ~ then ~ e ~ , where

cI~A ={x eXI{A,{x}} c ~} .

A pair (X, ~) is called a (semi-)nearness space provided ~is a

(semi-)nearness structure on X. A map f: (X, ~) § (Y, q) between

semi-nearness spaces is called nearness preserving provided ~e ~ im-

plies {fAIA ~} e q The category of all semi-nearness spaces and

nearness preserving maps is denoted by S-Near, its full subcategory,

consisting of all nearness spaces, is denoted by Near.

The categories S-Near and Near are known to be well-behaved cate-

gories ([25], ~7]). Here we need only the following facts:

(I) Near has products.

(2) If (X, ~) is a nearness space and S is a subset of X, then

S = {~C psl~ e ~} is a nearness structure on S and (S, {S ) is

called the nearness-subspace of (X, ~) determined by S.

3. T_opological Spaces and Nearness Spaces

If ~ = (X, cl) is a topological space, then

~= {~CPXI~{clAIA e~} # #} is a nearness structure on X. As is

easily seen this correspondence is functorial and, in fact, gives rise

to a full embedding of Top into Near . A nearness space (X, ~) be-

longs to the image of the above embedding iff it satisfies the follow-

ing axioml

(N6) If ~ ~ ~ then ~{cI~AIA s # ~ .

Nearness spaces, satisfying condition (N6), will be called topological

nearness spaces. We may identify each topological space with its as-

sociated topological nearness space, and from now on we will call such

spaces topological spaces. Also we will identify Top with the full

subcategory T-Near of Near whose objects are the topological (nearness)

spaces. Vice versa, we may associate with any nearness space X = (X,~)

269

a topological space T~ = (X, cl) = (X, ~t ) defined by cl = cl{ , resp.,

~t = {~CPXI/~{cI~ AIA e~} # ~ , which we may call the underlying

topological space (= the topolo@ical coreflection) of ~. The category

To~ is bicoreflective in Near and the bicoreflection has just been

described.

Consequently colimits in To~ are formed in the same way as in

Near, but limits are formed differently: a limit in Top is obtained

by forming it first in Near and then passing over to its underlying

topological space. Especially:

(I) If (~i)iei is a family of topological spaces, and ~ is their pro-

duct in Nea_~r, then the underlying topological space T~ of ~ is the

product of the family (~i)is in

(2) If X is a topological space, S is a subset of X, and S is the

nearness subspace of X determined by S, then the underlying topo-

logical space T~ of ~ is the topological subspace of ~ determined

by S.

As is well known and as the introductory examples demonstrate,

products and subspaces are ill behaved, when obtained in Top. The fol-

lowing results indicate that they are much better behaved when per-

formed in Nea__~r. It seems that in passing from a nearness space to its

underlying topological space too much valuable information gets lost.

4. Paracompact Spaces

We will use the terms paracompact and fully normal synonymously,

i.e. a topological space ~ is called paracompact provided every open

cover of ~ is star-refined by some open cover of ~. In order to de-

fine paracompactness for nearness spaces, we need a suitable equi-

valent for open covers in a nearness space. Let ~ = (X, ~) be a near-

ness space, then ~cpx will be called a ~-cover iff {X - AIA e ~}g ~.

Every X-cover is a cover of X. Moreover every ~cover is refined by

some open cover of X (with resp. to the underlying topology), and the

270

X-covers of a topological space are characterized by this condition.

A nearness space ~ is called ~aracompact provided it satisfies the

following condition~

(N7) Every X-cover is star-refined by some X-cover.

A topological space is paracompact in the nearness-sense iff it

is paracompact in the topological sense.

For any paracompact nearness space ~, the set of all ~-covers

forms a uniform structure on X in the sense of J.W. Tukey. This cor-

respondence is easily seen to be functorial, and, in fact, gives rise

to an isomorphism between the category of paracompact nearness spaces

and nearness preserving maps and the category of uniform spaces and

uniformly continuous maps. We may identify each paracompact nearness

space with its associated uniform space. Then the paracompact topo-

logical spaces are precisely those nearness spaces which are simulta-

neously topological and uniform. No wonder that many topologists have

found them so attractive. On the other hand they are badly behaved

with respect to any of the standard constructions. No wonder, topo-

logical spaces are bicoreflective in ~ear and paracompact (=uniform)

nearness spaces are bireflective in Near (as proved in [25]), and--

formation of the intersection of a bicoreflective subcategory with a

bireflective subcategory usually ruins all the constructions. But,

since paracompact nearness spaces form a bireflective subcategory of

Near, any product of paracompact nearness spaces (especially any pro-

duct of paracompact topological spaces), taken in Near, is again para-

compact (but generally no longer topological), and any subspace of a

paracompact nearness space (especially of a paracompact topological

space), taken in Near, is again paracompact (but generally no longer

topological). This proves that assertion (I) and the paracompact part

of assertion (3) are true, if interpreted in Near.

The reason why the nearness product of two paracompact topological

271

spaces is again paracompact, but the topological product generally

fails to be so, may be easier understood if we observe that it is

hardly possible to describe the open covers of a product in any decent

way by means of the open covers of the factors, whereas there is a

very simple description of the (~ ~ [)-covers by means of the ~-covers

and the ~-covers: ~C-P(X • Y) is a (~ x ~)-cover iff there exist a ~-

cover ~ and a ~-cover C such that{B • CIB e Z3,C s ~ } refines ~.

5. Extensions of Maps. Complete and Regular Spaces

Every uniformly continuous map from a dense subspace of a uniform

space into a complete uniform space has a uniformly continuous exten-

sion to the whole space. This well-known theorem of A. Weil seems to

have no direct topological counterpart. Translated into the nearness

language we obtain a straightforward "topological" application: Every

nearness preserving map from a dense nearness-subspace of a paracompact

topological space into a paracompact topological space has a continu-

ous extension to the whole space.

A. Weil's theorem has a natural generalization, whose topological

application is essentially assertion (7). It can be found in [26],

but was observed much earlier in a different context by K. Morita [41].

If ~ = (X, ~) is a nearness space, then ~CPX is called a ~-

cluster provided ~# ~ and ~ is a maximal element of ~, ordered by

inclusion. A nearness space ~ is called complete provided it satisfies

the following condition:

(N8) Every X-cluster has an adherence point.

If X = (X, ~) is a nearness space, A C X and B c--X, then

A < X B iff {A, X - B} / 6~ A nearness space X = (X, 6) is called

xe~ular provided it satisfies the following axiom:

(N9) Iff ~ c PX and {B ~ XlA < X B for some A s ~} ~ ~ then ~ c ~ .

A topological space is regular in the nearness sense iff it is

272

regular in the topological sense. Every paracompact nearness space

is regular. The following is the announced generalization of A. Weil's

result~ Every nearness preserving map from a dense nearness-subspace

of a nearness space into a complete regular nearness space has a near-

ness preserving extension to the whole space. As topological applica-

tion we obtain the following interpretation of assertion (7): Every

nearness preserving map from a dense nearness-subspace of a topological

space into a regular topological space has a continuous extension to

the whole space.

6, Normal

A topological space X is called normal provided each finite open

cover of ~ is star-refined by some finite open cover of ~. This can

be most easily expressed in the realm of nearness space by introducing

the concept of a contigual nearness space first. A nearness space (X,6)

is called continual provided it satisfies the following condition:

(Nlo) If ~ PX, and every finite subset of ~ belongs to 6, then

belongs to 6.

A topological space is contigual iff it is compact. A uniform

space is contigual iff it is totally bounded (=precompact) . Contigual

nearness spaces form a bireflective subcategory of Near. For any near-

ness space ~ = (X, ~) its continual reflection is defined by C~=(X,~ C)

with ~ e ~C iff each finite subset of /~ belongs to ~. A subset /~

of PX is a C~-cover iff it is refined by some finite ~-cover. Now,

normality of a nearness space X could be defined by requiring the con-

tigual reflection CX of X to be paracompact or, equivalently (see [25]),

to be regular. Since this property does not imply regularity of X,

we just add it, Hence a nearness space X is called normal provided it

satisfies the following condition:

(N11) The space X and its contigual reflection CX are regular.

Since the contigual reflector preserves paracompactness, every

273

paracompact space is normal. By definition, every normal space is

regular. A topological space is normal as a nearness space iff it is

normal as a topological space.

In order to study products of normal spaces we have to investi-

gate the behaviour of the contigual reflector C with respect to pro-

ducts. Unfortunately C does not preserve products. E.g. if ~=(X,~) is

a paracompact nearness space, which is not contigual, then

C(~ • ~) # C~ x C~. To see this, let ~ be a ~-cover, which cannot be

refined by a finite ~-cover, and let ~ be a ~-cover, which star-refines

. Then {(X • X)\ &X, U{B • BIB e~}} is a C(~ x !)-cover, but not a

(C~ • C~)-cover. But we have the following result:

Theorem: Let X and Y be nearness spaces.

c(x x y) = X x CY

If X is contigual, then

Proo____~f: Obviously, every (~ x C~)-cover is a C(~ x ~)-cover. Since any

nearness space ~ is uniquely determined by the set of all ~-covers, it

remains to show the converse. If A is a C(~ x ~)-cover, then ~ is

refined by some finite (~ x [)-cover ~. Hence there exists a finite

X-cover C and a Y-cover D such that {C x DIC s C, D e D } refines ~. m

For each B s ~and each C s C define E(B, C) = {y e YI C • {y} c B}.

Then, for each C e C , F C = {E(B, C) IB e~} is refined by ~ and finite,

hence a CY-cover._ Consequently, by axiom (N3), ~- = A{~--cIC e C }, de-

fined by ~-= {~ ~ i~e H FC}, is a C~-cover. Since CEC

{C x FIC e C ,F e ~--} refines Z~, and hence ~ , we conclude that

is a (~ x C~)-cover.

Let us call a nearness space ~roximal provided it is contigual

and regular. Observing that regularity is preserved under products

([2 5]), we obtain as an immediate corollary of the above theorem that

products of proximal nearness spaces with normal nearness spaces are

normal. An application to topological nearness spaces yields assertion

(2): Products of compact Hausdorff spaces with normal topological

274

spaces are normal. Observing that nearness-subspaces of regular spaces

are regular and that whenever ~ is a nearness-subspace of ~, then C~

is a nearness-subspace of C~, we obtain the missing half of assertion

(3): nearness-subspaces of normal topological spaces are normal.

7, Dimension Theory for Nearness Spaces

Topological dimension theory suffers from the fact that no di-

mension function has yet been found that behaves decently for a rea-

sonable class of topological spaces which is essentially bigger than

the class of metrizable topological spaces. If we consider e.g. the

Lebesgue covering dimension dim, which coincides with the large in-

ductive dimension Ind for metrizable spaces, we observe deficiencies

such as the following:

(I) As the Tychonoff plane shows, there exist zero dimensional com-

pact Hausdorff spaces having subspaces which are not normal and

hence of positive dimension, and as the example of Dowker [16]

shows there exist zero dimensional spaces having normal subspaces

with positive dimension.

(2) As an example of E, Michael [37] shows, there exist a zero dimen-

sional metrizable space and a zero dimensional paracompact space

whose product is not normal and hence of positive dimension.

(3) As a theorem of Noble [46] shows, for any zero dimensional non-

compact space, there exists some power which is not normal and

hence of positive dimension.

Because of the above deficiencies, several topologists have tried

with limited success to modify the definition of the covering dimen-

sion slightly, e.g. by considering cozero-set covers (e.g.L.Gillman

and M. Jerison [22] ) or normal open covers (e.g.M. Katetov [3~ , Yu.

M. Smirnov [53], K. Morita [43j) . But, as the above examples demonst-

rate clearly, it is not so much the dimension function, which is at

fault; it is the construction of subspaces and products in Top, in

275

other wor~s| it is the category ~ itself. This, in fact, has been

observed before, e.g, by Nagami ~ . The crucial misunderstanding

seems to be that topologists have usually tried to find a solution

i~sid__~e Top, whereas it seems that a solution can only be found outside

Top. In fact, there seems to be a rather decent dimension theory for

nearness spaces which extends the dimension theory for proximity spaces,

due to Yu. M. Smirnov 5~ , and the dimension theory for uniform spaces,

due to J. R. Isbell ~9-31~, which, indeed, are both highly satisfactory.

Definitions A nearness space X has dimension at most n provided every

~-cover can be refined by a ~-cover of order at most n + I . dim X is

the smallest natural number n such that X has dimension at most n, m

provided such a number exists, otherwise dim X = ~.

Except for the empty space, the above nearness-dimension coin-

cides

(I) for proximal nearness spaces (=proximity spaces) with the

~-dimension of Yu. M. Smirnov [53]

(2) for paracompact nearness spaces (=uniform spaces) with the large

dimension of J. R. Isbell [29]~

(3) for paracompact topological spaces with the Lebesgue covering

dimension (see C.H. Dowker [15], K. Morita [4o], J.R. Isbell [3o]

and B. A, Pasynkov [48]).

Before we present results, we like to mention some natural modi-

fications of the dimension function. If A is a (co-)reflective sub-

category of Near with (co) reflector Az Near-~ A then the A-dimension

of a nearness space ~ may be defined by dimA~ = dim A~ 0 Obviously,

for any space ~ in ~ we have dim A X = dim X . We mention especially

the contigual dimension dim C ~ and the proximal dimension dimp ~ �9

For a nearness space X we havez w

(I) if ~ is topological, then dim C ~ is the Lebesgue covering dimen-

sion of X

276

(2) if ~ is topological, then dimp ~ is the modified covering dimen-

sion of ~, introduced by M. Kat~tov [32] and independently by

Yu. M. Smirnov [53]

(3) if ~ is paracompact (=uniform), then dimc~ = dimp ~ is the uniform

dimension 8dX of X , introduced by Yu. M. Smirnov [53].

Moreover, dimc~ ~ dim ~ , for every nearness space ~ This follows

immediately from the fact that for every X-cover ~ of order at most

n , which refines a finite X-cover ~, there exists a finite X-cover

O of order at most n, such that ~ refines ~and C refines ~ . For

paracompact topological spaces ~ we have dimc~ = dim ~ . For para-

compact nearness spaces ~ with dim ~ < ~ we have dim ~ = dim cX , but

there exist paracompact nearness spaces X with dim X = O and dim X =

(J, R. Isbell [~o~ ). P. Alexandroff's long line is an example of a

normal topological space ~ with dimc~ = O and dim ~ = ~ .

,Proposition! If ~ is a nearness-subspace of [ , then dim ~ ~ dim [ .

Moreover, if X is a nearness-subspace of Y , then CX is a near-

ness-subspace of C~ , which implies dimc~ ~ dim C ~ . The topolo-

gical application of this result is our assertion (6). In the fol-

lowing, we will restrict our investigations to the dimension function

dim ,

There is a partial converse to the above proposition, asserting

that for sufficiently big nearness-subspaces ~ of [ we have

dim X = dim Y . For this we need some preparation. If (Y, n) is a

nearness space, and A ~ X ~Y , let OPxA = inty(A U (Y \ X)) denote m

the largest open subset B of [ with B /A X = intxA .

Definitionl Let X = (X, ~) and Y = (Y, q) be nearness spaces, and let m

X be a subset of Y0 Then the injection X ~ Y is called a strict

extension provided the following equivalent conditions are satisfied.

(I) ~ C py belongs to ~ iff {B C XlA ~ cl B for some A e ~ } q

277

belongs to

(2) ~ C PY is a ~-cover iff {BC X[OPxBC A for some A s ~ } is a

X-cover.

Every strict extension is a dense nearness-embedding and the con-

verse is true for regular nearness spaces (H~ Bentley and H. Herrlich

[6] ) , If ~ ~ is a strict extension and ~ is a ~-open ~-cover

then ~ = {OPxA]A ~ ~ } is a ~-cover, and ~ and ~ have the same

order. Therefore:

~osition (I) If ~ ~ [ is a strict extension, then dim ~ = dim

(2) If X is a dense nearness-subspace of a regular nearness space [,

then dim ~ = dim ~.

For every nearness space X there exists a complete nearness space

X ~ and a strict extension X ~+ X ~ , called the completion of X (see

[25] ), Therefore:

,P_ro~osition: If ~ ~ is the completion of ~, then dim ~ = dim ~* .

This proposition has a number of obvious corollaries:

(I) For a topological space ~, (C~) ~ is its Wallman--compactification

~X_ , hence dim ~X_ = dimcX_ .

(2) for a topological space X , (PX) x is its Cech-Stone compactifi- m

cation 8X Z , hence dim B~ = dimp~ 0

(3) for a uniform space ~, X ~ is its completion y~ , hence

dim yX = dim X . m

(4) for a uniform space ~ , (C~) ~ is its Samuel-compactification B~,

hence dim 8~ = dim C~ 0

(5) For a proximity space X, X ~ is its Smirnov compactification u~ ,

hence dim uX = dim X 0

Since the class of paracompact nearness spaces is closed under the

formation of products in Near the following proposition is just a re-

statement of a result stated as exercise in the book of J.R. Isbell

( [31] , p. 94):

278

~ I If ~ and ~ are paracompact nearness spaces, then

dim (X x y) < dim X + dim Y .

The following result is trivial:

Pro~ositionl Products of arbitrary families of zero dimensional

nearness spaces are zero dimensional.

Much more can be said and even more can be asked, but we leave

the matter here.

8. F_unction Spaces

For any pair (~, ~) of topological spaces, the set C(~,[) of all

continuous maps from X to Y can be supplied in several ways with a m

topological structure and thus made into a topological space, denoted

by Y~ . Among these structures, the compact-open topology is rather u

decently behaved, provided X is a locally compact Hausdorff space.

But for arbitrary topological spaces, none of the topological struc-

tures on function sets C(~, [) is sufficiently well behaved, e.g.

such that, for any triple (~, ~, ~) of topological spaces, the spaces

X5 x~ and (~5)s are naturally isomorphic (see R.F~ [3]). In

other words, the category Top fails to be cartesian closed. Because

of this deficiency, several authors have constructed better behaved

substitutes for Top, which are usually either subcategories or super-

categories of Top. Among the cartesian closed subcategories of Top,

the one which has been used most often especially in homotopy theory

and topological algebra is the coreflective hull of all compact

Hausdorff spaces in the category Haus of Hausdorff topological spaces,

i.e. the full subcategory of Haus, whose objects are the quotients of

locally compact Hausdorff spaces (see e.g.R. Brown [11], E. J. Dubuc

and H. Porta [17] , W. F. LaMartin [36] , E. C. Nummela [47], N. E.

Steenrod [553) resp. the slightly larger category of compactly gene-

rated spaces which is the coreflective hull of all compact Hausdorff

279

spaces in Top (R. M. Vogt [5 6] and O. Wyler [58]) . The main disadvan-

tage of this category is that it is rather awkward to describe it (e.g.

no convenient axiomatic description is known) and to prove even the

basic theorems. Among the cartesian closed supercategories of Top,

the category of quasi-topological spaces, introduced by E. Spanier

[541, is unnecessarily big, e.g. the quasitopolgies on a fixed set,

in general form a proper class. Also, no axiomatic description of

quasi-topological spaces is known.

Fortunately there exist cartesian closed supercategories of Top,

which not only can be described axiomatically in a very elegant man-

ner, but also can be obtained from Top in a rather natural way. First,

any topological space X is completely described by the set of all m

convergent filters in ~ (in other words: by its convergence structure),

and a map f: ~ § ~ between topological spaces is continuous iff it

preserves convergence, i.e. if for any filter F, which converges in ~,

the filter generated by fF converges in [.

Second, for any pair (~, ~) of topological spaces, the set C(X, [)

can be supplied in at least two rather natural ways with a "weakest

convergence structure", such that the evaluation map

e, ~ x C(~, ~) § [, defined by e(x,f) = f(x) preserves convergence.

Just call a filter ~- on C(~, [)

(a) convergent provided, for any convergent filter ~in ~,

the filter generated by e(~x ~--) = {e[G x F] G e~, F s ~--}

converges in Y

(b) Convergent to a point f e C(~, [) provided, for any filter

F converging to a point x in ~, the filter generated by

by e(C~x [~) converges to the point f(x) in ~ .

The latter convergence structure (b) on C(X,Y) has been introduced in

special settings and in the realm of sequences instead of filters al-

ready by K. Weierstrass [57J under the name "gleichm~Sige Konvergenz

in jedem Punkt", by P. Du Bois Reymond [Io] and H. Hahn [231 under the

280

name "stetige Konvergenz", and has been shown by C. Carath~odory [12]

to be the proper kind of convergence in the theory of complex functions.

In full generality and in the realm of Moore-Smith sequences it was

introduced by O. Frink [20] and analyzed by R.Arens and J. Dugundji [4].

Finally in the realm of filters it has been introduced by G. Choquet

[14] under the name "pseudo-convergence uniforme locale", by H. Schaefer

[5~] as "stetige Konvergenz", by A, Bastiani [5] as "quasi-topologie

de la convergence locale", by C.H, Cook and H, R. Fischer [13] as the

structure of "continuous convergence", by H. Poppe [49] as "stetige

Konvergenz", and by E. Binz and H.H. Keller [9] as "Limitierung der

stetigen Konvergenz". In case, X is a locally compact Hausdorff space, b

it is topological, in fact a filter ~ converges to f with resp. to

(b) iff / z~" converges to f in the compact-open topology on C(X, Y),

but in general the convergence structure (b) is not induced by any

topology on C(~, ~) . The former convergence structure (a) has been

introduced by M. Kat~tov [33]~ At first glance it may seem that the

latter structure (b) is more informative than the former (a), but it

is just the other way around: (a) cannot be recovered from (b), but

(b) can be recovered from (a). A filter ~-converges to f with

resp0 to (b) iff the filter {F e /----If e F} converges with resp. to (a) .

Hence we have found two natural constructions which, when applied

to topological spaces, yield structures more general than topologies.

Depending on the kind of convergence structure ((a) or (b)) and the

axioms, we want to impose on these structures, we obtain several car-

tesian closed supercategories of To~:

(I) filter-merotopic spaces (M. Katetov [33])

(2) convergence spaces (D. Kent [34], L.D0 Nel [45])

(3) limit spaces = espaces quasi-topologiques (H. J. Kowalsky [35],

H.R. Fischer [18], A. Bastiani [5], C.H0 Cook and H.R. Fischer

[13], E. Binz and H,H. Keller [9], E. Binz [8], A. Fr61icher and

W. Bucher [21], A. Machado [3~], and others)

281

(4) espaces pseudo-topologiques (= L~-R~ume) (G. Choquet [14 ], H.

Poppe [48 ], A. Machado [38], LoD. Nel [45 ])

(5) espaces ~pitopologiques (P0 Antoine [I ], A. Machado [3S]).

All of these categories are closed under the construction of func-

tion spaces (b), moreover the filter-merotopic spaces (I) are closed

under construction (a) . If we restrict our attention to spaces satis-

fying a weak separation axiom, which corresponds to the Ro-aXiom in

topology (if a filter ~converges and x ~ /~ ~-- , then ~-- converges

to x), then each of the above categories contains all subsequent ones

as full subcategories (W. A. Robertson [54]), and (5) is the smallest

cartesian closed topological subcategory of (2) containing Top (A.

Machado [38]). Moreover (I) , and hence all of the mentioned catego-

ries, can--by means of a very simple construction--be fully embedded

into S-Near. Hence the nearness concept provides a suitable framework

for the investigation of function spaces too.

Definitions~ (I) A filter-merotopic structure on X is a set of

filters on X (called convergent filters) such that the following

axioms hold:

(FI) If a filter ~-- Converges, and a filter ~ is finer than ~,

then ~ converges.

(F2) For every x e X, the filter {A C- Xlx c A} converges.

(2) A filter-merotopic space is a pair ~ = (X, y), where X is a set

and y is a filter-merotopic structure on X.

(3) A map f: X § Y between filter-merotopic spaces is called continu-

ous provided, for any convergent filter ~ in ~, the filter gene-

rated by f~- converges in ~.

(4) The category of filter-merotopic spaces and continuous maps is

denoted by Fil.

To obtain the embedding of Fil into S-Near, observe, that, for a

topological space X = (X,~), a subset ,~ of PX has an adherence point,

282

iff the subset sec ~ = {B C XIA ~ B # @ for each A s 4} of PX

converges. If y is a filter-merotopic structure on X, then

= {~ ~ PXlsec~ contains some Z~ e y} is a semi-nearness structure

on X, and this correspondence gives rise to a full embedding of Fil

into SUNear (in fact, it induces an isomorphism between the category

of all merotopic spaces and S-Near (see H. Herrlich [26])). By the

above observation this embedding leaves topological spaces fixed. The

image under this embedding of Fil has been characterized by W.A.

Robertson [5o]. A non-empty collection ~ C PX is called a grill on X

provided sec ~ is a filter, i.e. provided (I) ~ { ~ and (2)

A ~ B s ~ iff A e ~ or B e ~ . Then a semi-nearness space X =(X, ~)

is the image of a filter-merotopic space iff ~ is ~rill-determined,

i.e. iff ~ satisfies the following axiom:

(N12) For any non-empty /~ e ~ there exists a grill ~ s ~ on X with

AcB

Therefore the bicoreflective, full subcategory Grill of S-Near,

consisting of all grill-determined semi-nearness spaces, is a car-

tesian closed supercategory of Top, which not only contains the above

mentioned categories (2)-(5) as nicely embedded subcategories, but

also the category of all contigual nearness spaces and hence espec.

the category of proximity spaces (=totally bounded uniform spaces),

which is not contained in any of the other categories (2)-(5). That

Gril___~l is cartesian closed and has a number of other pleasant proper-

ties, e.g, that, in Gril___~l, arbitrary products commute with quotients,

and finite products commute with direct limits, so that especially

our assertions (8) and (9) are true if interpreted in Gril_____~l, can be

seen with little effort directly starting from scratch, as demonstra-

ted by H, L0 Bentley, H. Herrlich and W. A. Robertson [7] . There

it is also shown that Gril____~l, compared with Top, is not too big: every

grill-determined semi-nearness space is a quotient (in S-Near) of a

nearness-subspace of some topological space, or--the other way around--

283

a semi-nearness subspace of a quotient (in S-Near) of some topological

space. If the quotients (in S-Near) of topological spaces are called

conver@ence spaces-- and that is what they are-- and the subspaces (in

S-Nea___~r resp. Nea..__~r) of topological spaces are called subtopolo@ical,

then a semi-nearness space is topological ~ff it is a subtopological

convergence space. Again, much more can be said, but we leave the

matter here.

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Topological Functors Admitting Generalized

Cauchy-Completions

by

Rudolf-E. Hoffmann

Mathematisches Institut der

Universit~t DUsseldorf

In order to describe the ideas to be investigated in

this paper , let us start with two different aspects

(A,B) , which - at first sight - seem to be rather un-

related . Then (C) we shall briefly develop a program

of research in categorical topology , into which the

above aspects fit , thus exhibiting their inner relation-

ship In D we give a short conspectus of contents

Some of the assertions of the following introduction

are not repeated in the text ( in particular some

material from A ) : they are immediate consequences of

results stated explicitly

A

We are interested in those functors which are obtained

by restricting a topological functor to a full reflective

subcategory of its domain. The result we have does not describe

these functors completely, however it shows that this class of

functors is a very large class including many functors which

are of specific interest ~) :

287

A functor V: C ~D is a "reflective restriction" of a

topological functor, provided (1.8)

(I) V is faithful

(2) V is right adjoint

(3) the class Epi~ induces a "factorization of cones" in

Condition (3) can be replaced by (31 ) A (32), which implies (3)

(31 ) ~ is co-complete (i.e. ~ has small colimits)

(32) ~ is co-well-powered.

(I) and (2) above are necessary conditions. The same holds

for (31) , if D is co-complete . However this is not true

for (32) , even for ~ = Ens (being ... and co-well-powered) :

H.Herrlich [He2lgives an example of an epi-reflective subca-

tegory of Top being not co-well-powered; a result of the same

type ("an unpleasant theorem") was obtained by O. Wyler for

a category of limit spaces [Wy4]

As a consequence of the above result we obtain:

If V: ~ --~ satisfies (I) , (2), (3) or (31 ) A (32), and

- in addition:

(4) ~ is complete,

then also C is complete (1.8).

The proof of this result sheds some light on the influence

of the duality theorem for topological functors to the problem,

under which conditions a co-complete and co-well-powered category

is complete (this influence was for a special case already

observed in our doctoral thesis~ol] 4.2.7).Let ~ satisfy

(31)A(32) : Putting ~ = M-th power of Ens, i.e. assuming the

existence of a generating set M of objects in ~ (such that

the canonically induced faithful functor ~ ~Ens M has a left

adjoint, since C is co-complete), the above result becomes a

well known criterion (1.11, usually formulated in the dual

way) - c f . ~ b l ] 1 6 . 4 . 8 . T~) .

Condition (3) in the above result is not necessary,

288

even not in order to make sure that the functor constructed

from V in section I is topological. Proving an "iff"-statement

for this functor to be topological, (3) has to be replaced

by a condition saying something like "V admits a 'relative

factorization of cones'", i.e. one has to generalize appropriately

H. Herrlich's ideas[He3~ on factorization of morphisms B-~FA

to the situation with cones. (Similar ideas for the case of

morphisms in W. Tholen[T~ ) . However, instead of formulating

explicitly what means "V-relative factorization of cones",

we prove an implicit characterization (1.5) generalizing our

result inEHo ~ on factorizations of cones.

A tripleable functor with co-domain Ens is a reflective

restriction of a topological functor with codomain Ens by

means of a construction of M. Barr ~r~ , which is different

from ours:

The forgetful functor from Co__~ = {compact T2-spaces

and continuous maps~ ~ ~ns is tripleable; Comp is co-well-

powered. Barr's construction embeds Comp in to Top, whereas

our construction embeds Comp into the category of totally

bounded (not necessarily separated) uniform spaces and uniformly

continuous maps. (Barr's construction has recently been gene-

ralized to some base categories different from Ens by

S.H. Kamnitzer [Ka~ .)

Considering our characterization of topological functors

in [HolJ that the following conditions are equivalent:

(a) T: ~-~[ is topological,

is co-complete and co-well-powered,

(b) ~ is co-complete and co-well-powered

289

T is faithful, preserves colimits, and has a fully

faithful left adjoint,

it becomes clear, that we cannot expect to rediscover

such things like Ti-spaces (i=O,I,2,3) or Cauchy-complete

separated uniform spaces in the general setting of

all topological functors. However one can try to

restrict this class suitably to rediscover some of

these things on this general level.

P ~ InLHo ~ we have singled out those topological functors

having "(weakly) separated objects" (= To-objects). Here we

are interested in the general idea of Cauchy-completions. The

reconstruction process used in solving question (A) provides

the adequate framework for Chauchy-completions; our aximatization

emphasizes the aspect one was originally interested in in

point set topolgy: for a separated uniform space X there

is a u n i q u e Cauchy-complete separated space Y

(up to...) admitting a dense uniform embedding. Furthermore

the nice behaviour of uniformly continuous maps with respect

to dense extensions carries over to the general case. So the

universal property of the Cauchy-completion is - in a sense -

only a byproduct.

Most of the work, of course, has to be done in order

to verify the examples (cf. sections 3,4).

The interplay between the ideas of (A) and (B) fits into

a general program we had a presentiment of in the introduction

our Habill~ationsschrift [Ho5] , which follows the same of

program as the present paper.

I) In order to find the "reflection" of a property, say,

of some topological spaces, in the general setting of

topological functors, one should try to derive a cate@orically

formulated result on the behaviour of these spaces with

2)

3)

4)

5)

290

respect to all spaces.

Then one should axiomatize those functors obtained

by restricting topological functors to those classes to

be described in step I, i.e. g e n e r a 1 i z e the

concept of topological functor.

Now one needs a procedure reconstructing topological

functors from those functors described in step 2

In order to insure that the examples one had in mind

when starting the investigation fit into the framework

developped in step 3, one has to prove some results on

the "internal structure" of those topological functors

obtained by the reconstruction process in step 3.

Furthermore one is interested in the question how

to modify a given topological functor into a topological

functor obtained by the procedure of step 3. For

"(generalized) Cauchy-completions" we do not have any

answer to this, whereas inIHo5 ! we obtained a rather good

answer for (E,M)-universally topological functors with

co-domain Ens.

We plan to elaborate a more detailed exposition of this

program [Ho~ .

In section O we briefly give the basic definition

of the concept of topological functor. In section I we

establish the fundamental construction (1.4) and characterize

internally those topological functors obtained by this con-

struction (1.12). As "applications" we obtain the characteri-

zation of "factorizations of cones" inrHo31 (1.6) and the

well-known theorem of completeness of co-complete, co-well-

powered categories with a generating set (1.11). Furthermore

we prove an extension theorem (1.14). In section 2 we investi-

gate the relationship between those topological functors

characterized in section I and (E,M)-universally topolgical

functors introduced into5]. By the criterion thus obtained

29l

(2.4) it is easy to do the verification for the examples to

be discussed in section 3 (namely by means of results of[Ho~

and by some lemmata being "topological" in charactor - some

of them known, others until now unknown).

The most significant of these examples should be men-

tioned here:

The category of Cauchy-complete separated uniform

spaces "generates" (by the above mentioned procedure) the

category Unif of uniform spaces and uniformly continuous

maps (3.2).

The category of sober spaces "generates" the category Top

(3.1).

The category qBan~ of (quasi-)Banach-~-spaces "generates"

qn-ve~ of quasi-normed ~-vector the category

spaces ~ denotes a subfield of the field of complex numbers)

(3.5) .

In section 4 we discuss some examples which do not fit

into the above framework. In particular, we use an ad hoc-

version of Cauchy-complete (separated) object for topological

functors in order to verify (by an explicit characterization

of the~objects) that the topological functors in question

do not "admit (generalized) Cauchy-completions". We briefly

comment the problem to find for a given topological functor

a "best approximation admitting (generalized) Cauchy-comple-

tions" (which is for from being solved).

~) Cf. the footnote to 1.13

$~) Furthermore one obtains the co-completeness theorem

proved by Herrlich-Strecker[He~ and Pumpl~n-Tholen[P T] :

Let V:~--~ D be a faithful right adjoint functor , let

be complete , wellpowered and co-wellpowered ( this

implies : Epi~ induces a factorization of cones in ~ )

If ~ is co-complete , then so is ~ ( being a reflective

full subcategory of a "topological category over ~ " )

- cf. 1.8 .

292

w O

O.1 A cone (C,X:C7 --~T) in a category C is said to be

a V-co-identifyin~ ( = "V-co-idt." ) cone with respect to

a functor V:C~-~D, iff whenever (V,l) u E = V~n for some

cone (X,n:Xz--~T) an4some morphism u:VX--~VC, then there

is a morphism h:X --~ C in C being unique with respect to

the following properties

(I) Vh = u

(2) , = Xhr .

~-indexed cones (C,~) in C are interpre%c~as objects of C:

C is said to be a V-co-discrete object , iff (C,~) is

V-co-idt. *) .

A V-co-identifying cone indexed by a graph with exactly

one vertex and without arrows is identified with a C-morphism,

which is said to be a V-co-identifying morphism.

A diagram T:Z --~ C and a cone (D,~:D-~VT) form a

"V-datum" (T;D,~) . (C, I:C Z -~ T;h) is called a V-co-idt.

lift of (T;D,u), iff (C,I) is V-co-idt. and h:VC-~ D is

an isomorphism with ~hz = V~X A V-co-idt. lift is

"unique up to an isomorphism" (analogousto limits).

0.2 A functor V:C-~D is said to be a topological functor,

provided that

(I) every V-datum (T;D,~:D E ~> T) with E U-small and

discrete (i.e. without arrows ) has a V-co-idt. lift;

(2) V satisfies a "smallness condition" (for functors) :

whenever M ~ ObC consists of non-isomorphic objects,

which are mapped by V into objects isomorphic to some

Y & ObD, then M is U-small .

(I) imitates "existence of the 'initial' topology"

in Top - cf. Bourbaki

Topological functors abound in point set topology

and other branches of mathematics concerned with "elemen-

tary structures": E.g. the usual forgetful functors Top ~ Ens,

$) i.e. iff for every X 60b~ the mapping IX,C] ~ [VX,VC~

induced by V is bijective .

293

Unif ~ Ens,o-Alg ~ Ens, Preord ~Ens are topological. Of

course, there are also topological functors with a base

category different from Ens : E.g. TopGr (topological groups

and continuous homomorphisms)--~ Gr (group~and homomorphisms),

etc. . However, monadic functors are not topological except

for the identity( [Ho~ 6.8. I) .

A lot of examples are to be found in [HoI,Ho5,Ro,Wy~

etc., cf. also sections 3,4.

0.3 Topological functors are automatically falthful [HO~ .

If V:~-~ D is topological, then v~176 ~op is too

(duality theorem; [An , R~ ), i.e. V admits identifying

lifts.

0.4 A topological functor V:~ ~ D has a fully faithful

right ad~oint assigning to D & ObD a V-co-discrete object

"over" it; because of the duality (0.3) V has also a fully

faithful left ad~oint.

Let T be a diagram in ~, and let (D,~) be a limit of

VT in D: If (C,l ;h) is a V-co-idt. lift of (T;D,,), then

(C,I) is a limit of T. An analogQus statement holds for

colimits.

0.5 Let F:A ~ B be a topological functor. A full subcategory

of ~ is called an F-co-identifyin~ subcategory

( = a top subcategory [WFI]), provided that whenever

(C,l:C -~T) with Z U-small and discrete and T has values

only in X, then C e ObX (hence X is closed under isomorphisms).

Intersections of F-co-idt. subcategories are F-co-idt.;

the F-co-idt. hull of K ~ 0bA consists of those objects m

A of ~ admitting an F-co-idt. cone (A~:A Z ~ T) with Te ~ K

for every ee E - where E is U-small and discrete :

this is the smallest F-co-idt. subcategory of A containing

K . If B = Ens, then F-eo-idt. = bireflective and

F-idt. = bi-coreflective.

294

Of course, the restriction of F to X is a topological

functor into B.

A detailed investigation of the above concepts can

be found in our doctoral thesis ~o~ .

w

1.O We are interested in those functors which are obtained

by restricting a topological functor to a full reflective

subcategory of its domain. We cannot solve the problem of

describing these functors in full generality~).However, there

is a procedure reconstructing topological functorsfrom a fairly

wide class of functors of the above mentioned type. (A spe-

cial case of this construction can be found in ~nl].)

So we have changed the standpoint of our investigation:

We look for conditions on a functor V being necessary and

sufficient, such that the functor U obtained by this

c a n o n i c a 1 reconstruction is topological.

From the well known properties of topological functors

the following is immediate

1.1 Lemma: If V is the restriction of a topological functor

to a full reflective subcategory, then

I. V is faithful

2. V has a left adjcint.

1.2 Now we sketch the construction to be investigated:

Let V: C § D be faithful

(I) Objects of A are pairs (f: D ~ VC,C) , such that for

every pair g, h: C + X in C V(g)f = V(h)f implies

g = h. These pairs (f,C) are called "V-epimorphisms".

' ) c f . 1 . 1 3

295

(2) Morphisms (f,C) + (f' : D' § VC',C') in A are pairs

(a: D + D' , b: C + C') with f'a = V(b) f Composition

in A is defined componentwise.

(Of course, one has to make the hom-sets disjoint from

each other.)

(3) U: A § D maps (f: D + VC,C) to D and (a,b) to a.

(4) The embedding F: ~ + A is given by C ~---> (idvc,C),

(b: C + C') ~--> (Vb,b), since V is faithful; F is full.

(5) The universal morphism of the adjunction C + A is given

by the commutative square

f D > VC

VC VC

1.3 Lemma: Let V: C + D be a faithful functor, let L: D § C

be a left adjoint of V, and let ~: id c ~ VL be a unit of the

adjunction:

(f: D + VC,C) is a V-epimorphism, i.e. an object of A

(as in 1.2), iff the morphism h: LD ~ C satisfying

V(h)nD = f is an epimorphism.

It turns out that the question, whether U (in 1.2) is

topological or not, is strongly related to the problem,

whether EpiC , the class of epimorphisms in C, induces a

factorization of cones in C (cf ~o3], see below). This

suggests the following modification (generalization) of the

above construction - replacing EpiC_ by some class J.

1.4 Let V: C + D be a functor, let L: D § C be a left adjoint

of V, and let ~: id D + VL and e: LV ~ id C be a unit and,

resp., co-unit of the adjunction satisfying

(V . c)(n . V) = id V

(e * L)(L * n) = id L

Let J be a class of epimorphisms in C with

296

(i) Iso~ ~ J { EpiC,

(ii) J is compositive,

(iii) s ~ J for every C s ObC

Since J consists of epimorphisms, (iii) implies that V

is faithful (cf. ~bl] ?6.5.3)

(f: D § VC,C) with f ~ MorD, C 60b

is an object of Aj, iff the morphism h: LD + C in

with V(h)q D = f belongs to J.

A morphism (f: D + VC,C) + if' : D' + VC',C') in Aj is a

pair Ca,b) with a: D + D' in D, b: C + C' in ~ satisfying

f'a = V(b)f. Composition is defined componentwise. (Hom-

sets have to be made disjoint to each other.)

The full embedding Fj: ~ + Aj is given by C I----> (idvc,C)

(since e C e J and V(SC)NV C = idvc , (idvc,C) is admissible)

and by b ~--> (Vb,b).

The functor Uj: Aj + D is defined by (f: D § VC,C) ~--> D,

(a,b) ~--> a.

The universal morphism of the adjunction [ --> Aj is given

by the commutative square

f D > VC

VC VC

].5 Theorem:

Let U: C + D, L, J, Aj, Uj, Fj as above. Then the following

conditions (a), (b), (c) are equivalent:

(a) (i) Uj is topological;

(ii) "pq ~ J with q: L(D) + C in J and D 60b D"

implies p ~ J.

(b) (i) For every morphism f: D § X in D and every morphism

k: LD § Y in J there is a pushout in C

297

k

Lf LD

Y with 1 & J. g

> LX

> Q

(ii) For D 60bD and k. : LD + X in J (i 6 I;I E U;I ~ ~) -- 1 l

there is a multiple pushout li: X i § Q in ~ with i i 6 J.

(iii) The J-quotients of every LD form a U-set.

(e) Let Pj denote those cones (A,{mi: A § A i} ) i 6 I

in ~ with I s U, which do not factor over a non-isomorphiC

J-morphism x: A + X:

(i) Every cone (LD, [fi: LD ~ X i} ) with I ~ U factors I

fi = Pi k with k: LD ~ Q in J and (Q, {Pi}i in pj.

(ii) For every commutative diagram in

Lg

LD

LD'

1 Q

> X

P i > Yi

fi

with g: D § D' in D, k 6 J, j 6 J,

(Q' {Pi: Q + Y } i ) 6 Pj there is a morphism i is

h: X + Q with hk= jL(g) .

(iii) The J-quotients of every LD form a U-set.

(c) (ii) is some kind of "diagonal condition".

Proof: The fibre of Uj at the object D & ObD consis~of pairs

(f: D ~ VX,X), such that the induced morphisms LD + X in C

belong to J. This establishes a bijective correspondence

between a skeleton of the Uj-fibre of D and the J-quotients

of LD in C .

298

(a) => (b) :

Let ((Vk)qD ,Y) ~ Ob Aj and let f: D ~ X be a morphism in D with

domain D = Uj((Vk)N D ,Y). Since Uj lifts isomorphisms and Uj

is supposed to be topological, there is a Uj-identifying morphism

in Aj with domain ((Vk)Q D ,Y), which is taken by Uj into f: this

induces a commutative square

Lf LD > LX

Y > Q in C with i E J. g

It is easily shown by the universal property of uj-identi-

fying morphisms that this square is a pushout. Instead of

giving an explicit proof, we draw a picture (lying in D)

> X

VLD VLf > VLX

iv ii VY > VQ

-14 (idvp,P) has to be considered as an object of Aj .

Similarly, a family k : LD + X of J-quotients 1 1

(i ~ I; I & U; I # ~) is interpreted as a family of objects

(V(ki)qD,X i) of Aj in the Uj-fibre of D.

Since Uj lifts isomorphisms and Uj is topological, there is a

Uj-identifying con {~i }i e I in Aj with (Vki)qD , X i) =

domain of #i and Uj(~i) = id D .

�9 + Q} in C This cone obviously induces a cone {ii: X l i ~ I --

with ilk i independent from i 6 I and ilk i E J, since if

represents an object of Aj . Because of (a) (ii) we have

i i E J. It is shown by the universal property of the Uj-

299

identlfying cone in Aj that {li: X i + Q}i s I

pushout of {ki: LD § X i} i 6 I "

is a multiple

(b) > (c) :

(i) Let (LD, {fi: LD § Xi} i E I ) with I ~ U be a cone in C.

Let {kn: LD + Qn}n& N denote the family of all J-quotients

of LD such that every f factors over k 1 n

Then f factors over the co-intersection k: LD + Q of the 1

k n with k = hnkn, i.e. fi = Pi k for some Pi: Q + X i

Now let Pi = mie with e 6 J, then ek = kn s J(w.l.o.g.)

ek = h k = k, hence for some n E N. Consequently h n n n

hne = idQ , i.e. e is an epic coretraction, i.e. an isomorphism

(ii) Now let g: D + D' in D, and let j: LD' + Q in J, k s J,

and let

k LD > X

LD' 1 commute

J ~ Pi Q ~ y

1

with D s Ob D, i ( I (I 6 U), (LD', {Pi}i ) ~ Pj .

Applying (successively) (b) (i) and (b) (ii), we find

pushouts for (Lg,k) and (j,k') and induced morphisms

Lg

k LD >

LD' ~j'

Q > P i

f. l

X. 1

with k', k" s J.

Since (Q, {pi }

-I hence h:= k"

I ) 6 Pj, k" is an isomorphism;

j'h' satifies hk = jL(g).

300

(c) ---> (a) :

(i) We verify the existence of Uj-co-identifying lifts:

Instead of giving a detailed comment, we draw a picture.

§ X in J, gn: D § D in D, Let D n s Ob D, kn: LD n n n --

: X + X with f: E § D in D, k: LE ~ X in J, and let fn n

V(fn)V(k)n E = V(kn) n D gn f for every n6N. n

X f ~ ~ n ]"

k ",~ Q ~ X

~ h k n

Lgn LD LD > n

Let knLg n : pn h with h E J, (Q, {Pn}N ) s Pj. By (c) (ii)

we can fill in the diagonal - proving the universal pro-

perty of {(gn,Pn) : (D, h: LD § Q) ~ (Dn' kn: LDn + Xn)}ne N"

(The same proof applies to the case N = ~; only the

notation has to be changed slightly.)

(ii) Let D ~ Ob D, q: L(D) § C in J and pq ~ J, then

p = sj with j 6 J and s 6 Pj. Because of pq 6 J and jq s J,

we can fill in the diagonal h in

LD q > C P >

q 4 / h t / /

J . - I

Z" s

Since jq is epic, h is an epic coretraction, i.e. an

isomorphism, hence so is s.

This completes the proof.

301

1.6 Remark:

We want to apply 1.5 to ~ = D, V = idD:

(I) 1.5 (a) (i) states that the functor "domain" from the

category of J-morphisms to C is topological.

(2) 1.5 (c) gives a suitable formulation of the concept of

"factorization of cones" (j, pj) ([Ho3,4]) slightly modi-

fying the definitions given in [He2,Mar,Ca2;furthermore

cf. Ti ].

(3) 1.5 (b) becomes the characterization of factorizations

(E,M) of cones obtained in [Ho3] :

Let E be a compositive class of epimorphisms in C with

Iso C ~ E. E induces a factorization of cones, iff

(i) every "span"

has a pushout

A

el B

A > C

B > X

> C with e E E

with f s E,

(ii) for any collection {el: A + Ai} i 6 I of E-morphisms

with I 6 ~, I ~ ~ there is a multiple pushout

{ gi: Ai + B}i 6 I with g~ 6- E,

(iii) ~ is E-co-well-powered.

In the following we return to the case J = Epi~ (but we

shall apply 1.6). We refer to U ~n 1.1 as the canonical

extension of V.

302

1.7 Lemma:

Let ~, D, V, L, ~, U as in ].I, 1.2, 1.3:If U is topological,

then ~ is co-well-powered.

Proof:

Let C 60b C and let e: C ~ X be epic in ~, then there

is a morphism j: LVC ~ X with V(J)nvc = V(e). Since V

is faithful and @ is epic, j is epic. Thus an injection

from the quotients of C to the quotients of LVC ~s

defined . NOW 1,5(b) (ii~) applies �9

1.8 Coro!lar ~ :

Let V:C --~ D be a faithful right adjoint functor .

If the class EpiC of epimorphisms in C induces a

factorization of cones in C , then the above constructed D

"c a n o n i c a 1 e x t e n s i o n" U is

topological , and V is the restriction of U to a full

reflective subcategory

If D is complete ( resp. co-complete , resp. well-

powered ) , then so is C

Proof : By 1.6(3) and 1.5(b) U is topological . If D is

complete , resp. co-complete , resp. well-powered , then

so is A ( since it is the domain of the topological functor

U ) . Since C is a full reflective subcategory of A , the

same holds for C .

That Epi C induces a factorization of cones in ~ is

guaranteed by either of the following conditions (a) and (b) :

(a) ~ is co-complete and co-well-powered .

(b) ~ is complete , well-powered , and co-wellpowered

Cf. [HO~ , (a) is i~nediate from 1.6(3)

By virtue of 1.7 now we have

1.9 Theorem :

Let V:C -9 D be a faithful right adjoint functor , and M m

let U denote its canonical extension :

(a) Let ~ be co-complete : U is topological , iff ~ is

co-complete and co-well-powered .

303

(b) Let ~ be complete and well-powered : U is topological ,

iff ~ is complete , well-powered , and co-well-powered

1.1o Remark :

From 1.7 it is immediate that the "canonical extension"

does not solve the problem of reconstructing a

topological functor from a reflective restriction of

some topological functor in full genera~ty ( cf. 1.13 ) :

Let X be not co-well-powered ( e.g. the well-ordered

class of ordinals , which is - as a category - co-complete),

then id x is topological , in particular it is a reflective

restriction of a topological functor .

The result 1.8 above shows that the interrelationship

between "completeness" and "co-completeness" of categories is

somehow related to the"dualit~ theorem" (0.3) . In particular ,

the techniques developped here imply the following

1.11 Corollary (well known, [Sb1~ 16.4.8):

If C is co-complete, co-well-powered, and if there is a

generating U-set S in ~, then ~ is complete and well-po-

wered.

Proof: Let S denote simultaneously the set and the associated

discrete category. The collection

{Hom(C,-) : ~ + Ens} C 6 S of functors induces a functor

G: C + Ens ~ (being the ~-th power of En___ss) , which is faithful

and admits a left adjoint L: Ens ~ + C, L takes (MC)c ~ S

to ]~ I ~]c ) By 1 9 there is a topological functor C E S M C " "

U: A + Ens ~ and a full reflective embedding F: C § A

with G = UF. Since Ens ~ is complete and wellpo~ered, the

domain A of U is too, and so is the full reflective subcate-

gory C of A.

Finally we want to give an intrinsic characterization of

those topological functors U obtained by the above proce-

dure (with J = EpiC_ ).

304

1.12 Theorem:

A topological functor T: X § D is "induced" by a functor

V: C + D in the sense of 1.2, i.e. there is an equivalence

S: X + A with US = T,iff there is a full isomorphism -

closed reflective subcategory C of X, such that (I) and (2)

are satisfied:

(I) For every X & Ob~ the universal morphisms n x of the

adjunction C ~ X are T-co-identifying. m

(2) If f: X + C in X with C [ ObC is a T-co-identifying

morphism, such that gf : hf implies g = h provided that

the co-domain of g, h is in C (i.e. f is 'epic with

respect' to ~), then the morphism p with P~X = f is

an isomorphism.

Proof of 1.12

(a)Let T = US. W.l.o.g. we can assume S = id A .

(I) The universal morphism qX is given by

h D > VB

VB > VB

p r o v i d e d t h a t X = (h : D . V B , B ) . C o n s e q u e n t l y ~X i s U T-

c o - i d e n t i f y i n g by t h e e x p l i c i t d e s c r i p t i o n o f U j - c o -

identifying lifts in the proof of 1.5 (c) ~> (a) .

(2) That f: X + C in X with C 60b C is a T-co-identifying

morphism means that in

D g > VC

VB > VC Vp

with X = (h: D + VB,B) and with f = (g,p) p is in Pj, i.e.

does not factor over a non-isomorphic epimorphism.

The condition in 1.12(2) concerning f says that p is epic

in ~, hence p is an isomorphism and we have pn x = f.

305

(b) Now let T: X + D be a topological functor with C § X

satisfying the conditions in 1.12.Let V := TIC , then V

satisfies all of the conditions required in 1.5. S: X + A

takes X E Ob X to (Tnx,B) , where B denotes the codomain

of qX; S takes f: X § X' to (Tf, g) with g: B § B' satis-

fying g~x = ~X 'f" W.l.o.g. ~C = idc for C s Ob ~, hence

US = T. Since T is faithful, S is too.

Let

Tn x

h TX > TX '

TB > TB ' Tg

commute ;

Since NX' is T-co-identifying, there is a morphism

f: X § X' with Tf = k, hence S(f) = (h,g), i.e. S is full.

Let(N: D § VB,B) be an objebt of A. There is a T-co-iden-

tifying morphism u: X § B and an isomorphism i: D + TX

with h = T(u)i . Becauce of condition 1.12 (2) there is an

isomorphism p with P~X = u. By definition of S [~,p-1)

becomes an isomorphism in A from (h: D + VB,B) to S(X) .

1.13 Problems:

(a) Let T: X + Y be a topological functor, and let K be

a full reflective subcategory of X. We do not know

whether there is a suitable J in K satisfying the con-

ditions in 1.5 for V:= TIK.

(b) There seems to be a "natural" idea of "semi-topological"

functor . Let T: X ~ Y be topological, and let K be a

full reflective subcategory of X, then the functor

V:= T IK: K + Y satisfies (I) and (2) :

(I) The induced functors V/s: ~/S + [/TS are right

adjoint for every diagram S in K with a U-small discrete

domain.

(2) V satisfies the "smallness condition".

306

We do not know, whether functors V satisfying (I) and

(2) are always obtained by a "reflective restriction"

of some topological functor - cf . [HQ2] ") I

A uniformly continuous map from a dense subspace of a

uniform space into a Cauchy-complete separated uniform

space is known to have a unique uniformly continuous

extension. This result can be generalized (with respect

to 3.2) slightly in the framework of those topological

functors described in 1.12 (furthermore cf. H.Herrlich[He9]).

An object K of a category K is said to be "injective with

respect to a class Q of morphisms in K", iff Hom(q,K) is

surjective for every q 6 Q, i.e. every span

f Q2 > K

~ g q . ~ f

Q 1 t

is made commutative by some g: QI + K: "f is extendible

over q"

1.14 Theorem ("Extendibility"):

Let U: A + D be a topological functor and let ~ be a

full reflective subcategory of A satisfying the con-

ditions in 1.12. Let ~ denote the unit of the

adjunction ~ § ~ �9 If q: X + Y is a U-co-idt. morphism being 'epic with

respect to C' in A, then every morphism f: X § C with

C % ObC is extendible over q, i.e. gq = f for some

g: Y + C (which is uniquely determined):

C % ObC is injective with respect to q (which is not

supposed to be monic!) .

~) We are going to give a solution of these problems

in [Ho~ , which , however , is somehow "formal" in

character , since it "leaves" the universe ~ (!) .

307

Proof:

nyq is a U-co-idt. morphism being ' epic with respect to

the co-domain of which is in C, hence by 1.12 (2) there is

an isomorphism p with pn x = qyq. Since there is a morphism

h with hnx = f, we put g:= hp -InY, hence we have gq = hp-lnyq =

= h~x = f. Since q is ' C-epic ', g is unique.

1.15 Proposition :

Let U:A + ~ be a topological functor , and let ~ be a

full reflective subcategory of A with unit q m

such that all of the conditions in 1.12 are satisfied

Let Q denote the class of those U-co-idt. morphisms

in ~ which are 'epic with respect to ~ '

(I) Let g:X + Y in MorA : If every f:X ~ C in MorA with

C �9 Ob~ is uniquely extendible over g , then g e Q

(2) If A ~ ObA is Q-injective , then A is isomorphic

to ) an object of C

Proof :

(I) Since the "extensions" are unique , g is C-epic

Since n x :X ~ C is extendible over g , there is a

morphism h:Y + C with hg = n x . Because of the universal

property of n X , resp. qy :Y ~ C' there are

morphisms a:C + C' and b:C' + C with an x = ny g

and bny = h ban x = bny g = hg = nX implies

ba = id C . Since nyg is C-epic , abqy g =

= ahg = an x = n~g implies ab = idc,

Consequently g is U-co-idt. , since it is the first

factor in n~ = an x : a ( being an isomorphism )

and nX are U-co-idt. , hence qyg is , and - since

U is faithful - so is g .

(2) Since n A 6 Q , there is a morphism h:C ~

rendering commutative n A A ~C

II " A ~ / h

We have n~n A = D A ; this implies

hence n A is an isomorphism .

n~ = idc ;

308

w 2

In [He~ H. Herrlich has introduced the concept of (E,M)-topo-

logical functor T: ! ~ ~, where (E,M) denotes a factorization

of cones in Y; here we use our modification of the concept of

"factorization" proposed into3] , cf. 1.6; hence we have to

modify the concept of (E,M)-topological functor accordingly

(more detailed comments on this problem can be found in~ {o2] ).

T: ! ~ ~ is (E,M)-topolog~cal, iff

I. every T-datum (S;D,~) with (D,~) ~ M (i.e. the domain of

S is L-small and didcrete, and (D,~) does not factor over a

non-isomorphic E-morphism) has a T-co-idt. (~T-co-identifying)

lift, 2. T satisfies the "smallness condition" for functors

In ~e2] 9.1 it is shown that (E,M)-topological functors with

co-domain ~ are exactly those functors which are obtained

( up to an equivalence ) by restricting topological functors

with co-domain ~ to a full , isomorphism-closed , reflective

subcategory of their domain , such that the universal

morphisms of the adjunctions are mapped ( by the topological

functor ) into E . In our "Habilitationsschrift"~o~ we have

shown that those topological functors obtained by the

construction [He~]9.1 play an important role in categorical

topology : These data are the adequate framework to formulate a separation

axiom appropriate to many topological structures used in mathe-

matics, namely To: A.S. Davis [D] realized that the usual

separation axioms ( epi-reflective subcategories of Top being

not hi-reflective) are "intersections" of T and a "wider" o

property, namely a bi-reflective subcategory ("Davis' corres-

pondence" - the categorial interpretation was given first in

a4; cf 21 The universal property of the above mentioned construction

(proved in [Ho~ w I) has suggeste~ to call these topological

functors (E,M)-universally topological functors. In this

section we want to investigate whether the functors U described

in 1.5 are (E,M)-universally topological. This will give us

sufficient information on the construction in section 1 to study

individual examples (in section 3).

309

2.1

In the following we need the fundamental criterion for (E,M)-

universality (here it may be consSdered as a definition):

A topological functor R: K + L is (E,M)-universally topological -

where (E,M) denotes a factorization of cones in ~, iff there

is a full, isomorphism - closed, reflective subcategory B of

with universal morphisms O K (K ~ 0b ~) satisfying:

I. PK is R-co-identifying.

2. R (pK) ~ E.

3. If f: K ~ B in K with Rf E E, B 6 0b B is R-co-identifying

then there is an isomorphism k with kp K = f.

By these condition B is uniquely described:

K s Ob ~ belongs to ~, iff every R-co-identifying morphism

f with domain K and with Rf ~ E is a~isomorphism. The objects

of ~ are called "(U;E,M)-separated" : i.e. if (A,{fi: A § Ai}is I

in ~ with A 6 0b ~ is R-co-idt., then (RA,{Rfi} I) belongs to M.

2.2 Examples:

(a) Let L = Ens and E = {surjective maps}, hence M = {joint-in

jective families of maps, i.e. point-separating families

with U-small index sets} :

Now e.g. the usual forgetful functors U from Top (topological

spaces and continuous maps), Unif (uniform spaces and uniformly

continuous maps), Preord (preordered sets and isotone maps),

etc. to Ens are (E,M)-universally topological.

(U;E,M)-separated means in Top "T " in Unif "separated" (hence o '

the name '---separated") , in Preord antlsymmetric"

(b) Let T denote a "type" of universal algebras, then the forgetful

functor from topological T-algebras (and continuous T-hOmo-

morphisms) to T-algebras is universally topological with

respect to the factorization of cones induced by E = {sur-

jective T-homomorphisms}.

310

Much more examples can be found in ~o5], furthermore cf

section 3. In particular, the situation L = Ens is investigated

in ~Ho~ w 3: a basic "approximation theorem" is shown.

2.3 Theorem:

Let V: ~ + D be a faithful right adjoint functor, let Epi~ induce

a factorization of cones in C, and let U: A § D be the functor

obtained from V by the procedure described in 1.4; by 1.8 U is

topological; let y denote the unit of the adjunctien C ~ A.

Let (E,M) denote a factorization of cones:

(a) U is (E,M)-universally topological; the (U;E,M)- separated

objects of A are those

(f: D ~ VC,C) with f ~ M, i.e. those A ~ Ob~ with UYA 8 M.

(b) The full subcategory B of (U;E,M)-separated objects of A is

co-well-powered, provided that D is M1-well-powered (where

M I denotes the l-indexed cones, i.e. morphisms, in M) . The

embedding C + B preserves epimorphisms, provided that M I

consists of monomorphisms.

By the way, that M consists of mono-cones means exactly that D

has co-equalizers and that every co-equalizer belongs to E;

however, this condition is stronger than M I ~ Mono D (e.g. let

be a group ~ {O}) - cf. [Ho3].

Proof:

(a) Let A be an arbitrary object of A: YA: A + C factors YA = me

with U(e) E E, U(m) 6 M, m being U-co-identifying. Let PA := e

and let B denote the co-domain of e. Since m is U-co-identifying

and since m is 'epic with respect to C' according to 1.12 (2)

(because YA is too), there is an isomorphism p with PYB = m,

hence U(YB) s M, i.e. B 60bB. Now let B' ~ Ob~ , let u: A § B'

be a morphism in A, and let 7B,: B' + C', then there is % morphism

h: C + C' with hYA = YB,U. Because of the diagonal condition now

we find d: B ~ B' rendering

311

A e > B

l " J

I . d $ , - h

�9 > C' B YB'

commutative

(since YB' is U-co-identifying), i.e. we have dp A = u. Conse-

quently PA is the universal morphism of the adjunction B + A.

It is easy to see, that 2.1 (3) is satisfied: Let f: A § X be

U-co-idt. in A with X ~ ObB , U(f) ~ E; now y~ satisfies the

conditions of 1.12 (2), hence there is an isomorphism p with

YA = PY~" Since U~YX ) e M and U(f) ~ E, there is an isomorphism

q with qf = PA"

(b) Let B 60bB_ and let g: B + X be an epimorphism in B. There

is a morphism h with h7 B = YX g " Since y C is epic with respect

to ~, u g is too, hence h is an epimorphism in C. Since D is

M I -well-powered, and since YX is a U-co-idt. morphism

with U(YX) ~ M, those B-quotients g' of B satisfying

h7 B = yx,g' form a U-set. Consequently the class of B-quotients

of B is obtained as a union of a U-small number of U-small sets,

since ~ is co-well-powered.

Now let g: X ~ Y be an epimorphism in ~, let (f: D + VC, C) be

an object of B, i.e. f e M, and let (al,bl) , (a2,b2) be morphisms

(Ivy,Y) ~ (f,C) with (al,bl) o (Vg,g) = (a2,b2) ~ o (Vg,g)

Then we have fa I = V(bl), fa 2 = V(b2), and blg = b2g;

consequently b I = b 2 (g is epic in C) and f a I = f a 2. Since f

is monic, we have a I = a2, i.e. (Vg,g) is an epimorphisms in B.

Now we want to reformulate the inner description of U in 1.12

in the framework of(E,M)-universally topological functors. This

criterion will enable us to do the verification for the examples

in section 3 by means of lemmata being "natural" topological

statements.

312

2.4 Theorem

Let (E,M) be a factorization of cones in D, such that M consists

of mono - cones.

U: A + D is reconstructed as in 1.2 from some faithful right

adjoint functor V: ~ + D, such that Epi~ induces a factorization

of cones in C, iff

I. U is (E,M)-universally topological

2. Let B denote the full subcategory of (U;E,M)-separated objects

of A. B contains a full reflective subcategory C with:

(a) The universal morphisms IB: B + C for the adjunction

~ B are U-co-identifying B-epimorphisms (and mapped

by U into M)

(b) If f: Y ~ C in B is a U-co-identifying B-epimorphism

(with Uf ~ M) and if C 60b C,. then there is an isomorphism

j with jl B = f.

(c) Epi~ the class of epimorphisms in ~, induces a factoriza-

tion of cones in C.

(...) above contains consequences of the other assumptions -

because of 2.1.

Proof:

By 2.3 " ... ~> I., 2." is shown: Remember that 1 B = YB and

that the embedding C + B preserves ~-epimorphisms (!) .

In order to show the other implication notice that YA = IBPA '

where PA: A + B, YA: A § C denotes the universal morphism of the

adjunktion B § A and, resp., ~ § A. Consequently YA is U-co-

identifying. Let f: X ~ C in A be U-co-identifying and epic with

respect to ~, let C ~ Ob ~, then f = me , such that m: Y § C is

U-co-idt., U(m) 6 M, U(e) s E. Since B is stable under Mu-cOnes

(i.e. U-co-idt. cones which are taken by U into M-cones, cf.

[HoS] ), Y belongs to B (because of C s Ob B) . Since f is U-

co-idt., e is too, hence there is an isomorphism j: Y' + Y

with e = JPx . mj is U-co-idt. with co-domain C ~ Ob C and

313

domain Y' in B. Since M consists of mono -cones, f is not only

epic with respect to ~, but also epic with respect to B, hence

so is mj Consequently we have an isomorphism h with

hly, = mj , hence hYx = hly,px = mJPx = me = f. Now 1.12

applies.

The above criterion splits up the question whether a given

"structure" is obtained by the reconstruction process described

in 1.2 into several steps:

1) Verify that U: A + D is topological, i.e. construct U-co-idt.

lifts (or U-idt. lifts) and check "smallness condition"

2) Find the (U;E,M)-separated objects and look, whether they

fulfi~the conditions in 2.1

3) Describe the objects of ~ and check the conditions in 2.4 (2).

It will be evident from the examples in section 3 that this

splitting gives the natural approach to practical verifications.

The theoretical relevance of the reconstruction problem in

].12, 2.4 is in part based on the following observation:

2.5 Corollary:

Let (E,M) denote a factorization of cones in D satisfying

I) M] consists of monomorphisms

2) D is M1-well-powered.

(M I denotes the class of morphisms, i.e. l-indexed cones in M.)

If a topological functor U: A § D is reconstructed from

V: C ~ D as in 2.4, then [ is uniquely determined up to an

equivalence by U: A + D:

A 6 ObA is isomorph• to an object of ~, iff A satisfies

I. A is (U; E,M) -separated , i.e. A s ObB m

as in 2.4)

(where B is defined

314

2. Whenever f: A § B with B 60bB is U-co-idt. (hence Uf 6 M)

and epic in B (!), then f is an isomorphism.

The examples to be discussed in section 3 suggest to call the objects

of C "U-complete" : C does not depend on the special choice of

(E,M). U is said to "admit generalized C-completions" (instead

of "Cauchy - completions").

Proof:

(a) Let C ~ ObC_ If f: C § B with B 60bB is U-co-identifying

and epic in B, then the same holds for yB f, hence there is

an isomorphism j with JYC = YB f Since YC is an isomorphism,

yB f is too, hence f is an epic coretraction = isomorphism.

(b) Now let A E ObA_ satisfy I) and 2). YA: A + C is U-co-

identifying and epic with respect to C, hence epic in

(since [ is mono-reflective in B!). Consequently YA is an

isomorphism.

2.6 Remark:

Extendibility in the category Unif is sometimes (cf.~e9])

formulated with respect to not necessarily separated

Cauchy-complete uniform spaces:

A uniformly continuous map from a dense subspace of X

into a Cauchy-complete uniform space extends to X.

In order to recapture this in the framework of section

I, let R: A + Ens (!) be a topological functor admitting

C-completions. Let E = class of epimorphisms in Ens, and

let M be the (corresponding) class of joint-injective

cones (of discrete type).

Let ~ denote the unit as in 2.2. Let us call A ~ ObA

quasi-C-c o m p 1 e t e, iff the co-domain of PA is C-

complete. Now we have the following extension theorem:

Let f: A + B be a morphism in A, let B be quasi-C-complete.

Let q: A ~ A ~ be a R-co-idt. morphism with Rq being injective,

and let q be epic with respect to the class of C-complete

objects in A (or, equivalently, epic with respect to the

lass of (R;E,M)-separated objects in A), shortly, let

315

q: A + A ~ be a "d e n s e e m b e d d i n g":

Then f: A + B admits an "extension" g: A u + B with gq = f.

Proof :

By 1.14 there is an "extension" h: A ~ § C rendering

commutative

A ~ h

j q

A ~ f

since C is C-complete.

We define a map u: RA 9 § RB by u(Rq(x)) = Rf(x) (since

Rq is injective) and u(y) 6 (RPB)-I[ {Rh(y)}] (arbitrary

choice; since RPB is surjective, this set is non-empty)

for every y ~ RA ~ with y # Rq(x) for every x ERA.

Obviously uRq = Rf and R(PB)U= Rh, hence there is a

morphism g: A ~ + B with Rg = u. Since R is faithful,

we have gq = f.

There is an analogue of the extendibility theorem 1.14

for (E,M)-universally topological functors R: K + L:

Let K & ObK be (R;E,M)-separated, let d: Q2 ~ QI be R-co-idt.

with Rd 6 E, then every morphism f: Q2 + K is extendible over

d, i.e. there is a morphism g: QI ~ K with gd = f. (The proof

is analogous to 1.14.)

This suggests the following modification of the criterion 2.1.

2.7 Theorem

Let R: K + L be a topological functor, let (E,M) denote

a factorization of cones in L, and let Z denote the class

of all those R-co-idt. morphisms which are taken by R into E:

R is (E,M)-universally topological, iff for every object

K ~ ObK there is a Z-injective object A in K and a E-morphism

K + A. An object B % ObK is E-injective, iff it is (R;E,M)-

separated.

316

Proof:

By our remark above, (for some (E,M)-universally

topological functor R) every (R;E,M)-separated object is

~-injective. Now let B ~ ObB be Z-injective. Since QB s E

(p as in 2.1), there is a morphism g with gPB = idB'

hence PB is an epic co-retraction, i.e. an isomorphism.

Consequently B is (R;E,M)-separated.

Now let R: K + L be topological. Let ObB = {Z-injective

objects}, let B denote the corresponding full subcategory

of K, For K ~ Ob_K let PK denote a ~-morphism with Z-injective

co-domain. For a span B B~

K ~

with B ~ e ObB, we have a unique morphism g: B § B ~ with

gPK = f' since B ~ is Z-injective and since PK is epic.

If f is in ~, then there is also a morphism h: B ~ § B

with hf = PK' since B is Z-injective. The right cancellation

property of PK and f now implies hg = id B and gh = idB%,

i.e. g is an isomorphism. This completes the proof.

There is an analogous theorem for topological functors

U: A + D admitting generalized Cauchy-completions:

Thisis less satisfactory than 2.7, since it involves C

on both sides of the equivalence (in order to define

what is "~-epic").

317

w 3

3.0 In this section we want to give significant examples of

the construction explained in section I and 2. The cri-

terion 2.4 helps us to do the verification:

(E,M)-universality of the topological functors considered

below has been shown in [Ho~ - of course, characterizing

those objects "separated" with respect to (E,M) ; some of

the conditions in 2.1 and 2.4 are then guaranteed by well

known results from point set topology ~r,Sb2] .On the

other hand 2.1 and 2.4 have suggested new results in point

set topology, in particular on sober spaces.

We think that drawing pictures is more helpful for the

reader to understand the situation than giving sophisti-

cated explications. The pictures below are to be inter-

preted as

C > B > A

with C, B, A and the forgetful functors V,U, etc. as

in 2.3 and 2.4. In Ens we refer to the factorization (E,M)

with E={epimorphisms} = {surjective maps}, hence

M ={joint injective cones (of discrete type), i.e."point-

separating families of maps"}.

3 . 1 Sob > T > Top --o

E n s

Top denotes the category of topological spaces and

continuous maps, T its full subcategory of To-spaces, m O

Sob its full subcategory of 'sober spaces' (cf. ~r,Bl~ ).

A topological space X is "sober", iff

318

every irreducible, closed, non-empty subset A of X has a

unique "generic" point p, i.e. a point p with cl{p} = A.

Every topological space Y admits a universal "sobrification"

Sy: points of Sy are irreducible, closed, non-empty subsets

of Y; for an open set O in Y let So :={ A e Sy I A ~ O ~ ~} be

open in Sy, {Soio open in Y} is the topology (on) Sy; the uni-

versal morphism iy: Y + Sy is given by p ~--> cl{p} (cl is to

be interpreted in Y) , this morphism is an embedding and an

epimorphism in To provided that Y is T O . In [Ho6] we have

shown that every embedding X + Y being an epimorphism in T O with

X being sober is necessarily a homeomorphism. Thus by 2.5 the

U-complete objects are exactly the sober spaces - provided To~ +Ens

"admits generalized C-completions"

For the convenience of the reader we sketch the way the

statement 'U: To~ + Ens admits generalized C-completions'

has to be verified. We do so, in order to make clear,

what this statement means explicitly in an individual

situation.

3.1.1A space X is (U;E,M) - separated, iff x,y E X with

cl{x} = cl{y} implies x = y, i.e. iff X is T : o

(al) Let Y be a space : for a,b 6 Y let a ~b, iff

cl{a} = cl{b}. The quotient space Yo := Y/~ is To, the

canonical projection p: Y + Yo is open and closed, and p

induces the initial topology on Y, i.e. p is U-co-identi-

fying and U(p) 6 E.

(a2) Let g: Y ~ Z be a continuous map into a T -space Z, then o

Z with hp = g, h is continuous. there is a unique map h: Yo

(a3) Let g: Y + Z be a continuous surjection into a T -space o

Z inducing the initial topology on Y, then the morphism

h: Yo + Z with hp = g according to (a2) is a homeomorphism.

Consequently U: Top ~ Ens is (E,M)-universally topological

with E = {surjective maps} by 2.1.

(b I) A continuous embedding A + B into a T -space B is epic in o

To , iff A is b-dense in B, i.e B is the b-closure of A in B:

319

x 6 cI(A ~ cl{x}) for every x 6 B. T --o

co-complete. Cf.[Bar1,Sk,Ho6]

is co-well-powered and

(b 2) Every To-Space X admits a 'sobrification' Sx, with a canonical

e m b e d d i n g i x : X ~ Sx h a v i n g t h e u n i v e r s a l p r o p e r t y ; i x i s a T - - -o

epic embedding (ef. ~Ar,NI,Ho6]) .

The last statement to be verified - corresponding to 2.4.2 (b) -

seems to be a new result on the universal sobrification of

T -spaces : o

3.1.2 Proposition :

Let X be a b-dense subspace of the sober space Y, and let f

denote the embedding X + Y, then the (unique) continuous

morphism g: Sx + Y with g o i X = f is a homeomorphism. Roughly

speaking: Y is the universal sobrification of any b-dense

subspace.

Proof:

s x The induced morphism g: Y takes A 6 Sx to the generic

point of cl A:= cl f[A] ~ Y. cl is always interpreted in Y,

the embedding f: X + Y is sometimes neglected in the notation.

(a) Let A' # A, A' ~ Sx , then cl A # cl A, since A = X ~ B,

A' = X ~ B' for some closed sets B,B' in Y. Consequently g

is injective.

(b) Now let y ~ Y, then A:= X ~ cl{y} is closed in X and irreducible:

Let O,0' be open in Y with X ~ cl{y} ~ 0 # ~ and X ~ cl{y}~O'# ~,

then y s 0 and y ~ O', hence X ~ cl{y}~ O ~ O' } ~, since X is b-dense in Y; in particular X ~ cl{y} # ~. In order to

make sure that clA = cl{y}, i.e. g(A) = y, it is sufficient

to have y ~ clA, i.e. y ~ cl(X ~ cl{y}) ; this is immediate

from ' X b-dense in Y ' . Consequently g is surjective.

(C) Let U be open in X, i.e. U : X ~ 0 with O open in Y. We want

[ ~Su] then Su] = O i e. is open: If y s g to show that g , �9 g

there is an A ~ Sx with A ~ U # ~ and g(A) = y, i.e.clA = cl{y}.

Since cl{y} ~ O ~ X = clA ~ U # ~, we have cl{y} ~ 0 ~ ~, i.e.

320

y ~ O. If z ~ O, then cl{z} ~0 A X ~ ~, hence cl{z} ~ X is an

element of Su, cl{z}~ X is taken by g to z - according to (b) ,

hence z ( g[Su].

3.1.3 Remark

(a) We need not verify that (U;E,M)-separatedness means T , i.e. o

a space X satisfies To, iff a (family of) continuous map(s)

with domain X inducing the initial topology on X is (joint)

injective: this is a consequence of (al), (a2) , (a3) . Similary

the characterization of sober spaces in ~o6] "a To-space X

is sober, iff every b-dense embedding of X into a T -space o

is a homeomorphism" is a consequence of (bl), (b2) and 3.1.2.

(b) Checking individual examples, one often realizes a "natural"

separation axiom which turns out to be (U;E,M)-separatedness.

For Ens-valued topological functors U represented by a terminal

objeCt t (i.e. U ~ Hom(t,-)) we have given a criterion more

easily to be verified([Ho5Jw On the other hand generally

it seems to be difficult to find the U-complete objects.

(c) The theorem on the existence of the sobrification can be

combined with 3.1.2 to a statement being more "topological

in character": For a To-space X there is up to a homeomorphism

a unique sober space Y admitting a b-dense embedding ix: X + Y.

i is a unit of the adjunction Sob + T X --o

Thus the splitting up of the concept of "admitting generalized

C-completions" in section 2 becomes evident to be just the

adequate formulation of the categorical framework of the above

mentioned facts from point set topology.

In the following we shall omit details, but our above remarks

will carry over to the examples below (with obvious modifi-

cations).

321

3.2 C-Unif > Sep-Unif > Unif

E n s

unif denotes the category of uniform spaces and uniformly

continuous maps, Sep-Unif = (its full subcategory of) separated

uniform spaces, C-Unif = Cauchy-complete separated uniform spaces.

'Epic' in Sep-Unif means 'dense' [Pr] ; for a separated uniform

space X there is up to a uniform homeomorphism a unique Cauchy-

complete separated uniform space Y admitting a dense uniformly

continuous embedding ux: X + Y; Ux: X + Y is the universal morphism

of the adjunction C-Unif § Sep-Unif.

3.3 Comp - - > Sep-Prox > Prox

Prox denotes the category of totally bounded uniform spaces, i.e.

(uniformizable) "proximity spaces" and uniformly continuous maps,

Sep-Prox = separated totally bounded uniform spaces, Comp = compact

Hausdorff spaces (admitting a unique compatible uniformity,

continuous maps become uniformly continuous).

Since there is a full embedding Comp + C-Unif, the verification

is given by 3.2 realizing that V-epimorphisms are dense subsets

of compact spaces (one can verify this in the category CRe~ of

completely regular T2-spaces and continuous maps, since Comp is

mono-reflective in CReg : '~Reg-epi' means 'dense').

3 . 4 C-qMet > Sep-qMet > qMet

E n s

Let X be a set, a mapping d: X x X + [o,~] is said to be a quasi-

metric (= q-metric), iff

322

(I) d(x,y) : d(y,x) ("symmetric")

(2) d(x,x) = o

(3) d(x,y) < d(x,z) + d(z,y) ("triangle inequality")

for any elements x,y,z E X; (X,d) is said to be a q-metric

(= quasimetric) space; a mapping f: (X,d) + (X',d) is said to be

non-expansive, iff d(x,y) > d' (fx, fy) for any x,y E X.

qMet denotes the category of q-metric spaces and non-expansive

maps, Sep-qMet = separated q-metric spaces (d(x,y) = O implies

x=y), C-qMet = Cauchy-complete separated q-metric spaces : it

should be clear what "Cauchy-sequence" in (X,d) means. 'Epic'

in Sep-qMet means 'dense' (with respect to the induced topology)

- cf 3.4.1 below. For any separated q-metric space (X,d) there is

a unique Cauchy - complete separated q-metric space (Y,d') admitting

a dense e~edding of (X,d) into (Y,d') : the space of (equivalence

classes of) Cauchy - sequences in (X,d) where d' is defined in the

usual way. The canonical embedding (X,d) ~ (Y,d') taking x to the

constant, sequence (Xn) with Xn := x is the universal morphism of

the adjunction C-qMet § Sep-qMet.

3.4.1 Lemma:

A morphism f: (M,d) ~ (N,t) in sep-qMe t is an epimorphism,

iff f[M] is dense in (N,t) with respect to the usual topology

on N induced by t.

Proof:

Let f be not dense, w.l.o.g, f is an inclusion of M into N,

the closure cl is interpreted in the topology on N induced

by t.

From the sum Nil N := {(i,n) In e N, i=1,2 ] we obtain a

quotient set identifying "corresponding" points of

clM J~clM: (i,n) ~ (i',n') iff n=n ' and i=i ' ,

or n=n' and n e ciM.

On the quotient set Q = N~N/~ we define a separated

q-metric by

h [(i,n), (i,n'~ = t(n,n')

323

3 . 5

and, if i ~ i'

h[(i,n), (i',n')] = inf t(n,x) + t(x,n') . x E clM

For n ~ clM (or n' { clM) we have h [(i,n) , (i',n')3 = t(n,n')

because of t(n,x) + t(x,n') > t(n,n'), hence the definition

of h is compatible with ~, i.e. h becomes a map

For n 6 N - clM and i # i' we have

h[(i,n), (i',n')3 = inf t(n,x) + t(x,n') x E clM

t(n,clM) ~ o, hence (Q,h) is separated.

Let J1' J2: N + NIl N denote the canonical injections,

p: N~N + Q the canonical projection, then

PJ1' PJ2: (N,t) + (Q,h) are non-expansive, PJl ~ PJ2 and

PJl f = PJ2 f

The reserve assertion is immediate from the faithful functor

Sep -qMet + ~2 := {T2-spaces and c o n t i n u o u s m a p s } r e f l e c t i n g

epimorphisms, since 'epic' in T means dense. -2

qBa.~ > Sep-qn-ve~ ~ > qn-vec~

Vec~

Let ~ denote a subfieldt of C, e.g. ~ = ~. The factorization

(E,M) of cones in Ve~K = {vector spaces over ~ and X-linear

maps} we refer to is induced by E = {surjective ~-linear

maps}, hence M = {joint injective U-small-indexed cones

of ~-linear maps, i.e. cones (A,{fi: A § Ai} i 6 I ) in Ve~

with I C U and iOi~ kernel fi = {O} }.

~e,: X be a ~ < - l i n e a r s p a o e . ,< map 11? 11" X + [ 0 , : 1 i s called a quasi-norm (=q-norm) on X, provided that

itotl = o li~,xlI z i xl,~x

i ix+yii : l!xll + 11 y II.

324

(X, I I . 11) is said to be a quasi-normed (=q-normed)]<-space.

qn- Ve~ denotes the category of quasi-normed ]<-spaces

and ~. l-decreasing ]<-linear maps, Sep-qn-ve~< = separated

quasi-normed ]<-spaces ("separated" means: IIx~ = O implies

x = 0), qBan]< = quasi-Banach-]<-spaces, i.e. those separated

(!) quasi-normed ]<-spaces, such that the (canonically) induced

quasi-metric is Cauchy-complete.

If (X, I1.11) i s a separated quasi-normed ]<-space, then

the canonical Cauchy-completion of the induced quasi-metric

space (cf 3.4) becomes in a canonical way a ]<-space, and,

moreover, a quasi-Banach-]<-space.

The verification is similar to 3.4. We still have to

show 3.5.1.

3.5. I Lemma:

Let f: (X, II �9 ~) + (Y, ]I. II) be a morphism in Sep-qn-ve~< ,

i.e. Ilf(x)lly ! ~xI~.

f is an epimorphism in Sep-qn-ve~K, iff f is dense,

ie. s is dense in (Y,II. II ) with respect to the topology on Y

canonically induced by ll. ITy"

Proof :

Let us assume that f is not dense, and let w.l.o.g f be

an embedding, i.e. X ~ Y.

A := clX is a ]<-linear subspace of Y (cl is to be inter-

preted in the topology induced on Y by If. ~) �9 The factor

]<-space Y/A becomes a quasi-normed ]<-space by putting

II[ U == i n f b+all a 6 A

w h e r e [y ] := { y + a I a ~ ~} 6 Y / A

i f lily311 = o , t h e n t h e r e i s a s e q u e n c e ( a n ) n ~ = in A

325

with l~+anll tending to O, hence a n converges to -y, i.e.

-y and y ~ A, since A is closed (and a ]<-linear subspace) ,

i.e. ~ = [oi, i.e. (Y/A, ~. I[ ) is separated.

Of course, both the canonical projection

p: (Y,~. ~) § (Y/A,II . U) and the null homomorphism

n: (Y,U .II) + (Y/A,~, II) are U, ~ -decreasing, hence f is not

an epia~orphism in Sep-qn-ve~K because of n.f = p-f, n # p.

The reverse assertion is immediate from 3.4.1, since

there is a faithful functor Sep-qn-ve~ - Sep-qMet

(reflecting epimorphisms) .

3.5.2 Remark:

In the appendix (w of ~{o5~ we have shown that the

forgetful functor U: qn - Ve~ Vec~is not only

(E,M)-universally topological with (U;E,M)-separated =

separated, but U is also co-universally topological

with respect to the following co-factorization(E',M')

of co-cones in ~__~:

E' = {~-small-idexed co-cones in Vec~, such that the

union of the images "generate" the co-domain},

M' = {injective ~-homomorphisms}.

The (U;E',M')-co-separated objects of qn-vecz are

exactly those (X, II.~ ) with II x I # ~ for every x 6 X.

3.5.3 Remark:

The above examples 3.2 - 3.5 exhibit the common

features of Cauchy-completions in the framework

of uniform spaces,q-metric spaces, and q-normed

K-linear spaces.

(What about Cauchy-completions for nearness spaces

in the sense of H. Herrlich~le8]?)

The program, to find a common setting of the idea

of completion, goes back to an early paper of

G.Birkhoff ~i I . Some of the examples above suggest

326

that the results in section 2 are somehow related

to Rattray's ~al, but there seems to be no

straightforward implication.

3.6

3.7

The functor idEn s : En__~s + Ens is, of course, (E,M)-

universally topological: the separated objects are

the sets of cardinality at most I; moreover idEn s

admits generalized C-completions: the C-complete

objects are the sets of cardinality ].

More generally, id x admits generalized C-

completions, provided--X = ~ X. and every ~i has -- 1 6 I --l

a terminal object C.: Let C be the discrete) full 1

subcategory of X consisting of the C's and let 1

V: C + X denote the embedding.

Let ~ be a complete lattice (which is canonically

interpreted as a category). Applying the construction

of section I to the constant functor L + Ens which

takes everything to a one-element-set and, resp.,

to its identity, one obtains the following category

L-Ens:

Objects of L-Ens are pairs (M,i) with M % ObEn___ss,

i 6 L, L-Ens-morphisms f: (M,i) + (M',i') are maps

f: M + M' with i ! i' (in ~) . The forgetful functor

U: L-Ens + Ens takes (M,i) to M and f: (M,i) ~ (M',i')

to f. U is a topological functor admitting generalized

C-completions:

(M,i) is C-complete, iff cardM = ]; (M,i) is

separated, iff cardM < ].

Furthermore, U: L-Ens ~ Ens is also co-universally

topological with respect to the co-factorization

(E' ,M') of co-cones in Ens with M' = {injective maps}

and E' = {joint-surjective families of maps} =

= {U-small-indexed epi-co-cones in En__~s} : (M,i)

is co-separated, iff M = ~.

327

The above topological functors L-Ens + Ens

(for complete lattice ~) are characterized by the

fact that they are induced by constant (contravariant)

topological theories in the sense of O.Wyler [Wyl,21,

i.e. constant functors t: Ens Op + C-Ord (= complete

lattices and inf-preserving maps) taking everything

to the .lattice L and, resp., its identity.

We plan to prove elsewhere that every topological

functor with co-domain Ens, which lifts isomorphisms

uniquely, being both universal and co-universal

(with respect to the above co-factorization in E ns) ,

is necessarily of the above described type for some

lattice L (which is- up to an isomorphism - uniquely

determined).

328

w

In this section we want to give "unpleasant" examples for our theory of

"generalized Cauchy-completions". We shall use the same kind of general

notation and pictures as in section 3- The functors to Ens are to be

understood as the usual forgetful functors, the embeddings as the obvioUS

embeddings.

There are two different types of these examples.

The first kind of examples we mean fits too smoothly into the concept

of "topological functor admitting generalized Cauchy-completions":

Every separated object is already Cauchy-complete.

This turns out to be equivalent to the condition that epimorphisms

in the category of (U;E,M)-separated objects are mapped by U into E.

Thus C-completion does not give any "improvement" of these separated

"structures".

B The second kind of examples are (E,M)-universally topological functors

which do not admit generalized C-completions. However, using the descrip-

tion of C-complete objects given in section 2 as an ad hoc-definition, one

arrives at some kind of objects which in these particular situations are

of interest in themselves.

The examples given in the following are (E,M)-universally topological

functors with codomain Ens and E = Epi(Ens). We freely use the concrete

description of (U;E,M)-separated objects either given explicitly in ~o~

by direct verification (in particular, by means of the approximation

theorem [Ho Dw 3 - thus computation becomes trivial) or to be immediately

followed from the generalized Davis' correspondence theorem [Ho~w 2.

329

A

4. I Theorem :

Let U : A ~ D be a topological functor "admitting generalized Cauchy-

completions", and let (E,M) be a factorization of cones in D satisfying

M I ~ Mono

Then the following conditions (a) and (b) are equivalent:

(a) Every (U;E,M)-separated object is U-complete

(b) In the full subcategory B of A consisting of the (U;E,M)-separated

objects of A every ~-epimorphism is taken by U into Eo

(

Proof:

(b) ~ (a): Let ~ denote the unit of the adjunction ~ ~ ~. For B~Ob

TgB~E by wirtue of (b) (because ZB is epic in ~!). Conrequently ~B

is T-co-idt. morphism with T~B~E O M I = Iso ~, hence ~B is a

isomorphism.

(b) ~ (a):

Let f : X ~ Y be an epimorphism in ~ with Uf~E. Let Uf = me with

meM1, e6E, then there is a U-co-idt. morphism g : B ~ Y in ~ and a

morphism h : X ~ B with gh = f and Ug6MI, UhcE, in particular Ug@E

(otherwise Uf = UgUh(E).

In particular we have B~Ob ~, because ~ is "closed" in ~ under all those

(not necessarily U-co-idt.!) cones which are taken by U into M-cones. Now

we see that B is (U;E,M)-separa~ed, but not U-complete, since g : B ~ Y is

epic in ~ (in particular, with respect to ~) and U-co-idt.

4.2

Poset = Poset ~ Preord

Preord denotes the category of pre-ordered (= quasi-ordered) sets and

isotone maps. The objects of Poset are the partially ordered sets (pre-

order with antisymmetry).

Every preordered set gives rise to a topology on it, called the associated

Alexandroff-discrete space; thus Preord becomes a full bi-coreflective

subcategory of Top, namely the (bi)-coreflective hull of the Sierpinski

330

space (this is shown into7]); "antisymmetry" now means "T ". o In consequence, (E,M)~universality of U ; Preord ~ Ens is also a consequence

of~oS]3.14(once (E,M)-universality of To~ ~ Ens is verified).

4.2.1

Lemma:

Every Poset-epimorphism is surjective.

This result is due to [B B 1 ( I saw that after having worked

out the proof ) .

Thus our concept of "C-completion" is not anyhow related to some

"completion" of posers (which - usually - means a left adjoint from

Poset into a subcategory of some complete lattices which is n o t

a full subcategory of Poser).

4.3

~1 = ~1 ~ @ R --o

Ens

R m e a n s t h e c a t e g o r y o f R - s p a c e s a n d c o n t i n u o u s m a p s . A t o p o l o g i c a l ~ o o s p a c e i s s a i d t o b e R a ( D a v i s [ D ] ) , i f f t h e S i e r p i n s k i s p a c e (2 p o i n t s

and 3 open sets) is not embeddable into it , i.e.cl{x) ~ c~{y} implies

331

4 .4 If U : _A ~ D is (E,M)-universally topological, if E = Epi _D,

(M I s Mono D), and if - in addition - T = UIB : B ~ D is (absolutely)

topological (in particular, T preserves epimorphisms), then we arrive

at

C = B ~ ~ A

D

I.e. in particular, U admits generalized C-completions.

Proof;

(f : D ~ TB,B) is a T-epimorphisms, iff f is an epimorphism in ~.

Thus the (E,M)-universal extension of T coincides with the canonical I

extension of T. (M 1 6 Mono ~ guarantees that ~ is uniquely determined

Now we come to the second kind (B) of "unpleasant" examples.

4.5 Let ~3 a denote the category of T3a-Spaces and continuous maps; the

forgetful functor U : ~3a ~ Ens is (E,M)-universally topological with

E = Epi(Ens); the (U;E,M)-separated objects are exactly the completely

regular spaces (C Re~ ).

Using the description of U-complete objects as an ad hoc-definition

we arrive at:

A T3a-Space X is "complete", iff X is T o ("separated") and every dense

embedding of X into any separated T3a-Space is an isomorphism (recall

that "epi" in CRe~ means "dense"), i.e. iff X is compact (:= compact

A T2).

However, applying the canonical extension to Comp ~ Ens, we arrive at

the category Prox (cf. 3.3), which is different from ~3a:

Let ~ denote the discrete space of natural numbers, and let N* be its

Alexandrov-compactification, then N* is not the Stone-Cech-compactifi-

ca~ion ~N of the dense subspace ~ (in contrast to 2.4.2 b), since card

~N = exp exp card N.

In consequence ~3a ~ Ens does not admit generalized C-completions.

332

4.6

Let R I denote the category of R1-spaces and continuous maps, i.e.

those spaces whose universal To-quotient is T 2 (i.e."Hausdorffsch")

- cf [ D]. The forgetful funotor U : R I ~ Ens is (E,M)-universally

topological with E = Epi(Ens): the (U~E,M)-separated objects are

exactly the T2-spaces (= "separated spaces").

R1-complete = T2-closed since "epi" in T 2 means dense.

The T2-closed spaces do not form a full reflective subcategory of

T - cf.~Hel ] However for every T2-space there is a distinguished m 2

T2-closed extension, the so-called Kat@tov-extension (cf. [Ha]).

4.7

The examples above ( 4.5 , 4.6 ) easily generalize to

the following situation :

Let X be a full epi-reflective subcategory of Top ,

let ~ consist of To-spaces ( i.e. ~ is not bi-reflective in

Top ) , and let ~ denote the bi-reflective hull of ~ in

Top . The objects of ~ are the "To-separated" objects of

-according to [Ma~ or ~Ho~ w :

If "epi" in X means "dense" , then

"~-complete" = "!-closed"

( Sometimes it may be convenient to write "~-complete" instead

of "~-complete".)

(a)

(b)

This is true e.g. for ~ = regular T2-spaces .

Recently J.Schr6der [Sdl has pointed out that for

= Urysohn spaces ( = T2a-spaces ) "epi" is weaker than

"dense" We wonder , whether the R1a-COmplete spaces

coincide with the Urysohn-closed spaces , or , if they do

not , whether they behave more nicely than the

Urysohn-closed spaces do . Cf. [He7]

( ~la := bi-reflective hull of ~2a in Top )

333

4.8

Finally , let us briefly comment the problem of

finding an "approximating" topological functor

admitting generalized C-completions for those topological

functors described in 4.5 sqq.

(a) For ~3a---~ Ens (4.5) the answer seems to be

"natural" :

Prox --; Ens (3.3) , since Comp is reflective

in CRe~ , and since the canonical extension of

Comp --9 Ens yields Prox --~ Ens .

(b) In case of ~2 (4.6) the question is open :

What about the category of "T2-closed extensions

of R1-spaces *) " , is it topological over Ens , etc. ?

This would be analogous to the situation with

proximity spaces in (a) , which can be viewed at as

( not necessarily injective ) "compactifications"

of T3a-spaces

Cf. ~Blz,Ha,He7,Po~

What about a general theory of "approximation" with

respect to topological functors admitting generalized

Cauchy-completions ?

$) Such an "extension" f of an R1-space X is to be

understood as a continuous map f:X--~ Y furnishing X

with the initial topology , f ~] dense in Y , Y being

T2-closed . The morphisms of this category are , of

course , the extendible continuous maps . The forgetful

functor is supposed to be the functor "domain"

334

Some remarks on the bibliography :

I. Papers concerned with ( concepts strongly related to ) the

concept of t o p o 1 o g i c a 1 f u n c t o r or - at

least - working with some variation of this concept :

An I-2 , Brr (implicitly) , Be , Bor , Bod , BrG , B H ,

Ca I , Er I-2 , Fa , He 2 , He 1o , Ho I , Ho 2 , Ho 3 ,

Ho 4 , Ho 5 , Ho 7 , Ho 8 , Ho 9 , HuSh , Hu I-5 , Ka ,

Kn I , Mnl , Mn 2 , Mn 3 , Mar , M P , MG , N 2 , 0 , Ro ,

Sh I , Wi I-5 , Wy I-2 , Wy 3 ;

furthermore cf. Bob I , Bob 2 , Ce , Pu , Ta

We have tried to make this list as complete as possible

( the list is based on BrG , Ho I , Wy I , and , in particular ,

on Ho 5 ) .

II.

(a)

(b)

Papers concerned with f a c t o r i z a t i o n

of mo r p h i s m s :

Bar 2 , He 3 , He 5 , He 6 , Kel , Kn 2 , Ri , So , Th ;

of c o n e s :

Ca 2 , Fa (implicitly) , He I , Ho 2 , Ho 3 , Ho 4 , Ho 5 ,

Mar , Ti (implicitly)

III. Without any claim for completeness (1) we mention the

following papers ( including some material ) on

f i b r a t i o n s :

Eh , Ga I-2 , Go , La , Le , (So)

( Here , topological functors are , of course , excluded .)

I V .

335

Papers on " s o b e r " s p a c e s ( = " p c - s p a c e s " ,

i.e. "point closure spaces" ) :

Ar , Bar 2 (implicitly) , Bln , Ho 6 , Ho Io , N I , N W ,

Sk (implicitly)

V. Papers on e p i m o r p h i s m s in some "familiar"

categories :

Bar I , Bu , (Ko) , (Pr) , Sd , Sk .

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336

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( unpublished )

CATEGORY THEORETICAL METHODS IN TOPOLOGICAL ALGEBRA

Karl Heinrich Hofmann

INTRODUCTION

My contributions to this conference will consist of some comments

on the way category theoretical thinking has had an impact on the

mathematics of topological algebra. Traditionally, topological alge-

bra is understood to concern the study of algebraic structures

enriched by the presence of topology and the continuity of algebraic

operations; this includes the theory of topological groups, semigroups,

rings, fields. But I also include areas of what is generally called

functional analysis. In this way Banach algebra, operator algebras,

representation theory, harmonic analysis all become subsumed under the

concept of topological algebra. Right in the beginning, the name of

"topological algebra" is, therefore recognized itself as a very gen-

eral label. As far as category theory is concerned, many will main-

tain that it permeates everything mathematical. However, what matters

here is the analysis of functors, their preservation properties, the

theme of limits and that of monoidal structures with its variations,

and how all of this is to be applied and utilized in solving problems

inside topological algebra.

I am, therefore, not speaking as a professional category theore-

tician, since I cannot claim to be one, but as one whose experience

lies in topological algebra. In allusion to the title of MacLane%

book [23], I might say that I will concentrate on category theory for

the working topological algebraist and analyst.

It appears to me that there is a certain difference between pure

category theory and applied category theory (that is, the technique of

applying functorial methods to "concrete" problems arising in various

branches of mathematics) which in character resembles that which has

traditionally been felt as the difference between pure and applied

mathematics. The pure categorist will seek the highest level of gen-

346

erality and abstraction which will cover the largest possible area he

can conceive of (and then some). I visualize the problem of the

working mathematician who seeks out functorial methods to help him

solve his problems somewhat differently. His problem is to find the

"right pitch of generality", a term which I hope will be accepted as

an undefined concept. The degree of generality will in any case be

"right" if it adequately covers the problems at hand, hopefully allows

to attack related and even more general ones; but the concrete root of

the functorial concepts used should still be visible and serve as the

guideline for the abstract functorial setting which is used. I con-

sider it my task to explain by illustration what is meant.

Evidently, my selection cannot be encyclopedic even for the case

of topological algebra. It is clearly determined by some topics I

have been interested in myself together with my colleagues and students,

and I disclaim any attempt towards completeness.

ACKNOWLEDGEMENTS

To Ernst Binz and Horst Herrlich go my thanks for having invited

me to a conference which consolidates categorical topology as a new

discipline: on one hand it has emerged as a new field of applications

of functor theory and on the other it provides new impulses to

research in general topology, a branch of mathematics which, not so

long ago, was considered as rather stable (see Dieudonn~ [9]). I

further express my thanks to the National Science Foundation for hav-

ing supported through the years some of the research on which I will

report, and to Deborah Casey and Meredith Mickel for typing the manu-

script of this article.

CHAPTER I.

CHAPTER II.

347

TABLE OF CONTENTS

LIMITS IN TOPOLOGICAL ALGEBRA

Limits

Transformation of domain

Cofinality

Cone categories

Functor categories

Limits revisited

Pro-P-groups

Continuity of functors

Kan Extensions

MONOIDAL CATEGORIES AND FUNCTORS IN TOPOLOGICAL ALGEBRA

Coherence

Cartesian categories

Monoids

Bimonoids

Groups

Monoidal functors

Construction of monoidal functors

Duality theories in topological algebra

Extension theorems for monoidal functors

REFERENCES

348

CHAPTER I. LIMITS IN TOPOLOGICAL ALGEBRA

Through the work of von Neumann and Weyl it was known in the

twenties that every compact group could be "approximated" by compact

linear and therefore by compact Lie groups. In the early original

proofs of the duality theorem for locally compact abelian groups by

Pontryagin, van Kampen and Weil it was used, at least implicitly that

every locally compact abelian group could be, in the same vein, be

approximated by certain elementary abelian groups build up from vector

groups, finite dimensional torus groups, and finitely generated dis-

crete groups. As far as the general structure theory of (let us say,

connected) locally compact groups is concerned, half a century of dev-

elopment culminated in the fifties in the results of Gleason,

Montgomery, Yamabe, which showed that every such group is a projective

limit of Lie groups.

Much of the philosophy of structural investigations of locally

compact groups and transformation groups is based on the idea to

utilize the wealth of information available for Lie groups

(J. Dieudonn4 [9, p. 77]: Les groupes de Lie sont devenus le centre

des math4matiques; on ne peut rien faire de s4rieux sans eux), and to

lift this information to the general case by passing to the limit;

even before the final solution to Hilbert's Fifth Problem through

Gleason, Montgomery and Yamabe this philosophy was well established

through Iwasawa's key work on locally compact groups [21]. This tech-

nique has become absolutely indispensible for almost anything that has

to do with locally compact groups, be it structure theory, representa-

tion theory, harmonic analysis, probability theory, or topological

structure theory. Similar situations as those indicated for topologi-

cal groups arise, e.g., in the theory topological semigroups, for

which S. Eilenberg showed as early as 1937 [i0] that every compact

topological semigroup can be approximated by metric ones equipped with

subinvariant metrics. More generally, they arise in universal algebra

where in many varieties of compact universal algebras, all zero dimen-

sional algebras are profinite, such as is exemplified by rings,

semigroups and lattices [25]. It is noteworthy question concerning

349

compact topological algebras to determine why for certain varieties

all compact zero dimensional are profinite while this fails for others.

Banaschewski gave me an example of a variety defined by one unary

operation in 1970 in which not every compact zero dimensional algebra

was profinite; at this conference Linton showed me this behavior in

the variety of Jonnson-Tarski algebras (see Linton's paper in these

Proceedings); the first systematic studies in the general area are

recent and are due to Choe [3,4]. Related results are in [i].

While the formation of projective limits is a traditional tech-

nique, its fully functorial aspects were not fully utilized in topol-

ogy and topological algebra until the late sixties. A typical appli-

cation of the functorial scheme of projective systems in topology is .v

that of shape theory for which I refer to the contribution of Mardeslc

in these Proceedings. Some applications in topological algebra which

originated just a little earlier I will describe later. It should be

emphasized, however, that the principle of these considerations was

clearly formulated as early as 1945 in Eilenberg's and MacLane's lead

article on category theory [ll] (whose existence seems to be much

better recalled than its content).

One might ask at this point what might be so special about pro-

jective limits in topological algebra when category theory provides in

fact a general theory for all limits. The simple answer is that in

topological algebra numerous important functors arise naturally, often

as left adjoints which do not at all preserve arbitrary limits but

which do preserve projective limits. A prime example is the ~ech

cohomology functor on compact spaces or algebras [14,19]; in analysis

the functor which associates with a topological group its enveloping

von Neumann algebra preserves strict projective limits of locally com-

pact groups (we will speak of strict projective limits if all occurring

maps are proper and surjective) [13]; the mapping cylinder functor for

compact spaces and semigroups preserves projective limits, projective

limits of inverse systems with surjective maps [20]; Pommer's functor

is another example [26]. In fact, for the compact situation, Stralka

and I gave a general lemma which illustrates how the preservation of

(strict) projective limits may arise [20, p. 225].

In this chapter we analyse the functorial methods which are

necessary to deal with limit constructions in topological algebra and

describe some of the applications.

350

LIMITS

In order to fix notation, let us consider categories X and A;

for the most part we imagine X to be small although formally in the

definitions this plays not much of a role, but does so when it comes

to the existence of limits for functors defined on X with values in

A. Let A X be the category of functors f: X --~A with natural

transformations a: f ---~g between such functors as morphisms.

1.1 DEFINITION. i) The functor X---~ A whose value is constantly

equal to A eob A on objects and constantly equal to 1 e morph A A

on morphism is called the constant functor on X with values A and

it is written AX: X -->A.

ii) A functor f: X --->A is said to have a limit if there is an

object lim f ~ ob A and a natural transformation If: (lim f)x --~ f

such that for every natural transformation a: A X ---> f there is a f ,

unique morphism a': A --~ lim f in A such that h aX = a. We call

I f the limit morphism, and lim f the limit object. D

We note that the assignment

a natural bijection with inverse

ately

~-~a': AX(Ax,f) --~A(A,lim f) is

~--> lf~x . Thus we obtain immedi-

1.2 REMARK. If every functor f: X -->A has a limit, then the func-

tor A~-->Ax: A --~A X has a (right) adjoint lim: AX-->A. D

In particular, the limit construction is functorial. A typical

domain category X which may occur in topological algebra (and else-

where) is a partially ordered set. We recall:

1.3 CONVENTION. Every poser (= partially order set) (X,S) is in

particular a category with the elements of X as objects and an arrow

x --->y between x and y if and only if y s x. A function

f: X ---> Y between posets is a functor iff it preserves the partial

order. If f: X--> Y is such a functor, then lim f exists iff

sup f(X) exists and lim f = sup f(X). D

A pair (A,i r) consisting of a category A and a faithful

functor I I : A --~ Set is called a concrete or set based category.

Virtually all categories of interest in topological algebra are set

351

based. A morphism is a set based category is called injective [resp.

surjective, bijective] if I I is an injective [sum-, bijective]

function.

1.4 DEFINITION. A projective system is a functor f: X -->A where

is an upwards directed poset. If f is a projective system, then

lim f is a projective limit. If (A,I I) is a set based category,

then f is called a strict projective system if all morphisms

I f , x �9 X are surjective, and, accordingly, lim f is called a x

strict projective limit. D

•

Typically, one might have X = ~ , X = {1,2,3,...} with its

natural order. If, by way of example, A is the category Comp G

of compact groups, then A X = Comp G ~ is the category, whose objects

are the inverse sequences

G1 <---- G2<-- ...

of compact groups; morphisms are sequences of morphisms a : G --->H n n n

such that the infinite ladder with these as rungs commutes. The

projective limit of each of these sequences exists, and by 1.2 we know

that lim is in fact a functor, which of course one verifies immedi-

ately by direct inspection.

TRANSFORMATION OF DOMAIN

In practice however, it occurs frequently that we are given a

transformation of index categories, i.e., a functor f: X ---> Y and

a diagram, i.e., a functor F: Y --->A, and we would like to compare

the limits of Ff: X --->A and of F: Y ---~A, if they exist. We first

make the following simple observation:

1.5 LEMMA. The assignment F ~ Ff: ob A ~ ---> ob A X and

~--> ef: morph A Y --->morph A X (where (af) = af ) is a functor x (x)

A V --~ A X which we will denote by A f. In particular, we have a

na rural map of sets

AF,G:f AY(F,G) >AX(Ff,Gf). 0

If A 6 ob A, then, in particular, we note that Af(Ay) = Ayf =

If both F: Y ---~A and Ff: X ---~A have a limit, then by the A X �9

352

universal property of the limit of F there is a unique A-morphism

f: lim F > lim Ff such that IFf 1 Ff F = (Ff) X :

lim F > lim Ff f

F f

(Note that in the top row of this diagram we left off the designation

(-)X indicating that we have in fact a diagram in AX; we will con-

tinue to do this if no confusion is likely to arise.)

One observes directly the following Lemma

1.6 LE~. If f: X--~V and F: V--~A ave functors such that

and Ff have limits, then there is a unique morphism

f: lim F --~lim Ff characterized by any of the two properties: F (a) IFfx = IFf(Ff) X

(b) The diagram

A(A,lim F)

A(A,Ff)

A (A, lim Ff)

> AY(Ay,F)

A f

> AX (Ax, Ff)

commutes for all A cob A.

If g: V --~Z and G: Z --~A are functors such that G, Gg, Ggf

have limits, then G(gf) = GgfGg.

Let us briefly pause to illustrate the significance for the con-

crete applications:

If, e.g., GI<--- G2 <--- ... *-- G is a limit diagram of compact

groups, it may become necessary to consider a subsequence

G <-- G <--- ... <--- H with its limit, and to compare the limits. If n I n 2

the original sequence is formalized by the functor Ff: ~---~Comp G,

then the subsequence is described by the functor Ff: ~--->Comp G with

the indexing functor f: ~---~]N, f(k) = ~. The lemma says that

there is a natural map G ---~H with certain properties. Evidently,

one raises the question when this natural map is an isomorphism, as it

should be in this case.

In order to answer this question we return to Lemma 1.6. If

353

Ff: lim F ~ lim Ff is an isomorphism, then clearly

A(A,Ff) : A(A,lim F) ~A(A,lim Ff)

is an isomorphism for all A. By the dual of the Yoneda Lemma (which

one finds in any text on category theory) this condition is also suf-

ficient. By Lemma 1.6, the function A(A,Ff) is an isomorphism iff

Af(~,F) is an isomorphism. On the other hand, if we assume A f that

is an isomorphism, then, in view of the limit property, from the dia-

gram in 1.6 (b) we conclude that lim F exists iff lim Ff exists.

We summarize :

1.7 LEMMA. Consider the following statements

(i) Af: AY (Ay,F) --~ AX (Ax,Ff) is bijective for all A E ob A.

(2) lim F exists iff lim Ff exists and if so, then

f: lim F ---~ lim Ff is an isomorphism. F

Then (i) implies (2), and if one of lim F or lim Ff exists then

(i) and (2) are equivalent. D

It is now clear that one needs a handy sufficient condition for

condition (i) in 1.7 to be satisfied.

COFINALITY

1.8 DEFINITION. A functor f: X ~ V is called cofinal if the fol-

lowing two conditions hold:

(i) For each y eob V there is a morphism f(x) -->y.

(ii) For each pair of morphisms ~j: f(x )3 --->Y in Y, j = 1,2

there is a pair of morphisms ~. : x --~ x. in X, j = 1,2 3 3

such that ~if(~l ) = ~2f(~2 ) .

f(x)

f (x I) f (x 2 )

Y

1.9 EXAMPLE. i) Let X and Y be partially ordered sets and

f: X --->Y a functor (order preserving map). Then f is cofinal iff

354

for each y e Y there is an x with y ~ f(x), and if X is

(upwards) directed.

ii) If X has pull-backs and f has an adjoint g then f is

cofinal. (For each y there is the counit ~ : f(g(y)) --->y of the Y

adjunction; if ~.: f(x ) --->y, j = 1,2 are given, then the adjunc- 3 3 !

tion yields maps ~j: xj --->g(y) ; let

X

x I x 2

g(Y)

be the pull-back and verify 1.8.(ii).) 0

The crucial fact on cofinality is the following:

i.i0 PROPOSITION. If f: X--~V is cofinal, and F: V--->A is a

functor, then F has a limit iff Ff has a limit and if these exist,

then

f: lim F > lira Ff F

is an isomorphism.

Proof. By Lemma 1.7 it suffices to show that Af: AV(Av,F) --+

AX(Ax,Ff) is an isomorphism. Condition 1.8.(i) is readily applied to

show that A f is injective (if ef = Bf, then for each y s ob Y

there is a morphism ~: fx --->y, so that ay = afx = (af) x = (Sf) x =

= By) 8fx ; on the other hand, condition 1.8. (ii) suffices to show the

surjectivity of A f (if u AX --->Ff is given, use 1.8.(ii) to ob-

serve that for each y s ob Y and each morphism ~: fx -->y the

composition (F~)Xx is independent of ~ and defines an a: A~ --->F f

with A ~ = y.) 0

Any example of the type discussed after Lemma 1.6 can be com-

pletely answered by Proposition i.i0. We will now discuss other exam-

ples more systematically. By contrast with our first typical example

of limits occurring in topological algebra, the projective limits, we

355

encounter in here a potentially very large diagram whose limit we

wish to calculate. The tool to achieve this is the cone category.

CONE CATEGORIES

Suppose that we are given a functor J: A --~A (which most o

frequently is the inclusion functor of a full subcategory). We begin

by taking a fixed object A in A and by defining the cone category

(A,J) over J with vertex A: Its objects are pairs

(~,x) { morph A x ob A where ~: A --->Jx, its morphisms o

(~,x) ---> (~,y) are A -maps m: x --->y such that (Jm)~ = o

A

Jx > Jy

x ~y m

The cone category is a special case of the more general eo~a aategory

invented by Lawvere. The assignment which associates with an object

(~,x) s ob (A,J) the element x �9 ob A and with a morphism o

m: (~,x) ---> (~,y) in (A,J) the morphism m: x --->y in A is a o

functor PA: (A,J) --->A . Let us now suppose that 7: A--->A' is a o

morphism in A. Then there is a functor (~,J) : (A',J) ---> (A,J)

given by (~,J) (r = (~,x) and (~,J) (m) = m.

In order to understand how the assignment A ~ (PA: (A,J) --~ Ao )

is functorial we should expand our concept of functor categories. In

the first part of this chapter we considered functor categories AX;

for many purposes this is not sufficient; we have to allow the varia-

tion of the domain category X. Typically, the diagrams PA have the

variable domain (A,J). A similar phenomenon appears if we associate

with a compact group G the system of all closed normal subgroups N

such that G/N is a Lie group. For all of these purposes we need the

category of "all diagrams", barring set-theoretical difficulties.

FUNCTOR CATEGORIES

The objects of the category which we are to describe will be

functors F: X --->A with a fixed category A. In order to avoid set

3 5 6

theoretical qualms we assume that

transforming functors:

i) If f: Y-->X

have an object

X is small. There are two ways of

is a functor and F: X --~A an object, we

Ff: Y --->A and we consider

y Ff ; A

x > N F

as a transformation from

a vertical double arrow

F to Ff (!) which we denote with

Ff

F

ii)

A pair

If G I, G2: Y --->A

transformation G 1

are two objects, then we may have natural

---~G 2 .

(c~,f) = Ff

F

>G

of such transformation we will declare to be a morphism F --->G from

F to G. We have to explain, how these morphisms compose. For this

purpose let (e,f) : F --->G and (B,g) : G ---~H be two morphisms. The

following scheme explains the composition law (B,g) (~,f) =

(8(~g),fg) : F ---->H

Ffg

F

~g

g

> G

>H

357

It is routine to check that we have indeed defined a category. We

record:

i.ii DEFINITION. If A is a category, then A cat denotes the cate-

gory whose objects are functors F: X ---~A with small domains and

whose morphisms are pairs (a,f) : F --~G where f: dom G --~dom F is

a functor and ~: Ff --~G a natural transformation; the law of com-

position is defined by (~,g) (~,f) = (8(ag),fg). The category is

called the general functor category over A. 8

We note that there are variations of this theme obtained by re-

versing various arrows in the appropriate spots. For the purpose of

topological algebra, the version which we introduced is the most con-

venient one. One should note the similarity of the law of composition

in A cat with that of a semidirect product, taking into account the

contravariance in the second argument. This law of composition is

also familiar in sheaf and bundle theory.

We observe that the simple functor categories of i.i are sub-

categories of A cat. For the singleton category A = i we retrieve

the category cat a ~cat of (small) categories and functors.

We observed right in the beginning that the formation of limits

is functorial on the simplest instance of a functor category. Now we

have to see to what extent the limit functor (1.2) is functorial on

the general functor category (i.ii) or a subcategory thereof.

LIMITS REVISITED

Let C be a full subcategory of A cat

Ff > G

(~'f) = fJl in morph C

F

the limits lim F, lim Ff and lim G exist. As an example we might

take for C the subcategory A di-~r of all projective systems (where

dir is the category of up-directed sets and order preserving maps

(see 1.3)) or where C might be the category of all strict projective

systems (1.4).

Suppose that

Ff

(e,f) = f~

F

>G

such that for all morphisms

358

is a morphism F --->G in C. Then we define a morphism

lim(a,f): lim F --->lim G by

lim Ff

lim F

lim > lim G

/ / ~ m (a,f)

(see 1.2, 1.6). A commutative "diagram"

Ffg ag > Gg

Ff ) G

of functor transformations gives rise to a commutative diagram

lim Ffg

Ffgi ' lim Ff

lim a~

lima

> lim Gg

G g

lim G

This remark allows us to verify readily that

lim(B(ag),fg) = lim(B,g) lim(a,f) .

Thus we have:

1.12 PROPOSITION. Le~ C be a full subcategory of A cat such that

for each (f,a) : F -->G in C the limits lim F, lim Ff, lim G

exist. Then the prescription lim (a,f) = (lim a)Ff defines a functor

lim: C -->A.

1.13 PROPOSITION. Let J: A --+A be a functor (with A small). o o

Let cone be the full subcategory of cat generated by all cone

categories (A,J) , A �9 ob A. Then there is a functor A: A --~ A .c~ o

givenby A(A) =PA: (A,J) ---->Ao ' A(~: A --->A') = (iPA, ,(~,S)):

359

(A' ,J)

(~,J)

(A',J)

(A,J) D

1.14 LEMMA. If ~: A--~B is a functor, then there is a functor ~cat: Acat___~Bcat given by ~cat(F) = ~F, ~cat(a,f) = (~a,f) =

~Ff ~- CG

f

~F

If ~ is a full subcategory of catj then ~cat restricts and

corestricts to a functor ~D. D

Suppose that under the conditions of 1.13 the category A satis-

fies the hypothesis that all functors f: (A,J) --->A have a limit for

all A eob A, then the composition

A 'c~ jcone > Acone lim > A o

is a well-defined functor. If A s ob A, then we have a natural

transformation aA: JPA A A(A,j ) ~ given by a(~,x ) =

~: A ---> JP (#,x) = Jx. Hence there is a unique morphism A A

QA: A -->lim JPA = (lim jcone A) (A) such that (IJPA)p A = a , by the

universal property of the limit (i.i). The following is then directly

verified:

1.16 LEMMA. Q: Id A ~ lim jcone A (A) is a natural transformation. If

A ~ Ao then lim jcone A (A) = lim JPJA = JA and PJA = IjA' i.e.,

pJ = i: J --->J.

If J is an inclusion functor, then PA is an isomorphism iff

the cone (A,J) is a limit cone, i.e., iff A & lim JPA" This means

that every object A in A can be canonically approximated by ele-

ments in A . We therefore are led to the following definition: o

1.16 DEFINITION. A functor J: A --->A is called canonically dense o

360

(and if J is an inclusion functor then n is called a canonically o

dense subcategory) if p: Id A ---~lim jcone A is a natural isomorphism.

Here cone is the full subcategory of cat generated by all cones

(A,J), and A is assumed to have limits for all functors

f: (A,J)---~A.

Note that we do not require here that n be small. o

We illustrate this functorial set-up in terms of a basic concept

in topological group theory which we touched upon in the introduction,

the concept of approximating groups via projective limits.

PRO-P-GROUPS

We will speak of a "property" of topological groups while having

in mind properties such as being abelian, finite, a compact Lie group,

a Lie group and so on. In our present framework it is most convenient

to describe a "property" by singling out a full subcategory P of the

category Top G of topological groups. The statement G �9 ob P can

then be formulated as saying "G has property P"; we will also say

that G is a P-group.

1.17 DEFINITION. Let P

the following hypotheses :

(i) All singleton groups are in P.

(2) If G s ob P and H a G, then H E ob P.

Then P is called a property of topological groups, and a group

G s ob P is called a P-group.

For each G e Top G set Np(G) = {N: N is a normal subgroup of

G such that G/N is a P-group}. Note that G E Np(G) because of

(i) above. Evidently G is a P-group iff 1 e Np(G). we say that G

is a pro-P-group if the following conditions are satisfied:

(i) Np(G) is a filter basis.

(ii) If 4: G --->K is a Top G morphism and K is a P-group,

then G/ker ~ is also a P-group (i.e., ker ~ �9 Np(G)).

(iii) The family qN: G --~G/N, N �9 Np(G) is a limit natural

transformation (for the projective system

N --> G/N: Np(G) ~ ---> Top G) m

The essential condition (iii) may be rephrased somewhat briefly in the

form G = lim G/N, N �9 Np(G); by (i) this limit is a strict projec-

tive limit (1.4).

be a full subcategory of Top G. Assume

361

The full subcategory in Top G of all pro-P-groups is called

Ppro" 0

1.18 REMARK. If A is any variety of algebraic structures and Top A

denotes the category of all topological A-algebras and continuous A-

morphisms, one can equally well consider a property P in Top A and

define Np(A) = {R: R is a congruence on A such that A/R is a

P-algebra}; this allows to define pro-P-algebras through conditions (i), (ii), (iii) of 1.17 (with "P-group" replaced by "P-algebra"). D

Note. The category P of pro-P-objects is not to be mixed up with pro

the category of projective systems in a category T which in Mardesi~'

contribution in these proceedings is denoted Pro T; in our own nota-

tion this category would be denoted ~ir where dir is the

category of all up-directed sets (see 1.3, 1.4).

The following is a list of properties in topological group and

algebra theory and their associated pro-P-objects

compact groups

compact Lie groups

almost connected I finite dimensional Lie groups

2 elementary abelian groups

finite groups (rings, lattices, semigroups)

finite semilattices

finite algebras of a given type

compact matrix semigroups

P pro

compact groups

compact groups

contains all almost connected 1 compact groups

contains all locally compact abelian groups

compact O-dimensional groups (rings, lattices, semigroups)

complete algebraic s e m i l a t t i c e s 3

pro-finite algebras of a given t y p e 4

compact pro-matrix semigroups (Peter-Weyl semigroups)

1 A topological group G is almost connected if G/G o is compact, where G O is the component of the identity.

2 A topological abelian group G will be called elementary here if it is of the form (JR/Z) m x ~n • D with natural numbers m, n and a

discrete abelian group D. 3

See [18]. 4 Recall the question we raised in the introduction: Under which con-

ditions is a compact O-dimensional algebra pro-finite?

362

We now show that P is canonically dense in P Let pro ~

J: P--->P be the inclusion functor and P : (G,J) --~P the pro- pro G

jection functor which we introduced in the context of cone categories.

We now define a functor f = f : Np(G)~ ---> (G,J) as follows: g

For objects: f(N) = (qN ' G/N), where qN: G --->G/N is the

quotient map.

For morphisms:

f (M c N) =

G q/ N G/M ) G/N

We claim that f is cofinal (1.8). Firstly, let (~,H), ~: G ---~H

be an arbitrary object of the cone category (G,J). We factorize

(F)

G qker/\ G/ker > H

By 1.17 (ii) we know that ker ~ s Np(G) ; thus (F) constitutes a mor-

phism f(ker ~) --> (~,H). Secondly, suppose that f(Nj) --> (~,H),

j = i, 2 are two morphisms in (G,J). This means that we have a

diagram

It was assumed that Np(G) is a filterbasis; hence there is an

N s Np(G) with N ~ Nj, j = i, 2. Then there is a complementation of

the diagram

7< IN1 / . 91 2

363

which gives the desired diagram

f(N) / -.... f (N I) f (N 2)

By 1.17 (iii) we have G = lim JD G with the projective system

D G = PGf: Np(G) --->Top G. It then follows from i.i0 that lim JPG

exists and is (naturally isomorphic to) G. More precisely, the dia-

gram

PG G > lim JPG

JD = JP f G G

shows that PG is an isomorphism which we had to prove.

This example motivates the following definition:

1.18 DEFINITION. Let Q be a class of functors X --->A with some o

small X. A functor J: n --->n is called Q-dense (and if J is an o

inclusion functor then A is called Q-dense in A) if for each ob- o

ject A in A there is a functor D : X --->A in D and a cofinal A o

functor f: X ---> (A,J) such that D A = PA f and that lim JD A ~ A,

under an isomorphism induced by some natural transformation

~: AX --->JD A. If D is the class of all [strictly] projective systems

then J is [strictly] pro-dense. D

1,19 PROPOSITION. Any D-dense functor is canonically dense.

Proof. Adjust the diagram preceding 1.18 to the general case.

The significance of the concept of pro-density by comparison with

the simpler idea of canonical density is rooted firstly in the preva-

lence of projective limits in topological algebra, secondly and pri-

marily, in the occurrence of functors which are not limit preserving

but do preserve projective limits, thirdly, that in the version of

364

1.18 the question of the smallness of A does not present any worry. o

We illustrate this in the following proposition, which summarizes our

work on the example of P-topological groups:

1.20 PROPOSITION. Let P be a property of topological groups. Then

P is pro-dense in Ppro" In fact, P is D-dense in Ppro for the

class of all projective systems D = {G/N: N e Np(G)~, G e Ppro" ~

With no effort at all this Proposition may be formulated for any

class of topological algebras in place of topological groups by re-

placing normal subgroups with congruences. Recall that its scope is

illustrated by the table following 1.18.

We point out that in the definition 1.18 of pro-density we do Adir

not require that there is a functorial assignment A ~-->D : A ---> - . A o

It is, however, true (and not hard to prove) that in the case of a

property P of topological groups, the assignment G~-->D which we G

used in the discussion preceding 1.18 extends to a functor

p __>pdir in such a fashion that the composition pro

p > ~dir jdi% pdir lim ~ p pro pro pro

category

in A:

(with the inclusion J: P -->P ) is naturally isomorphic to the pro

identity. It was this fact which was used for cohomology calculations

in [14] and [19]. We will see however, that only the information

built into definition 1.18 is needed for these applications I have in

mind.

If A is a category given by the poset {x,y,z} with x, y ~ z

and no other non-equality relations, then the (discrete) full sub-

n containing x, y is canonically dense, but not pro-dense o

z

/\ Xo oy .

Before we conclude this section, we mention in passing a purely

topological situation which together with its dual fits into the

present framework: Let J: Comp ---> Top be the inclusion of the cate-

gory of compact Hausdorff spaces into the category of topological

v spaces. The natural map QX: X --> lim JPx is the Stone-Cech

365

compactification; it is an isomorphism (i.e., a homeomorphism) exactly

if X is completely regular. Dually, if we consider the co-cone

category (J,X) of all pairs (Y,JY---~X) (and the appropriate mor-

phisms), then the natural map colim JP -->X is a homeomorphism iff X

X is a compactly generated space (i.e., a k-space). Thus Comp is

canonically co-dense in k. If X is a weakly separated compactly

generated space (i.e., a t 2 k-space), then the upwards directed

family of compact Hausdorff subspaces of X and their respective

inclusion maps provide a direct (= co-projective) system which is co-

final in (J,X) and whose colimit is isomorphic to X. Thus Comp

is pro-co-dense in kt 2. I pose the question whether or not the t 2

k-spaces are preaisely those k-spaces which are colimits of a direct

system of compact Hausdorff spaces.

CONTINUITY OF FUNCTORS

Just as in topology, it is the preservation of limits under suit-

able functions which makes the limit concept particularly fruitful.

The appropriate functions in our present context are functors.

In 1.6 we considered situations

f

A

and compared the limits

consider the situations

lim F and lim Ff, if they existed. Now we

X

/ A >B

and compare

1.21 LEMMA.

lim D and

F(lim D) and lim FD, if the limits exist.

Let D: X--~A and F: A --~B be functors such that

lim DF exist. Then there exists a unique morphism

F : F(lim D) ~ lim FD D

366

such that ~FD(FD) N = F(I D)

F(lim D) "~ lira FD

FD

Also, the following diagram commutes:

n (A, lim D) F

X F

>~(FA, F lim D)

B (FA,FD) ~ B(FA, lim FD)

> ~X( (FA)! =, FD)

1.22 DEFINITION. Let Q be a class of functors X --~n with small

X. We say that a functor F: A --->B is D-continuous, if the follow-

ing two conditions are satisfied:

(i) For each D �9 D the existence of lim D implies that of

lim FD.

(ii) The morphism FD: F(lim D) --->lim FD is an isomorphism.

We say that a D-continuous F is continuous if O is the class of

all functors (with small domain) and that F is pro-continuous if

is the class of all up-directed sets (check 1.3!) If A is a con-

crete category (relative to a suitable grounding), then F is

strictly pro-continuous if O is the class of a strict projective

systems (see 1.4). []

D

Following 1.35 we will have a list of important functors which

are pro-continuous (or strictly pro-continuous) without being con-

tinuous.

The following Lemma is often useful to determine continuity

properties of functors:

1.23 LEMMA. Let f: X --~Y be cofinal and let D: V --~A be a

functor with a limit. If F: A --~ B is a functor such that

FDf: F lim Df ---~ lim FDf is an isomorphism. Then FD: F lim D --~

lim FD is an isomorphism.

Proof. We operate in the following commuting diagram

367

F D

F (lim D) ~ lira Fd

F ( l i m D~f) > l i m FDf FDf

By the cofinality of f we know that f and f (hence also D FD

F(Df)) are isomorphisms. By hypothesis FDR is an isomorphism. It

follows that F is an isomorphism. D D

As a corollary, we formulate the following proposition:

1.24 PROPOSITION. Let F: A -->B be a D-continuous functor. Then

F preserves the limits of all functors D: Y --~A which allow a

cofinal functor f:X § Y with Df �9 Q ~ D

For example, if F is pro-continuous, then lim FD ~ F lim D

for all D allowing a cofinal f: X --->dom D with X �9 dir.

A core question is now to what extent a continuous (procontinuous)

functor is determined on a dense (pro-dense) subcategory. This is a

uniqueness question. Subsequently one has to answer the question

whether a functor defined on a dense subcategory can be continuously

extended; this is an existence problem.

1.25 PROPOSITION. [14] Let J: A -~A be dense (D-dense). If 0

F,G: n--->B are two continuous (D-continuous) functors, a: FJ-->GJ

a natural transformation, then there is a unique natural transforma-

tion a': F ---~G such that ~ = a'a. In other words,

F ~-~ FJ: B A --~ B A~ induces a bijection

A A

B~ont (F'G)c --> B ~ (FJ,GJ)

Proof. We consider the following diagram involving an object A �9 ob A.

C~ ! A

FA > GA

1 A "~FJP > GJP A eP

A

Ge A

A

368

By the density hypothesis on J, the map QA: A ---~lim JPA is

an isomorphism (1.16 and 1.18), whence Fp, Gp are natural isomor-

phisms. By the continuity assumptions and 1.25 (JPA)F and (JPA)G

are isomorphisms (1.24). The diagram therefore defines a unique a'

whose naturality is readily checked. Since lim JPJA = JA for

A ~ Ao and PJA = IjA , we deduce a'J = a. For every natural trans-

formation a": F ---~G with a"J = a, we have a commutative diagram

I

aA FA 5 GA

FJP A a,,jpA=~p ~ GJP A

in which the vertical arrows are limit natural transformations. It

follows that ~" = a'.

1.26 COROLLARY. Let J: A --->A be dense (O-dense). If F,G: A--->B o are two continuous (D-continuous) functors such that FJ ~ GJj then

F~G.~

Once again, let us draw attention to the fact that many important

functors are [strictly] pro-continuous, but notcontinuous; therefore,

1.25 and 1.26 are a first indication of the usefulness of pro-density.

A couple of applications of this result in topological algebra are

typical. The functor H on the category of Hausdorff spaces into the

category of graded R-algebras over a commutative ring R given by v Cech cohomology and cup product is pro-continuous. In view of 1.20,

we have the following result:

1.27 LEMMA. If P is a property of topological groups satisfying,

then the r cohomology H of pro-P-groups is uniquely determined

(up to natural isomorphy) on the category of P-groups. D

For example, the ~ech cohomology algebra functor of compact

groups is uniquely determined on the category of compact Lie groups.

If B: Comp G --~k denotes the classifying space functor of

Milgram for topological groups into the category of compactly genera-

ted spaces, then B is pro-continuous. We let h = HB and call h

the algebraic cohomology functor; then we have

369

1.28 LEMMA. The algebraic cohomology functor h on compact groups

is uniquely determined (up to isomorphy) on the category of Lie

groups. 0

Another important example arises in duality of groups, if the

duality is implemented by a hom-functor.

The essential feature here is that for a given category n of

topological groups, say, there is a contravariant functor S: A -->B ~

into a category B with a grounding functor V: B -->set such that

VS(G) = A(G,K) with a distinguished object K of A and that sec-

ondly there is a functor T: B ~ --->n such that with a suitable

grounding functor U: n --->set and a distinguished object L of B

one has UT(H) = B(H,L). In many instances there is an isomorphism

U(K) ~ V(L) so that K and L represent "the same object viewed in

different categories A and B." The functor S is assumed to be

left adjoint to T. The prime example is the case A = B = category

of abelian topological groups with SG = Hom(G,~/Z), where the

hom-set is given pointwise addition and inversion and the topology of

uniform convergence on compact sets. In this instance, we take T = S.

(See [16], Chapter 0 for details.)

As a left adjoint, S will preserve all colimits; there is no

category theoretical reason why it should preserve any limits.

Nevertheless, it happens in important situations that it will preserve

certain projective limits. Let us discuss this situation more in some

detail:

1.29 LEMMA. If G is a pro-P-group and Np(G) is the associated

filter basis of closed normal subgroup, then for every neighborhood

u of 1 in G there is an N �9 Np(G) such that N ~ u.

Proof. We consider the morphism g ~-> (gN)N�9 G -->~N G/N. By

hypothesis, it is an isomorphism onto its image which is the set G'

of all (gN)N�9 �9 ~N G/N with ZNM(gM) = gN ' where ~NM(gM) = gN

for M ~ N. Thus the image U' of U is a neighborhood of the

identity; hence there is a finite set F c N such that

G' N T~N\ FG/M is contained in U' by the definition of the product

topology. Since N is a filter basis, there is an N �9 N which is

contained in N F. If g �9 N, then gM = 1 M �9 G/M for M �9 F,

whence (gM)Ms N �9 G' N ~-[N\ F G/M ~ u', and thus g �9 U.

370

i. 30 LEMMA. Let G be a pro-P-group and K

subgroups. Then for any morphism f: G --+ K

such that there is a factorization

f G >K

G/~ fE

a group without small

there is an N 6 NF(G)

Proof. The group K has a neighborhood V of the identity in which

{0} is the only subgroup. Let U = f-l(v). By Lemma 1.29 there is

an N s NF(G) with N s U. By the definition of U we then have

f(U) = {0}. This implies the factorization as asserted. D

Remark. Recall that a Lie group has no small subgroups. In fact, a

locally compact group has no small subgroups if and only if it is a

Lie group. In particular, K = ]R/Z is a group without small sub-

groups.

i. 31 DEFINITION. We say that a function f: X ---> Y between topologi-

cal spaces allows the lifting of compact sets iff f maps the set of

compact subsets of X s~l~jectively onto the set of compact subsets of

Y. B

Note that any proper map certainly allows the lifting of compact

sets.

* = Hom(qN K): Hom(G/N,K) --->Hom(G,K), where the 1.32 LEMMA. Let qN

horn-sets are equipped with the topology of uniform convergence on com-

pact sets. Suppose that K has no small subgroups. Then

Hom(G,K) = U {ira qN: N _c N}. If all q: G ---> G/N allow the lifting

of compact sets, then, Hom(G,K) has the colimit topology, i.e., a

set w is open in Hom(G,K) iff qN -l(w 0 im qN ) is open for all N.

Proof. The fact that Hom(G,K) is the union of the im qN is an

immediate consequence of Lemma i. 30. The topology of Hom(G,K) is

generated by the sets W(C,U) where C is compact in G and U

*-i W open in K. We note that qN ( (C,U)) = W(qN(C),U). If all qN

allow the lifting of compact sets, then every compact set C N of G/N

c qN 1 (CN) is one of the form qN(C) with a compact set C = . The asser-

tion then follows.

Let us apply this to the duality of abelian groups.

371

Let n = Top Ab [resp., k Ab] the category of topological abe-

lian groups [resp., abelian k-groups (i.e., group objects in the

category of compactly generated spaces and continuous maps)] and let

A:A ---~A Op be the functor given by Hom(G, ~/~) with the compact AA

open topology [resp. k(Hom(G, ~/~))]. Let ~G: G ---~G be the

natural transformation given by nG(g) (~) = ~(g). Let n d be the

full subcategory of all G ~ ob A for which n G is an isomorphism. A A

Then is left adjoint to : A ~ --*A and the restriction and co- A

restriction of to A d induces a duality of A d with itself.

1.33 PROPOSITION. If P is a property of A-objects and if all P-

objects are in A d, then any pro-P-object G is in Ad, provided

all qN: G --~G/N, N �9 Np(G) allow the lifting of compact sets

{1.31}.

Proof. Suppose that G is a pro-P-group such that all N e Np(G)

are compact. By i 32 the functor A �9 , : A --~A ~ preserves projective

limits of the type lim G/N, N �9 Np(G). since A: AoP --*A is an

adjoint, it preserves all limits. Hence AA A A AA

(i) (lim G/N) ---~ (colim (G/N) } --->lim (G/N)

is an isomorphism. But by hypothesis on P all morphisms AA

(ii) BG/N: G/N --~ (G/N) AA

are isomorphisms. Since both the identity functor and the functor

preserve projective limits of the type lim G/N (by (i)) we may apply

1.26 to the inclusion functor J of the full category of P-objects in

A into the category of all pro-P-objects G in A for which all

allow the lifting of compact sets and conclude that

(iii) qlim G/N: lim G/N ---> (lim G/N) AA

is an isomorphism. But by the definition of pro-P-objects we know

that G --->lim G/N is an isomorphism. Thus by naturality, from the

diagram

G ) lim G/N

Inlim G/N

nGAA AA G > (lim G/N)

we conclude that ~G is an isomorphism. D

For locally compact abelian groups in particular one applies this

372

with the property P defined by

G s ob P<==>G is a Lie group of the form

~m x (~/Z) n • D with a discrete

group D.

A proof of Pontryagin duality is then obtained by proving the follow-

ing two steps:

(a) Show that every locally compact abelian group is a pro-P-

group with all N E Np (G) compact.

(b) All P-groups G have duality (i.e., n G is an isomorphism

for G �9 ob P).

This program was carried through by Roeder [28]. We will return

to this matter at the end of the second section.

There is, once more, a version of 1.29-1.32 which is applicable

to universal topological algebra rather than to groups. The filter

of normal subgroups N is once again replaced by a filter of congru-

ences N the neighborhood of the identity U used in 1.30 has to be

replaced by a neighborhood of the diagonal in G x G. The object K

in i. 31 is replaced by any compact universal algebra in the class

under consideration which possesses a neighborhood of the diagonal in

K x K without any congruence other than the diagonal itself. With

these modifications in mind, we give the following example:

1.34 PROPOSITION. Let A = Top S1 [resp. kSl] be the category of topological semilattices (idempotent con~sutative monoids) [resp. k-

semilattices] and let ^: A ---~A ~ be the functor given by Hom (G,2)

with the compact open topology [resp. k(Hom(G,]R/~)) ], where

2 = {0,i} is the two element semilattice. Let ~G: G---~G A^ be the

natural transformation given by riG(g) (~) = t(g). Let A d be the

full subcategory of all G �9 ob A for which n G is an isomorphism.

Then ^ is left adjoint to ^: A ~ --~ A and the restriction and A

corestriction of to A d induces a duality with itself.

If P is a property of A-objects and if all P-objects are in

Ad, then any pro-P-object G is in Ad, provided all quotient maps

qN: G ---~G/N allow the lifting of compact sets (1.31). D

This has been applied to establish the duality between the cate-

gory of .discrete and that of compact zero dimensional semilattices by

Hofmann, Mislove and Stralka [18]. The method extends, however, to

373

all those locally compact topological semilattices which are pro-

discrete, among others. For further results on dualities in universal

algebra see e.g., Davey [8].

Let us mention a final example of a pro-continuous functor on

locally compact groups which was introduced recently by Greene [13].

Let A = Loc C G be the category of locally compact groups and

proper homomorphisms, W* the category of von Neumann algebras and

normal *-morphisms.

1.35 PROPOSITION. The functor w: Loc C G --~W* which associates

with a group G its enveloping w*-algebra (the double dual of its

c*-enveloping algebra, whiah in turn is the c*-enveloping algebra

of LI(G)) preserves strict projective limits with proper limit maps.

It is , therefore, determined (up to isomorphism) by its action on

Lie groups. B

Note that once again, W is a left adjoint and would not nor-

mally preserve any limits.

Let us record a list of pro-continuous functors

[respectively, strictly pro-continuous functors,

designated by [s]] which are not continuous:

domain category

spaces

compact spaces

compact monoids

compact groups

locally compact groups and proper morphisms

Top Ab [kAb]

compact semigroups

codomain category

(graded modules [rings]) Op

compact spaces

t 2 k-spaces

compact spaces

von Neumann algebras

Top Ab [kAb]

compact semigroups

functor

~ech cohomology

1 mapping cylinder [s]

universal and classi- fying constructions, E, B

space of closed (top- ologically) subnormal subgroups 2

W(G) = C* (LI (G)) ** [S]

Hom(-,~/~) [kHom(-,~/~) ] 3

Hom(-,{O,l})

i See Hofmann and Stralka [20].

2 See Pommer, [26] .

3 D-continuous in the sense of 1.32.

374

KAN EXTENSIONS

The second basic question which arises in the context is the

continuous extension of functors. The answer to this question is

provided by the formalism originally established by Kan.

One considers, precisely, as in the discussion of density (which

is in fact a special case of what follows) a functor J: A --->A o

which frequently is the inclusion of a full subcategory. Suppose that

we are further given a functor F: A ---~B. The question is whether o

we have an extension of F to a functor F*: n --->B with F*J ~ F.

In general, this is a bit too much to ask. One introduces therefore

the following universal concept:

1.36 DEFINITION. Suppose that there is a natural transformation

F F (~j) B n Bno e : F*J --->F such that the function ~ ~--> e : (G,F*) ---> (GF,F)

is a bijection. (Equivalently, for every functor G: A --~B and

every natural transformation 4: GJ --~F there is a unique natural

transformation 4': G --->F* such that ~ : EF(~'J).) Then F* is

called a (right) Kan extension of F (over J). D

Notice that, apart from set theoretic considerations, F ~-->F*

is a right adjoint to G ~--~GJ. The usual adjoint formalism shows

that a Kan extension, if it exists, is unique (up to natural isomor-

phism).

One has the following existence theorem:

1.37 PROPOSITION. Let J: A --~A and F: A -~B be functors. o

Suppose t h a t p

(lim) lim [(A,J) A > n F ~ B] exists for each A 6 ob n. o

Then there exists a right Kan extension F*: A --~B such that

F*A = lim FP(A,j ) and that eF: F*J --->F is an isomorphism. D

1.38 SUPPLEMENT. The following hypotheses are sufficient for (lim)

to be satisfied:

(i) A is equivalent to a small category and B is complete. o (ii) J is D-dense (i.18) and B is D-complete. D

We notice that a functor J: n -->n is canonically dense only o

i f the i d e n t i t y f u n c t o r 1A: n - -~A i s the Kan e x t e n s i o n of

J: Ao --->A. If lim JP(A,J) exists for all A, then this condition

375

is also sufficient.

The uniqueness of the Kan extension yields the following unique-

ness theorem immediately:

1.39 COROLLARY. Let J: A --~A and G: A ---~B be functors satis- o

fying at least one of the following conditions:

(i) A is equivalent to a small category J is dense, and G o

is continuous.

(ii) J is D-dense and G is D-continuous.

Then G is the Kan extension of GJ. 0

v As an example, we note that the pro-continuous Cech cohomology

functor on compact groups is the Kan extension of the (singular) coho-

mology functor on compact Lie groups. Note that on compact manifolds v

both Cech and singular cohomology agree. The functor W of 1.35 is

the Kan extension of its restriction to the subcategory of Lie groups.

As an example of an application to the existence theorem, we

note the following theorem on the existence of Lie algebras for arbi-

trary pro-Lie groups:

1.40 PROPOSITION. Let Lie be the category of Lie groups (finite or

infinite dimensional) and Lie pro the category of pro-Lie-groups.

Let Lie Alg be the category of locally convex topological Lie alge-

bras over the reals. Then the Lie algebra functor L: Lie --~Lie Alg

is the restriction of a unique functor L: Lie ---> Lie Alg such --?ro

that L is the Kan extension of L. 0

For locally compact groups the Lie algebra functor was directly

constructed by Lashof [22]. Some recent information on the category

of generalized Lie groups was given by Chen and Yoh [2] who developed

a theory I had outlined in my Tulane Lecture Notes on Compact Groups.

376

CHAPTER 2. MONOIDAL CATEGORIES AND FUNCTORS

IN TOPOLOGICAL ALGEBRA

It is natural that categories with some additional element of

structure should play a particular role in concrete applications. In

the first section we have seen this exemplified for categories equipped

with "projective limits". In the present chapter we discuss cate-

gories A with a multiplication i.e. an associative binary functor

: A • A § A; for the most part we will assume also that the functor

is commutative and has an identity, in which case we will speak of a

monoidal category. Monoidal categories open the door to such

applications in topological algebra as duality theories between groups

and operator algebras.

Unfortunately for the exposition, there are some delicate points

in the foundation of the theory of monoidal categories which have to

be explained, although in no application I know of do these fine points

cause serious difficulties. The problem arises if one wants to explain

what associativity, commutativity, identity element means for functors.

The relevant concept of equality of functors is that of natural

isomorphy. In order to have an idea what this means for the definition

of associativity or commutativity, one need only consider the cartesian

products of sets (A • B) • C and A x (B • C) which may be naturally

"identified" but which are not equal; similar things can be said on

the tensor product of vector spaces (A ~ B) | C and A | (B | C).

The background theory which takes care of this problem in the context

of multiplications in a category is the theory of aoherence which is

due to MacLane. We describe a recent presentation of coherence which

illustrated more clearly, that coherence is not a category theoretical

question at all, but one which belongs to the proper domain of univer-

sal algebra and combinatorics. The theory we present is due to

D. Wallace. [31]

COHERENCE

We first focus on the simplest case of an associative multiplica-

tion without commutativity of the presence of the identity; the basic

377

ideas are most easily explained in this case, while all conceptual

complications are already present.

Suppose that ~ is a category and | : ~ x M § ~ is a binary

functor. We assume that | is associative, i.e. satisfies

(M-l) There is a natural isomorphism

a : A | (B | C) § (A | B) | C. A,B,C

Each such morphism is called an associativity map. A reparen-

thesizing of a product of more than three factors can generally be

obtained by a composite application of associativity maps. We consider

the situation of four factors and postulate.

(M-2) The following diagram commutes for all A,B,C,D E ob

((A| | |

aA ~ B , /

(A~B) 8 (CSD)

aA,B,C~B~ A| (B8 (C|

A@a

~ B | ,C

(A| (B| |

ArB | C,D A| ((B| |

B,C,D

This diagram is called the pentagon diagram.

MacLane's first coherence theorem expresses the fact that all

diagrams formed from associativity maps in arbitrarily many factors

commute, provided (M-2) holds. The delicate point is to make precise

what is meant by "all diagrams in M formed by associativity maps".

Moreover, the particular nature of the category M evidently plays no

role in this at all. The problem therefore is to give a presentation

in the framework of universal algebra which does not involve any

particular category.

We begin by considering the free binary algebra <x~ in one

variable. The elements q E <x> are (non-associative) words of n

letters n = 1,2,3,... on the alphabet {x}:

378

n = i: x

n = 2: xx

n = 3: (xx)x , x(xx)

n = 4: ((xx)x)x, (x(xx))x, (xx) (xx),x((xx)x), x(x(xx)) ....

Words are multiplied by juxtaposition. Another important operation is

substitution: Let p,q 6 <x>; then q(i) (p) is the word obtained by

substituting the word p in place of the i-th letter of the word q.

Example: p = xx, q = x(xx), q(1) (p) = (xx) (xx), q(2) (p) = x((xx)x),

(p) = x(x(xx)). The algebra <x> describes formally all possi- q(3) bilities to place parentheses in a word of n letters in a meaningful

way. We now construct a directed graph: The vertices are the points

of <x>; the edges are the pairs (q(i) (ql (q2q3)) , ((qlq2)q3) q(i)

considered as arrows from the first component to the second. We write

q(i) (~ql,q2,q3) for such an edge, and we call the graph which we

produced the graph over <x>. We recall at this point that a groupoid

is a category in which every morphism is an isomorphism. One shows

that every directed graph generated a free groupoid whose objects are

the vertices of the graph and whose morphisms are words in the edges

and their formal inverses, whereby the words represent chains of

arrows which can be concatenated by virtue of the target of the first

arrow being the source of the next and so on. The free groupoid is

indeed characterized by a universal property which makes the assignment

of a free groupoid to a directed graph a left adjoint to the forgetful

functor which considers a groupoid as a directed graph. Evidently,

this description can be formalized (see [23], [31]).

Now let G be the free groupoid generated by the graph over <x>.

An object q(i) (ql(q2(q3q4))) of G has a particular type of auto-

morphism arising from the pentagon diagram, namely

-i

= q(i) (~(qlq2),q3,q4)q(i) (~ql,q2, (q3q4))q(i) (ql~q2,q3,q4)

)-i q(i) (~ql,q2,q 3 q4 )-I q(i) (~ql' (q2q3) 'q4

Every such automorphism is called a pentagon cyole. Evidently, in a

groupoid, a conjugate of an isomorphism ~ is any isomorphism of the -i

form ~ in the groupoid. Now we have Wallace's Theorem which is

explains which the pentagon diagram plays such an important role:

379

2.1 LEMMA. (Wallace's First Coherence Thoerem). Every automorphism in

the groupoid G is a finite composition of conjugates of pentagon

cycles.

The proof is long and technical.

We remark that the groupoid G is the disjoint union of the full

subcategories G n, n = 1,2,... spanned by the words in n letters as

objects. These subcategories are called the homogeneous components of

G; they are in fact the connected components of G. Now let X be n

an arbitrary set or class. Let X = X • ... • X be the set [class]

of n-letter (associative) words on the alphabet X. We form the cate-

gory G * x = (G 1 x x I) o (G 2 • x 2) o .... we can visualize the objects

of G n • x n as n-letter (non-associative) words on the alphabet X.

The category G , X is a free construction in a sense we will explain

presently. First a formal definition:

2.2 DEFINITION. A pair (M,| consisting of a category M and a

binary functor | : M • M § M is called a precmultiplicative category

if (M-l) is satisfied. We then write also (M,e,a) where specifica-

tion of the data is required. It is called a multiplicative category,

if, in addition (M-2) is satisfied.

If (M,| is a premultiplicative category and q ~ <x>,

A 1 ... An 6 (oh M n, then w = q(A 1 ..... A n ) is defined inductively:

If q = ql,q2 then w = ql(Al ..... A n ) M q2(Am+l ..... An). Similarly

§ A~ we define q(~l' .... ~n )" for a morphism ~J: A3 3

2.3 LEMMA. Let (M,e) be a pre-multiplicative category. Then there

is a unique functor F : G , ob M + M such that

(~ql,q 2 )' F(q,AIA2...A n) = q(AI,A 2 ..... An )' and that F(q(i) 'q3

AI'''An) = q(i) (AI ..... Ai_l, ~ql(A,),q(A,,),q(A,,,),Am ..... An) where

A',A",A'" represent appropriate subsequences of the sequence

AI,..-,A n �9

The functor F is called the functor associated with the pre-

multiplicative category (M,|

In our present discussion, MacLane's First Coherence Theorem

takes the following form

2.3 PROPOSITION. Let (M,| be a pre-multiplicative category.

Then the following statements are equivalent:

380

(i) (M,| is mul%iplicative.

(2) The associated functor F : G * ob M +

pentagon cycle to an identity.

(3) The associated functor F : G * ob M § M

automorphism to an identity.

maps every

maps every

Proof. (i) <----> (2) is immediate from the definitions, and (3) ~ (2)

is trivial. But (2) => (3) follows from Wallace's Lemma 2.1.

We have now cleanly formalized the idea of "all possible diagrams

formed by the associativity morphisms": They are represented by the

diagrams in the groupoid G * X; to say that all those diagrams

commute in M is precisely the statement (2).

We now sketch what has to be added if commutativity and identities

enter the picture.

2.4 DEFINITION. A pre-multiplicative category

pre-co~nutative, if

(M,| is called

It is called a commutative multiplicative category, if in addition to

(M-I,2,3) the following condition is satisfied:

(M-4) For all A,B,C eob M,

c~ A @ (B @ C) > (A • B) @ C

A~K i ;K

A | (C @ B) C @ (A @ B)

(A | C) | B - > (C | A) @ B

commutes.

The diagram in (M-4) is called a hexagon diagram.

We construct a groupoid G' whose objects are the elements of

x and whose arrows are those of the groupoid G constructed for

the proof of the first coherence theorem plus the ones generated

freely by these and the arrows q(i) (nql,q 2) : q(i) (q!q2) + q(i) (q2q!) and their inverses.

(M-3) there is a natural involutive isomorphism < : A | § B | 2 A,B

(involutive : < = i) . A,B

381

Analogously to the introduction of the pentagon cycles we introduce a

class of automorphisms of G called the hexagon cycles. Then we have

2.5 LEMMA. (Wal!ace's Second Coherence Theorem). Every automorphism

in the groupoid G' is a finite composition of conjugates of pentagon

cycles and of hexagon cycles. D

Once again, the proof of this Lemma is main burden of the

coherence d for commutative multiplicative categories.

As before, for any set X or class X, we form the category

G' * X = G 'I x X I) o (G '2 • X 2) o ... for each pre-commutative, pre-

multiplicative category (M,| we introduce the unique associated

functor F': G' �9 ob M + M, which extends F and satisfies

F' (q(i) (Hql,q2'Al'''An) = q(i) (AI ..... Ai-l' <A',A" 'Am ..... An) where

A',A" represent appropriate subsequences of the sequence AI,...,A n.

Then we have the Second Coherence Theorem:

2.6 PROPOSITION. Let (M,| ~ <) be a pre-commutative and pre-multi-

plicative category. Then the following statements are equivalent:

(l) ~,| is co~nutative and multiplicative.

(2) The associated functor E' maps every automorphism to an

identity. D

The final build-up for a monoidal category is as follows.

Let M be a category with a bi-functor ~ | ~ + ~. We say that

there are identity elements if

(M-5) there is an object E and there are natural isomorphisms

1 A : E | A --> A and PA : A | E ---> A.

Suppose that M is also pre-commutative and pre-multiplicative;

then we consider the following condition:

(M-6) For all objects A,B ~ M the following diagrams con~nute:

E | (A | B) > (E | A) | B

A| B ,

382

two similar diagrams with A | (E | B), rasp.

place of E | (A ~ B ) , a n d t h e d i a g r a m <

E ~ A > A | E

X / A

A | (B | E) in

2.7 DEFINITION. A category ~ together with a bi-functor satisfying

(M-I,3,5) is a premonoidal category (M,| A monoidal

category is a pre-monoidal category satisfying (M-I,...,6).

We hasten to remark that one might insist that a monoidal cate-

gory should not be expected to be commutative, and indeed there are

significant enough applications to warrant this more general concept

of a monoidal category (compare [23]). However, since the type of

monoidal category we encounter in the practive of topological algebra

are always commutative, we chose this definition as a convenience of

our definition.

In order to formulate the coherence theorem for monoidal

categories, we consider the free groupoid G' constructed above. Let

(X,e) be a set with a distinguished base point. We enlarge the

groupoid G' * X by adding the following morphisms and their inverses

plus all the arrows they generate freely together with the existing

arrows:

(a) : q(i)

t r ) : q(i)

(q(i) (xx),~ .... Xi_lexi...x n) § (q,x I .... xi_ixi...x n)

(q(i) (xx~x 1... x.exl i+l'''Xn ) + (q'xl . . . . ..xixi+ 1 Xn )

Denote the groupoid so obtained by G(X,e). In addition to the penta-

In addition to the pentagon and hexagon cycles in G(X,e) we

introduce four types of additional automorphisms arising from (M-6)

and we call these triangle cycles. Then we have

2.8 LEMMA. Every automorphism in G(x,e) is a composition of

conjugates of triangle, pentagon, and hexagon cycles.

To any monoidal category we associate a unique functor

F" : G(ob ~, E) ~ M which extends G' and maps q(i) (o} to the

383

morphism q(i) (hA) and q(i) (T) to the morphism q(i) (PA.)" 1 1

The coherence theorem for monoidal categories then reads

2.9 PROPOSITION. (Coherence for monoidal categories). Let

(M,| be a pre-monoidal category. Then the following are

equivalent

(i) M is monoidal.

(2) F" maps all automorphisms to identities. D

CARTESIAN CATEGORIES

As a first remark we note

2.10 PROPOSITION. If A is a category with finite products, then

there are natural isomorphisms (A x B) x C § A x (B • C) and

A • B + B x A relative to which A is a commutative multiplicative

category. If A has a terminal object E then projections

E • A § A, A x E + A make A into a monoidal category

Dual statements hold in a category with coproducts [respectively,

with coproducts and initial objects]. D

For the purposes of this exposition, we will call a monoidal

category (A,x) arising from a category with products and terminal

objects a cartesian category (which is not to be mixed up with the

concept of a cartesian closed category, in which A ~> A • B has an

adjoint).

384

We will give a list of monoidal categories which are of

significance in topological algebra:

Category

spaces

pointed spaces

[pointed] k-spaces

R-modules

commutative R- algebra with identity

Banach spaces

C*-algebras

W*-algebras

multiplication

x (product)

v (coproduct)

same

| (tensor product)

8 R

(projective tensor product)

8* (C*-tensor product I )

8W, (W*-tensor product 2 )

identity object

singleton

singletons

same

R

R

C

cartesian

yes

co-cartesian

same

no

co-cartesian

no

no

no

Considering the rather formidable formalities which are involved

to even talk about coherence it may come as a relief to the applied

category theoretician to notice that almost all pre-monoidal categories

arising naturally are automatically monoidal. In other words, the

automatic presence of coherence is the natural phenomenon in concrete

situations.

MONOIDS

Monoidal categories are the abstract setting in which one may

define monoids comonoids and groups. We discuss this in the following:

2.11 DEFINITION.

Then a mono{d in

i) Suppose that (M,X) is a monoidal category.

M is a pair of morphisms

ASA--9--~m A< ~------~

i) Several choices are possible and reasonable; the one with the expected universal properties is defined by A 8*B = C*(A ~ B), where C*(--) is the C*-enveloping algebra.

2) Several choices are possible and reasonable; the one with the expected universal properties is due to Dauns [ 7 ] ; the analysts

generally prefer a spatial version.

385

such that

and that

m is associative, i.e. the following diagram commutes:

A8 (A|

(A | A/I) | A

m | v

A | m

A| "> A@A

> A

u is an identity, i.e. the following diagram commutes:

E | > A| < A|

A

A monoid is com~T~tatiue iff

A~gA AA

~ A@A /m A

commutes.

A morphism of monoids is a morphism f : A§ in ~ such that

m u A| - > A <

f @ f

B | B "> B < n v

E

E

commutes. The monoids in together with the monoid morphisms form a

category Mon M. The full subcategory of commutative monoids is

called MonAb M. A monoid in M ~ is called a comonoid. One defines

Comon M = (Mon Mop)op

386

2.12 LEMMA. If (M, x) is a cartesian category then

A x A < dia~ A const > E

is a (co-)co~utative comonoid, and every comonoid is of this form. D

The associativity and commutativity map in the category allow the

definition of a unique isomorphism

]]A,B,C,D : (A ~ B) | (C @ D) > (A | C) | (B | D)

called the middle two exchange. If m and n are multiplications on

A and B, respectively, one shows that

m~n (A ~ B) ~ (A ~ B) "> A ~ B

(A ~ A) | (B ~ B)

is an associative multiplication on A @ B and that for identities

u : E + A, v : E § B the diagram

PE

u~v E ~ A|

E | E

defines an identity. Thus we have obtained a monoid (m,u) ~ (n,v) =

(m ~ n, u ~ v) on A ~ B. Specifically, one can prove

2.13 PROPOSITION. Mon(M,~) is a monoidal category. D

(Proof e.g. in [17]).

Let us illustrate what we obtained in a couple of examples.

A monoid in the monoidal category of R-modules (over some commutative

ring R) is given by R-module maps

m u A O A > A < R

R

and thus is nothing but a ring with identity. In a similar vein, a

monoid in the category of Banach algebras with the projective tensor

product is a Banach algebra. Thus, if B denotes the monoidal

387

category of Banach spaces with the projective tensor product, then

Mon B is the category of unital Banach algebras (and identity pre-

serving morphisms), and MonAb B is the category of unital commuta-

tive Banach algebras.

Proposition 2.13 seems to open up a cornucopia of monoidal

categories arising from a given one by iteration of the formation of

monoids. We discuss in the following that in reality this situation

is harmless. m u

Form a monoid A @ A > A < E in a monoidal category,

m is a morphism of monoids iff the monoid is commutative. If m u

A | A > A < E is a commutative monoid in M then n v

A 8 A > A < E is a monoid in MonAB M iff m = n, u = v.

In fact we have the following result

2.14 PROPOSITION. [17] Let M be a monoidal category. Then the

categories Mon Mon M, MonAb M, Mort MonAb M are all isomorphic

(which is a bit stronger than equivalent). D

For instance, a monoid in the category of rings with identity and

a commutative ring are one and the same thing. In topological algebra

one encounters a situation where the first link seems to be missing:

Consider the category C* of C*-algebras with identity: Then Mon C*

is isomorphic to the category C*Ab of commutative C*-algebras with

identity and that of commutative monoids over itself, but there does

not seem to be a monoidal category M such that C* & Mon M. A

similar statement applies to the category W* of W*-algebras.

388

LIST of monoids

M

Cartesian

Set

Top

k

Comp

R-modules

Banach spaces

C*-algebras

W*-algebras

x

x

x

| R

(proj .

|

|

Mon Usual monoids in

monoids

top. monoids

k-monoids

comp. monoids

unital R-algebras

unital Banach algebras

k

Mon Mon M M Comm. monoids in

0v

u!

lu

vl

Comm. Unital R-algebras

Comm. unital Banach algebras

V

Comm. unital C*-algebras

< Comm. u n i t a l W * - a l g e b r a s

/

BIMONOIDS

For applications it is important to recognize that monoids and

comonoids occur simultaneously. For this purpose let us formulate the

Lemma

2.15 LEMMA. Let

m u A | A - - ' > A < - - E

be a monoid in M and

c k A| < A > E

be a comonoid. Then the following statements are equivalent:

(ii The following diagram commutes

E E

A | A ~ A ~ A | A

c | m |

A S A | 1 7 4 > A | 1 7 4 1 7 4

389

(2) c and k are morphisms of monoids (i.e. c,k c Mon ~)

(3) (c,k) ~ ob Comon Mon

(4) m and u are morphisms of comonoids (i.e. m,u e comon ~)

(5) (m,u) eob Mon Comon ~. 0

This leads to the following definition

2.16 DEFINITION. A configuration

k c E < A > A ~ A

m u > A < E

in a monoidal category is called a bimonoid if it satisfies the con-

dition in 2.15. We call A the underlying object. If (k',c',m',u')

is a bimonoid with underlying object A', then f : A § A' is a

morphism of bimonoids if it is simultaneously a morphism of monoids

and comonoids. The class of bimonoids and their morphisms is a

category Bim ~.

2.17 PROPOSITION. The categories M on Comon ~, Comon Mon ~ and Bim

are isomorphic. D

Let us observe for the purpose of applications, that in the case

of cartesian categories we obtain nothing new:

2.18 PROPOSITION. If (M,• is a cartesian category, then Bim M

and Mon M are isomorphic.

m u Indeed, for any monoid M x M > M < E the data

const diag m u E < M > M • M > M < E

constitute a bimonoid, and every monoid morphism is a bimonoid

morphism. However, by 2.12 every bimonoid is of this form.

Let us consider an example in a purely algebraic context:

If M is the category of commutative rings with identity then

the tensor product is the coproduct. By 2.18 any comonoid is auto-

matically a bimonoid in M, hence in particular a bimonoid in the

category of abelian groups and the tensor product. Such bimonoids are

also called bigebras.

It is a noteworthy phenomenon that in most natural concrete

examples of bimonoids I know either the comultiplication or the

multiplication is commutative; however, I do not know which conclusion

390

along these lines would follow from the axioms. We will find it easy

after the introduction of multiplication preserving functors to pro-

duce in the monoidal category of (finite dimensional) vector spaces

with the tensor product, bimonoids A which are cocommutative but

not commutative ((finite) monoid algebras), and bimonoids B which

have the reverse property (their duals). Since the category of

bimonoids is monoidal, we can form tensor products A ~ B and obtain

bimonoids which are neither commutative nor cocommutative.

We note that each bimonoid has a canonical involutive endomor- k u

phism of bimonoids p : A § A given by A > E > A. (In the

case of a cartesian category this is the constant endomorphism.)

GROUPS

Very frequently, a bimonoid is enriched by an additional element

of structure which, in tlhe case of the cartesian category Set, makes

the difference between a monoid and a group.

2.19 DEFINITION. Let

k c m u E < A > A@A >A < E

be a bimonoid. An inversion is a morphism s : A § A which satisfies

the following conditions:

i) s is an endomorphism of the comonoid (c,k), i.e.

E <

E < k

k c A > AOA

s 0 s

A ~ > A | c

commutes.

2) A @ A

c I A

s | 1 > A OA

m

> A P

commutes.

One proves the following facts:

391

2.20 LEMMA.

(i) s

[17] Let s be an inversion on a bimonoid 2

is an involution (i.e. s = i).

A. Then

(ii) los

A ~ A > A ~ A

c I I TM

A > A P

(iii)

commutes.

s is uniquely determined (i.e. if

then s = s'.)

s is an antimorphism of the monoid

s, s ' are inversions

(m,u), i.e.

A AY A

s| A| ~ A|

m

> A D

2.21 LEMMA. [17] If A and B

bimonoid morphism, and if s and

respectively, then the diagram

are bimonoids and f : A § B is a

t are inversions on A and B

s A > A

I 1 B > B

t

commutes, i.e. f preserve inversion automatically.

So we are ready for the definition of a group:

2.22 DEFINITION. Let M be a monoidal category. Then a group in

is a bimonoid with an inversion. The full subcategory of groups in

Bim ~ will be called G__rr~. Cogroups are defined dually.

M

PROBLEM. It now seems indeed plausible that the comultiplication in a

group would have to be commutative as a consequence of the axioms, but

I do not know whether this is the case.

If M is the cartesian category of sets, topological spaces,

k-spaces, differentiable (analytic) manifolds, etc. then G__rr M is the

392

category of groups, topological groups, k-groups, Lie groups, etc.

A group object in the monoidal category (R-Mod,| R) of R-modules

over a commutative ring R with identity is called a Hopf-algebra;

the inversion is then called an antipode. Other types of group

objects we will see after the introduction of monoidal functors.

MONOIDAL FUNCTORS

We saw the significance of continuous or procontinuous functors

in the first section; in a similar vein, the true importance of the

concepts which were introduced so far is revealed in the consideration

of functors between monoidal categories preserving the monoidal struc-

ture in one way or another.

2.23 DEFINITION. Let (M,| and (N,o) be monoidal categories (with M N

associativity morphisms a , a etc.). A functor F : M § N is

called left monoidal if there are natural transformations

~A,B : F(A | B) § FA O FB and ~ : FE M § E N

which are compatible with coherence in the sense that various diagrams

involving a, <, h, k as well as ~ and

following is representative:

commute for which the

F(A 1 | (A 2 | A 3)) ~-~-> FA 10 F(A 20 A 3) 10 V > FA 1

F((A 1 | A 2) ~ A 3) ~> F (A I | A 2) O FA 3 ~ O i

O (FA 20 FA3)

> (FA 10 FA 2) O FA 3.

We call F right monoidal, if there are natural transformations

~A,B : FA O FB § F(A | B), ~ : F N § FE M for which the corresponding

diagrams commute. Finally, and most importantly, we call F monoidal

if F is left monoidal and ~ and ~ are isomorphisms.

We observe that if natural transformation ~ and ~ arise in

concrete situations, then they will always be compatible with

coherence. In particular we note

2.24 REMARK. Every functor between cartesian categories preserving

finite products is monoidal. In particular, every additive functor

between abelian categories ~s monoidal. D

393

LIST of monoidal functors

Domain Category

(Set, x)

(Set, ")

(R-Mod, @)

vv

(R-LieAlg, x)

Codomain Category

(R-Mod, | )

(R-Mod, ~)

(Grad R-Mod, |

(R-AssocAlg, | )

Functor

F(=free functor)

iv

A(= exterior alg.)

S (= symm. alg. )

U (= enveloping alg.)

(Comg, x)

(C*Ab I , |

(C*, |

(Comp G, x)

(Mo__~nTop, x)

([Top], x)

(Comp, x)

(R-Mod, |

(R-Mod, 8)

(C*, |

(Comp, x)

(W*, | )

(C-Vect, |

(Top, x )

(Srp, x)

(Grad R-Mod, )

(Set, x)

(Grad R-A! 9 |

x ~> c(x)

Spec

( )**

R(= representative functions)

E(= universal space Milgram)

n V H(= Cech cohom.)

over a field

U(= forgetful f.)

T(= tensor alg.)

Left I Right Monoidal

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ + +

- +

+

CONSTRUCTION OF MONOIDAL FUNCTORS

There are several canonical ways how monoidal functors arise from

other, often more simple ones. The first arises in adjoint situations.

2.25 PROPOSITION. Let M and N be monoidal categories and let

F : M § N be left adjoint to u : N § M. Then the following are

equivalent:

(i) F is left monoidal.

(2) v is right monoidal.

The proof is natural but requires lengthy diagram chasing for the

verification of all of the details. (See e.g. [17]).

The last two lines in the list of monoidal functors exemplify the

situation described in Proposition 2.25 and show that in general one

may not expect much more, although in specific situations one of the

two adjoints may in fact be monoidal (as in the first eight lines of

the list).

394

The second way to produce monoidal functors is by lifting

monoidal functors to monoid categories.

2.26 PROPOSITION. Let U : N § M be a right monoidal functor between

monoidal categories. Then there is a unique right monoidal functor

Mon u : Mon N § Mon M such that for the grounding functor

If: Mon N § N one has IMon u I = u. In a similar way, and left

monoidal functor F : M § N defines a left monoidal functor

CoMon F : CoMon M § CoMon N.

m u For the proof one takes a monoid (re,u) = (B @ B --> B <--EN)

and defines Mon (re,u) = (UB | UB § U(B | B) § UB + UE M § EN). This

assignment is functorial. The assertions of the proposition then have

to be verified in detail by diagram chasing [17].

2.27 COROLLARY. Any monoidal functor F : M + N induces unique

monoidal functors

Bim F : Bim M § Bim N,

Gr F : Gr M § Gr N,

CoGr F: CoGr M § CoGr N.

This set-up is exemplified by the free functor F : Set § R-Mod

which is monoidal as functor (Set, x) § (R-Mod, | ) . The category

Mon Set = Bim Set is the category Mon of ordinary monoids, and the

category Bi___mm(R-Mod) is the category R-Big of R-bigebras. Thus F

induces a functor Bim F :Mon § R-Big which is nothing else than the

monoid algebra functor. Similarly Gr Set is just the category Group

of groups and Gr R-Mod is the category R-Hoof or Hopf algebras,

and the functor G_~r F : Group § R-Hopf is the group algebra functor.

However, the applications demand stronger results. One is the

following:

2.28 THEOREM. Let (M,| and (N,| be monoidal categories and

F : M § N a left adjoint of u : N § M. If F is monoidal, then

Mon F : Mon M § Mon N is left adjoint to Mon U : Mon N § Mo___nnM.

Notice that U is right monoidal by 2.25, whence Mon U is well

defined by 2.26. The proof of the theorem [17] is by verification

through diagram chasing.

A parallel theorem treats the situation that CoMon F is left

adjoint to CoMon U; however, here the situation is more complicated

395

and the proofs are more difficult.

2.29 THEOREM. [17] Let (M,| and (N,| be monoidal categories and

F : M § N left adjoint to u : N § M. Suppose that F is left

monoidal and that the following hypotheses are satisfied: i) N has

pull backs and intersections of countable towers (this is clearly

satisfied if N is complete) ii) | : M • M + M preserves monics

and intersections of countable towers, iii) the natural morphisms

v : UA | UB § U(A | B) and q : E § UE are monic. Then M N

CoMon F : CoMon M § CoMon N has a right adjoint Pr : CoMon N §

CoMon M. Specificallyj there is a function p : CoMon ~ § ~ and a

natural transformation ~ : P § U such that for any

(c,k) = (A | A < c-~- A k> E) 6 ob CoMon ~ We ~ave

P_.r_r(c,k) = (PA | PA <~-~--PA k--> E)

and there is a commutative diagram

PA 8 PA <

UA ~ UA

u(a | a ) < Uc

PA > E

UA U--~---->UE

The functors P and Pr are constructed [17] by a pull-back

procedure, which is modelled after a construction I used in the

duality theory of (compact monoids and) groups to derive functorially

the Mostow-Hochschild Hopf algebra R(G) of a compact group from the

C*-Hopf algebra C(G) (see [15]). The same process was used by

Michor in his contribution to these Proceedings.

Let us illustrate the theorem by a few other examples which are

purely algebraic and therefore are simpler.

a) Let F :Mon § Alg be the monoid algebra functor (which

itself is the lifting of the free functor

fixed ground field K. Then by 2.28, F

underlying monoid functor U : AIg §

F = CoMon Mon§ CoMon Alg , ~ = CoMon F.

F : Set § Vect) for a o

is left adjoint to the

We induce a functor

But Mon is cartesian,

whence CoMon Mon= Bim =Mon by 2.17 and 2.18. Further,

CoMon AI~ = CoMon Mon Vect = Bim Vect = Big is the category of

396

bigebras over K. Thus ~ : Mon § Big is the monoid bigebra functor.

According to Theorem 2.29 it has an adjoint. For a bigebra c k

A ~ A <--A --> K we define PA = {a e A: c(a) = a ~ a, a ~ 0}. Then

PA is a submonoid of UA (the underlying multiplicative monoid of A)

giving an inclusion ~ : PA § UA of monoids, and Pr(c,k) = A

~A x PA < diag PA 9pnst> i), is a comonoid in Mo___n_n, i.e. a bimonoid

in Set (which, in this case, is the same as a monoid by 2.18). The

Monoid PA is called the monoid of monoi~l elements of the bigebra

(c,k) (sometimes also the monoid of primitiue elements, but this is in

conflict with the standard notation in example b) which follows).

b) U : LieAlg § Alg be the universal enveloping algebra functor.

Warning: This functor is traditionally called U, but is a left

adjoint with right adjoint L : AI@ + ~ which associates with an

associative algebra A the Lie algebra defined on the vector space of

A by [a,b] = ab - ba; therefore, the present U corresponds to F

in Theorem 2.29, while L corresponds to U in Theorem 2.29! The

functor U is monoidal as functor (LieAl~, • § (Alg, | and

(LieAlg, x) is cartesian, whence CoMon LieAl~ = LieA!g. Recall

CoMon Alg = Big. We induce a functor U : LieAl~ § Big, which by c k

Theorem 2.29 has a left adjoint. For a bigebra A | A <-- A --> K we

define PA = {a s A : c(a) = a ~ 1 + 1 ~ a}. Then PA is a Lie sub-

algebra of LA, giving an inclusion z : PA § LA of Lie algebras, A

and Pr(c,k) = (PA | PA <~ PA~ k> K) is a bigebra for a suitable

comultiplication ~ and augmentation ~ induced by the diagram

A

PA | PA < PA

LA | LA ~A

L(A | A)< LA Lc Lk

> K

> K

The Lie algebra PA is called the Lie algebra of primitive elements

of the bigebra A.

DUALITY THEORIES IN TOPOLOGICAL ALGEBRA

The framework which we have described in 2.28, 2.29 is at the

root of virtually all those duality theorems in topological algebra

which establish a duality between some category of topological monoids

397

or groups on one hand and some sort of topological bigebras, respec-

tively Hopf algebras on the other; in turn, some of the classical

duality theories such as Pontryagin duality for compact abelian groups,

Tannaka or Hochschile-Mostow duality for compact groups may be deduced

from the former.

As a typical example we note the duality for compact monoids (see

[15]). There is a duality C : Co__~ § C*Ab ~ Spec : C*Ab ~ + Comp

between compact spaces and commutative unital C*-algebras given by the

Gelfand-Naimark formalism. Both functors C and Spec are monoidal

between (Comp, x) and (C*Ab, 8"), Then by 2.25-3.28 there are

dualities

(i)

C : Mon Comp + Mon(C*Ab ~ = (CoMon C*Ab) ~

Spec : (CoMon C*Ab) ~ + Mon Comp,

C : Gr Comp § Gr(C*Ab ~ = (CoGr C*Ab) ~

where, in order to simplify notation, we also write C in place of

Mon C etc. In [15] I called the category CoMon C* the category of

C*-bigebras, and CoMon C*Ab the category of commutative C*-bigebras.

The category CoGr C*Ab of cogroups in C*Ab should be called the

category of C*-co-Hopf algebras. A more general variant of this theory has now been developed by Cooper

and Michor (see [5], [6], and Michor's contribution in these

Proceedings).

Another example, which needs to be fully developed from this

viewpoint departs from the free functor F : TopG § W* which

associates with a topological group the "W*-group-algebra"; this

functor is defined as the left adjoint to the grounding functor

U : W* § TopG which associates with a W*-algebra A the group UA

of all unitary elements of A in the ultraweak topology. I believe

that the functor F is monoidal (TopG, • § (W*, | where | is

the Dauns tensor product for W*-algebras [7]; in the absence of any

proof on record let me formulate this as a conjecture. Given this

conjecture we can carry out an analogue of the algebraic example a)

above: Since (TopG, • is monoidal we have CoMon TopG = TopG. One

then obtains a functor ~ = CoMon F : TopG + CoMon W*. Theorem 2.29

(ii) Spec : (CoGr C*Ab) ~ + Gr Com_p_,

398

m

should apply to show that F has an adjoint. One has to concretely

identify PA in this situation. The objects in CoMon W* have been

called W*-Hopf algebras, some indications are in Dauns' paper; the

full features of this program need to be worked out. Since the all

W*-algebras carry a canonical predual along, the theory is particu-

larly rich in this context since the predu~l A, of an

A E ob CoMon W* is a unital Banach algebra; there are extensive

studies of the "duality" between A and A, on the part of operator

theoreticians notably by Takesaki [29], and Vainerman and Kac [30],

Enock and Schwartz [12], but the functorial aspects have not been

fully investigated.

EXTENSION THEOREMS FOR MONOIDAL FUNCTORS

We conclude our sampling of applications of monoidal functors in

topological algebra by indicating a parallel to the continuous exten-

sion of functors which we discussed in Section i.

For a monoidal category M, the category CoMonAb MonAb M is

denoted BimAb M. we remark that BimAb M shares certain features

with abelian categories.

2.29 PROPOSITION. If M is a monoidaZ category and B = BimAb M,

then for each pair A, B of objects in B the set B(A,B) is a

commutative monoid w.r.t, to the addition defined by

A cl ASA

f+g > B

m

>B~gB fSg

k and to the identity given by A ---> E u_~ B. In other words,

(A,B) ~-->B(A,B) : B ~ • B --->MonAb Set is a functor; i.e., (B,e)

a semiadditive monoidal category. B

with semiadditive categories, we have a matrix calculus (see

Mitchell [24]). We formulate the following definition:

2.30 DEFINITION. If B

matrix category Matr B

(A 1 ..... An ) �9 (ob B) n,

(A 1 ,..., Ag---> (B 1 ..... Bn )

is

is any semiadditive category, then the

is the category, whose objects are n-tuples

n = i, 2, ... and whose morphisms

are m by n matrices

399

fjk ) , f A --->B k ( j=l ..... m, k=l ..... n jk: j plication as composition. D

in B with matrix multi-

The matrix calculus in a semiadditive monoidal category B is

then expressed by the following Lemma:

2.31 LEMMA. If G is a full subcategory of B, then there is a

functor s: Matt G --->B given by S(A 1 ..... A ) = A | ... | A and n 1 n

by a suitable definition for morphisms (which is modelled after Ab). D

DEFINITION. We say that B is freely generated by G ~ B

S: Matr G -->B is an equivalence.

if

Let us tabulate a couple of examples.

List of Additive Categories Which Are Freely Generated

= Bimab

R-Modfin I for R a prin-

cipal ideal domain

Abfin 1

K-Vectfin 1

Comp. conn. ab. Lie groups

loc. comp. ab. Lie groups

ob G

{R/I: I ideal of R}

{cyclic groups}

{K}

{R/z}

{~,~/~, discrete abelian groups}

We then have the following Kan extension theorem whose rudiments

I developed in [14] and which in this form was given by Mostert and

myself in [19]:

2.32 THEOREM. Let (B,e) be a semiadditive monoidal category

B = BimAb M for a monoidal category M which is freely generated by

G s B. Let J: G--~B be the inclusion functor. Then every functor

F: G --~ into a semiadditive monoidal category (C,| has a unique

extension F*: B --~C such that there is a natural isomorphism

~: F*J --~F and that F* is monoidal. ~e function

I index signals finitely generated objects

400

~ ~ (~J) : C B (H,F*) --~ C G (HJ,F) is bijective (i. e., F* is the

left Kan extension of F over J (1.36).

In particular, a monoidal functor from B into some category C =

BimAb N is uniquely determined by its action on G, and is in fact,

the Kan extension of its restriction.

We may combine the extension and uniqueness theorems of Sections

1 and 2 and obtain the following Corollary:

2.33 PROPOSITION. Let M and N be monoidal categories and let

G ~ B be full subcategories of BimAb M such that B is monoidal

freely generated by G and prodense in BimAb B . If BimAb N is

pro-complete, then every functor F: G ---~BimAb N has a Kan extension

F*: BimAb M ---~BimAb N over the inclusion G---~BimAb M, and every

procontinuous monoidal functor G: BimAb M --->BimAb N is the Kan

extension of its restriction to G. The assertion also holds if

BimAb is replaced by GrAb. D

The proof is a simple application of 1.37-1.39 and 2.32.

Let us take for M e.g., the category CompAb of compact con- - - O

nected abelian groups. Then BimAb M = GrAb M = M since CompAb is O

cartesian and cocartesian. For N we take the opposite of the cate-

gory of graded abelian groups. Relative to the tensor product of

graded groups, N is monoidal when given the commutativity involution

: <A,B(a 8 b ) �9 KA, B A O B -->B | A defined by P+q P q = (-l)Pqb q~a p With

this convention, the commutative monoids N are precisely the anti-

commutative graded algebras (which are characterized by aPb q =

(-iPqbqaP). Typical examples are the exterior algebra A M generated

by a module M in degree 1 and the symmetric algebra SM generated

by a module M in degree 2. The category GrAb N is the category of

commutative and co-commutative graded Hopf-algebras Hopf. v

We consider the Cech cohomology functor H: M ---> N and the in-

duced functor GrAb M = CompAb --->GrAb N = Hopf which we will still O

denote with H. Similarly, we have the algebraic cohomology functor

h: CompAb --->Hopf. Both H and h are monoidal and procontinuous - - O

[19]. By the table preceding 2.32, the category CompAb is freely O

generated by the full subcategory containing the single object JR/2,

i.e., by End ~/~. By 2.33, this means that H and h are uniquely

determined by their action on the single object ~/~ and the single

morphism 13R/Z which generates End ~/Z as an additive group. It

401

is not too difficult, to show that the action of H on the circle

group is the same as that of the monoidal procontinuous funetor A A

G ~--~^ G (with G in degree i). It follows from 2.33 that A

H(G,Z) M ^ G as Hopf algebras. Similarly one treats h and arrives

at

2.34 EXAMPLE. The ~ech cohomology integral graded Hopf algebra HG A

of a compact connected abelian group is naturally isomorphic to ^ G A

the exterior algebra generated by G in degree 1. The algebraic

cohomology Hopf algebra hG = HBG of a compact connected abelian

group is naturally isomorphic to s~ the symmetric algebra generated

by ~ in degree 2. D

The details are given in Mostert's and my book on cohomology

theories, where the much more complicated non-connected case is

treated also by similar methods [19]. Example 2.34, however, illus-

trates the remarkable situation that relevant functors may be deter-

mined completely by their behavior on one single object. A similar

situation occurs in one of the more recent proofs of Pontryagin duali-

ty of locally compact abelian groups [28]. Here we consider the

cartesian and cocartesian category of locally compact abelian groups

LCAb which then agrees with GrAb(LCAb). The functor

A: LCAb --->LCAb is additive (i.e., monoidal). Let P be the monoid-

al subcategory of LCAb containing all G ~ ]R TM • (~/~)n x D with

D discrete. By the table following 2.31, P is freely generated by

the full subcategory G containing ~, ~/~, and discrete abelian

groups. In order to see that all groups in F have duality, it then

suffices to know by 2.32, that ~G: G --~G is an isomorphism for

G = IR, ~/Z or discrete. The first two cases are straightforwardly

verified, as is the case that G is cyclic. Hence UG is an iso-

morphism for finitely generated G. The category of finitely gener-

ated abelian groups is co-pro-dense in the category of all abelian

groups Ab. The functor AA: Ab ---~Ab preserves direct limits with

injective maps (proof via 1.32); one concludes through the continuity

arguments of Section 1 that n G is an isomorphism for all discrete

G. Thus by 1.33, the duality theorem for LCA is completely reduced

to the proof of the statement that B is strictly prodense in LCA.

402

REFERENCES

I.

2.

3.

4.

5.

6.

7.

8.

9.

i0.

ii.

12.

13.

14.

15.

16.

Bulman-Fleming S., and H. Werner, Equational compactness in quasi-primal varieties, preprint 1975, 22 pp.

Chen S., and R. W. Yoh, The category of generalized Lie groups, Trans. Amer. Math. Soc. 199 (1974), 281-294.

Choe, T. H., Zero-dimensional compact association distributive universal algebras, Proc. Amer. Math. Soc. 42 (1974), 607-613.

Choe, T. H., Injective and projective zero-dimensional compact universal algebras, Alg. Univ. 1976.

cooper, J. B., Remarks on applications of category theory to functional analysis, preprint 1974, 17 pp.

Cooper, J. B., and P. Michor, Duality of compactological and locally compact groups, preprint 1975, 19 pp.

Dauns, J., Categorical W*-tensor product, Trans. Amer. Math. Soc. 166 (1972), 439-440.

Davey, B. A., Duality theory for quasi-varieties of universal algebras, Dissertation, U. Manitoba 1974.

Dieudonn~, J., Orientation g@n~rale des math@matiques pures en 1973, Gazette des Math~maticiens, Soc. Math. France, Octobre 1974, 73-79.

Eilenberg, S., Sur les groupes compacts d'hom~omorphies, Fund. Math. 28 (1937), 75-80.

Eilenberg, S., and S. MacLane, General theory of natural equiva- lences, Trans. Amer. Math. Soc. 58 (1945), 231-294.

Enock, M. and F. M. Schwartz, Une dualit@ dans les alg~bres de von Neumann, c. R. Acad. Sc. Paris 277 (1973), 683-685.

Greene, W. A., W* preserves projective limits, Preprint.

Hofmann, K. H., Categories with convergence, exponential functors, and the cohomology of compact abelian groups, Math. Z. 104 (1968), 106-140.

Hofmann, K. H., The duality of compact semigroups and c*-bigebras, Lecture Notes in Math. 129, Springer-Verlag, New York, 1970.

Hofmann, K. H. and K. Keimel, A general character theory for partially ordered sets and lattices, Memoir Amer. Math. Soc. 122, 1972, 121 pp.

403

17. Hofmann, K. H. and F. LaMartin, Monoidal categories and monoidal functors, Seminar Notes Tulane University 1971, 103 pp. (limited circulation).

18. Hofmann, K. H., M. Mislove, and A. Stralka, The Pontryagin Duality of Compact O-Dimensional Semilattices and its Applica- tions, Lecture Notes in Mathematics 396, 1974.

19. Hofmann, K. H., and P. S. Mostert, Cohomology Theories for Com- pact Abelian Groups, Dt. Verl. d. Wiss., Berlin and Springer- Verlag, Heidelberg, 1974.

20. Hofmann, K. H., and A. Stralka, Mapping cylinders and compact monoids, Math. Ann. 205 (1973), 219-239.

21. lwasawa, K. On some types of topological groups, Am. Math. 50 (1949), 507-558.

22. Lashof, R. K., Lie algebras of locally compact groups, Pac. J. Math. 7 (1957), 1145-1162.

23. MacLane, S., Categories for the working mathematician, springer- Verlag, New York, 1971.

24. Mitchell, B., Theory of Categories, Academic Press, New York, 1965.

25. Numakura, K., Theorems on compact totally disconnected semigroups and lattices, Proc. Amer. Math. Soc. 8 (1957), 623-626.

26. Po~er, H., ~ojektive Limites kompakter R~e, Topolo~ iO (1971), 5-8.

27. Roeder, D. W., Functorial characterizations of Pontryagin duality, Trans. Amer. Math. Soc. 154 (1971), 151-175.

28. Roeder, D. W., Category theory applied to Pontryagin duality, Pac. J. Math. 52 (1974), 519-527.

29. Takesaki, M., Duality and yon Neumann algebras, in Lectures on Operator Algebras, Lect. Notes Math. 247, Springer-Verlag, New York 1972, 665-786.

30. Vainerman, L. I. and G. I. Kac, Nonunimodular ring groups, and Hopf-von Neumann algebras, Dold. Akad. Nauk SSSR 211 (1973), 1031-1034; Soviet Math. Doklady 14 (1973), 1144-1148.

31. Wallace, D., Permutation groupoids, Dissertation Tulane University, 1976.

LATTICES OF REFLECTIONS AND COREFLECTIONS IN CONTINUOUS STRUCTURES

by

Miroslav Hu~ek, Praha

This contribution reflects recent investigation of reflective

and coreflective subcategories of Unif and Top due to Z.Frolfk,

A.W.Hager, the author and others. The first part is a general study

of reflections and the remaining part deals with special applications

in Top and Unif.

In our approach we shall not look at reflectivity from the stand-

point of subcategories but from the standpoint of functors. The back-

ground for our procedure will be any theory where we may speak about

category of functors between two given categories. (It is true that

a more careful approach will avoid using such categories.) For termino-

logy see [HS2], [~], [If]. We shall use the term topological cate-

gory [He 3] instead of S-category defined in [HUl,2]; we suppose

for the corresponding forgetful functor to have small preimages

{ (i.e., F (A) are sets) . But for initial or final objects from [He 3]

we will use the term projectively or inductively generated objects,

[~], [Hui,2].

I. General part

If not stated otherwise, a subcategory means a full subcategory

(thus reflections and coreflections are onto full subcategories) .

Clearly, every reflector F : A > A determines (up to iso-

morphism) the corresponding natural transformation ~ : 1A > F

and, in the sequel, we shall not distinguish between F and n in

this case. Since all the natural transformations of the identity

functor i A are quasi-ordered by the reflexive and transitive rela-

tion M < M' (i.e., e �9 M = M' for a natural transformation E) ,we get

405

that all the reflectors are quasi-ordered. As one can easily show, this

quasi-order on natural transformations becomes an order if the trans-

formations are reflections or epitransformations. If M : 1A > G,

~' : 1A > G' are natural transformations such that, for a func-

tot U,UM = UM ~ = i, then the relation M < M' is a special case of

an order G < G' from [Hu3].

The main aim of this paper is to construct reflectors and co-

reflectors by means of the above order-structure. To avoid difficul-

ties we shall restrict our consideration to epireflectors. Usually,

there are no difficulties with coreflectors because in most continuous

structures they are lower modifications (i.e., if ~ : F ~ 1A

is the coreflection, then F is concrete and all ~X are identity

mappings of the underlying sets).

Suppose now that n is co-well-powered, has cointersections and

a terminal object

in A and by T (A) (or T ) e e

(thus R c T and T e e e

P r o p o s i t i o n 1. T e

is meet-stable in T . e

T; denote by R (A) (or R ) all the epireflectors e e

all the epitransformations of 1A

is ordered by < ) .

is complete with respect to < and R e

Proof: The identity transformation 1 : 1A > 1A is the

first transformation in T and the constant transformation e

1A > {T} is the last one. If ~ c T , then for each object X e

~X of A, there is a representative set {x ~ FX} of all ~X,~ 6 ~.

Put ~x,GX to be a cointersection of this set. The obvious extension

to morphisms defines a functor G, and ~ : 1A > G is sup

in Te " If ~ 6 Te, ~ : 1A > F, we shall denote

= sup {n~l~ is an ordinal}, where ~o =~' ~+i = F o ~ and_

for limit ~ ' ~5 is the cointersection of {~I~ < ~}. Then ~ is

an epireflection smaller than any epireflection bigger than ~, which

entails the second assertion of our Proposition.

The epireflective modification of ~ will be denoted by ~ or F

406

as in the preceding proof, and similarly ~ , F for monoreflective mo-

difications. Instead of epireflectors we may consider other reflections

as bireflections or upper modifications, with a necessary change of

conditions put on A in Proposition i, e.g. for upper modifications

we need almost that A is a topological category (in this case sup

is just a supremum of structures on the same underlying sets) .

Various constructions of reflections and coreflections are known:

epireflective or coreflective hull of a subcategory, F-coarse or

F-fine structures for a functor F, A-fine structures for a subcate-

gory A , and just recently found reflections F+ and coreflections

F for a functor F. We shall try to bring these constructions to-

gether, to show that they have a common basis. First we shall look at

the above constructions in more details.

(A) The epireflective hull of a subcategory

is the smallest epireflective subcategory in

A of a category

containing A(equi-

valently: the corresponding e~ireflection is the biggest epireflection

in B bein~ identity on A) . It may be proved more: the correspon-

ding epireflection is the biggest epiadjunction being identity on A.

The details of this last assertion and of other results about "biggest"

and "smallest" adjunctions and about factorizations of adjunctions

(decompositions into coreflections and reflections) will appear in

Comment. Math. Univ. Carolinae.

(B) The second above mentioned construction are F-fine structures:

If F : A > B is a concrete functor of concrete categories, we

say that an object A in A is F-fine if A(A,X) = B(FA,FX) for

all objects X in A. In many cases, F-fine objects form a co-

reflective subcategory of A.

If each nonvoid fiber F-I [B] contains an F-fine object, then

F has a concrete left adjoint H; HF is the coreflector onto F-fine

structures and simultaneously the smallest coreflector G 1 with FGI=F

and the biggest coreflector G 2 such that G 2 = F'F for a functor F'

407

It may happen that for a given F both G 1 , G 2 exist even if F

has no left adjoint (then G 1 , G 2 are different). The functor G 2

gives rise the above F-fine coreflector and G the functor F 1

(see (D)).

Example 1 Put A=Unif, ~=Prox,

Both functors G 1 , G 2 exist: G

F the canonical functor A

= i A (see Theoreml , G 2 = pf

> B.

(the proximally fine coreflector - [Po], [Hu2]) .

Example 2 Put A=Unif and let ~ be the category with all uniform

spaces as objects and with morphisms B(X,Y) = Top (X,Y) U {f 6 Set

(X,Y) I card f[X] ~ 2}. Let F be the natural embedding of A into ~.

Then, on X with card X > 2, G 1 is the topologically fine coreflec-

tor, and G 2 is the uniformly discrete coreflector. This example

contradicts the assertion 4.1 (c) , part (iii) in [Ha4].

The F-fine and F-coarse uniform spaces were investigated

mainly by Z.Frol~k and A.W.Hager (see Literature) . From categorical

�9 , . Iv2] point of view they were studied by J.Vlllmovsky in

(C) The most known and probably most important nontrivial example

of A-fine structures are metric-fine uniform spaces [Ha3], i.e.

those uniform spaces X for which Unif(X,M)=Unif (X,tfM) for all

metrizable spaces M (tf is the topologically fine coreflection) .

These spaces form a coreflective subcategory of Unif with the co-

reflector which is the biggest coreflector agreein ~ with tf o__~n

metric spaces. One can easily generalize this procedure to other

classes of spaces and other coreflections - see [Ha 4] for Unif and

[V I ] for general categories. (As just a construction, this method

first appeared in [I3] , [Ke2].)

(D) The last construction mentioned at the beginning was found by

Z.Frolik several months ago. It seems to be important because of

interesting connections with other known reflections. If F : A ~

is a concrete functor of concrete categories, then F is the smallest

coreflector in A such that F F_=F and F+ is the biggest reflec-

408

tor such that F F+=F . In [F 7]

examples, e.g. if A=unif, F=coz (i.e., FX = (X,coz X) , coz X

all preimages of open sets in reals R by morphisms X

then F is the metric-fine coreflector from (C) .

Now the promised general constructions:

Definition Suppose that F : B > K is a functor,

category of A. Then we denote

F (Re) = inf {H 6 R (A) IH B : F' F for an F' , - e

F'>I if F>I},

one can find several important

are

~R),

B is a sub-

F+(Re) = sup {H 6 Re (A) IF = F'H B

F' > 1 if F > 1 },

for an F ' ,

and similarly for monocoreflections

F+(C m) = sup {H 6 Gin(A) I H B = F' F for an F'

F' < 1 if F < 1 },

F_ (Cm) = inf {H 6 Ore(A)IF : F'H B

F'<I if F < I } .

for an F ' ,

In special cases one can give to the functors F+ (R), F+ (C) other

equivalent forms. To make our exposition simpler we shall suppose now

that all the categories and functors investigated are concrete and A

is a topological category; we shall look at upper and lower modifi-

cations instead of epireflections and monocoreflections (briefly

F• (R) , F• (C)) .

Assume first that F > 1 (i.e., FX > X for each X from B,

or equivalently, the identity mapping of the underlying set of X is

A-morphism x ) FX) . Then

F (R) = min {H 6 RIH B = HE} = min {H 6 RIH B > F}

409

2 If, moreover, F = F, then

F (R) = min {H 6 RIHX > sup F-IFX for each X from B}

If F is surjective on objects and B = A, then

F_(R) = min {H 6 RI A(X,H Y) = K(FX,FHY) for

each X,Y }.

Dually for F (C) : +

F+(C) = max {H 6 CIH B = HF} = max {H 6 CIH B < F }

if F < I ,

F+(C) = max {H 6 Cl A(HX,Y) = K(FHX,FY) for

each X,Y} if n = B, F[obj A] = obj K .

we see that if A = B, then F_(R) , F+(C) generalize F-coarse and

F-fine modifications from (B) . If, moreover K = A,F > i, then

F_(R) = F, F+(C) = F (for F,F see the remark following Propo-

sition I) .

If B is a subcategory of A and F a restriction of a modi-

fication in A, then F (R) describe the situations of (A) , (C) .

Put F = ~B ' where ~ 6 R; then

F (R) = min {H 6 RIH B = F } ,

F+(R) = max {H 6 RIH B = F }

and similarly for coreflections. Thus in this case, F (R) is the

smallest upper modification agreeing with F on B and F+(R) is

such biggest upper modification. Specially, if ~ = 1A then F+(R)

is the reflection corresponding to the bireflective hull of B in A.

As for the remaining case n = B, F+(R) , F_(C) , we easily see

that F+(R) = sup {H 6 R I FH = F}

F_(C) = inf {H 6 C I FH = F} ;

no other expressions are known (see (D)).

We shall see in the next part that we can compute almost all the

410

F+(R) , F+(C) provided F is a modification in Top or Unif.

At the end of the first part only several words to nonconcrete

functors F on a topological category A~ In that case F+(C) =sup ~;

if F 6 R and R is a nice subconglomerate of R (e.g. R are all e e

bireflections in A) , then F (R) inf ~ if F ~ R, F (R) = F if

F 6 R and F+(R) is the R-reflective hull of F.

II. UNIF and TOP

In this part we shall try to show how to use the first general

part in special categories. The procedures in Unif will be described

in more details.

First we need a nice class of uniform spaces generating all uni-

form spaces. The class of all metrizable spaces is not convenient for

our purposes because metrizable spaces may be very wild (this is true

for any class epireflective hull of which is Unif) . We shall use spaces

mentioned in [If], [5] the coreflective hull of which is Unif. For

a set P and a filter X on P denote by PX the following uniform

space: the underlying set is (O,i) • P and the uniformity has a base

of covers { (i,p) li 6 (O,i) , p 6 P - X} U { ((O,p) , (l,p)) Ip 6 x},

X 6 X (i.e., PX = inf {PxlX E X}, where PX = Z (0,1) + Z (0,i) p-X X

and (O, i) is uniformly discrete, (O,I) indiscrete two-point space).

Every uniform space (x,U), U the filter of uniform neighbor-

hoods of diagonal, is a quotient of (X • X) U along the map

f(O, (x,y)) = x, f(l, (x,y)) = y, ([Ii], [5]) .

The usage of spaces PX is very wide; we will show their appli-

cations but first their properties:

(i) PX is uniformly O-dimensional (base of decompositions) complete

space; it is Hausdorff iff X is free, the topology of PX is a sum

of indiscrete spaces;

(2) Any strictly finer uniformity on PX is complete uniformly

411

O-dimensional and induces another proximity than PX"

To have the property that ind {Px} = Unif, it suffices to take

only ultrafilters X; as P. Simon noticed, these spaces are atoms in

the ordered set of all uniformities on the set (O,i) • P. There exist

also atoms of another character; the problem of these atoms was inves-

tigated by J.Pelant and J.Reiterman in [PRI, 2] and was shown to be

nontrivial (in contrast to Top) - e.g., if X is an ultrafilter on N

and the uniformity on N has a base of covers (X) U { (n) In 6 N - X},

x 6 X, then this uniform space is an atom iff X is selective.

ter X on a p with

PX 6 ind (Qy) implies

because PX 6 ind (Qy)

f : Q > P.

Before we apply the spaces PX we must know some facts about the

order structure of C and R in Unif. There is the largest bire- e

flector F in Unif smaller than indiscrete I : F = pz = zp, where

p is the precompact modification and z the uniformly o-dimensional

modific~tioi~, and the largest epireflector F different from I,

const : F = h p z, where h stands for Hausdorff; p is the largest

upper modification in Unif H. Between both p and pz, there is a

proper class of different bireflectors. The least bireflector IUnif

is infimum of a decreasing class of bireflectors (e.g. cardinal re-

flectors or point-cardinal reflectors).

As for C, there is the smallest coreflector F = k bigger than

discrete D : F(X,U) , (here U are uniform neighborhoods of diagonal)

has a base nU Again iUnif is supremum of an increasing class

of coreflectors, moreover, there is no maximal coreflector in

C - (iunif) . Indeed, for such a maximal F there is an ultrafil-

FP x = DP X and if Qg is another atom, then

Qy 6 ind (Px) ; but there is no such PX

means exactly that fy = X for an

(a) The spaces PX may be used to give simple examples of pro-

ximities having no finest compatible uniformity: if (X,U) is not

412

proximally fine then the proximity of (x x X) U has such a property

because (X • X) U is a minimal uniformity compatible with its proxi-

mity (use the property (2)) . Even if (x,U) is topologically fine,

it may happen that iX • X) U is not proximally fine (first notice

that if X is e.g. a free ultrafilter on N • N not containing

diagonal and such that PrlX = Pr2X , then PX is not proximally

fine; now take N x N with all its points isolated except (O,O) with

neighborhoods (O,O) U X,X 6 X ; then (N x N • N • N) U , U the

topologically fine uniformity on this topology of N x N, contains

IN x N) X as a retract) . It is proved in [PR 2] that e.g. for selec-

tive ultrafilters on N,N X are proximally fine. Of course, if (x,U)

is metrizable or has a linearly ordered base then (X x X) U has the

same property and is thus proximally fine.

(b) The property (2) implies that there is no coreflector except

identity preserving proximities (answer to a question by J.Vil{movsk~)

Moreover, there is no coreflector except identity preserving uniform-

ly O-dimensional precompact Hausdorff modification.

Next, a coreflector F is not the identity iff FP X = DP X for

an atom PX" These two facts imply

Theorem i. If F 6 R , then F (Re) = F (R) = F and F (C)=I e + - e -

if F # I, const, F_(C) = D if F = I or F = const.

If F 6 C, then F+(C) = F (C) = F and F (R) = I if F # i, - - e

F (R) = i if F = I. - e

The remaining cases F (C) for F 6 R and F (R ) for F 6 C + e + e

can have various values. The only general assertions about them are

the following:

if F is a bireflector, then F+(C) = 1 if F = I, F+(C) = D

if F is not smaller than p, and F+(C) > pf (proximally fine)

if F is smaller than p;

if F is a coreflector, then F+iR ) = I if F = D, e

413

F+ (Re) < p if F ~ D.

For instance, k+(Re) = p and thus the precompact modification can

be defined by means of a trivial cereflection k.

For the category Unif H, the results are almost the same

(one must exclude I from the consideration) .

(c) The spaces PX may be used to a simple proof of the fact

that the only subcategory of Unif which is both reflective and co-

reflective in Unif is the whole category (clearly, every PX must

belong to the subcategory) . In fact, we can prove more:

Theorem 2. Let A be a reflective subcategory of Unif such

that either n contains all uniformly discrete spaces or A is

closed-hereditary. If B is both reflective and coreflective sub-

category of A, then B = A.

The proof of the last result is more complicated and uses modi-

fied spaces PX constructed by means of inverse limits and the pro-

cedure contains special construction of inverse limits of a sink

(not of a source) . Instead of to be closed-hereditary it suffices to

suppose for A that any its object has a base of uniform neighbor-

hoods of diagonal composed of objects of A and, if A c Unif H ,

that a discrete two-point space belongs to A. For details see [HUs] o

I do not know whether these last properties of A imply that A

contains all uniformly discrete spaces or that A is closed-heredi-

tary. Perhaps, it will be interesting to recall examples from [Hu 5]

showing that there are coreflective or reflective subcategories of

Unif containing nontrivial both reflective and coreflective sub-

categories:

Example 3. Let C be the de Groot's strongly rigid metrizable non-

compact space and u a collection of uniformities on C that is

meet-stable in the collection of all uniformities on C. The full

subcategory A(u) of unif composed of all products of uniformities

from u form a reflective subcategory of Unif (the proof is simi-

414

lar to the Herrlich's proof that all the products of C in Top form

a reflective subcategory of Top, [Hell) . The subeategory A(topolo-

gically fine uniformity on c) is both reflective and coreflective

in A(all uniformities on c ) .

Example 4. Let n be the full subcategory of Unif composed of uni-

form spaces (x,U) , U uniform neighborhoods of diagonal, with the

property that the intersection of equivalences from U belongs to U.

If B is the full subcategory of A generated by (X,U) with

n U 6 U then B is coreflective and bireflective in A. Also all

uniformly discrete spaces are cozeflective and epireflective in A.

Example 4 shows that a nice topological category may contain

a nontrivial both coreflective and reflective subcategory (see also

Example 6) . It also implies that results similar to Theorem 2 but for

coreflective A are not so general, e.g. if n is a coreflective

hull in Unif of some complete uniformly Q-dimensional spaces, then

the result of Theorem 2 holds for this A.

(d) We shall look at the similar problem as in (c) for nonfull

subcategorieSo One can deduce from results in [V 2] that if a non-

full subcategory B of A is epireflective in n then there is a

full cor eflective subcategory C of B which is also a full epi-

reflective subcategory of A (this also follows from factorizations

of adjunctions - see (A) in Part I) .

Theorem 3. Let Unif be both coreflective and epireflective

object-full subcategory of a concrete category n. Then either

A ~ Unif or A ~ Set .

Proof: Suppose that Unif is both coreflective and epireflec-

tive in A with F and G the corresponding bireflector or bi-

coreflector in Unif. If F = I then G = D and A ~ Set. If

F ~ I then G = IUnif because FG = F (see Theorem I) and, hence,

F = iUnif ~

The similar result for Unif H was earlier proved by J.Vil{movsky

415

with the only alternative n ~ unif (in the proof always F ~ I ) . H

(e) If F is defined on Unif, then each F-fine space is the

finest member in Unif of all "F-isomorphic" structures on the same

set (i.e., if X is F-fine then X = min F-IFX) . If F is the cano-

nical functor Unif ) Prox, then, conversely, X = min F-IFX

implies that X is proximally fine. That this converse result is

not true in general shows the answer to the question by P.Pt~k for

F to be the modification assigning to each X the uniformity

with the base of all finite-dimensional convers of X . One can

easily prove that all atoms PX have properties FPx = PX ' PX =

= inf F-IFPx If any PX is F-fine then also any its quotient is

F-fine, thus any uniform space is F-fine - a contradiction.

(f) At the end quite different application of spaces PX

For a long time it was an open problem whether product of proximally

fine spaces is always proximally fine. We shall show here an example

that such a product need not be proximally fine. For details of all

results in this section se [HU6].

Example 5. Let (x,U) be a uniform space, S the uniformly

discrete space with the underlying set X • X - (i X) and T be

the set X x X - (IX) U (t) , t ~ X x X , endowed with the fine

uniformity of the topology having only one accumulation point t

with the base of neighborhoods {u - (I X) U (t) ] U 6 U}.

Then (X x X] U is a retract of S x T and, consequently,

S x T is not proximally fine provided (x,U) is not proximally

fine.

We shall only mention positive results. The proofs use facto-

rizations of proximally continuous maps defined on subspaces of pro-

ducts of uniform spaces.

Theorem 4. Suppose that X,Y are proximally fine spaces. Then

X x y is proximally fine provided one of the following conditions

hold:

416

(i) X,Y have linearly ordered bases;

(ii) X is precompact;

(iii) X has a linearly ordered base and intersection of ~ uniform

neighborhoods of 1 is again a uniform neighborhood of 1 Y Y

(here ~ is the least infinite cardinal such that any uniform cover

of X has a subcover of cardinality less than ~ ) .

The part (ii) generalizes the result from [I 2] (both X,Y

and for compact x it was earlier proved by another are precompact)

method in [Ku] .

Theorem 5. A product of proximally fine spaces is proximally

fine iff any finite subproduct has this property.

Thus any uniform space can be embedded into a proximally fine

space (product of pseudometrizable spaces) and, consequently, injec-

tive spaces are proximally fine.

We shall look now at similar procedures in Top. The role of

PX will be played by R X , where R is a set, X a filter on R,

all the points of R except one, say r , are isolated, neighbor- o

hoods of r are (r ) U X,X 6 X Again, if X are ultrafilters o o

then R X are atoms in Top (all the atoms in this case) and

ind {Rx> = Top. The spaces R X are paracompact spaces; they are

Hausdorff iff X is free.

In Rb(TOp) there are the biggest bireflector z in R - (I) e

( z for zerodimensional) and the smallest bireflector s in

R b - (ITo p) ( s for symmetric) . In C (Top) there is the smallest

coreflector F in C - (D) (FX has as a base all the intersections

of open sets from X) and no biggest coreflector in C - (iTop)

For more details see [He2].

By the same methods as in Unif we can prove:

Theorem 6. If F is a bireflector, then F (R) = F (R ) = F, + e e

F (C) = D if F ~ 1 and F (C) = i if F = i, F (C) = I if + + -

F ~ I and F_(C) = D if F = I.

417

If F 6 C , then F+(C) = F_ (C) = F, F+(Re ) = 1 if F ~ D

and F (R) = I if F = D, F (R) = I if F ~ 1 and F (R) = i if + e - e e

F = i0

Similarly for TOPH , TOPunif, etc.

We see that unlike Unif, in Top even F (C) for F 6 + e

and F+(R e) for F 6 C have trivial values.

Theorem 7. Let n be a reflective subcateogry of Top contai-

ning a two-point space and such that any X in A is locally A.

Then any both reflective and coreflective subcategory B of A coin-

cides with A.

Theorem 8. Let Top be both reflective and coreflective object-

full subcategory (in general, not full) of a concrete category A.

Then either A ~ Top or A ~ Set.

Theorem 7 improves the corresponding result from [Ka] (A a

bireflective subcategory of Top ) , where an example of a full subcate-

gory n of Top is given containing a nontrivial both reflective and

coreflective subcategory - but A is neither reflective nor coreflec-

tive subcategory of Top. We shall show that such A may be found

coreflective in Top or epireflective in ToPH (for the proofs and

details see [Hu5]) :

Example 6. Let A be the category of locally connected spaces

and B its full subcategory composed of all spaces the collections

of open and closed sets of which coincide. Then n and B are co-

reflective in Top and B is bireflective in A. If B I are all

discrete spaces then B' is coreflective in Top and epireflective

in n.

The condition on A in Theorem 7 is satisfied if A c TOPReg

and A is closed-hereditary. The next example shows that TOPH

contains an epireflective subcategory with a nontrivial coreflective

and epireflective subcategory.

Example 7. Let n be the epireflective hull in ToPH of

418

S~I+I (S i+i is the space of countable ordinals T 1 together with

~i and neighborhoods of e I are sets (~i) U [~,~i] N T ~ ~i '~ < ~i '

where T ~ are isolated numbers) and let B be the epireflective

hull in TOPH of T 1 Then B is both coreflective and epireflective

in A .

The corresponding Theorem 8 in TOPH is trivial because there is

no nontrivial bireflection in Top preserving the Hausdorff property

[HSl].

We have seen in the previous part that it is important to be

familiar with atoms, coatoms in concrete categories, whether the ca-

tegory is atomic, coatomic, because these facts then imply cortes-

ponding properties in C , R , etc. I would like to mention now one e

i n t e r e s t i n g q u e s t i o n a b o u t t h e o r d e r s t r u c t u r e o f R ( T O P H ) , T h e r e e

is the biggest epireflection const (onto singletons) , the biggest epi-

reflection ~ in R - (const) (onto compact O-dimensional spaces) , o e

a maximal epireflection ~N in Re (const, ~o ) (onto N-compact

spaces) - see e.g.[He2] . There was a problem on existence of other

maximal epireflections in Re (const, ~o ) . R.Blefko, [B] , proved

that the epireflection onto T - compact spaces is not maximal. e l

J.Pelant, [Pe], proved that any epireflection onto T - compact e l

spaces is smaller than a maximal epireflection in R - (const, ~ ) e o

and in [HP] it was proved in addition to it that any epireflection

in TOPH onto a subcategory containing a uniformizable space which is

not strongly countably compact (closure of a countable subset is not

compact) is smaller than a maximal epireflection in R e (const, ~o ) .

Moreover, these maximal epireflections were characterized as those

onto X-compact spaces where N c X ~ B N, ~X N = X (just take for X

the given epireflection of N) . There is at least 2 2~~ of such

maximal epireflections. For higher cardinals we were able to prove

only one implication (----~) - one must add here a condition that X

contains closures in B D , D discrete, of all its subsets A with

419

card A < card D. The other implication was proved only in special

cases because in our proof we need that Souslin numbers of certain

subsets of X are card D, which is not true in general.

with T , there is connected also another problem: Is any w 1

O-dimensional'perfect image of a T compact space again T - w I w 1

compact? For the motivations and connections see [Hu4]. The authors

of [RSJ] disproved my conljecture that any such image of T is again w 1

T ; they characterized all perfect images of IT ] and one can w I w 1

prove from their result that all are T - compact. Perhaps it will be w 1

of help to notice that a O-dimensional • is T - compact iff any w I

maximal filter of clopen sets in X with linear wl-intersection

property is fixed ( X has linear wl-intersection property if any de-

creasing subcollection {X ID < w I} of X has a nonvoid intersection) .

At the end we mention some facts about the order structure of the

conglomerate R of all reflections in Top . The situation at the

bottom is the same as in R (Top) , i.e., the epireflections onto e

symmetric spaces of To-Spaces are minimal reflections in R-(ITop) ,

moreover, any reflection different from the identity follows one of

the two minimal reflections. Further, there is no reflective subcate-

gory strictly between TOPT 1 and ToPT or TOPT and TOPT o 1 o

The top of R was described by H.Herrlich in a letter to the author

(May 1974) . The situation is quite different from the top in R e

(in R there is a counterpart to the bottom: {singletons} is strictly e

followed by {indiscrete spaces} and {O-dimensional T -spaces} and o

these two classes are strictly followed by {O-dimensional spaces}).

There is a proper class of maximal reflections in R , (const) : any

strongly rigid T2-space gives rise to such a reflection (onto powers

of X - see [Hel]) . The corresponding categories are composed only of

connected T2-spaces. That there is a proper class of strongly rigid

T2-spaces was proved in [KR] , [T] ; perhaps it is worth to mention

some of the Trnkova's results because not all of them are published:

420

There is a strongly rigid proper class of paracompact connected and

locally compact T2-spaces or of unions of compact and metrizable

spaces; under the assumption (M) there is a strongly rigid proper

class of metrizable spaces or of compact T2-spaces. Now back to R

There is only one maximal reflection in R - (const) onto a subcate-

gory containing a nonconnected space: onto compact O-dimensional

To-Spaces. There is only one maximal reflection in R - (const) onto

a category containing a non-To-space: onto indiscrete spaces. H.Herr-

lich added a question whether there are other maximal reflections in

R - (const) . I can add only those onto powers of a strongly rigid

T I- space, because the proof in [He I ] works also for Ti-spaces.

An example of a strongly rigid Tl-space which is not Hausdorff was

communicated to me by V.Trnkov~ (take four disjoint subcontinua of

the Cook continuum, in any of them pick out two points

i ,i ai,bi, i=l,...,4, and double a 1 into a 1 , a 1 ; now put to-

gether a I' and a 2 , al" and a 3 , b 1 and b 4 , b 2 and b 3 and

a4) . Almost all the preceding results follow from the following easy

considerations (n is a reflective subeategc~y of Top with the re-

flection ~) :

(a) If there is an X 6 A , x , y 6 X such that the subspace (x,y)

is indiscrete, then ~ is bireflection and A contains all indis-

crete spaces;

(b) If there is an X 6 A, x, y 6 x such that (x,y) is connec-

ted To, then all ~y are projectively generating surjections and

A ~ ToPT ; o

(c) If there is an X 6 A, x, y 6 x such that (x,y) is discrete,

then all ~y , Y O-dimensional, are embeddings and are dense if

(x,y) 6 A ; then n contains all compact O-dimensional T -spaces. o

421

[B]

[6]

C F 1 ]

IF 2 ]

[ F 3 ]

[F 4 ]

[F 5 ]

[F 6 ]

[r 7 ]

[ Ha I ]

[ Ha 2 ]

[ Ha 3 ]

[ Ha 4 ]

Blefko R.:

v Cech E. :

Frolfk Z.:

Frol{k Z.:

Frolfk Z.:

Frol~k Z.:

Frol~k Z.:

Frol{k Z.:

Frol{k Z.:

Hager A.W. :

Hager A.W. :

Hager A.W. :

Hager A.W. :

L I T E R A T U R E

Some classes of E-compactness, Austr.Math.J.

(1972) , 492-500.

Topological spaces, Academia ]Prague 1966 (revi-

sed edition by M.Kat~tov, Z. Frol~k)

Basic refinements of uniform spaces. Topology

Conf. Pittsburgh 1972, Lecture Notes in Math.

378 (1974, 140-158.

Three uniformities associated with uniformly

continuous functions, Symposia Math.

Interplay of measurable and uniform spaces,

Top. and its AppI. Budva 1972 (Beograd 1973) ,

98-101.

Locally e-fine measurable spaces, Trans. Amer.

Math. Soc. 196 (1974) , 237-247.

A note on metric-fine spaces, Proc. Amer. Math.

Soc. 46 (1974), 111-119.

Measure-fine uniform spaces, Seminar Abstract

Analysis 1974 (preprint) .

Cozero refinements of uniform spaces, Seminar

Uniform Spaces 1975 (preprint) .

Three classes of uniform spaces, Proc. 3rd

Prague Top.Symp. 1971 (Academia Prague 1972),

159-164.

Measurable uniform spaces, Fund. Math. 77

(1972), 51-73.

Some nearly fine uniform spaces, Proc.London

Math. Soc.28 (1974), 517-546.

Vector lattices of uniformly continuous

functions and some categorical methods in uni-

form spaces, Topology Conf. Pittsburgh 1972,

Lecture Notes in Math. 378 (1974), 172-187.

[ i ] 2

[ z 3 ]

[ Ka ]

[ K R ]

[ Ke 1 ]

[ Ke 2 ]

[ ~ ]

[ Pe ]

[ PR I ]

[ PR 2 ]

[ p o ]

[ RSJ ]

422

LITERATURE

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Isbell J.R. : Structure of categories, Bull. Amer. Math.

Soc. 72 (1966), 619-655.

Kannan V.: Reflexive cum coreflexive subcategories

in topology, Math. Ann. 195 (1972),

168-174.

Kannan V. : Constructions and applications of rigid

Rajagopalan M. : spaces I (preprint) .

Kennison J.F. : Reflective functors in general topology

and elsewhere, Trans. Amer. Math. Soc. 118

(1965), 303-315.

Kennison J.F. : A note on reflection maps, Ill. J. Math.

11 (1967) , 404-409.

K6rkov~ V. : Concerning products of proximally fine

uniform spaces, Seminar Uniform Spaces

1974 (Prague 1975), 159-171.

Pelant J.: Lattices of E-compact spaces, Comment.

Math. Univ. Carolinae 14 (1973) , 719-738.

Pelant J.:

Reiterman J. :

Atoms in uniformities, Seminar Uniform

Spaces 1974 (Prague 1975), 73-81.

Pelant J.:

Reiterman J. :

Which atoms are proximally fine?

Seminar Uniform Spaces 1975 (preprint) .

Poljakov V.Z.: Regularity, products and spectra of

proximity spaces, Doklady Akad. Nauk

SSSR 154 (1964) , 51-54

Rajagopalan M. : On perfect images of ordinals, Report of

Soundararajan T.:Memphis St. Univ. 74/16

Jakel D.:

[ R 1 ] Rice M.D.: Metric-fine uniform spaces (to appear)

[ Be 1 ]

[ He 2 ]

[ He 3 ]

[ HS I ]

[ HS 2 ]

[ Hu I ]

[ HU 2 ]

[ Hu 3 ]

[ Bu 4 ]

[ Hu 5 ]

[ Hu 6 ]

[HP ]

[ I i ]

Herrlich H. :

Herrlich H.:

Herrlich H.:

Herrlich H.: Strecker G.E. :

Herrlich H. : Strecker G.E.:

Hu~ek M.:

Hu~ek M. :

Hu~ek M. :

Hu~ek M.:

Hu~ek M.:

Hu~ek M.:

Hu~ek M.:

Pelant J.:

Isbell J.R. :

423

LITERATURE

On the concept of reflections in general

topology, Proc.Symp. Berlin 1967 (Berlin

1969), 105-114.

Topologische Reflexionen und Coreflexio-

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Topological structures, Math.Centre

Tracts 52 (Amsterdam 1974), 59-122.

H-closed spaces and reflective subcate-

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Category theory, Allyn and Bacon (Boston,

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S-categories, Comment. Math. Univ. Caro-

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Categorical methods in topology, Proc. 2nd

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190-194.

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applications, Comment. Math. Univ. Caro-

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Perfect images of E-compact spaces, Bull.

Acad. Polon. Sci. 20 (1972), 41-45.

Reflective and coreflective subcategories

of Unif and Top, Seminar Uniform Spaces

1974 (Prague 1975), i13-126.

Factorizations of mappings on products of

uniform spaces, Seminar Uniform Spaces

1974 (Prague 1975), 173-190.

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dence 1964) .

424

[R 2 ]

IT]

[v I ]

[v 2 ]

Rice M.D.:

Trnkov~ V.:

Vll• J. :

"" " ~vl-lmovsKy J. :

LITERATURE

Metric-fine, proximally fine, and locally

fine uniform spaces (to appear).

Non-constant continuous mappings of metric

or compact Hausdorff spaces, Comment.

Math. Univ. Carolinae 13 (1972) , 283-295.

Generation of coreflections in categories,

Comment. Math. Univ. Carolinae 14 (1973) ,

3O5-323.

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to reflective subcategories, Seminar

Uniform Spaces 1974 (Prague 1975), 83-111.

Pro-categories and shape theory

by

~, I) Sibe Mardeslc

Shape theory is a modification of homotopy theory created

with the scope of obtaining a theory more applicable to spaces

with bad local properties. In systematic manner the theory was

initiated by K.Borsuk in his talks delivered at the Symposium

on infinite-dimensional topology, Baton Rouge, Louisiana, 1968

[4] and the Topology symposium, Herceg-Novi, 1968 [I]. His

first technical paper was [2]. Borsuk wanted a coarser clas-

sification than the homotopy type which would make e.g. the

"Polish circle" equivalent to the circle. Instead of maps be-

tween metric compacta X, Y embedded in the Hilbert cube Q ,

Borsuk considered fundamental sequences of maps

(fl,f2...) :X~-~Y . These are sequences of maps fn:Q--~Q,nE N,

such that every neighborhood V of Y admits a neighborhood U

of X and an integer n v with the property that fnlU ~fml U

in V for n,m~ n V Fundamental sequences are composed coor-

dinatewise. Two fundamental sequences (fn),(gn) :X--~Y are

considered homotopic provided every neighborhood V of Y ad-

mits a neighborhood U of X and an integer n V such that

fnl U~gnl U in V for n~ n V . The relation~is an equi-

valence relation.

1)Presented at the Conference on Categorical Topology,

Mannheim, 21.-25. VII, 1975.

426

Every map f : X-~ Y admits an extension ~ : Q--gQ and

(f,f,...) is a fundamental sequence X--~Y whose homotopy

class is independent of the extension f. In particular, the

identity I : X--~ Y determines a class of fundamental se-

quences (I) : X--*X. Two compacta X, yc Q are said to

have the same shape, sh(X) = sh(Y) , provided there are fun-

damental sequences (fn) : X--~Y , (gn) : Y~-+X such that

(gn) (fn)-~ (I) , (fn) (gn) ~--- (I) . If X and Y have the same

homotopy type, X--~Y , then sh(X) = sh(Y) . Borsuk has also

shown that for ANR's sh(X) = sh(Y) implies X~Y . The

notion of shape does not depend on the embeddings of X and

Y in Q .

In 197o the author and J.Segal have noticed that the main

notions of Borsuk shape theory admit a rather elegant de-

scription using inverse systems of ANR's [14J , [153 . The essen-

tial reason for this is that X and Y are intersections of

closed neighborhoods in Q which can be chosen in such a

manner that they are ANR's. This approach was also developed

for compact Hausdorff spaces.

W. Holszty{ski gave the first axiomatic description of the

shape category of Hausdorff compacta h8j. This is a category

having all Hausdorff compacta for objects, the morphisms are

shape maps and correspond to Borsuk's homotopy classes of

fundamental sequences. Two Hausdorff compacta have the same

shape provided they are isomorphic objects in the shape

category.

427

A shape theory for metric spaces patterned after Borsukls

approach was deviced by R.H. Fox [7].

In 1973 the author has described the shape category ~ for

topological spaces [1o I . The same category has been descri-

bed independently by G.Kozlowski (unpublished), J.H Le Van

~9] and C. Weber [19].

The objects of ~ are all topological spaces. In order to

define the morphisms of ~ called shape maps, one considers

the category ~whose objects are all spaces having the homo-

topy type of a CW-complex and the morphisms are homotopy

classes of maps. One considers the functors EX,.~ : ~-~ Enss,

EY,.3 : ~J~--~ En~s, where [X,~ , P~Ob ~J~, denotes the set

of homotopy classes of maps X-~P . A shape map F:X--~Y

is a natural transformation [Y,.~---+ EX,.3 In other words,

f assigns to every homotopy class ~ ~ ~Y,~ a homotopy

class f(n) ~ [X,~ . If ~' ~ [y,pl~ , ~ [p, ,~ , ~' : n ,

then ~f(n I ) = f(n).

Recently K. Morita E16] has noticed that the notion of a

shape map in the above sense can be described using inverse

systems in the category ~ essentially in the same way as

in the ANR-system approach to shape of Hausdorff compacta

of J.Segal and the author (~I~ and [I~ , Section 7). This

can be described very conveniently using the notion of pro-

category of a given category. We follow here E11~ where

these notions are described in sufficient generality for our

purposes.

428

Let ~ be an arbitrary category. With ~ one associates

a new category pro (~) whose objects are all inverse

systems ~ = (Xl,pllj ,i ) in ~ over all directed sets

(i ; { ) A map of systems X-~Y = (Yu,quul , M) con-

sists of a function f : M--~ A and of a collection of

morphisms f

~ ~ there is a

= qua' fz' Pf (z') 1

(f' f' ) : X --~ Y l

for each u & M

---~ Y , ~ ~ M , in ~7~ such that for : Xf(u)

I >~ _{(u) , f(u~ ) such that f~pf(~)l=

Two maps of systems (f; f ),

are considered equivalent provided

there is a I >i f(~) , f/ (~) such that

f pf = f' pf Morphisms f : X--~Y in pro (~)x ~ (~)~ �9 _ _ _

(~) are equivalence classes of maps of systems

(f) f ) : __X--~ Y__ �9 If --g : _Y--+~-- = (~, rv~ ,N ) is given

by (g]gv) , then the composition _gf : X-~ ~ is given by

(fg] g fg(v)) The identity ! X : X-~ X is given by

(I A '~ Ixx )

Generalizing the situation encountered in [14] and [8]

Morita calls an inverse system X = (Xl,pll, , A) in ~0 ~,

i.e. an object of pro (~) , associated with a topological

space X provided there exist homotopy classes of maps

Pl : X--~X 1 such that I ~ l' implies Pl =plll pl a and

the following two conditions hold for every P~ Ob (~.e)

(i) For every homotopy class m~ ~X,P~ there is a I E A

and a homotopy class mle ~XI,P 3 such that m = mlp 1

(ii) Whenever mlp I = m'lpl ' ml ' m/le-~Xl' P~ ' then there

is a I'~ I such that mlpll , = mllpll I

429

In other words, the mapping

Dir lim (EXI,P~ , pl~i ~ , A) --+ ~,P]

induced by (PI' I ~ A) is a bijection.

It is not difficult to see that there is a natural bijection

between shape maps X--~Y and morphisms ~ : X-~Y in pro

(~) , where ~ and ~ are systems associated with X and Y

respectively E16~.

Every topological space X admits an associated system X

namely the ~ech system which consists of nerves of locally

finite normal coverings D6]. It is however important to be

able to use other associated systems as well. E.g., if X

is the inverse limit of an inverse system ~ of compact ANR's

in the category Top, then X is associated with X D4]. Also

if X is embedded as a closed subset in an ANR for metric

spaces, then the open neighborhoods of X form a system asso-

ciated with X [16]. This is the reason why the Fox approach

yields the same notion of shape for metric spaces.

In a similar way one can define the shape category of pairs

of spaces, of pointed spaces or pointed pairs.

With every inverse system X = (XI' PlI' , A) in ~ one

can associate homology pro-groups. These are the inverse

systems of groups Hm(X) = (H m (Xl) ,Plli~ , A) , hence, objects

of pro (Grp). If X and X' are inverse systems in ~ asso-

ciated with a space X, then Hm(X) and Hm(X') are

430

naturally isomorphic pro-groups, i.e. isomorphic objects

of pro (Grp). Therefore, one can define homology pro-groups,

of spaces X as homology pro-groups of associated systems

X and they are determined up to a natural isomorphism.

Clearly, isomorphic pro-groups have isomorphic inverse

limits but the converse is not true. E.g., the pro-group

2 2 ~c ~ ~-- .....

where 2 denotes mulitplication by 2 is not isomorphic to

{o} although both have the inverse limit o. The inverse

limit of the homology pro-group Ilm(X) is the usual ~ech

homology group Hm(X). Homology pro-groups are finer in-

variants than homology groups. E.g., H I of the dyadic

solenoid vanishes but the corresponding pro-group is non-

trivial.

In a similar way one defines homotopy pro-groups ~m(X,x)

and their limits called shape groups. In shape theory the

homotopy pro-groups play the role of homotopy groups in

homotopy theory of CW-complexes.

For these reasons it is of interest to study the category

pro (Grp) of pro-groups. This is a category with zero-objects,

i.e. objects which are simultaneously initial and terminal.

Such is the system _0 = {0} consisting only of one trivial

group. In general the pro-group G = (G I, plkl ,A) is a

zero-object if and only if each IeA admits a 1 ~ ~ 1

such that pkl~ = 0 In pro-groups there exist kernels and

cokernels but pro (Grp) is not an exact category.

431

Nevertheless, one can speak of exact sequences of pro-groups.

f A sequence G ~> H ) K is exact at H provided gf = 0

and in the unique factorization f = if I , where i is the

kernel of g, the morphism f' is an epimorphism.

One can prove that for pointed pairs of spaces (X,A,x) the

corresponding homology and homotopy sequences of pro-groups

are always exact [11].

The author [12] and K.Morita [17] have proved independently

that for pro-groups isomorphisms and bimorphisms coincide.

In various situations it is important to be able to decide

when is a morphism of pro-groups a monomorphism or an epi-

morphism. In [12] the following necessary and sufficient

condition for ~ : G--~H generated by

(f,f~) : (G~,p~, ,A) ~ (Hp,q~p, ,M) are given:

(i) f is a monomorphism if and only if

(V ~GA)(~ ~ ~ ~)(~ ~'~ ,f(~))

-I PlI' (f~Pf(n)l ') (I) = I

(ii) f is an epimorphism if and only if

( V D E M) (V ~ ~ f(u)) (~ ~'~ )

qua, (H , ) c f pf (G) - ~ (~)~

One of the most important applications of pro-groups is the

Whitehead theorem in shape theory:

Let ~ : (X,x o) --+ (Y,yo) be a shape map of connected topo-

logical spaces having finite covering dimension.

432

If f induces an isomorphism of homotopy pro-groups

~m(X,x)--~m(Y,y) for all m, then ~ is a shape equi-

valence.

The theorem was first proved by M.Moszy{ska for metric

compacta [I~ . Her proof was simplified and also extended

to cover the case of topological spaces and shape maps

generated by continuous maps by the author ~11] . Finally,

the general result was obtained by Morita [I~ .

The assumptions that X and Y be finite-dimensional cannot

be omitted as shown by a counterexample due to J.Draper

and J.Keesling [5] . In their example X and Y are infinite-

dimensional metric continua and f is generated by a conti-

nuous map.

Recently D.A.Edwards and R.Geoghegan [6] have proved an

infinite-dimensional Whitehead theorem for shape theory.

Their result asserts that f is a shape equivalence pro-

vided it induces isomorphisms of homotopy pro-groups, X

and Y are metric continua, Y has the shape of a CW-complex

and X is movable. Movability is an important shape inva-

riant notion introduced by K.Borsuk in ~ (also see ~I Z ) .

An important corollary asserts that a map f : (X,Xo)--*(Y,y o)

of metric continua such that sh (f-1(y)) = sh (point), for

every y~ Y , is a shape equivalence provided (X,x o) is

movable and (Y,yo) has the shape of a CW-complex.

Institute of Mathematics

University of Zagreb / Zagreb, Yugoslavia

References:

433

[ I ] K. Borsuk: Concerning the notion of the shape of compacta.

Proc. Intern. Symp. on Topology and its Applications.

(Herceg-Novi 1968), Belgrade 1969, pp. 98-1o4.

[2] ........ : Concerning homotopy properties of compacta.

Fund. Math. 62 (1968), 223-254.

[3] : On movable compacta. Fund. Math. 66 (1969),

137-146.

[~ : On homotopy properties of compact subsets

of the Hilbert cube. Ann. Math. Studies 69 (1972),25-36.

J. Draper and J. Keesling: An example concerning the

Whitehead theorem in shape theory. To appear in Fund.Math.

[~ D.A. Edwards and R. Geoghegan: Infinite-dimensional White-

head and Vietoris theorems in shape and pro-homotopy.

To appear in Trans. Amer. Math. Soc.

[7] R. H. FOX: On shape. Fund. Math. 74 (1972), 47-71.

[8] W. Holszty~ski: An extension and axiomatic characteriza-

tion of Borsuk's theory of shape. Fund. Math. 70 (1971),

157-168.

[~ J.H. Le Van: Shape theory. Thesis, Univ. of Kentucky,

Lexington, Kentucky, 1973.

[ lo] S. Marde{i6: Shapes for topological spaces. General Topo-

logy Appl. 3 (1973), 265-282.

: On the Whitehead theorem in shape theory I.

To appear in Fund. Math.

[12] : On the Whitehead theorem in shape theory II.

To appear in Fund. Math.

434

[i 3]

[14]

[1 5]

[I 6]

[I 7]

%" . i

S. Mardeszc and J. Segal: Movable compacta and ANR-systems.

Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 18

(197o) , 649-654.

: Shapes of compacta and ANR-systems. Fund. Math.

72 (1971), 41-59.

: Equivalence of the Borsuk and the ANR-system

approach to shapes. Fund. Math. 72 (1971), 61-68.

K. Morita: On shapes of topological spaces. Fund. Math.

86 (1975), 251-259.

: The Hurewicz and the Whitehead theorems in shape

theory. Sc. Rep. Tokyo Kyoiku Daigaku, Sect. A 12 (1974),

246-258.

[1 8-]

[I 9]

M. Moszy~ska: The Whitehead theorem in the theory of

shapes. Fund. Math. 8o (1973), 221-263.

C. Weber: La formed' un espace topologique est une

compl~tion. C.R. Acad. Sci. Paris, S&r. A-B 277 (1973),

A 7-A 9

A note on the inverse mapping theorem of F. Berquier

P. Michor

We show that the notion of strict differentiability of [I], w IV is

rather restrictive. In fact, we give a complete characterization of

strictly differentiable mappings and use it to give a short proof of

the main theorem of [I]. Notation is from [I], we only remark, that

X is a finite dimensional C ~ manifold and C(R,X) is the space of

continuous realvalued functions on X with the Whitney C ~ topology.

Theorem I: Let r C(R,X) * C(R,X) be strictly differentiable at

o s C(R,X) . Then there exists an open neighbourhood V o of ~o in

C(R,X) and a continuous function f: ~ * R , where O is a suitable

open neighbourhood of the graph of ~o in X~R such that

r = f(x,~(x)), x 6 X for all �9 6 V and furthermore the map o

f(x,.) is differentiable at ~(x) for all x s X and

(D~(~o)h)(x) = df(x,.)(~o(X)).h(x) , x ~ X for all h ~ C(R,X).

If ~ is furthermore differentiable in V ~ (cf. [I], w then

f(x,.) is differentiable in O O {x}• and df(x,.) is continuous on

each point of ~o(X).

Remark: The theorem says, that each strictly differentiable mapping

9: C(R,X) * C(R,X) looks locally like pushing forward sections of

the trivial vector bundle X~R by a suitably differentiable fibre

bundle homomorphism. Of course each such map is strictly differen-

tiable, so we have obtained a complete characterization.

Proof: First we remark that the topology on C(R,X) can be described

in the followig way: C(R,X) is a topological ring and sets of the

form Vr = { g E C(R,X) : Ig(x)l < ~(x) , x 6 X} are a base of open

436

neighbourhoods of 0 , where ~: X * R is strictly positive and con-

tinuous.

Now by definition IV-I of [I] we may write in a neighbourhood of O o

~(g+h) - ~(g) = D~(Oo)h + R(g,h) where R satisfies the following

condition: For each V~ there are V 8 , V~ such that R(g,hk) E h.V c

for all g E O ~ + V 5 , h E V~ and k E C(R,X) with Ik(x) l ~ I, x E X .

Let V e = V I , k = I , then there are V 5 , V~ such that R(g,h) E hV I

for all g E O ~ + V~ , h E V~. Let V o = �9 6 + ( V~ 0 V~/2 ).

We claim that if O1, O 2 E V ~ and x E X such that ~1(x) = O2(x)

then ~(O1)(x) = ~(O2)(x). This follows from the equation

~(O I) - ~(O 2) = D~(Oo)(O I - 02 ) + R(~2, O I - 02 ) , since

[Dr I - O2)](x) = (01 - O2)(x).[Dr = 0 and

R(O2,O I - 02) E (01 - O2).V I , so R(O2,~ I - ~2)(x) = 0 .

If �9 E C(R,X) denote the graph of �9 by X~ = { (x,~(x)): x E X }.

Let O = U { X~ : ~ E V ~ }. By the form of V o it is clear that 0 is

�9 For O E V ~ define fo: X~ * R by an open neighbourhood of X~o

fo(x,~(x)) = {(~)(x). By the claim above we see that we have

fol X~ 0 X~ = f~l X~ N X~ if ~ and ~ are in Vo, so we have got a

mapping f: 0 * R , and {(O)(x) = f(x,O(x)) for all �9 E V ~ and x E X.

We show that f is continuous�9 If (Xn,t n) * (x,t) in 0 ~ X• we may

choose a sequence O n * ~ in C(R,X) such that (Xn,On(Xn)) = (Xn,tn),

O(x) = t (remembering that a sequence converges in the Whitney C ~

topology iff it coincides with its limit off a compact set K of X

after a while and converges uniformly on K ). But then r ) * r

uniformly, and x n * x , so r n) = f(Xn,t n) * @(O)(x) = f(x,t).

Now we show that f is differentiable at each point of X~ if ~ is

differentiable at �9 (strict differentiability implies differentia-

bility, see [I]). We have r h) - ~(O) = D@(O)h + r~(h), where r O

is a "small" mapping ([I], w i.e. for each V~ there is V 6 such

437

that r~(h) E h.Vr for all h E V 5. Evaluating this equation at x we

get f(x,~(x) + h(x)) - f(x,~(x)) = [D~(~)(1)~(x).h(x) + r~(h)(x) .

It is clear that the map h(x) * r~(h)(x) is o(h(x)) by the "smallness"

of r~ , so f(x,.) is differentiable at $(x) and [D$(~)h](x) =

= df(x,.) (~(x)).hCx) .

It remains to show that df(x,.) is continuous at each point of X~o.

This follows easily from Proposition IV-2 of [I~ wi~kh the method we

just applied to show that f is continuous, qed.

Theorem 2: Let $: C(R,X) * C(R,X) be differentiable in a neighbour-

hood of ~o E C(R,X) and strictly differentiable at ~o and suppose

that D~(~ o) is surjective. Then there exists a neighbourhood V o of

~o and a neighbourhood W ~ of ~(~o ) in C(R,X) such that ~: V o * W o

is a homeomorphism onto. Furthermore the map ~-I: Wo , Vo is differen-

tiable on Wo, strictly differentiable at $(~o ) and for each ~ E V ~

we have D(@-I)($(~)) = (D~(~)) -I.

Proof: By theorem I we have that @(~)(x) = f(x,$(x)) and D@(~)(S)(x) =

df(x,.)(~(x)). Since D@(~ o) is surjective we conclude that

df(x,.)($o(X)) ~ O for all x E X , and since df(x,.) is continuous

at ~o(x) it is ~ O on a neighbourhood of ~o(X) in R. Writing fx = f(x,.)

we see that f~1 exists and is differentiable on some neighbourhood

of ~($o )(x) in R by the ordinary inverse function theorem. So the

map (x,t) * (x,f(x,t)) is locally invetible at each point of the graph

X~o in XxR such of ~o ; one may construct a neighbourhood O of X~o

that this map is invertible there (considering neighbourhoods

U x x V~o(X ) of (x,$o(x)) where Id x f is invertible and taking m

O = ~ U x • V~o(X ) ). Then @-1(~)(x) = fx1(~(x)); all other claims

of the theorem ase easily checked up. qed.

438

Remark: Theorem 2 is a little more general than the result instil.

The method of proof is adapted from [2], 4.1 and 4.2 where we

treated an anlogous smooth result.

References m

[I~ F. BERQUIER: Un theoreme d'inversion locale, to appear in the

Proceedings of the Conference on Categorical Topology,

Mannheim 1975.

[2~ P.MICHOR: Nanifolds of smooth maps, to appear.

P. Michor

Mathematisches Institut der Universit~t

Strudlhofgasse 4

A-IO90 Wien, austria.

CARTESIAN CLOSED TOPOLOGICAL CATEGORIES

L. D. Nel

In section 1 we take stock of categories from general topology which

admit straightforward axiomatic description and are cartesian closed. Several

new ones have recently come to light, all of which are definable by filter

axioms. Section 2 discusses topological categories, in the sense of Herrlich.

The axioms, which blend initial completeness with simple smallness conditions,

allow a rich theory including an efficient charaterization of cartesian closed-

hess. Categories of spaces satisfying a separation axiom cannot form a topo-

logical category but may be included in a more general theory of initially

structured categories. This is what section 3 is about. The next section

discusses sufficient conditions for a reflective or coreflective subcategory

of a cartesian closed topological category to inherit cartesian closedness.

We conclude with a consideration of possiblities for the embedding of a given

concrete category into a cartesian closed topologicai category.

Generally speaking our terminology will follow the book of Herrlich and

Strecker [29]. Subcategory will mean full and isomorphism-closed subcategory.

Recall that a category with finite products is called cartesian closed when for

any object A the functor A• has a right adjoint, denoted by (_)A. The cate-

gories in which we are interested always have structured sets as objects and

for them cartesian closedness means that for given spaces A,B,C there is al-

ways a function space B A available, structured strongly enough to make the

natural evaluation function A• A § B a morphism and at the same time weakly

enough to ensure that for any morphism f:A• § B the associated function

f*:C § B A is also a morphism. For the definitions of the categories Con, L~n,

PsTop and F~ see the appendix; for PNea~ and SNeaA see Herrlich E63].

440

Cartesian closed categories with simple axiomatic description

The lack of natural ruction space structures in Top makes it an awkward

category for several theories such as homotopy theory and topological algebra.

Steenrod [55] and MacLane [48] have advocated its replacement by the cartesian

closed category k-Hour5 of compactly generated Hausdorff spaces (=k-spaces =

Kelley spaces). Dubuc and Porta [20] demonstrated convincingly how topological

algebra (particularly Gelfand duality theory) benefits from being cast in k-Haul.

For related work see also Binz ~7] where the cartesian closed category LJJn is

used and Franke [23] where an approach via abstract cartesian closed categories

is studied. The advantage of a cartesian closed setting is already illustrated

by the formation of function algebras with suitable structure. Whereas in Top

the search for a suitable topology on an algebra of functions A § B would not

always be successful, the availability of a categorically determined power

object B A ensures that the "right" topological structure for the function

algebra is obtained by embedding into B A.

In recent theories of infinite dimensional differential calculus Top

has largely been replaced by L~ or k-Hau~ , see the papers by Frolicher and

Seip at this conference and also Frolicher and Bucher [24], Keller [40],

Machado [46], Seip [53]. For use in topology cartesian closed replacements

for Top have been suggested by Spanier [54], Vogt [56], Wyler [61]. It is a

pity that some of these suggested replacements of Top are awkward to describe

axiomatically (e.g. k-Haul) while Spanier's Quasi-topological spaces have the

smallness problems to be discussed later. So it seems of interest to list a

few cartesian closed categories with simple axiomatic description and no atten-

dant smallness problems: Con (Kent [42], Nel [49]), Lim (Bastiani [3], Cook

and Fischer [14], Binz and Keller [8], Fischer [22], Kowalsky [44land others),

PsTop (Choquet [16], Machado [47], Nel [49]), F~ (Katetov [39], Robertson [52]).

441

Cartesian closed categories within the realm of nearness spaces were

recently discovered and studied by Robertson [52] and Bentley, Herrlich and

Robertson [5]. Gr/// is the subcategory of SNear formed by the objects whose

near families are all contained in grills (recall that a family of subsets is

a grill if all are non-empty and a union of two sets belongs to the family iff

at least one of the two sets do). Now Grill is a cartesian closed coreflecti~e

subcategory of SNear. It is equivalent to the category F// and contains the

category of proximity spaces as a bireflective subcategory. It also contains

suitably restricted convergence spaces as a coreflective subcategory, namely

those that satisfy the following axiom:

R 0 If a filter F converges to x and y belongs to every member of F,

then F converges to y.

This axiom by the way, reduces to the usual R 0 axiom (x is in every neighbour-

hood of y iff y is in every neighbourhood of x) when restricted to topological

spaces. The category RoCon thus defined is again cartesian closed and being

bireflective in Con and coreflective in Gr/~/ it provides a link between con-

vergence and nearness structures. In similar fashion one obtains two further

cartesian closed categories RoLim and RoPST, bireflectively embedded in L/m

and PsTop respectively and also bireflectively embedded in RoCon.

R.M. Vogt ~6] remarked that "many topologists dislike working with things

that are not topological spaces". The nice properties of the above categories,

in particular the simple form that the usual categorical constructions take

in Con, Lim and Gr/// make it seem possible that Vogt's remark will become less

true in future.

2 Topological categories and cartesian closedness

The categories Con, Lim, PSTop, Top, SNear, Grill along with a multitude

of others share many categorical features. These can usefully be studied in

terms of an abstract category satisfying certain axioms.

442

A is called a topological category if it comes equipped with a faithful

functor U:A § Set such that

T1 A has initial structures for all sources to UA and

T2 for any set X the fibre U-Ix has a representative set of objects

and when X has cardinality 1 its fibre is represented by just one object.

The first axiom is a straightforward abstraction of the well-known existence

of a smallest topology on a domain making a given source of functions into

topological spaces continuous. Its fundamental role has been recognized and

exploited by Bourbaki [9] and a host of others e.g. Antoine[l], Bentley [4],

Brummer [13], Hoffmann [30,31], Kamnitzer [37], Wischnewsky [57,58], Wyler [59].

The smallnes condition (T2) formulated by Herrlich [28] seems to be a very

suitable companion for TI. It is simple to check in special cases, does not

exclude any category of interest in general topology and yet is strong enough

in conjunction with initiality to yield a rich theory. A striking example of

how successfully T1 blends with T2 is the following result.

Theorem (Herrlich [28]) For a topological category A the following statements

are equivalent:

1 A is cartesian closed

2 the functor A• always preserves colimits

3 the functor Ax- always preserves coproducts and quotients

4 the functor Ax- always preserves final epi-sinks.

("always" means for any ~object A; a quotient is a final epimorphism)

Topological categories are stable under formation of bireflective and

coreflective (automatically bicoreflective) subcategories. Examples of an

apparently non-topological origin include the category of bornological spaces

(bounded sets are axiomatized, see Hogbe-Nlend [32]) and of pre-ordered spaces,

both of which are in fact also cartesian closed. Topological categories are

(co-)complete with U preserving both limits and colimits and (co-) well-powered.

443

They automatically have final (i.e.coinitial) structures.

ties see Herrlich [28] and the next section.

For further proper-

3 I n i t i a l l y structured categories

The category of Hausdorff topological spaces is not a topological

category since it lacks initial structures for sources that are not point

separating. Nevertheless this category has many features in common with topo-

logical categories. In view of the importance of categories formed by spaces

satisfying a separation axiom it seems worthwhile to have a similar abstract

theory for them. To this end we relax axiom T1 by demanding instead:

TI* A has initial structures for mono-sources to UA.

In the terminology of Herrlich [26] this means that U is an (epi, monosource)-

topological functor. If we think of A-objects as structured sets it means that

for any class of objects (Xi,~i) I and functions f.:Xl § X.I the quotient set Q,

obtained by collapsing points not separated by the fi' has a smallest A-struc-

ture available for which all the induced functions Q § X. are morphisms. i

A significant portion of the theory of topological categories genera-

lizes although some statements are complicated by the necessity of passage to

a quotient set. It is no longer true that embeddings coincide with regular

monomorphisms and also with extremal monomorphisms, but final epi-sinks do

still coincide with extremal epi-sinks and quotients with extremal epis and

also with regular epis. Well-poweredness, completeness and co-completeness

are still with us but U no longer preserves co-limits. Factorization properties

abound: every initially structured category is an (epiu, embedding)-category,

(epi, extremal mono)-category, (epiu, initial monosource)-category, (quotient,

mono)-category and a (final episink, mono)-category. Here epi U denotes the

class of e such that Ue is an epimorphism and (E,M)-category is used as in [29].

444

By using the above facts one can show that Herrlich's characterizations of

cartesian closedness given in section 2 remains valid for initially structured

categories.

Initially structured categories have stability under formation of sub-

categories that is better than that of topological categories. They are not

only closed under coreflective subcategories (which incidently are characterized

by being closed under formation of colimits or equivalently under final epi-sinks)

but also under all epireflective subcategories. For the results of this section

and further details see Nel [49].

closed?

(a)

Subcategories that i nhe r i t cartesian closedness

When is a subcategory B of a cartesian closed category C again cartesian

For reflectLve subcategories we have the following:

(Day [18]) Suppose B is reflective in the monoidal category C and that

the reflector R:C + B preserves finite products, Suppose also that B is dense

in C (i.e. adequate in the sense of Isbell [38]). Then B inherits cartesian

closedness from C.

(b) (Robertson [52]) Suppose C is a topological category and B is a

bireflective subcategory whose reflector R satisfies R(B• = B• for all B

in B and C in C. Then B inherits cartesian closedness from C.

(c) (Nel [49]) Suppose C is initially structured and B is a quotient-

reflective subcategory. Then B inherits cartesian closedness from C.

For coreflective subcategories there is a similar result:

(d) (Nel [49]) Suppose C is initially structured and B is a coreflective

subcategory closed under finite products in C. Then B inherits cartesian closed-

ness from C.

As a useful corollary of (d) we note that if K is any finitely productive sub-

category of C, then its coreflective hull inherits cartesian closedness from C .

445

5 Embedding into cartesian closed topological categories

The categories Con, Lim, PsTop discussed in section 1 were apparently

not intrduced with cartesian closedness in mind: in each case this property

was discovered several years later. But there have been deliberate constructions

to create cartesian closed categories for use in topology. The first of these

was the category ~uasi-Top introduces and studied by Spanier [54]. Quasi-Top

does not satisfy axiom T2 and thus is not a topological category. In fact an

object whose underlying set has more than one point may be undefinable in terms

of sets. The same disadvantage is present in similar later constructions by

Antoine [i] and Day [18].

Other constructions of cartesian closed categories were carried out

within a given special category. Thus Vogt [56] and Wyler [61] studied the

embedding of compact Hausdorff spaces into cartesian closed subcategories of Top.

Antoine [I] and Machado [47] studied the embedding of Top into cartesian closed sub=

categories of L/m. ln particular Machado showed that Antoi~e's epltopologieal

spaces formed the smallest such category between Top and L/m. Bourdaud [10,62]

obtained corresponding results for an embedding into L~m of pretopological

spaces (see also Bourdaud's paper at this conference).

Embeddings of suitably restricted abstract categories into cartesian

closed topological categories are being studied by H. Herrlich and myself. We

conclude with a preliminary report about this. Suppose A to satisfy T2 and the

following conditions: A has quotients, finite products preserved by U and in A

the product of two quotients is a quotient. Let us call such A preconvenient.

The category A* is now constructed as follows. Its objects are pairs (~,X)

where ~ is a set of pairs (A,a) such that a:UA § X is a i-I function subject

to (I*) if UP is a singleton, then (P,p)c~ for any p:UP § X; (2*) if (A,a)~

and a = coUq where c is I-I and q:A § C is a quotient, then (C,c)~. Morphisms

from (~,X) to (n,Y) are functions f:X § Y such that for any (A,a)~ we have

446

(B,b)E~ where (B,b) is the unique pair such that foa = boUq with q:A § B

a quotient .

Then A* is a cartesian closed topological category into which A can be

embedded as a subcategory so that existing initial structures and powers are

preserved. The coreflective hull of A is all of A*. If A is already a

topological category then it is bireflective in A*.

The preconvenient categories to which this construction applies include

PNear, SNear, the category of finite topological spaces, the category of Top-

quotients of compact Hausdorff spaces.

By considering a concrete category A with embeddings and by using onto

functions a:X § UA one can also construct an embedding A § A' where A' now turns

out to be a preconvenient category containing A (if A is topological it is in

fact coreflectively embeddedl. Thus any topological category can be fully

embedded (in two steps) into a cartesian closed topological category. Unfortu-

nately this embedding need not preserve initial structures.

However if some embedding of A into a cartesian closed topological

category C is known to exist such that C coincides with the coreflective hull

of A, then A is contained in a smallest cartesian closed topological subcategory

B of C. In fact B can be constructed as the bireflective hull in C of all

C-powers formed out of A-objects. Thus the embedding of A into B preserves

initial structures and moreover it preserves powers. In the special case A =

Top, C = Lx~n one obtains as a corollary Machado's result mentioned above.

Appendix. A convergence space [42] is a pair (X,q) where q is a function which

assigns to each x in X a set qx of filters on X "convergent to x"; moreover

the following conditions must hold: (FI): every principal ultrafilter ~ is in qx;

(F2) : if F is in qx and G refines F, then G is in qx; (C): if F is in qx, then

FAk is in qx. A limit space [44], [22] is a convergence space in which (C) is

strengthened to (L): if F,G are in qx, then FAG are in qx. A pseudo-topological

447

8paoe El6] is a limit space in which (L) is strenghtened to (PsT): if F is such

that all its ultrafilter refinements are in qx, then F is in qx. Further

strengthenings of this axiom lead to pretopological and topological spaces.

By taking these spaces as objects and continuous (i.e. convergence preserving)

functions as morphisms we obtain respectively the category Con, L~11, PsTop.

If (FI) and (F2) are modified by axiomatizing only a family of convergent fil-

ters (not convergent to points) the two modified axioms give rise to the cate-

gory F~ [39].

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et Geom. Diff. 14 (1973) 291-308.

A. Bastiani, Applications differentiables et variet~s differentiables

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H.L. Bentley, T-categories and some representation theorems, Portugaliae

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H.L. Bentley, H. Herrlich and W.A. Robertson, Convenient categories for

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i0

ii

12

13

14

15

16

17

18

19

20

21

22

23

24

448

G. Bourdaud, Structure d'Antoine associ~es aux semi-topologies et aux

topologies, C.R. Acad. Sci. Paris Set. A279 (1974) 591-594.

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R. Brown, Function spaces and product topologies, Quart. J. Math. Oxford

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G.C.L. Brummer, A categorical study of initiality in uniform topology,

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C.H. Cook and H.R. Fischer, On equicontinuity and uniform convergence,

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Berlin (1969).

B. Day, A reflection theorm for closed categories, J. Pure Appl. Algebra

2 (1972) i-ii.

B. Day, An embedding theorem for closed categories, Category Seminar,

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E.J. Dubuc and H. Porta, Convenient categories of topological algebras

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Algebra, La Jolla (1965) Springer, Berlin (1966) 421-562.

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449

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28 H. Herrlich, Cartesian closed topological categories, Math. Colloq. Univ.

Cape Town 9 (1974) 1-16..

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(1973).

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31 R.E. Hoffmann, (E,M)-universally topological functors (preprint)

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33 W.N. Hunsaker and P.L. Sharma, Proximity spaces and topological functors,

Proc. Amer. Math. Soc.

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42 D.C. Kent, On convergence groups and convergence uniformities, Fund. Math.

450

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301-340.

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II (1970) 231-279.

47 A. Machado, Espaces d'Antoine et pseudo-topologies, Cahiers de Top. et

Geom. Diff. 14 (1973) 309-327.

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(1971).

49 L.D Nel, Initially structured categories and cartesian closedness,

Canad. J. Math. 27 (1975) 1361-1377.

50 H. Poppe, Compactness in general function spaces, VEB Deutscher Verlag

der Wissenschaften, Berlin (1974)..

51 R. Pupier, Precompactologie et structures uniformes, Fund. Math. 83 (1974)

251-262.

52 W.A. Robertson, Convergence as a nearness ~on~ept, Thesis, Carleton

Univ. (1975).

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Notes in Math. 273 (1972).

54 E. Spanier, Quasi-topologies, Duke Math. J. 30 (1963) 1-14.

55 N.E. Steenrod, Aconvenient category of topological spaces, Michigan Math.

J. 14 (1967) 133-152.

56 R.M. Vogt, Convenient categories of topological spaces for homotopy

theory, Archiv. Math. 22 (1971) 545-555.

57 M. Wischnewsky, Partielle Algebren in Initial-kategorien, Math. Zeitschr.

127 (1972) 17-28.

451

58 M. Wischnewsky, Generalized universal algebra in initialstructure catego-

ries, Algebra- Berichte (Uni-Druck, Munchen) i0 (1973) 1-35.

59 O. Wyler, On the categories of general topology and topological algebra,

Archiv. Math. 22 (1971) 7-17.

60 O. Wyler, TOP Categories and categorical topology, Gen. Topol. Appl.

1 (1971) 17-28.

61 O. Wyler, Convenient categories for topology, Gen. Topol. Appl. 3 (1973)

225-242.

Closely related talks given at this conference:

62 G.Bourdaud, Some closed topological categories of convergence spaces.

63 H. Herrlich, Some topological theorems which fail to be true.

64 W.A. Robertson, Cartesian closed categories of Nearness structures.

65 O. Wyler, Are there topoi in topology?

Carleton University

Ottawa, Canada

This research was aided by NRC grant A5297.

EPIREFLECTIVE CATEGORIES

OF HAUSDORFF SPACES

Peter J. Nyikos, University of Illinois, Urbana, Iii.

0. Introduction. The study of epireflectlve categories of

Hausdorff spaces is a natural outgrowth of a considerable body of

v topological theory. In fact, the Stone-Cech conpactification is the

archetypal example of an epireflection, and one hardly needs to be

reminded of its central role in such "classics" as [GJ]. I will be

mentioning many more examples in the course of this talk, whose

principal theme is the various methods, old and new, that can be used

to obtain epireflective subcategories of the category T2 of

Hausdorff spaces.

The methods employed fall naturally into two theories, to

which Sections i and 2 of this talk are devoted. The first is based

upon the pioneering work done by MrSwka, Engelking, and Herrlich on

-compact and g-regular spaces [MI], [EM], [HI]. The second is of

much more recent vintage and has to do with the various types of

"disconnectedness" as defined by G. Preuss and most recently

summarized and clarified by Arhangel, skii and Wiegandt [AW]. There

is no hard and fast division between these theories, and Section 3

gives one way of bringing them together.

Here follow some of the definitions and results basic to this

area of categorical topology.

0.1 DEFINITION. Let

is a reflective subcategory of

be a category. A subcategory 2 of

C if for each object C c r there

and a map r C : C -+rC with the following exists an object r C ~ B

453

property: if B is an object of B I then rC for each map f : C -~B there exists a ~ J"

~,r unique map fr : r C -+B in B such that

fro rc = fl i.e. such that the diagram C ~ ~ l~'

at right commutes. The pair (rc,rC) is

called the reflection of C in B. B

We can speak of "the" reflection of C in B because of

the elementary categorical fact that universal objects are determined

up to a unique isomorphism. In the case of topological spaces, the

"isomorphisms" are the homeomorphisms.

0.2.

follows :

TOP

T2

FT2

T~

D2

D3

CHSP

NOTATION. The following categories will be symbolized as

= the category of topological spaces and continuous

maps between them.

= the category of Hausdorff spaces

= the category of functionally Hausdorff spaces, i.e.

spaces X such that for any two points

Xl,X 2 ~ X, there exists a continuous function

f : X ~ such that f(xl) # f(x2).

= the category of Tychonoff (completely regular

Hausdorf~ spaces.

= the category of spaces whose quasicomponents are

single points, where the quasicomponent of a point

is defined to be the intersection of all clopen

sets containing it.

= the category of zero-dlmensional spaces, i. e.

Hausdorff spaces with a base of clopen sets.

= the category of compact Hausdorff spaces.

We will use the convention that all categories are full

subcategories of TOP (that is, if two spaces are in a subcategory,

454

so are all continuous functions between them) and that they are replete

(that is, if X is in a subcategory, so is every space homeomorphic

to X). Thus, by abuse of language, a class of spaces will be spoken

of as a "category" even if there is no explicit mention of the maps

between them. Mostly, we will be restricting our attention to

Hausdorff spaces, but sometimes we will be forced to step outside and

look at some non-Hausdorff construction.

0.3. DEFINITION. A reflective subcategory B of a categorye

is epireflective if the reflection map r C is an epimorphism for all

C e C ; monoreflective if r C is a monomorphism; extremally

epireflective if r C is an extremal epimorphism.

For the definitions of [extremal] epimorphism and

monomorphism, see [H2]. For the present, it is enough to know that

the monomorphisms of all the above subcategories of TOP are the I-i

maps [not just the embeddings, although in the case of CHSP the two

concepts merge] and the extremal epimorphisms are the quotient onto

maps. In TOP, the empimorphisms are the onto maps, while in all the

other cases, they are the maps with dense range [H2, pp. I14-S]; of

course, in the case of CHSP, these two concepts coincide.

0.4. THEOREM. Let B be a subcategory of TOP [resp. T2] which

includes a nonempty space.

(a) B is epireflective in TOP if, and only if, B is

productive and hereditary.

(b) B is epireflective in T2 if, and only if, B i_~s

productive and closed hereditary.

(c) B is extremally epiref!ective if, and only if, B i ss

productive, hereditary, and closed under the taking of finer

topologies.

For a proof of (a) and (b), see [H2]. The proof of (C)

is similar, and involves writing the reflection map r c as a compo-

455

sition of a quotient map and l-1 map of the quotient space into rC.

Part (b) holds true of all the categories of spaces listed

in 0.2, other than TOPj while part (C) holds not only for TOP

and T2 but also for FT2, D2, and CHSP. (Of course, in the case

of CHSP, parts (b) and (c) say the same thing. In fact, every

surJective map in CHSP is quotient, even closed.)

Theorem 0.4 shows that it is not enough to call as

category "epireflective"; one must specify what it is epireflective

in. For example, by (b) the category of compact Hausdorff spaces

is epireflective in T2, but not in TOP. The same is true of the

class of realcompact spaces.

On the other hand, the T i - spaces for i = 0,1,2,3, and 3~2

are epireflective in TOP, and the first four are extremally

epireflective as well. The categories D2 and FT2 are extremally

epireflective in TOP (and T2 - there is no distinction here as

long as the category is contained in T2) and the category D3 is

epireflective in TOP.

As a result of Theorem 0.4, we know that the intersection

of a class of epireflective subcategories of T2 is itself

epireflective in T2. So, given a subcategory ~ of T2, one can

speak of "the epireflective subcategory T2(g ) generated by g ",

viz. the intersection of all epireflective subcategories of T2

containing ~ Similarly, we define TOP(~ ). The same goes for

extremally epireflective subcategories, and we will let Q(g ) stand

for the extremally epireflective subcategory (of TOP or T2,

depending on whether ~ C T2) generated by

One might expect, given Theorem 0.4, that (say) T2 ( ~ )

can only be obtained by taking closed subspaces of products of spaces

in ~ , repeating the process on the new category, and so on

456

ad infinitum. Actually, a bit of topological insight is enough to

show that one step of this process is ample:

0.5. THEOREM. Let ~ be a subcategory of TOP.

(a) TOP (g) is the category of all subspaces of products

of spaces in ~ .

(b) Q (~) is the category of all spaces which admit i-i

maps into a product of spaces in

(c) Let g ~ T2. Then T2(g )

closed subspaces of products of spaces in

A categorical proof of parts (a)

is the category of all

.

and (c) may be found

in [H2], and one of (b) may be constructed along the same lines.

Of course, since T2 is extremally epireflective in TOP,

Q (~) ~ T2 whenever ~ ~ T2. Afortiori, TOP (g) ~ T2 whenever

C T2. However, even in this case, TOP(~) usually contains

T2(~ ) properly. For example, when ~ = (~ }, TOP (~) is the

category of completely regular spaces, and T2(~ ) is the category

of realcompact spaces. It is also important to note that T2(~) is

defined only if ~ ~ T2.

Section i. ~ - regular, ~ - compact, and ~c-compactlike

spaces.

With Theorem 0.5, we have already entered the realm of

- regular and ~- compact spaces. As a class, these were first

studied systematically by S. Mr~wka and R. Engelking [MI], [EM],

[M2]. The credit for fully realizing the categorical significance of

-regular and ~- compact spaces goes to H. Herrlich [HI], [H2].

For example, it was he who dropped the insistence that ~ consist of

a single space (or, equivalently, a set of spaces).

457

I.i. DEFINITION. Let g be a class of spaces. A space X

is ~- Hausdorff if for each pair of points x, y ~ X, there exists

f : X -~E eE such that f(x) ~ f(y). A space X is ~ - regular

[Mr6wka: ~- comPletely regular] if it is ~ - Hausdorff and, for

each closed set A~ X, and each x c A, there exists f : X ~Y,

where Y is a finite product of spaces in ~ , such that

f(x) ~ ~f(A),

1.2. THEOREM. Let X be a topological space, ~ ~ ~ TOP.

(a) [M I] X i_~s ~ - regular if, and only if, it can be

embedded as a subspace of a product of spaces in

(b) X i__~s ~ - Hausdorff if, and only if, it admits a 1-1

map into a product of spaces in

In other words, X s TOP (g) if, and only if, it is

- regular, and X c Q(~ ) if, and only if, X is ~ - Hausdorff.

(Of course, ~ - Haudorff implies Hausdorff if, and only if, ~ ~ T2.)

The definition of ~ - compact spaces given in [EM] is a bit

technical, and the most convenient definition is simply:

1.3. DEFINITION. A space is ~ - compact if it can be

embedded as a closed subspace in a product of spaces in

1.4. NOTATION. We will let ~ stand for the closed unit

interval [0,!] and ~ for the real line, both with the usual

topology. The countably infinite discrete space will be identified

v~ith the natural numbers and denoted ~ We let 2 stand for the

two-point discrete space.

Here is a table listing some of the best-known examples in

this area:

458

TABLE 1

g T 2 (g) TOP (g) q (g)

[ n ~ cHsP T ~ r r 2

[~q ] realcompact spaces T ~ FT2

[ 2 ] D3(~ CHSP (=D2 6h CHSP) D 3 D 2

[~ } N - compact spaces D 3 D 2

Several characterizations of the spaces in T2( ~ ) for

these examples may be found in [HI] , [H2] , [H4] , and [M2]. Also,

necessary and sufficient conditions exist on ~ for TOP ( ~ )

and T2 (~) to be the above categories. Perhaps the most striking

of these conditions is that, if ~ ~ CHSP, then CHSP = T2 (~)

iff T3~2 = TOP(~ ) iff there exists in ~ a space which contains a

copy of TT. [Proof: Of course, the latter condition implies the

former; conversely, for CHSP to equal T2 ( E ) or T3~2 to equal

TOP(~ )I some product of spaces in ~ must contain XI. There

exists a projection map with respect to which the image of Z! is

non-trlvlal, and this image contains a copy of Xi - in fact [HS] it

is a Peano continuum, hence arcwise connected.] We will make use of

similar results in Section 3.

Less familiar than the above examples is the case of T2(a~),

where ~ is the Z - th cardinal, looked upon as the space of all

ordinals less than itself, with the order topology. If ~ is of

countable cofinality, this is T2( ~ ) again: on the one hand,

contains a closed, discrete, countably infinite space; on the

other, a~ is strongly zero-dimenslonal and realcompact, hence [H1]

it is ~N - compact. The situation in the case where ~ is of

uncountable cofinality is different. Every such space is countably

compact and not compact, hence not in T2 (~). This is also true

459

[B], [M 2] of any product of copies of a~. More generally, as

Blefko has shown [B], the categories T2(~h) and T2(~ ) are

incomparable (with respect to containment) for any two regular

cardinals a~ and ~ , including the case where one of them is

~0 =IN Additional results of Blefko, s concerning these spaces,

and the open problems they leave, may be found in [M2].

In principle, Theorem 0.5 (c) solves the problem of

characterizing the epireflective subcategories of T2, and (a)

characterizes those which are apireflective in TOP as well. But

this theorem alone can only be expected to yield essentially

haphazard results such as the above. I will now describe one approac~

worked out by S. Hong [H 5] [H 6] which makes the search for

epireflective subcategories a bit more systematic. To introduce it,

let me recall the following generalization of real compact spaces.

1.5. EXAMPLE. Let ~ be a cardinal number. Herrlich has

referred to a Tychonoff space X as being ~- compact if every

Z - ultrafilter with the ~- intersection property I is fixed. For

example, the ~0 - compact spaces are the compact Hausdorff spaces,

while the ~l - compact spaces are the realcompact spaces. For each

infinite cardinal ~, the ~- compact spaces form an epireflective

subcategory of T2. Moreover, the category of ~- compact spaces

is distinct for distinct ~ : if ~< ~l, then the category of

~l _ compact spaces contains the category of ~- compact spaces

properly. Specifically, Hu~ek has shown [H 7] that the category of

iIn this definition, the ~-intersection property refers to a

collection such that the intersection of fewer than ~ sets is

nonempty; thus what is commonly called the "countable intersection

property" is here referred to as the ~l - intersection property.

460

~- compact spaces is T2(P& ) where P~O II ; P~ where

= ~v+l is II v _ [p} where p is any point; and ~ for a

limit cardinal is H [~ I ~<~ ].

It is a third characterization of these spaces which Hong

utilized in his system.

1.6. DEFINITION. A subset A of a topological space X is

~- closed if for every point x c X - A, there exists a G~ - set

disjoint from A and containing x. [A G~- set is a set which is

the intersection of fewer than ~ open sets.]

1.7. THEOREM [Hs] A space X i~s ~- compact if, and

only if, it is Tychonoff and ~- closed in GX. For any Tychonoff

space X, the reflection of X in the category of ,~- compact

spaces is its ~- closure in 6X.

Now, given any epireflective subcategory ~ of T2 (in the

case of ~- compact spaces, CHSP plays the role of ~ ) one can

define the category ~ of all spaces X for which the reflection

map r x to ~ is an embedding, such that the image rx(X ) is

- closed in the reflection space rX.

1.8. THEOREM [H5] For any epireflective subcategory B

o_~f T2 and any~ ,~ is an epireflective subcategory of T2

containing B.

Because spaces in S~ are embeddable in spaces of Bj it

follows from Theorem 0.4 that ~ C TOP (~). In fact, from

Theorem 0.4 we obtain:

I. 9. COROLLARY. Let B be an epireflective subcategory of

T2. The reflection map r X is an embedding if, and only if,

X ~ TOP ( E ) .

461

Simple cardinality arguments show that every member of

TOP( B ) is in ~ for some #~, so that (although foundationists

shy away from such statements) one may informally say that TOP (B)

is the union of the spaces in ~ for all ~ .

Now, obviously, if B is epireflective in TOP, then

= B for all ~. At the opposite extreme, if all the spaces in

are ~mpact Hausdorff, and B contains a space with more than

one point, then the categories ~ are distinct for all ~ . Indeed,

the space D~ obtained by replacing lI with ~ in Example 1.5 is

in B~ , but not in B~ for any ~<~ . See Example 1.18,

below.

So even this system is enough to give us a wealth of

epireflective subcategories of T2, one hierarchy for each

epireflective subcategory of CHSP [and we will see in Section 3 how

to obtain a number of such epireflective subcategorles], the

members in each hierarchy in one-to-one correspondence with the 1

infinite cardinal numbers. Moreover, if B and B are con-

tained in CHSP and B ~ B 1 then also ~ and B~ I are

distinct, as can be easily shown from:

1. lO. LEMMA. If ~ and B 1 are epireflective subcate-

gories of CHSP and B ~ B l, then ~ ~ B~ 1 for any

Proof. Suppose ~ were contained in B~ Then this

would imply that TOP (B~) is contained in TOP( B~ ). Since

TOP(BA ) = TOP( ~ ), this would imply that every member of B can be

embedded in a member of B l, and since each member of B is

compact, we would have B C T2(B l) = B l, contradiction.

But S. Hong has carried the system a step further. His

generalization is motivated by the observation that one can form a

new topology ~A on the space (X, ~) by using the G& - sets as

462

a base. Now (X, ~ ) is the c_oreflection of (X,~) in the categary

of spaces in which the intersection of fewer than ~ open sets

is open, and S~ is the category of all spaces in TOP( 8 ) which

are closed with respect to the ~ - coreflection of their

B - reflection. This can be generalized to any coreflective

subcategory of T2. The word "coreflective" is defined dually to

"reflective":

I. ii. DEFINITION. A subcategory ~ of a category ~ is

coreflective if for each X c

c X : cX ~ X such that for every

map f : C -+X, where C c C ,

there exists a unique map fce

such that the diagram at right

commutes.

there exists cX ~ ~ and a map

The following theorem is basic to the theory of topological

coreflections;

1.12. THEOREM [H2]o Let ~ be a coreflective subcategory

o_~f T2 o_~r TOP. Then the coreflection map c X is bijective.

In other words, cX can be thought of as X with a finer

topology. H. Herrlich has given four axioms for a so-called

idempotent closure operator on all topological (or Hausdorff) spaces

simultaneously, in such a way that there is a l-1 correspondence

between coreflective subcategories of TOP [resp. T2] and idempotent

closure operato~on TOP [resp. T2]. Using this characterization,

S. Hong showed [H6]:

1.13. THEOREM. Let C be a coref!ective subcategory of

TOP, B an epireflective subcategory of T2. Le__~t B e be the

category of s~aces in TOP(B ) which are closed in the

C - coreflections of their B - reflection spaces. Then B e is

463

an epireflective subcategory of T2.

The proof of Lemma l.lO can be generalized to show the first

part of :

I. 14. THEOREM. Let B and B 1 be epireflective subcate-

gories of CHSP, ~ a eoreflective subeategory cf TOP.

if B and B i are distinct, then B 0 and B~ (a) are distinct.

1 (b) if B C B i, then B e C B~

(c) if r C C j - - , then B 0 D BCl.

The diagram below is the key to b). We assume X e ~e

The maps f and h are the reflection maps to rlx and rX

respectively, and the unlabeled maps are the canonical reflection and

coreflection maps. The map f induces f rl which in turn induces

(frl)c The maps h o g and f o g are monomorphisms, hence so are

(h ~ and (f ~ g)c

, f

X ~ ~ cX

H% c (~" X)

The map (frl)c maps the image of cX under (h o g)c biJectively

onto the image of cX under (f ~ g)c" Since "X is closed in the

coreflection of its B - reflection" [more exactly, the image of

cX under (f ~ g)c is closed in e(rX)] the same is true, by

continuity of (frl)c, of the coreflection of its B 1 _ reflection.

The proof of (c) is much simpler and relies upon the

existence of the bijective map ctX-+ cX induced by c f c" X : X-~X.

464

In contrast to Theorem 1.14, distinct e may give rise to

the same B e

1.15. EXAMPLE. Let ~ and e' be distinct coreflective

subcategories of TOP, each of which contains CHSP. Then the

C - coreflection of any compact Hausdorff space is that space itsel~

so B ~ = B e , = B .

However, the following problem seems to be open:

1.16. PROBLEM. Let C and C' be distinct coreflective

subcategories of TOP, and let B be an epireflective subcategory

of CHSP, contained neither in ~ nor ~r; are B e and B~,

always distinct?

Once we step outside the realm of compact spaces, B c and

B~ may be identical. (A trivial example is where C is the

category ~el ' B = CHSP, and B 1 is the category of real-

compact spaces.) So it would be nice if every epirefleetive

subcategory of T3~2 (= TOP(CHSP)) were of the form B6 for some

class B of compact spaces. Unfortunately, that is not true; for

example, any epireflective subcategory which consists of totally

disconnected Tychonoff spaces, some of which are not zero-dimensional,

cannot be of this form. This is obviously true of the class of totally

disconnected Tychonoff spaces, itself. Nevertheless, the order which

these classes B~ bring to the study of epireflective subcategories

is still considerable.

In closing this section, we have the following applications

of a generalization of Theorem 1.14.

1.17. EXAMPLE. If E is a Tychonoff space with more than

one point, then TOP ([E}) contains T2 ({E}) properly. Here is the

reasoning. Since the two-point discrete space embeds in

465

E, D3 C TOP([E)). Therefore the space ~ obtained by replacing TT

with ~ in Example 1.5 is in TOP((E}). However, E is ~- compact

for some cardinal ~ , whereas D~ is not 2- compact for any

A >g �9 So, while T2([E}) C (CHSP)~ , D~ is not in (CHSP)~f

That T2([E]) C (CHSP)~ follows from the generalization of

(b) of Theorem 1.14 to epireflective subcategories ~ of T2;

its proof, which was strictly categorical, goes through without change.

1.18 EXAMPLE. Let B be epireflective in CHSP. Let ~)

denote the category of compact zero-dimensional spaces. If B

contains a non-trivial space, then by Theorem 1.14, C B~

(CHSP)~ for all ~. Thus, since D~ is in ~ (in fact [H 7]

we have ~ = T2(D& )), but is not ~-compact for any ~<~ , it

follows that D~ c ~ , D~ ~ B~ for any Z<~.

Section 2. Extremally epireflective subcategorles and

disconectednesses.

A great broadening of horizons takes place when we move up to

extremally eplreflective subcategories of T2: the smallest nontrlvlal

such category, D2 , already spills out beyond the class of Tychonoff

spaces. We could, of course, restrict our study to Tychonoff spaces

or even compact Hausdorff spaces, considering only those spaces in e.g.

D2 which are Tychonoff or compact Hausdorff. But in the latter case,

as Preuss has pointed out [P2 ], we obtain nothing new: every

epireflective subcategory of CHSP is extremally eplreflectlve (in

CHSP). On the other hand, the extremally epireflective subcategorles

of T3~2 are just the intersections of T3~2 with the extremally

epireflective subcategories of TOP, and this is true also of the

extremally epireflective subcategorles of T2.

A special kind of extremally epireflective subcategory is a

disconnectedness. To define it, it is probably best to begin with a

466

continuously closed category, that is, one closed under the taking of

continuous images of its members. (Of course, it makes a big differ-

ence which subcategory of TOP these images are allowed to lie in;

for the moment we will use all of TOP itself.) Let % be a

closed subcategory of TOP; then ~TOP ~ is the continuously

category IX c TOPIX does not contain non-trivial ~- subspaces]

Then ~TOP ~ is a disconnectedness, and every TOP -

disconnectedness is by definition a category of this sort~ We define

T2 - disconnectedness, T3~2 - disconnectedness, etc. anagolously.

2.1. EXAMPLE. Let ~ be the class of connected spaces;

then ~TOP~ is the class of totally disconnected spaces.

2.2. EXAMPLE. A Peano continuum is defined as a continuous

Hausdorff image of If. The category g~ of Peano continua is

continuously closed in TY, and (of course) in any of its subcate-

gories. Since every Peano continuum is arcwise connected [HS], it

follows that ~T2 ~ is the class of arcless spaces. Sometimes

referred to as "arcwise totally disconnected," these are the spaces

containing no subspace homeomorphic to a nondegenerate interval of ~R.

This is one of the rare cases where ~ can be replaced by

in the definition of ~T2 g~ which we have chosen. a single space

Since T3~2 = TOP([~R]), an eplreflective subcategory of T2

which is properly contained in T3~ is also contained in ~T2 ~

1. In [P1 ] and lAW] there is a more general definition, to include

the case where g~ is not continuously closed: ~TOP ~ for any

subcategory a of TOP is the category of all spaces such that every

map f : X --Y, X c~, is constant. But while ~TOP is made

more general, the classes of disconnectednesses do coincide with the

disconnectednesses defined as in this paper.

467

The concept of a connectedness can be defined dually to that

of a disconnectedness, but it is more natural to deviate slightly from

strict duality. Let B be a hereditary class of spaces. The

connectedness e ToP B is

[X~TOPIX cannot be mapped continuously onto a non-trivial space in B]

A category of the form ~TOP B for a hereditary B is called a

connectedness.

"connect d The appropriatedness of the word e ness" is

debatable since e.g. the class of arcwise connected spaces is not a

not even of the form ~T2 B or CCHSP B connectedness,

1 Consider the graph of sin ~ together with the portion of the y - axis

between 1 and -l: this space is not arcwise connected, but any

connectedness which contains X! also contains this space (see

Theorem 2.4 ). Nevertheless, this is the only logical dual to the

concept of a disconnectedness, and it works together well with it.

For instance, the two operators ~ and ~ form a Galois corespond-

ence, so that ~g~(~) = ~(~) and ~C(B ) = e(B )

provided the same subscript (TOP,T2, etc.) is used throughout.

Moreover, we have the following fundamental theorem, applying equally

well to TOP-disconnectedness or T2-disconnectedness.

2.3. THEOREM. [AW, Theorem 3.7]. Let B be a

disconnectedness, and let ~ = ~ (B) be the connectedness dual to

B . Then B is a extremally epireflective subcategory of TOP .

Moreover, the reflection map f : X-~rX has the properties:

(i) f-l(y) c~ for each y ~ rX

(ii) If X' is a subspace of X belonging to ~ , then

X u is contained in f-l(y) for some y ~ rX.

So the reflection map can be thought of as a breaking up of

X into its ~- " the sets f-l(f(x)). Further " components ,

468

clarification of the situation is provided by:

2.4. THEOREM. (a) Given a subcategory B o_~f TOP

[resp. T2], there is a smallest disconnectedness containing it, viz.

~ (BI), where B I is the category of all subspaces of spaces in

B .

(b) Given my s ubcategory (~ o_~f TOP [resp. T2], there

is a smallest connectedness containing it, viz. C~(~I), where

~i is the category of all continuous images of spaces in

Proof is immediate from the Galois correspondence.

2.5, THEOREM. [AWl A nonempty subcategory B of TOP is a

TOP-disconnectedness if, and only if, the following conditions hold

(where X e TOP)

(i) B is productive.

(ii) B is hereditary.

(iii) If f : X-~Y is a continuous function such that

Y e B and f-l(y) c B for all y c Y, then

X e B.

Note that if we replace ,f-l(y) ~ B" by ,.f-l(y) is a

singleton," we get a necessary and sufficient condition for "B is

epireflectlve , while if we drop (iii) altogether, we extremally " "

get the equivalent of "B is eplreflective in TOP".

If we replace TOP by T2 , or T3~, or CHSP, the

three conditions are necessary for B to be a disconnectedness, but

are they sufficient? In [AW] a negative answer for T2 is given,

but the "counterexample" is faulty: it is claimed in lAW] that the

category D2 satisfies (iii), but it does not.

2.6. EXAMPLE. Let T be the Tychonoff corkscrew as

described in [SS, Examples 90 - 91]. The two extreme points, a +

and a-, are in the same quasicomponent of T. Let Y be the space

which results from identifying a + and a-, and let f be the

quotient map from T to Y. Now Y c D2, and f-l(y) is in D2

for all y c Y (in fact, it is a singleton for all but one y, and

a doubleton for that onel).

469

Not only is T not in D2, it is not even in FT2; on the

other hand, T is totally diconnected (in fact, scattered) so that

every subspace of T admits a map onto a two-polnt discrete space and

thus a non-constant map to the real line. This answers Problem (a) of

[AW] in the affirmative.

2.7. EXAMPLE. A Tychonoff space exhibiting the same phencm-

enon as T with respect to D2 can be constructed using Dowker.s

example M. This space is constructed by taking a collection

[S~I~ < e 1] of subsets of il such that S~ C s~ whenever ~ < ~l

such that the complement of S is dense for all ~, and such that

g~l~ < ~l ] = II. Now let M be the subspace of IX x ~l consi~ing

of the points S~ • [~] as a ranges over ~l" It can be shown (cf.

[GJ], exercise 16L, where M is referred to as A l, and [E], pp.

254-255, where it is referred to as Y) that M is O-dimensional

and that, given any clopen subset A of M, either A or A c must

contain a cofinal subset of M. In other words, there exists ~ such

that {S~ • ~I~ > ~] is a subset either of A or A c. Thus if we

enlarge M by "two points at ~l" say (0,~l) and (1,~l) , giving

these the product topology neighborhoods, then the resulting space is

still totally disconnected, but the two extra points are in the same

quasicomponent. Now if we identify these two points, we obtain a

space in D2. Thus D2 does not satisfy (iii) even within the

category of Tychonoff spaces.

One can even obtain a metrizable example, e.g. the Knaster-

Kuratowski "Cantor fan" (denoted Y* in [SS, Example 129]).

Identifying the points within each quasicomponent gives a homeomorph

of the Cantor set, and each quasicomponent is zero- diemnsional.

So, apparently, the following question is still open.

2.8 PROBLEM. Let B be a subcategory of T2, or T3~2, or

CHSP, satisfying the three consitions of Theorem 2.5 (with "closed

hereditary" in (li) in the case of CHSP). Can it ever happen that

B is not a disconnectedness?

In the case of CHSP, we can ask a bolder question: is

every epireflective subcategory a CHSP-disconnectedness? This splits

up into Problem 2.8 on the one hand and a kind of left-fittlng

problem on the other:

470

2.9. PROBLEM. Let ~ be an epireflective subcategory of

CHSP and let f : X-~Y be a mapping with X e CHSP, Y ~. If

f-l(y) e $ for all y c Y, is X in ~?

Whatever the answers to these last two problems, it is sti~

interesting to study categories satisfying the three conditions of

Theorem 2.5. Suppose, for example, we are given a subeategory ~ of

T2 and asked, "what is the smallest subeategory B of T2

that contains g~ and satisfies those three conditions? Of

course, ~ will contain TOP(~) and even Q(a ) - - see the

comment after Theorem 2.5 - and we can define, by transfinite

induction, an ascending sequence extremally of epireflective

subcategories of T2 which together will constitute ~.

To simplify matters, we can assume that ~ is extremally

epireflective to begin with. There is no loss of generality, of

course, and it allows us to use the label ~ without danger of

confusion with the categories ~ defined earlier.

2.10. LEMMA. Let ~ and g~l be extremally epireflective

i_~n T2, with r X the reflection map with respect to 6~ for each

X e T2. Let r(~l,g~) be the category whose objects are the spaces

X ~ T2 such that rx-l(z) c ~l for all z e rX. Then r(~l,~)

is extremally epirefleetive.

Moreover, r(~l,g~) is the category of spaces X ~ T2

which admit a map f into some member Y of g~, such that

f-l(y) ~l for all y ~ Y.

Proof. The two descriptions of r(~ l, ~) coincide:

clearly the first implies the second; on the other hand, r X will

compose with fr to produce f, so that each set of the form rx-l(z )

is contained (as a closed subset) in a set of the form f-l(y),

hence is an object in gg I

471

By the second description, it is clear that r(6~l,gg)

is closed under subobjects in T2, so it remains only to show that

r(g&l,~) is productive. Let [r : X 7 ~ Y717 c s be a set of

reflection maps, with X c ~ for all 7. The induced map

f : H X ~ H Y ( = Y) has the property that f-l(y) (= H r -l(yT))

is in ~l for all y c Y, since (~ is productive. So by the

second description, r(~l,~) is productive.

Had we wanted to show merely that r(~l,o-) is epireflecti~

in TOP, it would have been enough to assume that (~l is

epireflective in TOP. The same goes for "eplreflective in T2",

mutatis mutandis, because the inverse image of each point is closed.

We can apply Lemma 2.10 inductively by starting with an

epireflective subcategory (~, r ) and letting ~l = (~ . Then

(~l,rl) = r(~,~) will be epireflective and contain (~, r). We

seem to have a choice of ways to continue the induction: we could

define (~2' r2) to be either r(~, ~I) or r(al,~ ). Actually,

the two are the same, but the proof requires a good deal of machinery

to state with suitable generality. For purposes of characterizing

the category B for which we are aiming, we can (fortunately) skirt

the whole issue by instead defining (~2' r2) as r(~l,~l).

~2 contains "both" of the epireflective subcategories Clearly,

Just named.

Supposing ( as, r ) to be defined, let ~+i be

r( ~, as). If ~ is a limit ordinal, let ~ be the category of

all spaces which admit a 1 - 1 map into a product of spaces from any

of the ~ with ~ < ~.

It is clear that the category B of all spaces belonging to

some ~ is productive, hereditary, and inversely closed under maps

f : X -~ Y with point - inverse~in B. [The last named condition can

472

be satisfied e.g. by letting ~ be the supremum of the g~ to which

the point-inverses and the space Y belong; then the domain X will

be in ~ ~+i - ]

This whole procedure is susceptible to a wide categorical

generalization. In particular, the proof of Lemma 2.10 can be

modified for complete categories in which the extremally epireflective

subcategories are those closed under products and subobjects - -

provided we can give meaning to the concept of "inverse image". But

this is best handled in a separate paper.

Section 3. Generalization of a famous problem.

Given a Tychonoff space X belonging to a disconnectedness,

it is natural to ask whether GX also belongs. In this section I

will handle some conditions on the disconnectedness for which the

answer is affirmative if X is realcompact. The general idea

behind the conditions is that the closed subsets of ~X - X are

"pathological" for a realcompact space X.

First, we will look at an example where the answer is very

much in the negative: the category of totally disconnected spaces.

We define the following classes of spaces, each containing the one

before:

: realcompact X such that ~X is totally

disconnected.

: N - compact spaces

: zero-dimensional realcompact spaces

~ : realeompact spaces, every compact subspace of which

is totally disconnected.

It is easy to see that O and ~ are distinct: the

Cantor fan, for example, is in ~ but not ~ . A famous problem

473

that stood for twelve years was whether O =~Z . I solved it in

1970 by showing that Prabir Roy's space A is not N-compact [Ny].

As for the relationship between ~ and ~ , S . Mr6wka settled

it in 1972 by describing a space in ?% which is not in ~ [M3].

These classes all have categorical interpretations in terms

of the class B of totally disconnected Hausdorff spaces. In fact,

if B is an epireflective subcategory of T2, we can define:

~l(S) = [XIX realcompact, ~X e B]

~ l ( S ) = (BP, CHSF)~l ~I(B) = [X]X realcompact, X c TCP B]

I(B) = [XIX realcompact, every subspace of X

closed in ~X is in B]

From now on, we will assume B ~ CHSP, so that the '~ CHSP"

in the description of ~I(B) becomes redundant, and we can say

~i(8) = [XIX realcompact, every compact subspace

of X is in B]

If B = ~CHSP ( ~ ) for some continuously closed ~ ,

then this is just the category of realcompact spaces belonging to

~T2((t). In any case, since B is closed hereditary, we have

~I (B) ~ ~i (B). Nex~ since the class of realcompact spaces is

CHSP~ where ~= ~, it follows from Theorem 1.14(b) that

~I(B) C ~I(B). Finally, if X is rea!compact, then X is

~ - closed in ~X (~ = ~i), so that if ~X s B, then X c ~i (B) .

Summing up,

and the case where B is the class of totally disconnected compact

Hausdorff spaces is an example where all containments are proper. We

will now give some example where the four categories coalesce.

474

In each case, B will be of the form ~CHSPa , raising afresh the

question of whether every epireflective subcategory of CHSP is a

disconnectedness.

3.1. EXAMPLE. Let B be the class of arcless compact

Hausdorff spaces (see Example 2.2) To show ~l(B) = ~l(B) --- so

that all four classes are equal - - it is enough to show that if a

realcompact space X contains no arcs, then ~X contains no arcs

either. This will follow immediately from Theorem 9.11 of [GJ],

which I will restate as follows.

3.2. LEMMA. Let A be a closed subset of ~X. If A

does not contain a copy of ~, then Ag~(~X - oX) is closed and

discrete in the relative topology of ~X - ~X.

And, since no arc contains G~, it follows that if Y is

realcompaet and ~Y contains an arc, all but a discrete subspace of

that arc must lie inside Y, and so Y itself contains an arc.

Equality between the four classes follows immediately.

Here is a related result which is even easier to apply.

3.3. THEOREM. Let A be a closed subset of BX. If A

does not contain a copy of ~ , then the points of A lying outside

oX are isolated in the relative topology of A .

Proof. By Lemma 3.2, any point x of A which is not in

oX has a closed neighborhood N such that N~A is contained in

oX, except for x itself. Since x is not in oX, there is a

zero-set Z containing x which misses X[H6]. Let E = (Nr]A)\[xl.

If x is not isolated in A, then E and Z fulfill the conditions

of Lemma 9.4 in [GJ], and so E contains a C* - embedded (in ~X)

copy of ~ . In other words, the closure of E in ~X contains a

copy of G~. But this contradicts the assumption that A (which

475

contains E) is closed in BX.

3.4. COROLLARY. Let A be a closed, dense-in-itself

subspace of ~X. l_~f A does not contain a copy of ~ , then

AC ~x.

I will refer to a compact, connected Hausdorff space as a

" c o nt inuum".

3.5. COROLLARY. Let A be a continuum in ~X.

does not contain a copy of B~ , then A ~ oX.

If A

For example, if A is metrizable, or first countable, or

even sequential; or if A is orderable; or hereditarily separable;

or of cardinal less than 2 @ then A~ oX.

3.6. COROLLARY. Let X be a realcompact space, and let A

be a continuum contained in ~X. If A does not contain a copy of '~,

then A ~ X~

And so we come to:

3.7 THEOREM. Let ~ be a class of continua, none of which

contains a copy of ~, and let B = ~CHSP~. Then

i(~) = ~i(B)= ~I(B)= ~i(~) �9

Proof. If ~I is the class of all continuous images of

spaces in ~, then B = ~CHSP (ai). Suppose some member Y of 0_1

contains a copy of G~. Let f : X -- Y be a map with X c~. Then,

by projectivity of ~[H4], there is a copy of ~ in X mapping

i-i onto the copy in Y, contradicting the hypothesis on ~ . There-

fore, if X is a realcompact space, any nontrivial member of ~i

contained in 8X must be contained in X, and X cannot be in

~i(~). In other words, if X is not in ~I(B), then X is not in

~I(B) either.

476

3.8. EXAMPLE. Let d 0 be the class of all continua

containing no copy of ~. From the proof of Theorem 3.7, ~0 is

continuously closed. Let B 0 = ~CHSP~O. Clearly, B 0 is the class

of all compact Hausdorff spaces, every nontrivial subcontinuum of

which contains a copy of ~ An example of a connected space

belonging to this class is ~ ~+\ ~+[GJ].

~0 is obviously the largest class of continua (and hence

B 0 the smallest epiref!ective subcategory of CHSP) to which Theorem

3.7 can be applied directly. For any disconnectedness containing B 0,

the four categories defined above will coalesce. Of course, the

largest disconnectedness to which 3.7 can be applied is the class

ATDC of arcless compact Hausdorff spaces, since any epireflective

subcategory of CHSP which contains a space outside ATDC is all of

CHSP.

Here are some intermediate examples.

3.9. EXAMPLE. Let ~ be the class of orderable compact

Hausdorff spaces. Since ~ cannot be embedded in an orderable

space (in fact, it is not even hereditary normal), ~ satisfies the

hypotheses of Theorem 3.7.

Characterizing the continuous images of spaces in ~ is an

unsolved problem. They must all be locally connected, since the

members of ~ are locally connected and compact: every closed,

continuous image of a locally connected space is locally connected

[HS]. But we lack a convenient necessary and sufficient condition;

certainly we have nothing like the Hahn-Mazurkiewicz Theorem, which

characterizes the continuous images of XI, the Peano continua, as

those continua which are locally connected and metrizable.

3.10. EXAMPLE.

struct an analogue XI&

For each infinite cardinal ~ we can con-

of IX as follows. Let ~ be the first

477

ordinal of cardinality ~ Points of Z~ are transfinite

sequences of the form (b)~ < ~ where each b is either 0 or l,

and for each ~ < ~ there exists 7 > ~ such that b = 0. Give

Z~ the lexicographical order topology. Think of the members of

Z~ as transfinite binary decimals, and it is not hard to show that

~& is Dedekind complete. [Let B be a subset of Z~ For each

ordinal ~, say that B eventually agrees up to c if there

exists a member b of B such that all members of B greater than

b agree with b up to and including ~. The only complication

occurs if there exists c such that B does not eventually agree

up to ~. In that case, take the smallest such ordinal; the

supremum of B will have a 0 at ~ and all ordinals beyond.] Thus,

Z~ is connected and locally compact. Since it contains a greatest

and least element, it is compact.

It is easy to show that CHSP~ contains CHSPZ~

properly whenever ~ > ~ .

3.11 EXAMPLE. Let ~ be the class of all perfectly nor-

mal continua. Since a regular space is perfectly normal and Lindel~f

if, and only if, it is hereditary Lindel~f, it follows that

coincides with the class of hereditarily LindeISf continua. Thus

is continuously closed, and no member of ~ contains a copy of ~.

In fact, every nontrivial member of ~ has the cardinality of the

continuum, being countable on the one hand and connected on the other.

The disconnectedness of all compact Hausdorff spaces

containing no metric continua contains ~CHSP~ ; but is the

containment proper? Or is it true that every perfectly normal

continuum must contain a nontrivial metric continuum?

We can ask similar questions about other easily describable

classes ~. For example, must every sequential continum contain a

nontrivial first countable subcontinuum, or even a metrizable

subcontinuum? The big problem seems not the finding of different * see Errata at the end of the paper!

478

to plug into Theorem 3.7, but the determining of whether the

resulting disconnectednesses coincide.

What about classes not covered by Theorem 3.7? We have,

for example, the class ~ of locally connected continua. It is

possible for a locally connected continuum to contain a copy of ~

for example, !I d where ~ is the cardina!ity of the continuum.

But we can apply Corollary 3.5 to show that lId , if it occurs in

~X, must already be in X if X is realcompact: every point of

]Z~ ~ is contained in an arc A, which must lie completely inside X.

In line with this, we may ask:

3.12. PROBLEM. Let X be a locally connected continuum.

Does X have a locally connected subcontinuum which does not contain

a copy of BIN ?

If the answer is always "yes", then the four classes

coincide for B = ~CHSP~ (~= all locally connected continua) also.

In fact, Theorem 3.7 could be broadened to include all continuously

closed classes ~ such that each member of ~ contains a sub-

continuum with no ~]N- isomorphic subspace.

If we pass from realcompact spaces to ~2 - compact spaces,

there is no possibility of extending the results in this section. Say

we let

~2(B) = [XIX is ~2 - compact and ~X c B]

and define ~2(B) to be ~& with ~= ~2 " Now ~2(2) already

contains Dowker, s example M, because every free clopen ultrafilter

on M has at most the ~I - intersection property [see H6, Lemma 2.8].

Hence by Theorem 1.14, ~2(B) contains M for all nontrivial B.

On the other hand, ~M contains a copy of l-T, so that the only B

for which M c_ , ,~2(B~ is CHSP itself.

479

ERRATA

(i) for each a < ~ there exists y > ~ such that by : 0

(ii) there exists a limit ordinal 8 such that b =i for al

~ 8, and for each ~ < 8, there exists y > ~ such

that b =0 u

(iii) b =i for all a < ~.

480

BIBLIOGRAPHY

[AW] A.V. Arhangel,skii and Wiegandt, Connectednesses and disconnectednesses in topology, Gen. Top. Appl. 5(1975)9-33.

[B] R. Blefko, Doctoral dissertation. University Park, Pennsylvania 1965.

[E] R. Engelking, Outline of General Topology. Amsterdam, North

Holland, 1968.

[EM] R. Engelking and S. MrSwka, On E-compact spaces, Bull. Acad.

Pol. Sci. Math. Astr. Phys., 6 (1958) 429-436.

[GJ] L. Gillman and M. Jerison, Rings of Continuous Functions. Princeton, 'Van Nostrand, 1960.

[H I ] H. Herrlich, ~- kompakte R~ume, Math. Z. 96 (1967) 228-255.

[H 2] H. Herrlich, Topologische Reflexionen und Coreflexionen. New York, Springer-Ver!ag, 1968.

[H 3] H. Herrlich, Limit - operators and topological coreflections, AMS Transactions 146 (1969) 203-210.

[H 4] H. Herrlich, Categorical topology, Gem. Top. Appl. ~ (1972)

1-15.

[H 5] S.S. Hong, Limit-operators and reflective subcategories, in: TOPO 72 - General Topology and its applications (New York, Sprlnger-Verlag, 1974).

[H 6] S.S. Hong, On ~- compactlike spaces and reflective

subcategories, Gen. Top. Appl. ~ (1973) 319-330.

[H 7] M. Hu~ek, The Class of ~- compact spaces is simple, Math. Z., ii0 (1969) 123-126.

[HS] D.W. Hall and G. L. Spencer, Elementary Topology. Wiley,

New York, 1955.

[M I] S. MrSwka, On universal spaces, Bull. Acad. Pol. Sci.

~(1956) 479-~81. [M 2] S. Mr$wka, Further results on E-compact spaces I, Acta

Math. 120 (1968) 161-185.

[M 3]

[Ny]

[Pl]

[P2]

[ss]

481

S. Mr6wka, Recent results on E-compact spaces, in: TOP 72 -

General Topology and its A~plications (New York, Springer-

Verlag, 1974).

P. Nyikos, Prabir Roy's space A is not N-compact, Gen. Top.

Appl. ~ (1973) 197-210.

G. Preuss, Trennung und Zusammenbang, Monatsh.

Math. 74 (1970) 70-87.

G. Preuss, A categorical generalization of completely

Hausdorff spaces,h~General Topology and its Relations to Modern Analysis and Algebra IV. New York, Academic Press,

1972.

L. A. Steen and J. A. Seebach, Counterexamples in Topology.

New York, Holt, Rinehart and Winston, 1970.

CATEGORICAL PROBLEMS IN MINIMAL SPACES

BY

JACK R, PORTER

Abstract. A space X with a topological property P is called

minimal P if X has no strictly coarser topology with property P

and is called P-closed if X is a closed set in every space with

property P that contains X as a subspace. This paper surveys,

from a categorical viewpoint, a number of results recently obtained

in minimal P and P-closed spaces where P includes the properties

of regular Hausdorff, extremally disconnected Hausdorff, and the

separation axion~s S(~) for each ordinal ~ > 0. Particular

attention is focused on some of the categorical problems in these

areas.

483

CATEGORICAL PROBLEMS IN MINIMAL SPACES

BY

JACK R, PORTER 1

For a topological property P, a minimal P space is a set X with

a topology that is minimal among the partially ordered set of all

P-topologies on Xo If P is a property that implies Hausdorff, then a

c~act P space in minimal P. Closely associated with minimal P

spaces is the property of P-closed--a P-space X is P-closed if X

is a closed set in every P-space containing X as a subspace. For a

large number of properties P (cf. [BPS] ), a minin~l P space is

P-closed.

The theory of categorical topology has touched the area of minimal

P and P-closed spaces, as many other areas of general topology. This

paper surveys a mrnber of categorically related results recently obtained

in the minimal P and P-closed spaces where P is the properties of

regular Hausdorff, extremally disconnected Hausdorff, and the separation

axioms S(a) for each ordinal ~ > 0 (S(~) defined below); en~3hasis is

placed on the problems in these areas. This survey paper is a contin-

uation, in spirit, of [BPS, SI].

Let TOP (resp. HAUS) denote the category of spaces (resp. Hausdorff

spaces) and continuous functions. The full subcategory of HAUS of

The research of the author was partially supported by the University of Kansas General Research Fund.

484

regular (includes Hausdorff) spaces is denoted by REG. For each ordinal

> 0, a space X is said to be R(~) (resp. U(~)) [PVl] if for every

pair of distinct points x,y ~ X, there are subfamilies {F~:B < e} and

{G~:B < ~} of open neighborhoods of x and y, respectively, such that

F o AG o = ~ (resp. Cl~o A clxG O = ~) and for ~ + 1 < ~, clxG~+ 1 ~%

and ClxF+ 1 ~ F. For ~ ~ ~ (the first infinite ordinal), it is easy

to verify that R(~) and U(~) are equivalent concepts. For notational

convenience, the symbols R(~) and U(~) for ~ > ~ are replaced by

a single symbol S(~); for n E IN, R(n) is replaced by S(2n-l) and

U(n) by S(2n)o Thus, a space is Hausdorff (resp. Urysohn) if and only

if it is S(1) (resp. S(2)).

spaces is denoted by S(~).

The notation A c B

The full subcategory of HAUS of S(~)

is used to denote that the category A

is a subcategory of the category ~; by the largest subcategory of B

with a certain property, we are referring to this inclusion. As usual,

a replete subcategory means an isomorphi~n-closed subcategory.

A subset A of a topological space X is regular-open if !

int(clA) = A. The family of regular-open sets form an open basis for a

topology, coarser than the original topology; X with this new topology

is denoted by X and called the semiregularization of X. A space X

is said to be semiregular if X = X s. Let SR denote the full sub-

category of HAUS of semiregular spaces.

A function f: X -~ Y where X and Y are spaces is g-continuous IF]

if for each x E X and open set U containing f(x), there is open set V

containing x such that f(clV) C clU. The proof of the following facts

485

are straight-forward.

(l.l) If f: X -~ Y is P-continuous where X is a space and Y is a

regular space, then f is continuous.

(1.2) For a space X, the identity function X s ~ X is P-continuous.

2. P = Hausdorff. The term "Hausdorff-closed" is shortened to "H-closed".

There are two well-known methods to densely embed a Hausdorff space X

in an H-closed space--(1) using the Kat~tov extension [K] (denoted as

~X) and (2) using the Fomin extension [F] (denoted as aX). The full

subcategory of HAUS of H-closed spaces is denoted by HC. Three

situations motivate the first problem.

(I) In 1968, Herrlich and Strecker [HSI] showed that HAUS is the epi-

reflective hull of HC, however, they did find a subcategory, denoted

HAUS , of HAUS such that HC C HAUS , ob(HAUS ) = ob(HAUS), and HC is ,

epireflective in HAI~ with ~ as the epireflector. In 1971, Harris

[Hal, Ha2] proved there is a largest subcategory, denoted as pHAUS, of

HAUS such that HC C pHAUS and HC is epireflective in pHAUS with

as the epireflector. Thus, HAUS C pHAUS and ob(pHAUS) = ob(HAUS).

This situation does not fall under the scope of the usual theory

of epireflection; the point of failure is best understood with the

following characterization theorem of the theory.

Theorem 2.1. [HS2, Th. 37.2] If A is a full, replete subcategory

of a complete, well-powered, co-(well-powered ) category B , then A

is epireflective in B if and only if A is closed under the formation

486

of products and extremal subobjects in B.

Now HC is a full, replete, epireflective subcategory of pHAUS and

IIC is closed under the foz~ation of products and under the formation of

extren~l subobjects in pHAUS. Also, pHAIB is well-powered and co-(well-

powered). By Theorem 23.8 in [HS2], con~leteness is equivalent to being

closed under the formation of products and finite intersections. But

pHAUS is neither closed under the fornmtion of products as noted in

Fz~an~ple 2.2 nor under finite intersections as noted in Example 2.3.

Exarg01e 2.2. Let C denote the Cantor space, i.e., C = H{~: n E IN}

where X is {0,2} with the discrete topology. Since HC is a full n

subcategory of HAUS and X E ob(HC) for each n ~ ~, it follows n

that C is categorical product of {~: n E ~} in HC and, thus, is

the only candidate for the categorical product of {Xn: n ~ IN} in

pHAUS. Let ~ : C ~ X be the usual coordinate projection function. n n

It is straightforward to show that C contains a countable subspace

X = {ai,bij: i,j E IN} such that (1) there is a family {Ui: i ~ IN}

of pairwise disjoint open sets such that a i E Ui, (2) {bij: j ~ IN} _C Ui

for i E IN, and (3) for each i E IN, the sequence {bij: j @ IN}

converges to a.. Let f: IN ~ X be a bijection. For each n E IN, 1

~n ~ f: IN ~ Xn is a p-map (recall that an open cover of a space Y

is a p-cover if there is a finite subfamily whose union is dense and a

continuous function f: Y ~ Z is a p-map if for each p-cover U of

487

Z, f-l( U ) = {f-l(u): U E U} is a p-cover of Y) since each open cover

of X is finite. Now, V = X\ (ai: i ~ ~} is open subset of X and

V : {Ui: i E IN} U (V} is p-cover of X. Now f-l(v) has no finite

subcover. But the only p-cover of a discrete space is one with a finite

subcover. Thus, f-l(v) is not a p-cover and f is not a p-n~p. So,

C is not the categorical product of {Xn: n E IN} in pHAUS. This shows

that pHAUS is not closed under the formation of products.

Exan~01e 2.3. In this example, pHAUS is shown not to be closed under

finite intersections with a space X E ob(HC). Let X = ((i/n,I/m):

n,m,-mE ~} U {i/n,0): nE IN} U ((0,i),(0,-i)} where X\ {(0, i),

(0,-I)} has the usual subspace topology inherited from the plane and a

basic neighborhood of (0,I) (resp. ,(0,-i)) is a set containing

{(0,1)} U {(I/n,i/m): n >k, m E IN} (resp. {(0,-i)} U {(i/n,-i/m):

n_> k, mE IN}) for some k C 1N, cf. Ex. 3.14 in [BPS]. The space

X is H-closed. Let A 1 (resp. A_I ) be the subspace {(0,i)} U

{(i/n,i/m): n,mE IN} U {(i/n,0): n E IN} (resp. {(0,-I)} U {(I/n,-I/m):

n,m E IN} U ((I/n,0): n E IN}). Both subspaces A 1 and A_I are

H-closed and the inclusion functions if: A 1 ~ X and i_l: A_I -~ X are

p~n~aps. Let A = A 1 ~ A_I and i: A -~ X be the inclusion function. It

is straightforward to show that if the subobjects (Al,il) and

(A_l,i_l) have an intersection in pHAUS it must be (A,i). But

(A,i) is not a subobject in pHAUS as i is not a p-map for (I) by

Theorem D in [Hal], i is a p-map if and only if cl~ is H-closed

488

and (2) el~ = A is an infinite discrete subspace and not H-closed.

One of the inherent flaws of the class ob(HC) is not being closed

hereditary; in fact, by results in [L,SI~], every Hausdorff space is a

closed subspace of some H-closed space. However, the class ob(HC) is

regularly closed hereditary (cf., Prop. 3 in [HSI]), i.e., if

X E ob(HC) and A = c~intxA C X, then A E ob(HC).

(II) The existence of a largest subcategory of HAUS in which HC is an

epireflective subcategory with ~ as the epireflection occurs in a

more general setting as noted by the following theorem.

Theorem 2.4. [Pol] Let 8 C HAUS and A be a full, replete sub-

category of HAUS. Let F: oh(B) ~ ob(A) such that for each X ~ ob(8),

FX is a topological extension of X and for X ~ ob(A), FX = X. For

X E ob(B), let FX: X ~ FX denote the inclusion function. Then there is

a largest subcategory C of HAUS such that A is an epireflective

subcategory of C with F as the epireflection. Also, ob(C) = ob(B).

Let aHAUS denote the largest subcategory of HAUS such that HC is

an epireflective subcategory of oHAUS with o as the HC-epireflection.

An old problem has been to characterize the morphi~rs of oHAUS which is

equivalent to characterizing for X,Y in ob(HAUS), those continuous

functions f: X -~ Y that have a continuous extension af: oX ~ oY;

D. Harris communicated to me, at the Memphis Topology Conference during

March 1975, that he has solved this problem.

In [Pol], it is shown that aHAUS c pHAUS, i.e., if a continuous

function between Hausdorff spaces has a continuous extension to their

Fomin extensions, then the continuous function has a continuous extension

489

to their Kat~tov extensions. Similar to the category pHAUS, the extremal

subobjects of IIC in aHA[N are the H-closed subspaces and oHAUS is not

closed under finite intersections (same spaces and argunent in Example

2.3) and is not closed under formations of products (use the same spaces

as in Example 2.2 and the following fact to show that ~ is a n

morphism in GHAUS). Thus, the category eHAUS, like the category pHAUS,

is not con~lete and, hence, does not satisfy the hypothesis of Theorem 2.4.

Proposition 2.5. A continuous function f: X ~ Y where X is Hausdorff

and Y is c~10act Hausdorff has a continuous extension F: GX-~ Y.

Proof. By Theorem 2.1 in [K], f has a continuous extension

g: KX -~ Y. By Theorem 8 in [F], there is a 0-continuous function

h: oX ~ ~X such that h(x) = x for x E X. Let F = h o g. Thus,

F is an extension of f and is 0-continuous since the composition of

0-continuous functions is 0-continuous. By I.i, F is continuous.

(III) Banaschewski proved [B] that a Hausdorff space has a minimal

~Lausdorff extension if and only if it is semiregular. Each semiregular

Hausdorff space X has a largest (in a partition sense, cf. [PV2,3])

minimal Hausdorff extension, denoted as ~X and called the Banaschewski-

Fomin-Shanin (abbreviated to BFS) minirr~l Hausdorff extension of X.

Let N1 denote the full subcategory of HAUS of minimal Hausdorff spaces.

By Theorem 2.4, there is a largest subcategory of SR, denoted ~SR,

such that MH is an epireflective subcategory of ~SR with ~ as the

epireflection. The category ~I is closed under the formation of products

[0], but the class ob(~4) is not regularly closed hereditary (the

490

regularly closed subspace A 1 of the minimal Hausdorff space X in

Exan~le 2.3 is not minimal Hausdorff). The extremal subobjects of MH

in ~SR are unknown. D. Harris ccranunicated to me at the Men~phis

Topology Conference that he has characterized the morphisrns of ~SR,

i.e., those continuous functions between objects in SR that can be

extended to their BFS minimal Hausdorff extensions. The category

~SR is not closed under the formation of products (use the same spaces

as in Example 2.2 and the following fact to show that = is a n

morphi~n in ~SR for each n E IN ) and is not closed under finite

intersections by the following example. So, the category ~SR, like

the categories pHAUS and aHALB is not complete and does not satisfy the

hypothesis of Theorem 2. i.

(2.6) Let X @ ob(SR) and Y be a compact Hausdorff space. A

continuous function f: X ~ Y has a continuous extension F: ~X ~ Y.

Proof. Use the ssn~ proof as in 2.5 with his modification: Let

h:(~X) s ~ ~X be the identity function and note that h is 8-continuous

by 1.2 and the fact that ~X = (~X)s by Theorem 5.5 in [PT].

Example 2.7. Let X be the space in Example 2.3 and X* = {x*: x E X}

be a copy of the space X. Let Y be the topological sum of X and

X with the points (i/n,0) and (I/n,0) identified for each n ~ IN. ,

The spanes X, X , and Y are minimal Hausdorff. Let i." X ~ Y and

* * i*(X*) i : X -+ Y be the inclusion functions. Let A = i(X) M and

j: A -~ Y be the inclusion function. Now, i and i are morphisms

in MH, and it is straightforward to show that if (X,i) and (X ,i ) have

491

an intersection, it must be (A,j). Hov~ver, A is infinite discrete,

non-H-closed, closed subspace of Y. So, if j has a continuous

extension from #A to ~Y(=Y), then ClyA(=A) would be H-closed.

So, j is not a morphism in ~SR and ~SR is not closed under finite

intersect ions.

In situations I, II, and III, we have three cases of epireflections

(~,a, and ~) not falling under the scope of the usual theory of

epireflect ions.

Probl~n A. Generalize the theory of epireflection to cover the three

epireflections in situations I, II, and III.

It must be remarked that G. Strecker, at this conference, has

extended the theory of epireflections to cover the epireflection

in situation I. The possibility still exists that Strecker's extended

theory of epJmeflections rmay also cover the epireflections o and

in situations II and III.

Also, it must be remarked that the existence of the largest sub-

category ~SR of SR on which ~I is epireflective with ~ as the epire-

flection solves part of problem 5 in [Hel] and a part of the problem

on page 308 in [HSI]. The solution to the other part of the problem

in [HSI] uses this variation of Theorem 2.4.

Theorem 2.4' . Let B C TOP and A be a full, replete subcategory of

TOP. Let F: ob(B) ~ ob(A) such that for each X E ob(8), there is a

continuous function FX: X -~ FX with this property: for each

f E M~X,Y), there is at most one morphi~n g E MA(FX,FY ) such that

X '

I f Y

492

a_ FX

L .Q

commutes. If AC B and FX = X for each

a largest subcategory C of B on which F

Also, ob(C) = ob(B) and A C C .

X E ob(A), then there is

is an A-epireflection.

Proof. The proof is similar to the proof of Theorem 2.4 in [Pol].

To apply Theorem 2.4' to second part of the problem on page 308

in [HSI], let B = HAL~, A= MH, and for X E ob(HAUS), define

FX = ~(Xs). So, FX: X -~ ~(X s) is the inclusion function (not

necessarily an embedding). By Theorem 2.4' , there is a largest

subcategory, denoted ~HAUS, on which ~I is an epireflective subcategory

with ~( )3 as the epireflection.

Our next problem is also motivated by three situations.

(IV) In 1930, Tychonoff IT] proved that every Hausdorff space can

be embedded (not necessarily densely) in an H-closed space. Actually,

he showed for an open basis B of a Hausdorff space X, X can be

embedded in a decrmT~sition of H~ with a special topology where I

is the unit interval with the usual topology.

(V) Frolik and Liu [FL] proved that every H-closed space can be

embedded as a maximal separated subspace of its closure in the product

of the unit intervals with tl~e upper semicontinuity topology.

493

(VI) Parovicenko [Pa] proved that if {y : ~ E A} is a set of

H-closed extensions of a Hausdorff space X and e: X ~ H {Y : ~ ~ A} is

the embedding map defined by e(x)(~) = x, then cl(e(X)) is an H-closed

extension of X which is the supremum of {Y : ~ ~ A} relative to

the usual partial ordering defined between extensions.

A popular construction of the Stone-Cech compactification of a

Tychonoff space is tsking the closure of an embedding into a product

of unit intervals. Situations IV, V, and VI motivate our next problem

by presenting some evidence that a variation of the e~nbedding into a

product n~ay be possible for the KatVetov, Fomin, and BFS extensions.

Problem B. For X E ob(HAUS) (resp. X E ob(SR)), construct, hopefu/ly

along the lines of (V), ~X or oX (resp. ~X) in telm~ of products.

S. Salbany has communicated to me, at this conference, that he has

a method of constructing the KatVetov extension ~X of a Hausdorff

space along the lines of the Frolik-Liu technique.

3. P = S(~) for e > 0. KatVetov [K] showed that a space is minin~l

S(1) if and only if it is S(1)-closed and semiregular. The corresponding

fact for S(~) spaces, ~ > I, is not only false but no reasonable

substitute has been found for the property of semiregular (cf. Ex. 4.8

in [BPS]). In fact, for limit ordinals ~, minimal S(~) spaces are

regular [PVI]. Even though it is false that the class of S(2)-closed

[He2] and the class of minimal S(2) [$2] spaces are closed under the

formation of products, it is true [PVI] that every S(~) space can be

densely embedded in a S(~)-closed space. For ~ > 0, let S(~)C (resp.

~S(~)) be the full subcategory of HAUS of S(~)-closed (resp. minimal

494

S(~)) spaces.

(3.1) There is nontrivial subcategory A of S(~) such that S(~)C

is epireflective in A and ob(A) = ob(S(a)).

Proof. Since every S(c) space can be densely embedded in a

S(~)-closed space [PVI], then by using the axiom of choice for a class

of nonen~0ty sets, for each X E ob(S(~)), assign a S(~)-closed extension

FX of X. By Theorem 2.4, there is a largest subcategory A of S(~)

on which S(~)C is epireflective with F as the epireflection. Also,

ob(S(a)) = ob(A).

For X ~ ob(HAUS), let SX: X-+X s be the identity function.

By Theorem 2.4' , there is a largest subcategory A of HAUS on which SR

is epireflective with s as the epireflection; actually, in this case

s is a monoreflection. Also, s is a surjection from ob(S(1)C) onto

ob(~S(1)) [K]; hoverer, in general, a S(2)-clsoed space m~y have no

coarser minimal S(2) topology (cf. [Po2] ). It is still unknown for

> 1 (the ~ = 2 case is Problem 16(b) in [BPS]) if an S(a)-closed

space with at least one coarser minin~l S(~) topology has only one

such topology; in the case ~ = i, this is true. Let S(~)C' be

the full subcategory of S(~)C whose objects have at least one coarser

minimal S(~) topology. For each X ~ ob(S(a)C'), using the axiom

of choice for a class of nonempty sets, let X be X with some m

coarser minimal S(a) topology. Let reX: X-+X m be the identity

function; m X has the uniqueness property of Theorem 2.4' . An

application of Theorem 2.4' yields the following fact.

495

(3.2) For ~ > 0, there is a largest subcategory A of S(~)C' on

which ~(~) is a monoreflective subcategory with m as the mono-

reflection. Also, ob(A) = ob(S(a)C').

We conclude this section by noting that two of the unsolved

problen~s (Probl~s 7 and 16(b)) in [PBS] for S(2) spaces should be

extended to S(~) spaces for ~ > I.

Probl~ C.

(i) [BPS, Prob. 7]. For ~ > I, find a necessary and sufficient

condition for a S(~) space to be en~edded (densely embedded) in a

minimal S(~) space.

(2) [BPS, Prob. 16(b)]. For ~ > I, prove or disprove that each

object in S(~)C' has at most one coarser minimal S(a) topology.

4. P = regular and S(~). Recall that our definition of regularity

includes Hausdorff. The relationship between S(~) and regularity

is given by this theorem.

Theorem 4.1. [PVl]

(a) A regular space is S(~), but the converse is false in general.

(b) A space is regular-closed if and only if it is regular and

S(~)-closed.

(c) A space is minimal regular if and only if it is minin~l S(~).

Even though it is false that every regular space has a regular-

closed extension [He3], it is true that every S(~) space has S(~)-

closed extension [PVl]. The category REG is an epireflective subcategory

of HAUS [He4, Th. 2.12]; denote the REG-epireflector by r. By 3.1(e)

496

of [PVI] if X is a S(~)-closed space, then rX is regular-closed.

We are quite interested in those S(~)-closed spaces with coarser regular-

closed topologies, and, of course, these are precisely the S(o~)-closed

spaces on which r is one-to-one.

Problem D. Internally characterize those S(m)-closed spaces on which

the FZC--reflector is a monomorphism.

By tracing the construction of the REG-reflector r, it is straight-

forward to show for X E ob(TOP), r: X -~ rX is a monomorphism if and

only if for every pair of distinct points x, y E X, there is Y ~ ob(REG)

and continuous function f: X ~ Y such that f(x) ~ f(y). This type of

external characterization does not seem to be useful in identifying

those S(o~)-closed spaces with coarser regular-closed topologies.

The full subcategory of REG of objects possessing regular-closed

extensions is denoted as RC-REG. D. Harris [Ha3] has developed a

proximity theory (called RC-proximities) that generate, in a bijective

manner, the set of regular-closed extensions for each X ~ ob(RC-HEG).

So, a topological space X is a object in RC-REG if and only if the

topology of X has a compatible RC-proximity. This characterization

of objects in RC-HEG reveals a global nature of such spaces; however,

an internal characterization of such spaces might provide valuable

insight into the internal structure of regular spaces.

Problem E. Find an internal characterization of those regular spaces

possessing regular-closed extensions.

Since every regular space can be densely embedded in a S(0~)-closed

extension, then a solution to Problem D might lead to a solution to

497

Problem E.

5o P = Extr~T~lly disconnect Hausdorff. In this section, the property

of extr~s/ly disconnected Hausdorff (resp. extremzlly disconnected

Tychonoff) is abbreviated to EI~ (resp. EDT). In [PW], an EDH space

is shown to be EDH-closed if and only if it is H-closed. If X is

EDH space, then X is EDT space [PW, 2.1]. So, a minimal EDH space

is EDT, and is seeking a characterization of minimal EDH spaces, it

suffices to restrict ourselves to the class of EIYf spaces.

Theorem 5.1. [PW] An EDT space X is not minimal EDH if and only

if there is a nonempty clopen subset B C X and a continuous injection

B -~ clBx(X\ B) \ X.

In the case that X is a locally con~0act, EDT space, the "continuous

injection" can be replaced by "6~nbedding" into BX \X. The role of

the Stone-Cech con~0actification in Theorem 4.1 hints of the possibility

of a categorical theory that would characterize minimal P spaces

where P is EDH plus, hopefully, many other properties. An inter-

mediate step in this direction would be a solution to the next problem.

Problem F. [PW] Find an internal characterization of minimal E~I

spaces.

Since the absolute (or projective cover) of a Hausdorff space is

EDT, it is natural to inquire which Hausdorff spaces give rise to

absolutes that are minimal EDH. The absolute of an H-closed space is

compact EDH and, hence, minimal EDH; however, there are minimal regular

spaces whose absolutes are not minimal EI~I. The exxnplete solution is

498

provided by the next theorem which extends Theorem 5. I.

Theorem 5.2. [PW] Let X be a Hausdorff space and EX the absolute

of X. EX is not minimal EDH if and only if there is a non~Tpty

regularly closed subset A _C X (i.e., A = cl~ntx(A)) and a continuous

injection EA -~ Clax(X \ A) \ X (recall that aX is the Fomin H-closed

extension of X).

There is s~r~ evidence that a minimal EDH space is pseudocxx~act,

e.g., a separable minimal EDH space is countably compact. An affirmative

answer to the conjecture that a minimal EEH space is pseudocompact

should be useful in obtaining a categorical theory that characterizes

minin~l ~ spaces.

Problem G. [PW] Prove or disprove that minimal EDH spaces are

pseudoc~npact.

In conclusion, we remark that the regular-closed extension

problem (Problem E) and the problem of finding a categorical theory

that characterizes minimal EDH spaces are global problems (in the

latter case, note the reference to a nonempty clopen set in Theorem

5.1). Much of the theory of minimal spaces accc~nplished during the

1960's was of a local nature, i.e., certain properties had to hold at

each point. It is now clear that many of the unsolved problems re-

n~aining in the area of minimal spaces are of a global nature and seem

to be ripe for an attack by global machinery, such a~, categorical

theory.

Department of Mathen~tics University of Kansas

Lawrence, Kansas 66044 U.S.A.

499

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B

BPS

F

FL

Hal

Ha2

Ha3

Hel

He2

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SOME OUTSTANDING PROBLEMS IN TOPOLOGY

AND THE V-PROCESS

by

M. Rajagopalan

(Dedicated to Swami Gnanananda and Professor M. Venkataraman)

INTRODUCTION.

We mention some well-known problems in topology:

(a) Is a product of sequentially compact,

compact in general?

(b) Is it true that a completely regular,

T 2 spaces countably

T2, scattered, count-

ably compact space X cannot be mapped continuously onto

the closed interval [0,i]?

(c) Is every completely regular, T2, scattered space 0-dimen-

sional?

(d) Does every scattered completely regular, T 2 space admit

a scattered compactification?

All these problems have been raised by well-known mathema-

ticians and have proven to be hard problems. The problem (a)

was raised by C. Scarborough and A. H. Stone [9]; (b) by P. Nyikos

and J. J. Schaffer [5]; (c) by Z. Semadeni [i0] and (d) by

R. Telgarsky [ii] and Z. Semadeni [i0].

Though the above problems seem to be unrelated to each

other at first sight all can be answered in the negative by con-

structing suitable counterexamples and all these examples can be

obtained by a single method which we call V-PROCESS (which is a

short form for VENKATARAMAN PROCESS). Some of them can be

answered under weak set-theoretic axioms as well. Under suit-

able axiom of set theory this V-process yields very strong

examples also. For example, using the Jensen's axiom called

502

and the V-process we can show the existence of a hereditarily

separable, perfectly normal, locally compact, locally countable,

first countable, normal, sequentially compact, scattered space

which is not Lindel6f.

Many persons have asked the author to explain this v-process

clearly. In this paper we explain the V-process. Then, we also

produce a family of sequentially compact spaces whose product is

not countably compact using the axiom ~ (mentioned below) and

ZFC only. Thus we give a negative solution to the problem (a)

of C. Scarborough and A. H. Stone under fairly weak set-theoretic

axioms.

NOTATIONS AND DEFINITIONS:

All spaces are assumed to be T 2. N denotes the discrete

space of integers > 0. If X is a completely regular space

V

then 8X denotes the Stone-Cech compactification of X. Let

Y be a topological space, A CY a subset of Y and z a

partition of Y. Then we use int A to denote the interior of

A and Y/~ to denote both the quotient set and quotient space

and IYI the cardinality of Y. R denotes the set of real

numbers and IRI is denoted by c. We put ~ for INf.

ZFC denotes the Zermelo-Fraenkel axioms of set theory to-

gether with the Axiom of Choice. (CH) denotes the continuum

hypothesis which states that there is no cardinal number l so

< ~ < c (MA) denotes the Martin's Axiom which states: that o

(MA) denotes the Martin's Axiom which states: Let X be

a compact Hausdorff space in which every pairwise disjoint family

F of non-empty open sets is countable. Then X cannot be ex-

pressed as a union U A of a family G of closed sets A so AeG

that int A = ~ for all A e G and IGI = c.

503

(Axiom ~ is the following statement: Let J

be a set so that IJI < c. Let 0 6 be a compact open

set of 8N-N for each e e J. Let the family {0 6 I ~ c J}

have the finite intersection property. Then ~ 0 has non-empty ~eJ

interior in BN-N.

We use A to denote the initial ordinal of cardinal c.

We use ~ to denote the first uncountable ordinal. We may not

use (CH) in every one of our assertions. So A may not be

equal to ~. We note that given a countable set ~l<e2 < ...

<e < ... of ordinals so that ~ < A for all n e N there n n

is an ordinal 7 < A so that ~ < u for all n e N. n

In the sequel, we generally use Greek letters like 6, 8, Y,

... to denote ordinals < A and English letters like m, n

etc. to denote the elements of N. It is known that (CH) im-

plies (5~) and (MA) implies the axiom ~. (See [7] for this).

We follow [i] and [12] for notions in General Topology, extremal

disconnectedness and aN. We follow [2] and [4] for properties

of scattered spaces. We note that a topological space is scat-

tered if every non-empty subspace has a relative isolated point

in it. In [4] it is shown that a compact, T 2 space is scat-

tered if and only if it cannot be mapped continuously onto [0,i].

SECTION i: A Description of V-Process.

DEFINITION i.i. Let AI, A2,...,An,... be a sequence of subsets

of 8N. The growth of the sequence (A n ) is defined to be the

set n n"

504

DEFINITION 1.2. Let P be a property of closed subsets of 8N

such that {n} has P for all n e N. P is called a good

property if the following hold:

(a) If F is a closed subset of 8N which does not have P,

then there is a subset A ~F which is relatively open

as well as closed in F and A ~ ~ and A has P.

(b) 8N cannot be expressed as the union of a family F of

closed sets with property P and also with IFI < c.

DEFINITION 1.3. Let P be a good property of closed subsets

of BN. Let Y be a dense open subset of 8N and ~ a parti-

tion of Y by closed subsets of BN. The pair (Y,z) is said

to satisfy the condition ~ if the following hold:

Y (i) ~ is locally compact, T 2 and first countable.

(ii) Each member of ~ has P.

(iii) {~{ < C

(iv) Given A e ~ we have that there is a compact open subset

M of 8N so that M is saturated under ~ and

ACM~Y,

(V) {n} e ~ for all n e N.

Now we proceed to describe the V-process which is used to

solve the problems (a), (b), (c) and (d) mentioned in the intro-

duction.

STEP I OF THE V-PROCESS.

The step I of the V-process begins by choosing a suitable

good property P0 of closed subsets of 8N to attack the prob-

lem concerned.

STEP II OF THE V-PROCESS.

This step consists of proving the following lemma.

505

LEMMA 1.4. Let P0 be the given good property of Step I. Let

Y b_ee ~ dense open set of 8N and ~ a partition of Y by

compact subsets of 8N. Let (Y,~) satisfy the condition ~ .

Let AI,A2,...,An,... be a sequence of distinct members of

so that Y N A = ~ where A is the growth of the sequence (An). m

Let A ~ ~ and closed.

a partition T 0 of Y0

hold:

(a)

(b)

(c)

Then there exists an open set Y0 and

by compact sets so that the following

(Y0,~0) satisfies the condition (Vp0) �9

Y0 ~ Y and ~0 ~ ~.

YO~ A~ ~.

STEP III OF THE V-PROCESS.

This consists in proving the following lemma.

LEMMA 1.5. Let y be a limit ordinal which i__ss < A. Le___tt P0

be as in Step I above. Suppose that Y is a dense open subset

of 8N and ~ is a partition of Y by compact subset of 8N

and (Y ,~ ) satisfies the condition (Vp0) for all ~ < y. Let

Ya ~Y8 and za ~ ~B for all ordinals ~,8 < Y so that ~ ~ 8.

Then the pair (Y,~) satisfies the condition ~ where L_Y

Y = UY and ~ = U~ . a<y ~<y

We remark that this lemma is easy to prove.

STEP IV OF THE V-PROCESS.

We now prove the following lemma.

LEMMA 1.6. Let PO be as in Step I above. Let (Y,~)

in Lemma 1.4. Let F be a family of closed subsets of

that the following hold:

(a) IFI < c.

be as

8N so

506

(b) I_~f A e F and A ~ % then there is a sequence AI,A2,... ,

An,... of distinct members of z s__oo th~___~t A is the growth

of the sequence (An).

Then there is an open set Y0 o_ff 6N and a partition ~0 o_~f

Y0 by compact subsets of 6N so that the following hold:

(i) Y0 ~ Y and 70 ~ ~.

(ii) (Y0,z0) satisfies ~ .

(iii) Y0 ~ A ~ ~ i_ff A e F and A # %.

Proof: The proof of this lemma is obtained by well-ordering F

and using transfinite induction. To be more precise, let the

non-empty sets in F be written as FI,F2,...,F ,... where the

suffix ~ is understood to be an ordinal < A. If Y N F 1 ~ 9,

put Y1 = Y and 71 = ~. If Y ~ F 1 = r then replace Y, 7, A

of Lemma 1.4 by Y, ~, F 1 of this lemma and get (YI,ZI) so

that Y1 ~ F1 ~ ~ and (YI,71) satisfies the conclusions (a)

and (b) of Lemma 1.4. Now assume that 7 is a successor ordinal

and y = 8+1 and 8 < A and we have defined (Y6,~6). If

Y6 ~ F6 ~ ~ then put Y6+2 = Y6 and ~6+i = ~" If Y6~ F B =

then replace Y, z, A of Lemma 1.4 by Y6' ~6' F6 and get

(Y6+I,~6+I) so that this pair satisfies the conclusions (a), (b)

and (c) of Lemma 1.4 when the obvious substitutions are made in

those statements.

Now suppose y is a limit ordinal and we have defined

= M Y and (Y ,~ ) for all ~ < u and y < A. Put Yy_ ~<Y

= U Fy 7y_ z . If Y ~ ~ ~ then put Y = Y and ~ = zy_. ~<y ~ y - Y y - Y

If Yy_ N Fy = ~ then use Lemma 1.5 to observe that

satisfies the hypotheses of Lemma 1.4. Then replace

of that lemma by Y _, ~y_, Fy and get an (Yy,Zy)

(Yy_, Zy_ ) Y, ~, A

so that

507

this pair satisfies the conclusions of Lemma 1.4 with the obvious

substitutions made in them. Thus by transfinite induction we get

a well ordered sequence of pairs (Y ,z ). Finally put Y0 = U Ye

and n 0 = U~e. Then this (Y0,n0) will be the required pair.

STEP (V) OF THE V-PROCESS.

This is essentiallythe last step in the V-process. This

will construct an open set Y of 8N and a partition ~ of

Y by compact sets of 8N. The construction will be such that

Y/n will always be a scattered, locally compact, first countable,

non-compact, T2, space that is countably compact. Hence the

quotient space Y/z will always be sequentially compact, too.

Let us give the details of this step which is done by transfinite

induction using all ordinals ~ < A.

CONSTRUCTION 1.7. Put Y1 = N and 71 = {{n} I n e N}. Put

G 1 to be collection of all non-empty sets A so that A is

the growth of a sequence AI,A2,...,An,... of distinct elements

of ~ and A is compact. Well order G 1 as AII,AI2,...,AI~,...

where ~ < A. Now suppose that ~ is a successor ordinal and

y = 8+1 and ~ < A and we have defined (Ys,~B) and A ~ for

all ordinals e and ~ so that 1 < e < ~ and 1 < ~ < A. Put

F = {A ~ I l<e<B, I<$<8}. Then replace (Y,z,F) of Lemma 1.6

by YS' ~8' F above. Then get (Ys+I, 78+ 1 ) so that this pair

satisfies the conclusions (a), (b) of Lemma 1.6 with the obvious

substitutions made in those statements. Let GS+ 1 be the set of

all non-empty closed sets A of BN so that A is the growth

of a distinct sequence (AI,A 2 .... ,An,..) of elements of ZB+l"

Write GS+ 1 as a well ordered sequence A(B+I)I,A(B+I)2 .... ,

A(8+I)~,... where ~ < A.

508

Now suppose that y is a limit ordinal < A an4 that we

have defined Y , ~ , A 8 for all ordinals ~ < y and 6 < A.

Put Yy_ = U Y and ~ = U~ . Put F to be the collec- ~<y ~- ~ y- ~<y

tion {A ~ I l~&<y and l~8<y}. Now use Lemma 1.6 taking

Yy_, ny_, Fy_ in the place of Y, ~, F of that lemma. We get

(Yy,~y) so that Yy ~ A ~ ~ ~ for all e, 8 so that 1 ~ e < y

and 1 < 8 < y. Now put G to be the collection of all non-empty -- y

closed sets A which are growths of a distinct sequence of mem-

bers of ~y. Well order Gy as AyI,Ay2,...Ays,... f.or 1 ~ 6 < A.

Thus the transfinite induction gives open sets Y& and partition

of Y for all ~ < A. Finally put YA- = __~3Y and ~<A

ZA- = U n . This YA- and ZA- is the goal of the Step (V) ~<A ~

of the V-process. If we put Y for YA- and z for ~A- for

short, then the quotient space X = [ is the required counter-

example.

THEOREM 1.8. The space X obtained in Construction 1.7 at the

end of the V-process described above, is locally compact, T 2,

first countable, separable and scattered.

Proof: Let Ye, ~ be as in Construction 1.7. Put Ya/n = X a

for e < A. Then X is an ascending union of the subspaces

X . Since n ~ and Y is open in fiN and saturated under

7, we have that X is open in X for all ~ < A. Since

(Y ,~ ) satisfies condition ~ for all a < A we get that

X is locally compact, T2, first countable. Let q :'YA- + X

be the quotient map. Then q(N) is dense in X and hence X

is separable. Now X is locally compact and T 2 and IXal < c

for all ~ < A. So the one-point compactification X U {~} of

509

X cannot be mapped continuously onto [0,i]. So X U {~} and

hence X is scattered for all e < A. So if F CX is a non-

empty set and F ~ X 0 # ~ for some e0 < A then F N X~0 has

a point x 0 which is isolated in F ~ X 0. Then {x 0} is open

in F because F ~ X is open in F. Thus X is scattered. s 0

THEOREM 1.9. Let X be the space constructed at the end of the

V-process i__nn Construction 1.7. Then X i__{s sequentially compact.

Proof: Let Y , z , YA-' ~A- be all as in Construction 1.7.

Put q : YA- + X to be the natural quotient map and

q(Y ) = Yel~ = X for all ~ < A. Now it is enough if we

show that X is countably compact because X is first countable

by Theorem 1.8. For this it is ~nough to show that if {al,a 2,

...,an...} is an infinite set of distinct elements in X and

{a n } is isolated in {al,a2, .... an,...} for all n e N then

{al,a 2 .... a n .... } has a cluster point in X. So we take such

a subset {al,a 2 ..... an,...} in X. Put q-l(an) = F n for all

n e N and M = [ )F n. Now M cannot be compact because n~--i

q(M) = {al,a2,...an,...} cannot be compact. Moreover F n is

open relative to M for n E N. So the growth F = n~=iFn - n~=iFn

of the sequence (F n) is non-empty and closed in BN. So there

are ordinals ~, ~ < A so that F = A ~. If ~ is an ordinal

< A so that y > ~ and ~ then Yy ~ A ~ ~ ~. So YA- ~ A~6 ~ ~"

Let Y0 e YA- ~ A~6" Then q(y0 ) is a cluster point of the set

{al,a2,...a n .... }. Thus we get the theorem.

THEOREM i.i0. The space X of Construction 1.7 is not compact.

510

Proof: Keep the notations of Theorem 1.9. Since Y is a dense

open set of 6N and IY~ / ~I < c and P0 is a good property

of closed sets of BN and every member of ~ has P0' we

that Y~ ~ 8N for all ~ < A. So X ~ X for all ~ < A.

the family {X I e<A} is an open cover of X from which a

finite subcover cannot be extracted.

have

Then

REMARK i.ii. Whatever be the axioms of set theory we use, if

we can prove Lemma 1.4 under those axioms then the remaining steps

in the construction of the space X of the V-process follow and

that final space X is always a locally compact, T2, first

countable, scattered,separable, sequentially compact, non-compact

space. The role of the set-theoretic axioms we use is only to

prove Lemma 1.4 for a particular property P0 and the space X

that we finally obtain by the V-process will be the required

example.

In the next section we show how we can use the V-process

to solve the problem (a) of C. Scarborough and A. H. Stone of the

introduction. We solve it using only the axiom ~ beyond ZFC.

SECTION 2: A Solution to a Problem of C. Scarborou@h and A. H. Stone.

C. Scarborough and A. H. Stone [9] asked whether the product

of sequentially compact spaces is always countably compact.

Rajagopalan and R. G. Woods [8] showed the answer to be negative

to this problem using continuum hypothesis. They used the V-pro-

cess. J. Vaughn announced in a private communication that he

also solved the problem of C. Scarborough and A. H. Stone above

in the negative by using [CH]. Eric van Douwen informed the

author that he also got a solution to the same problem under the

511

axiom BF and ZFC. Neither the solution of Eric van Douwen nor

that of J. Vaughn appeared yet in print. We produce a family of

sequentially compact, locally compact, T 2 spaces whose product

is not countably compact.

In this section we use only the axiom ~ which is weaker

than both Martin's Axiom and continuum hypothesis.

DEFINITION 2.1. Let x be a given element of 6N-N. We say

that a closed set F C 6N has property Px if x ~ F.

REMARK 2.2. It is clearly seen that P is a good property. x

Now we t r y t o p r o v e Lemma 1 . 4 f o r t h i s P u s i n g o n l y (~) a n d X

ZFC.

THEOREM 2.3. Let Y b_~e ~ dens______e subset of 6N and z ~ parti-

tion of Y by compact sets of 8N. Let (Y,~) satisfy the con-

dition ~ . Let AI,A2,...,An,... be a sequence of distinct

members of z so that the growth A = UAn - UAn of the n=l

sequence (A n ) i__ss closed and non-empty. Let A N Y = ~. Then

there exists an open set Y0 of 8N and a partition ~0 of

Y0 b__y_y compact sets of 6N so that Y0 ~Y and ~0 ~ and

Y0nA~.

Proof: Given n e N choose a compact open set W n of 8N so

that A n~ W n~ Y and W n is saturated under 7. (This is

possible because (Y,z) satisfies the condition ~ . ) Put

n-i M n = W n - UW i for all n e N and n > 1 and M 1 = W I. Now

i=l

oo oo

AnC MnC and A= A - n= 1 n ~iAn is non-empty and

512

cO

disjoint with Y. So n~=iAn cannot be compact. So there is

an increasing sequence n I < n 2 <...< n k <... of integers so

that Mnk~ Ark for some integer r k for all k E N. Let E

be the set of even integers. Let G 1 = U M and G 2 = ~ Mnk. keE nk keN-E

Then G I, G 2 are disjoint open sets of BN. Since 8N is

extremally disconnected x can belong to atmost one of the sets

G1 or G2 (See [12,1]). So there is a subsequence (Mnk.) of

1

(Mnk) so that x ~ i~l M Put D. = M and C. = A nk. z nk. 1 rk. 1 1 1

for i e N. Then i~=ICi is not compact because CiC D i for

i e N and {D i I i E N} is a pairwise disjoint collection of

open sets of 8N. So C = ~]C i - ~_iCi ~ % and C CA. By

y Dk assumption ~ has cardinality < c. So -{- is a compact, T 2

space of cardinality less than c and hence totally disconnected

for all k e N. Since [ is first countable by assumption, it

follows that given k ~ N there is a descending sequence

Vkl ~Vk2 ~ ... ~Vkn ~ ... of compact open sets Vkn of 8N

so that Vkl = D k and DiVkn = C k and Vkn is saturated

0o

under ~ for all n ~ N. We put k~iDk= = D. We note that

C C D and x ~ D and D is a compact open subset of BN. Now

let F e n and F ~ D = ~. Then using the normality of ~N

we get a compact open set V F of ~N so that C CU F and

F ~ U F = ~. Let F be the family {U F I F e ~ and F ~ D = ~}

513

U {D-D n I neN} Then C C A for all A e F. So the axiom

gives us that there is a set G C D-Y which is non-empty and

compact and open relative to BN-N. Then there is a compact open

set M of aN so that M ~ (aN-N) = G. Then M ~ D i ~ ~ for n

an infinite set ZI' Z2,...,in,... of integers s Then given

n e N there is an integer s n so that M ~ Vs ~ ~. Put

R = n~=iVs . Then (R-R) ~ Y = ~ and (R-R) ~ C # # and

A Y = R is saturated under ~ and R-R has the property Px"

Also R is compact open in aN. Put Y0 = Y U R and

~0 = ~ U {R-R}. Then (Y0,~0) is the required pair of the

theorem.

LEMMA 2.4. Let X be a topological space. Then X is count-

ably compact if and only if given a i-i map f : N + X from

N into X there is an element p e BN-N and a continuous ex-

tension g : N U {p} § x of f.

Proof: Let us assume that given a i-i map f : N § X there

is an element p e BN-N and a continuous extension g : N U {p} § x

of f. Let {al,a 2 .... ,an,...} be an infinSte subset of X.

Let f0 : N § X be the map given by f0(n) = a n for n e N.

Then there is a p e 8N-N and a continuous extension

go : N U {P} § X of f0 to N U {p}. Then g0(p) is a cluster

point of {al,a 2 .... ,an,...} in X. Conversely, let us assume

that X is countably compact. Let f : N + X be a I-i func-

tion from N into X. Then f(N) is an infinite subset of X.

Since X is countably compact, f(N) has a cluster point '~'

Let us now consider two cases:

514

Case (i) : Let ~ % f(N). Let V be the filter of all neighbor-

hoods of s in X. Let F be the collection of all sets of the

form f-l(v N f(N)) where V E V. Then F is a filter on N

and extends to an ultrafilter U on N. Then there is a unique

element p e BN-N such that p e ~ (see [1,12]). Now put AeU

g : N ~ {p} § X as

g(x) = ~f(x) if x e N

L s if x = p

Then this g : N U {p} § x is a continuous extension of f to

N U {p}.

Case (ii) : Let Z e f(N). Without loss of generality we can

take s = f(1). Then ~ e f(N-[l}) and i % f(N-{l}). So, by

Case (i) above we have an element p e BN-N and a continuous

extension g : (N-{I}) U {p} + X of f / N-{I}. Then put

X : N U {P} § X as

= ~f(x) if X e N X(x)

L g(p) if x = p

This gives a continuous extension of f to N U {P}- Thus we

get the result.

LEMMA 2.5. Let p e BN-N be given. Let X be the space con- p

structed by the Construction 1.8 of the V-process using the re_r_ ~-

perty Pp of Definition 2.1. Le__~t qp : N + Xp be the restric-

tion of the natural quotient map q : YA- + Xp of Theorem 1.9.

Then qp is a i-i map of N into Xp which cannot be extended

to a continuous map g : N U {p} § Xp.

Proof: Let Xp U {~} be the one-point compactification of Xp.

Then the map ~p : BN § Xp U {~} given by

515

|q(x) if x e YA-

qp(X) = ~ ~ if x ~ YA-

is easily seen to be a continuous map from 6N onto XpU{~}.

So the function ~ : (N U {p}) + Xp U {~} which is the restric-

tion of qp to N ~ {p} is continuous and maps p on ~. Since

N is dense in N U {p} there can be only one continuous exten-

sion of f from N U {P} + X U{~} and ~ is already one such.

So there cannot be a continuous extension g : N ~ {p} + X of

f because in that case g(p) e X and hence g and ~ will be

two continuous extensions of f from N ~ {p} to X U {~}. p

This gives the lemma.

THEOREM 2.6. Assuming the Axiom ~ and ZFC there is a

family of locally compact, first countable, T2, scattered, sequen-

tually compact, seperable spaces X whose product is not count-

ably compact.

Proof: P

Then the collection {Xp I P e BN-N}

as in the statement of Theorem 2.6.

Given p e 8N-N let X be the space as in Lemma 2.5.

as in Lemma 2.5 for each p e 8N-N.

that H is not countably compact.

is a collection of spaces

Let qp : N + Xp be the map

Let H = H X . We claim peBN-N p

For let H qp : N + H be

the map that so that the pth coordinate of (~ qp) (n) is qp(n)

for all p e 6N-N and n e N. Then an easy application of

Lemma 2.5 gives that there is no 'p6 e BN-N so that H qp ex-

tends to a continuous function g : N U {p0} § H. So Lemma 2.4

gives that H is not countably compact. Thus we have the theorem.

REMARK 2.7. TO make the paper short we are not giving the good

properties here which are used to solve the problems (b), (c), (d)

of the introduction. The interested reader can look into [3] for

516

details about the solution to (b) and into [6] for details about

the problems (c) and (d). In [3] and [6] CH is used. Jerry

Vaughn has shown in a private communication that a method due

to ostazewski can be used to solve the problem (a) of Scarborough

and Stone using CH. The same method of ostazewski and an axiom

different from ~ was used by Eric van Douwen (in a private com-

munication) to solve the same problem (a).

Finally we raise the following problems:

OPEN PROBLEMS:

I.

2.

3.

Can any of the spaces X obtained in Lemma 2.5 be made nor- P

mal? How about if we use either [CH] or [MA]?

Can any of the spaces X obtained in Lemma 2.5 be made P

hereditarily separable if we assume [CH]?

Can the family of spaces X of Lemma 2.5 be so constructed P

that X is not homeomorphic to X if p, a ~ 8N-N and p a

p ~ a?

4. Is there a nicely described procedure to generate good proper-

5.

ties of closed subsets of 8N? In particular, is there a

nice way to find all the good properties?

Is it possible to find p s BN-N and X as in Lemma 2.5 P

9 SO t h a t ~(Xp) i s the o n e - p o i n t c o m p a c t i f i c a t i o n of Xp.

Is this possible under stronger set-theoretic axioms like

or [CH] ?

517

REFERENCES

[i] DUGUNDJI, R.; Topology, Allyn and Bacon, Boston (1966).

[2] KANNAN, V. and M. RAJAGOPALAN; On scattered spaces, Proc. Amer. Math. Soc., 43 (1974), 402-408.

[3] KANNAN, V. and M. RAJAGOPALAN; Scattered spaces II, Ill. J. Math (To appear).

[4] MROWKA, S., M. RAJAGOPALAN, and T. SOUNDARARAJAN; A charac- terisation of compact scattered spaces through chain limits (Chain compact spaces), TOPO 72, General Topology and Its Applications, Second Pittsburg International Conference, Dec (1972), Springer-Verlag, Berlin (1974), 288-297.

[5] NYIKOS, P. and J. J. SCHAFFER; Flat spaces of continuous functions, Stud. Math., 42 (1972), 221-229.

[6] RAJAGOPALAN, M.; Scattered spaces III, J. Ind. Math. Soc. (To appear).

[7] RUDIN, M. E.; Lecture notes on set-theoretic topology, CBMS Series, No. 23, AMS, Providence, R.I. (1975).

[8] RAJAGOPALAN, M. and R. GRANT WOODS; Products of sequentially compact spaces, (To appear).

[9] SCARBOROUGH, C. and A. H. STONE; Products of nearly compact spaces, Trans. Amer. Math. Soc., 124 (1966), 131-147.

[i0] SEMADENI, Z.; Sur les ensembles clairesem~s, Rozprawy Math., 19 (1959), 1-39.

[ii] TELGARSKY, R.; C-scattered spaces and paracompact spaces, Fund. Math., 73 (1971), 59-74.

[12] VAIDYNATHASWAMY, R.; Set topology, Chelsea Publishing Co., New York (1960).

[13] VAN DOUWEN, ERIC; First countable regular spaces and ~ , (To appear).

[14] VAUGHN, J. E.; Products of perfectly normal sequentially compact spaces, (To appear).

Nearness and Metrization

H.C. Reichel, Vienna

O~ Introduction:

From an heuristic point of view the general ideas of "nearness"

and "~istance functions" in topology seem to be strongly connected.

This paper deals with some part of these concepts; it tries to

compare distance-functions which satisfy all the usual "metric"

axioms, with several "topological structures", especially with the

concept of nearness-spaces recently developped by Horst Herrlich

(viz. w 3) and, since then, studied by several authors. - Real, as

well as non-numerical, distance-functions have been studied under

several aspects since the beginning of general topology. Especially

those (generalized) distance-functions taking their values in an

ordered group yield a satisfactory theory. (Compare the bibliography

of this paper and ~227 e.g.). - Working with such vector - (or, more

generally, group-) valued metrics on X, one trivially uses only the

positive cone of this group G. Thus one may try to replace G by a

partially (or totally) ordered semigroup S from the very beginning

of the theory. This is also motivated by several different problems

in metrization theory and the theory of general "convergence

structures". (For example, think of statistical metric spaces or

the interesting question to what extent "countability" inherent in

metrization theorems can be generalized using the well-ordering of

natural numbers instead of their cardinality). - But using semigroups S,

the S-valued metrics d S need not induce uniform structures (even

if all metric axioms are satisfied included the triangular inequality),

neither must the convergence structure induced by d S on X be compatible

with a topology on X. (For a detailed study, see a paper of Reichel

and Ruppert ~0] and w of this paper). The "reason" for this disad-

vantage is that a totally ordered abelian group is a topological

group in its order-topology which need not be true for totally ordered

abelian semigroups S. In w we study uniform structures generated by

"semigToup-valued metrics" d S. Here, d S induces a (semi)uniform

structure on X in the sense of E. Cech and A. Weil, respectively, from

which a topologically equivalent one can be "rediscovered" again.

Surprisingly, the latter is not true if we use structures in the sense

of Tukey and Isbell (see example 3.1. and w

519

w indicates some facts showing to what extent the topological

theory of metrizable spaces becomes more general if we admit also

(non-numerical) "semigroup-value~' metrics and quasimetrics. (Com-

pare [21], [31] and especially [30]).

The situation becomes more transparent if we consider nearness-

spaces (in the sense of H~ Herrlich) which provide an incisive frame

for studying general metrization problems (w 3). Since metrizability

of nearness spaces clearly is connected with the study of nearness-

structures having linearly ordered bases, w 3 can be interpreted as

a general study of such structures. The definition of metrizability

of N-spaces given by H. Herrlich ~7~ and Hunsaker and Sharma ~I0~,

is rich in meaning only for uniform nearness spaces with linearly

ordered bases (as they have been investigated in many papers); so

we supplement the theory by another definition (compatible with the

old one for uniform N-spaces) and propose several problems in this

context. Here, the crucial point is to find the correct generalization

of the concept of the diameter of bounded subsets of S-metrizable

spaces. ~ote, that there need not exist suprema of bounded subsets in S.)

(See page 15).We conclude by characterizing regular, semiuniform,tmiform

and paracompact nearness-spaces with linearly ordered bases. At the

same time the latter yields conditions for topological nearness spaces

to be also uniform, and thus includes Herrlichs characterization of

Nagata-spaces. As a byproduct we generalize a theorem of Atsuji, ~I~,

by characterizing those uniform spaces X with linearly ordered bases

where all continuous mappings from X into any other uniform space are

uniformly continuous. And finally, ~-compactness of a completely

regular space X is discovered as a necessary condition for the fine

uniformity (Isbell [11]) on X to possess a linearly ordered base.

w I. Semi~roup metrics and uniform structures

In this paper we consider only commutative semigroups (S,+) with

identity O. (S;~) is a ~artiall2~~ ~ (p.o.) sem~ou~ iff (S;<)

is a partially ordered set and a<b implies a+c<b+c for all a,b,cE S.

Moreover we assume a< 0 or a> 0 for a@O and, if a has an inverse -a,

-a< 0 iff a> O. A ts (t.o.) sem~grou~ is a p.o. semigroup

where a<b or a> b if a~b.

520

S + denotes the set of all s> O, st S. S is said to be of character

w iff there exists a decreasing ~H-sequence converging to 0 in the

order topology of S. (Here w denotes the wth infinite cardinal

number. The power of ~ is denoted by ~ . - Iff �9 is a topology on S,

(S,T) is called a ~t~o~olog~cal~semigrou~ iff the binary operation is

a continuous function on the product space S • S. -

- An S-metric on a set X is a function ds: Xx X * S + (i.e. ds(x,y)~

for all x,yE X) such that

(i) d(x,x)= 0

(ii) d(x,y):0 (iii) d(x,y)=d(y,x) (iv) d(x,y)S d(x,z) +d(z,y)

for all xt X

implies x = y

for all x~y~ X

for all x~y,zE X

(X,d S) is called an S-metric space or a space metrized by the semigroup S.

A special case are the (usual) metric spaces (X,~) which are R-metric

in our notation.

- Various "topological structures" compatible with S-metric spaees~ as

well as their relations to the structure of the linearly ordered semi-

group S will be studied in a paper by H.C. Reichel and W. Ruppert [30!

appearing in "Monatshefte fur Mathematik" 1976/77. Concerning nearness-

structures compatible with S-metrics see w 3.

For a p.o.-semigroup-metric space (X,d S) it is possible to define

"topological" structures in several ways. R. de Mart and Fleischer, [4]

for example, study p.o. semigroups S such that, for every nonempty Mc S,

inf M exists; and they associate with d S the following convergence-

structure:

a net {xa/x ~E X, a~ I} is said to converge to x~ X iff lim sup d(x~,x)=O;

[4]. Nevertheless, there need not be a topology on X for this convergence:

521

let S be the semigroup of all non-negative bounded real-valued

measurable functions on [0,I], p.o. by f< g a.e.; and let X be

the set of all real-valued measurable functions on [0,I], then

d ( f , g ) = l f - g l " (1 - I f - g 1 )-1

is an S-metric on X which - in the above mentioned sense - yields

convergence almost everywhere for sequences (fn) . But there is no

topology for this convergence.-

An other possibility is to study the entourages induced on X • X by

d S in the sense of A. Weil. Obviously, every p.o.-semigroup-metric

d S induces a semi-uniform structure Uds by le t t ing

Ur ={(x,y)/ds(x,y)<~}, g~ S +,

be a subbase for Uds. Hereby, as usual, a semiuniform~st~u~~u~~~ U

on X is defined analogously to a diagonal-uniformity on X with the

exception of the "uniform axiom":

u U~ U 3 Vu U such that V o V~U.

For details see e.g. the book of Cech [2]. - By a theorem of

G.K. Eallisch [13], Uds is a uniform~ 2 on S if S is a p.o. vector

group and, conversely, any uniform space (X,~) can be induced

by a p.o.-vector-group-metric on X in this way.

For arbitrary sem~r~s S, the theory shows remarkable differences.

So, for example, the semiuniformity Uds need not be a uniformity,

even if S is totally ordered. See example 1.5. below. (Theorem 1.1.

shows the "reason" for this disadvantage). - A semiuniformity U on X

is called S-metrizable iff there is a p.o. semigroup S such that

U=Uds. The question arises to characterize those semigroups S which

yield uniform structures:

522

Theorem 1.1.: The following assertions are equivalent:

(ii)

(iii)

(i) U is a separated uniform structure with a

linearly ordered base ~(B I<B 2 iff B I DB2;BI, 2

is G-metrizable by a t.o. (abelian) g~u~ G

is S-metrizable by a t.o. semigroup S such

that S,with the order topology ~o,becomes a

(iv) U is S-metrizable by a totally ordered semi-

group S such that the function s * s+s,s6 S,

is continuous at 0 with respect to the order

topology on S.

Proof: The equivalence of (i) and (ii) was shown explicitely by

F.W. Stevenson and W.J. Thron in [26]. To prove [ii) ~ (iii)

we only have to show that every t.o. group G is a topological

group with respect to its order topology ~o (which is not true

for semigroups as we shall see). Without loss of generality,

let G be abelian (compare also P. Nyikos and H.C. Reichel [21])~

Inversion is continuous which follows directly from the definitions.

To show for each neighbourhood W of 0 the existence of a neighbour-

hood V with V+Vc W, proceed as follows: take s E G, s> O, such that

the interval (-s,s)cW, pick t such that t < s, and let {si/i<a} be

a net converging to 0 from above. (If no such t or no such net exists,

then G is discrete and we are done).

Now {t + si/i<a} converges to t and we need only pick s i such that

s i< t and t+sl < s, whence s.z +s <l t+sz < s. Thus V= (-si,s i) satisfies

V + Vc W.

Since (iii) ~ (iv) is trivial, let us show (iv)~ (i):

Let S be a t.o. semigroup such that s* s+s, ss S, is continuous at 0

with respect to ~o" If S + has a smallest element a, the S-metric d S

523

obviously induces the discrete uniformity on X. Otherwise, let

{r /~ ~ } be a cofinal decreasing well-ordered net converging

to 0 with respect to ~o" The~ for every r , there exists an

index ~I such that r + c o< r if o> ~I" Similarly, start from

~I + I and construct ~2 such that r + ~ < r for all o> ~2" i +I

Then, for all a>~2, r162162176162162 +I +r I +I< r

Now the system of all

U e = { ( x , y ) / d s ( x , Y ) < C r } ,'r < w u , 'T

is a linearly ordered base for a separated uniformity U on X,

since, ~or every ~<~U' U~2+I ~ U2+lCU" �9

There is another assertion equivalent to (i)- (iv), showing that

the theory of non-archimedean (n.-a.) semigroup-metrics deserves

an extra study. An S-metric d S is n.-a. iff S is totally ordered

and d S satisfies the "strong" triangular inequality:

ds(x,y)~ max (ds(x,z),ds(z,y)),

for all x,y,zE X.

(N.-a. ~-metrics play an important rSle in general valuation

theory (Krull [ 15], Schilling [ 23] )).

Theorem 1.2.: (X, U) is a separated uniform space with a linearly

ordered base ~ if and only if I~ can be metrized either

by a suitable a-metric or by a non-archimedean

~A~2P~c~ d S.

Proof: Let S be a t.o. semigroup and d S a n.-a. S-metric on X.

For e E S +, let Be(x)={y/d(x,y) < e} be the e-ball with

centre xE X; then the covering ~r (Br X} essentially

is a partition of X, because, by double application of the

strong triangular inequality, two such balls either have

524

empty intersection orG~ysreidentical. Now the coverings ~r162 E S +,

can serve as a base for a (Tukey~ uniformity U on X which is

equivalent to Uds. (Clearly, each ~ is a star-refinement of

itself). Conversely, we sharpen the result of Stevenson and Thron,

mentioned before. (Compare also the paper of P. Nyikos and H.C.

Reichel [21] on w -metric spaces). If �9 is countable we are done;

so let ~ = (Bi/iE I} be a linearly ordered base for ~ such that

Bi<Bj iff B iDBj. By a construction of A. Hayes [g ], to each

BiE �9 we can associate a sequence (Bn), n=1,2 .... ,

B I =Bi, Bn+ I o Bn+ ICBn,

such that

~(i) N B = n=1 n

has the property

B(i)o B(i)=B (i), and ~' = {B(i)/i E I}

is a linearly ordered base of U. The uniform cover

~i ={B(i)(x)/xE X}, where B(i)(x) = {y/(x,y)E B (i)},

is a partition of X and we obtain a B ~ Z ~ z ~ ~ ~ by a

construction which - although different - is essentially equi-

valent to the proof of the above mentioned theorem of Stevenson and

Thron: Let S=N Z i (iE I) where Z i is the group of the integers,

S being ordered lexicographically, and set di(x,y)= 0 iff x and y

belong to the same set in ~i' di(x'Y) = I otherwise. Now

dsCx,y): = (di(x,y)) iE I

is the desired non-archimedean metric inducing U.

525

With the theorem of Stevenson and Thron in mind (and Kallisch's

result) one could conjecture a similar characterization of semi-

uniform spaces (X,U) using p.o. and t.o. semigroups, respectively.

Indeed, one can show an analogue (theorem 1.4. ), but the proof

has to be completely different, since both theorems mentioned above,

indispensably use the fact that every uniform space is unimorphic

with a subspace of a product of R-metric spaces, for which of course

there is no analogue in the theory of semi-uniform spaces. CThere

would be an analogue only if we did not require the distance functions

to satisfy the triangular inequality).

~emma_~• A semiuniform space (X,U) induces a T1-topology if and

only if U is "separated", i.e.:

N U i (UiE U)=A, the diagonal of X x X.

Remark: The straight forward proof is omitted, however, remark

that a separated uniform space induces a T2-topolog~, but

to prove the converse you indispensably have to use the

"uniform-axiom" or Tukeys eqtuivalent "star-refinement axiom"

either. Example 1.5. presents a separated semiuniform space

which induces a non-Hausdorff topology.

Theorem 1.4.: A separated semiuniform space (X,U) has a linearly

ordered base if and only if U can be metrized by a

t.o. semigroup S; *) i.e.: U=Uds for the semigToup-

metric d S .

Proof: We only have to show S-metrizability of (X,U) where U is

a semiuniformity having a linearly ordered base. Let �9 de-

note such a base of least power.

*) Without loss of generality, we can restrict ourselves to semi-

groups S being two-sided canvellative. (Of course, if moreover

S is holoidal (see ELjapin; Semigroups, AMS Translations 19631,

i.e. "naturally ordered" (see ~Fuchs; Partially ordered algebraic

systems, Pergamon Oxford 1963~), then S is the positive cone ef

some t.o. group G. However, the latter would be a rather strong additional condition.

526

= {Bi/i<w u} for some ordinal WU" Here i< j need not be

equivalent with B. DB . But, if, for every k <w , we let i 2 U

V k=n B i (i~ k), we obtain a well ordered base ~ = {Vi/i<w u}

for U such that i< j is equivalent with V D V . Indeed, ~ is l j

a base for U, since otherwise there would be a k<w such that U

V k is not refined by any B i. But �9 is linearly ordered so that

in this case, for every i> k, there is a j< k such that B iDBj.

Therefore, the system {Bj/j< k} would be a linearly ordered base

for ~ with a power less than w , which yields a contradiction. H

- Now let us construct the t.o. semigroup S by which (X,U) can

be metrized (S will be used in 1.5. again):

Let M be a set of arbitrary elements x i (i<~) and set x i<xj

iff i> j. Now let S be the free abelian semigroup over the set M.

which, of course, can be visualized as the semigroup of all formal

"linear combinatlons Z m x (m a natural number or O) where only iI 1

finitely many m i~O. Identify the empty word with 05 S. Further,

for every pair a=E mixi, b=~ njxj with Z m i~Z nj let a< b iff

Z mi<E nj. If a~b and E mi=E nj, pick the minimum index k such

that mk~n k and set a<b iff mk< n k. For those a=Z m x l 1 where Z mi=1,

this order coincides with the given order of the "basis set" {xi/i< j}.

It is easy to show that (S,<) is a totally ordered (and therefore

cancellative semigroup: a< b = a+c<b+c, for all a,b,cE S, is proved

by direct computation. - Now let ds(x,y)=x k iff (x,y)E V i for all

i<k and (x,y)@ V k. Moreover, let ds(x,y)=O iff (x,y)E V i for all

i<w . Thus we obtain an S-valued metric d S on X, since the triangular

inequality is satisfied for all x,y,zE X by the special order of S.

And obviously, this S-metric induces ~, i.e.: U=Uds. Note that in our

semigroup S, {xi/i<w u} is a net converging to 0 in the order topology

of S, but lim (x i+x i) =x I ~0. - (For our metric ds, every set consisting

of elements ds(x,y) has a greatest lower bound). -

527

We supplement theorem 1.4. by an example of a semiuniform space

(X,U) with a linearly ordered base for which there is no equi-

valent uniform structure. Note that, by theorem 1.2., no such

semiuniformity can be compatible with a non-archimedean metric d S.

Examp_le_1.~• Let S be as in the proof of the proceeding theerem,

and let X=M, metrized by ds(x,y)=min {x,y} if

x~y; x,yE X; and ds(x,~=O for all xE X.~Because

of the special ordering of S, d S satisfies the tri-

angular inequality, and, as usual, the system �9 of

all U = {(x,y)/ds(x,y)<r r S +, is a base of a

separated semiuniformity U on X. But certainly,

there is no uniform structure U' equivalent with ~,

since, for every r S +, U o U =Xx X (of course, for

every r and every (x,y) there is a z<r so that (x,z)

and (z,y)t U c, yielding (x,y)E Ur Ur Thus no

entourage U~ can have a "uniform refinement" Ur -

(The semigroup metric ds(x,y) =max {x,y), x~y,

and ds(x,x)= O, for all x~ X, would induce a

uniform structure on X inducing the discrete

topology).

w 2. Semigroup metrics and topologies:

With similar methods as used in theorem 1.4. we can prove a theorem

concerning S-metrizability of toplogical spaces; hereby a topology

on X is metrizable by a p.o. (t.o.) semigroup S iff

is a base for ~~ Since we want to exclude trivial cases we assume

that vor every ~> O, there exists an ~ with 0<~<r Then for every

t.o. semigroup $, let {r , r ~0' be a well-ordered net of

528

minimal power that converges to 0 with respect to the order topology

of S. Then, if the system ~ of all balls (B~ (x) / i~ ; x E X } 1

is a base for a topology �9 on X, then �9 will be a T1-topology such

that every non-isolated point has a well-ordered local base {Ui(x)/i~ }

UicU j iff i > j. (Note that ~ is a base for a topology on X if S

is (the positive cone of) an abelian group. If S is only a totally-

ordered abelian semigroup with minimal element 0 E S the system of

all ds-balls need not form a base for a topology on X (see [30]).

On the other hand side, example 1.5. shows readily that ~ could be

a base for a topology T on X even if addition in S is not continous

at 0 E S). -

Conversely, every T1-space (X,~) such that every non-isolated point

p ~ S has a totally-ordered neighbourhood base of open sets and of

of minimal cardinality w can be metrized by a distance-function d S

over a t.o. semigroup S, satisfying (i), (ii) and (iv), i.e. the

triangular inequality (page 3). By several authors such distance-

functions are called W-(Wilson)-quasimetrics, see e.g. [29]. The proof

of this assertion is modelled by the proof of theorem 1.4.:

construct the t.o. semigroup S in the same manner as in this proof (1.4.~,

�9 E S iff q E Uj(p) for and for all p~q ~ X~ let ds(p,q) = x I

all j < i and q @ Ui(P); and ds(P, q) = 0 iff q E U i for all i< w~,

Since, by the construction of S, the triangular inequality is satis-

fied for all u,p,q ~ X, we obtain the desired distance-function

that generates the topology ~.- IZ p is isolated let ds(p,q)=x I VqEX.

Omitting all details which are similar to the proof of theorem 1.4.,

we obtain

Theorem 2.1.: A T1-topology ~ on a set X can be generated by a

W - quasimetric d S taking its values in a totally-

ordered abelian semigroup S of character wp, if

and only if every non-isolated point p E X has a

well-ordered open neighbourhoodbase of minimal car-

dinality ~ {Ui(P) / i ~ w ~ UicUj iff i~j.

529

Remark 2.2.: Note that by this theorem, every first-countable

space (X,~) can be quasimetrized by a t.o. semigroup S

of countable character. This, of course, does not mean

that (X,~) is quasimetrizable by R as it would follow

if S were a group, e.g. (see w I). On the other side,

W.A. Wilson [29] showed that every ~s countable

T1-space is R-quasimetrizable. In E22aj H. Ribeiro has

shown that (X,~) is R-quasimetrizable iff every pE X

has a countable neighbourhoodbase ~Ui(P)/iE N) such that

for qE Ui+1(p) , Ui+1(q)cUi(P) , i=1,2,...

Other conditions for R-quasimetrizability of (X,~) can

be found in [24] or E29] e.g. -

Similary we can ask under what conditions in Theorem 2.1., S can be

chosen as a t.o. group of character w . This question will be answered u fully in the above mentioned paper by H.C. Reichel and W. Ruppert.[30]

This paper contains also many examples and applications of the theory

of S-quasimetrics.

- In [207 P. Nyikos characterized spaces of the type considered in

our theorem 2.1. by distance functions of a different type, by so

called protometrics. Compare also his concept of m-semimetrics ([20~).

Let us go back to semigroup-metrics as defined in w I. An R-metrizable

space is always paracompact and - as it is well known (see J. Juh~sz

[12] or A. Hayes [6]) - the same is true for spaces which can be me-

trized by t.o. groups. (Compare also [21]). However, working with

semigroup-metrics we obtain from theorem 1.1. and the preceeding

assertion:

_~_o~osition 2.~.: Spaces (X,~) which are metrizable by t.o. semigroups

S need not be paracompact. However, (X,~) is para-

compact if the function s~ s+s, sE S, is continuous

at 0E S with respect to the order topology on S (or,

more specially, if S is a topological semigroiAp with

respect to the order-topology on S). Note that S-

metrizable spaces are only T I in general, therefore

it is convenient to define a space to be paracompact

if it is T I and every open covering has a locally-

finite open refinement.

530

Proof: We only have to present a non-paracompact space (X,d S)

metrized by a t.o. semigroup. Therefore, consider example

1.5., and let D be an open covering of X. Let OED and,

for xE O, pick a ball Bc(x)c O. Note that ~~~el~2~~

and every y~ X, B~(y)n Be(x)~. So D cannot have a star-

refinement. Moreover, (X,~) is not T 2 although all "metric

axioms" (page 3) are satisfied; neither is (X,T) T3, since

a set AcX is closed if and only if there is an cE X such

that e ( a for all a~ A. - Concerning paracompactness in the

realm of nearness-spaces, see w 3, theorem 3.9.

w 3. Metrizability of nearness-structures:

Nearness spaces have been def ined by H. Her r l i ch ( [7] , [8 ] , [9]) as a unifying concept for the theories of topological spaces, uniform

spaces, proximity spaces, contiguity spaces, merotopic spaces and

other well known "topological" structures. - (Compare also [191 ).

H. Herrlich and several other authors (see the bibliography in [7~)

developped a rich theory of nearness-spaces ("N-spaces") and their

applications. So for example, N-spaces provide an incisive and adequate

method for studying extensions of spaces. - In the following we study

some metrization problems connected with nearness-spaces. For detailed

definitions, we refer to [7]. (See also a forthcoming paper of the

author on metrizability of nearness-spaces). - Usually, N-spaces (X,~)

are defined as an axiomatization of the concept of "nearness of

arbitrary ~ollections of sets", i.e.:

is a collection of subsets of the power set PX of a set X described

by some system of axioms. ~ is then the collection of those systems

of sets Ac X which are "near". In a topological Ro-space X, for example,

a nearness structure ~ is defined by ~E ~ w Ncl (A)~, A~ ~.

For our purposes, it seems to be more useful to use another, equivalent,

definition (see [7j): a system ~ of coverings of X is called a nearness

structure on X if

531

(i) ~E~ und ~<~ (9 is a refinement of ~) implies ~e~.

(ii) ~,~P(PX).

(iii) if ~,~ then (~^~)E~

(iv) if ~g ~ then {int AIA~ ~) E~ where int~ A={xE XI{A,X- {x}} EW).

c~ is a base of ~ iff, for every ~e~, there is a ~E~ such that ~<~.

is a subbase of ~I iff {~IA ...^ ~nl~i~ ~} is a base of ~.

(For an equivalent definition involving the ~-structures see [7] and

[10]-~ is a seminearness-structure iff it satisfies (i)- (iii). -

If W is the system of all uniform covers of a uniform space (X,U),

is a nearness-structure. Conversely, the category of those nearness-

spaces (X,~) which can be derived from a uniform structure U on X,

is called U-Near ("uniform" N-spaces). U-Near is isomorphic with Uni~

and a full bireflective subcategory of Near. The morphisms of Unif and

Near are the uniformly continuous mappings and the nearness-l~reserving

maps [8], respectively. By Tukeys theory, ~ is a uniform nearness-

structure iff every ~E ~ has a star refinement ~ W.

Analogously, for every topological Ro-space (X,v), the system W of all

open covers is a base of a nearness-structure on X. And the full

subcategory T-Near of Near consisting of those N-spaces which are

derived from topological spaces forms a bicoreflective subcategery

of Near. - For more details, see [7] e.g.

By a definition of Hunsaker and Sharma [102 (see also Herrlich [7], a

nearness space (X,~) is metrizable iff there exists a metric d on X

such that ~E ~ iff, for any positive real number r there exists a

point x~ X with the ball Br sec ~, where sec ~={B~XIu AE

AN B~). Equivalently, (X,u) is metrizable iff there is a metric d

on X such that ~r ={B~(x)/x~ X}, r yields a base of ~. It is easy

to show that ~ is metrizable iff ~ is uniform and has a countable

base. Naturally, for arbitrary (semi)nearness spaces we can define

metrizability over (t.o.) groups S in an analogous manner, and a

reformulation of w I, shows that metrizability by t.o. groups cha-

racterizes exactly the category of uniform nearness-spaces with li-

nearly ordered bases (ordered with respect to refinement). However,

as we shall see, the theory of arbitrary nearness-spaces with linearly

ordered bases seems to be more difficult. - In the following, an in w I,

532

every semigroup is abelian and has an identity-element O. - To

prevent trivialities, the zeroelement should not be isolated from

above with respect to the order.

Exam~le_~.1~: Analogous to the theory of (semi) uniform spaces, for

any semigroup metric d S on ~ let ~R be the (semi) nearness structure

on X induced by d S by taking ~r tBc(x)Ix~ X}, cE S +, Be(x )=

= {Ylds(x,y)<r as a (sub) base of ~R" Now consider ~R(ds) induced

by the semigroup-metric d S constructed in example I .5. {~r r E S~ is

then a base for the set of all "uniform coverings" of the semiuniform

space (X,Uds) considered in w I, and certainly, Uds is not a uniform

structure.

Nevertheless, ~R is a uniform nearness-structure: every ~8 is trivially

a star-refinement of every ~r since every ~c contains the whole space

X~(Br =X if e>x!). More exactly, HR is the trivial nearness-

structure on X which has {~ as a base. In other words: the metric d S

cannot be "rediscovered" from the nearness-structure ~R induced by it.

The situation is completely different if we study another (semi-)

nearness-structure ~D induced by ds, letting~r ={AcXldiam A=r cE S +,

be a (sub-) base for ~D" (Notice the following problem in defining

"diam A" which will be discussed in all details in a forthcoming

paper of the author on metrizability of nearness-spaces by linearly

ordered abelian ~em~groups: if the cofinality of S at OE S is > w o

inf and sup of bounded subsets of S need not exist in general. Therefore,

in order to define "diam A", we choose a net {xi/iE I) with wellordered

index-set I converging to OE S with respect to the order topology of S

and with minimal cardinality (i.e. the well-defined cofinality of S

at oE s), and define a set AcX to be bounded if d(x,y)<xl, for all

x,yE A. For a bounded set A consider i = min {JEI /3 x,yE A such

that d(x,y)~ xj} and define

diam A = ~ xi_ I if i is not a limit ordinal, and

x i if i is a limit ordinal.

Then we can show that d(u,v)~ diam A, for all u,v~ A. (Moreover,

d(u,v) < diam A in the first case which appears if A is bounded and

sup {d(x,y) / x,yE A} in S, e.g.).- For the theory of S-metriza-

bility of nearness-spaces, it does not matter that the terms "bounded"

and "diam A" is defined only with respect to a O-cofinal net { xi/iE I}

as chosen above. For more details see the paper mentioned before.

533

Note that in our example, a ball Be(x) either is equal to X (if x<e)

or, for x ~ e and e= x i E X, diam Be(x) = xi+ I ~ r Besides, for

every A c X with diam A ~ r e ~ O, there exists x E X and e',

x ~ r > e such that the ball Be,(x) covers A and diam Br ~ e.

Thus the system of all ~e= ~B~(x) / x ~ e~, s E S +, is a base for

~(ds). Letting e = x i (i ~ ), we see that ~D is a nearness-

structure with a linearly ordered base, and PD is not uniform, since

the ~r of every Be(x) ~ Ge is the whole space X! So no Ge can

have a star-refinement here. (Neither is ~JD a regular nearness-structure

as it can be deduced straight forward from the definitions in ~7~,w

Compare theorem 3.8. - A more detailed study of PD will be found in

the forthcoming paper of the author cited above.

~_em_ar~k=~ The advantage of considering ~D instead of ~R is that -

at least in our example - the metric d S (or, generally,

an "equivalent" one) can be "rediscovered" from ~D(ds)

whereas ~R in our example is the trivial nearness-

structure:

for x,y E X, let d(x,y) = x i ~ S + iff, for all

a ~ i, there is one set in ~ containing both, x and y,

and there is no set in ~ containing x and y~ if no X, 1

no such x i exists, set d(x,y) = O. This procedure

yields the J-metric we started with. (Compare remark 3.5.).-

Given two semigroup-metrics ds, d S' on X, call them R-equivalent

(D-equivalent) if UR(ds) = UR(ds') , and ~D(ds) = ~D(ds') respec-

tively. Consider again example 3.1. and let d be the trivial pseudo-

metric on X: d(x,y) = 0 for allx,y E X; then d S and d are R-equiva-

lent whereas they are not D-equivalent.

534

This could be another motivation to prefer the structures ~D instead

of ~R in studying the nearness-structures induced by semigroup-

metrics on X.

Resum% ~. �9 Let S be a totally-ordered semigroup, then every

S-metric d S on X induces the (semi) nearness-structures

~R and ~D" In general ~R ~ ~D; however, if S is a

totally-ordered abelian group (or a semigreup where

the function s * s + s , s ~ S +, is continous at

0 ~ S with respect to the order-topology of S), then

~D = ~R by the considerations in w I.

Another condition assuring ~D = ~R will be given in

theorem 3.4.-

Summarizing, the following definition of metrizabi-

litiy of nearness-spaces seems to be the most adequate

one. By the comments above, it coincides with

Hunsaker and Sharma's definition if we consider only

real metrics.

Definition: Let $ be a totally-ordered abelian semigroup as above,

then a nearness-space (X,~) is metrizable by the S-metric

ds, ds: X ~X ~ S, iff ~ = ~D(ds), where the latter

is described in example 3.1. -

Equivalently, we can describ~ nearness-spaces by their

merotopic structure u (see r7J); then V induced by d S

on X is equal to ( ~ c PX / for every r E S +, there

is an A ~ 9! such that diam A < r }. The covering

structure ~u X belonging to this u coincides with Ud(ds).

535

Theorem ~•177 If UDCd S) is a uniform nearness-structure then

~oCds) =~(~s ) �9

Proof: Let ~eE UD and let ~E S + such that ~ is a star-refinement

of ~r Now let A EG , then, for every xE A and a <~, the

ball Ka(x) is a subset of the star St(A,~ ), since for every

yE Ka(x), there is a set B such that x,yE B and diam B~a<~.

therefore BE ~ and yE Bc St (A,~). Thus the covering ~a

consisting of all a-balls Ba(x), xE X, is a refinement of ~r

so UR is finer than UD. Conversely, every covering ~r is re-

fined by ~ if ~ e; so UD is finer than UR, hence U R=u D.

Remark_~• If (X,u) is a uniform nearness-space with a linearly

ordered base ~ then, by the preceeding text, there is

a t.o group S and an S-metric d S on X such that

=UD=~R(ds). If ~ is an arbitrary nearness-structure

with a well-ordered base ~ of least power~u, we can

immitate the technique of remark 3.2. using the t.o.

semigroup S with generators (xi/i< ~u} as it was constru-

cted in the proof of theorem 1.4. Then we obtain an

S-metric d S on X such that ~ is finer than ~D(ds) and

this structure being finer than UR(ds). Clearly, for

every BE ~i E~ we have diam B~ xi; therefore every

covering (A/diam A< x i} of X is refined by ~i+I E ~.

- Let (X,~) be the N1-nearness-structure of example 3.1., then, by

remark 3.2., ~=~D(ds)@UR(ds). So it seems interesting to deal with

the following problems:

536

Problem I: For an arbitrary N1-space (X, ~) with a linearly ordered

base ~ let d s be the S-metric constructed in (3.2.).

Hereby S is the semigroup constructed as indicated in

theorem 1.4. Under which conditions is U=UD(ds)? More

generally, find necessary and sufficient conditions assuring

U =UD(d S) for a given N1-space iX,u) and a suitable S-metric

d s �9

Problem 2: If S is p.o. semigroup and d S an S-metric on X, we can derive

prenearness-structures u R and UD as before but using subbases

instead of basis. Under what conditions is U R or ~D a

nearness-structure, which nearness-structures can be described

in this way by S-metrics?-(Added in proof: a forthcoming

paper of the author is concerned with partial solutions of 1,2).

Another problem arises from the following observation:

let (X,u) be any N1-space with a (linearly ordered) base ~ ={~i/i6 I} ;

then the system U =U (B• B) B~ ~ - iE I - is a base of a separated 1 ' 1

semiuniformity U j on X (in the sense of Cech-Weil; w I ), Conversely,

given any separated semiuniformity U on X, the system of all uniform

coverings ~i = (Ui(x)/xE X) where Ui(x)= (Yl (x,y)E U i}, Ui~ U, is a

base for a N1-structure UU on X. Example 3.1. shows that, for a given

N-structure u ,u need not be equivalent with ~U : U

in example 3.1., UU(UD ) =UR(ds) is the trivial nearness-structure on X

whereas U D is not trivial!

Of course, U is a uniform N-structure iff ~ =UU , since - by a U

straightforward arguement - U is uniform if every W-cover ~/ has a

barycentric refinement ~.

Call an N-structure u on X semiuniform iff U = U u for a suitable

537

semiuniformity U on X, then all uniform N-spaces are also semi-

uniform; and we ask for

(i) a characterization of all semiuniform N-spaces within the

realm of the theory of nearness-structures; and

(ii) a characterization of those semiuniform N-structures which

have a linearly ordered base. (See also theorem 1.4.). - -

Regular and paracompact N-spaces with linearly ordered bases:

As we have seen the concept of metrizability of nearness-structures

is definitely rich (and unique) in meaning if we restrict our-

selves to metrics d S over t.o. groups S (or, equivalently by

theorem 1.1., to t.o. semigroups S which are topological semigroups

in their order topology). Then the category of metrizable N-spaces

in this sense is isomorphic with the category of uniform N-spaces

with a linearly ordered base. Hereby these categories are equipped

with the usual morphisms. - More generally, let us now study regular

N-spaces (for the exact definition see [7j, w 11, or ~87). These

structures are equivalent with K. Moritas regular T-uniformities,

[17], called "semi-uniformities" by A.K. Steiner and E.F. Steiner

in [25]. - Note that every uniform N-space is regular.

Definition: A nearness-space (X,u) is submetrizable by an ordered

abelian group G iff there is a G-valued metric d G on X such that

the N-structure ~R(dG) induced by d G is finer than ~ (i.e.: WR(dG)~).

Moreover, two N-structures ~I and ~2 on X are topologically equi-

valent iff, for their topological coreflections U~,~, we have

T T ~I =W2" Of course, ~J1 and ~2 are topologically equivalent iff they

induce the same topology on X: ~I = ~2' where - for any N-structure

-~ is induced by the system of all stars st(x,~), xE X, WE ~ as

a base for ~ . (For regular N-spaces (X,u), the system of all sets

AE ~, ~E ~, is also a base for �9 ).

538

Theorem ~8~: Every reg~ar nearness space (X,u) with a linearly

ordered base ~ is submetrizable by a t.o. group G.

More exactly:

For every regular N-space (X,u) with a linearly

ordered base ~ there is a topologically equivalent

uniform N-structure H on X with a linearly ordered

base ~ which is finer than ~. (~ and ~ are of the same

cofinality).

Remark: In [25] we find an example of a non-uniform regular nearness-

space with a countable base. - By theorem 3.8. every such

N-space is submetrizable by a realvalued metric d R on X.

@

Proof: We have to show that there is a uniform nearness-structure

with a linearly ordered base ~ such that ~ is finer than ~, and

which is topologically equivalent with ~. This will be done by

applying a theorem which generalizes a metrization theorem of

A.H. Frink to group-valued metrics (P. Nyikos and H.C. Reichel [21],

theorem 4).

Let (X,u) be a regular N-space and ~ a linearly ordered base of ~l,

say of least power w (u ~ 0). Then by the same method as used in

theorem 1.4., we can assume ~ = {Ni/i<w ) to be well ordered by

refinement. By regularity of u, for every i < u~U, there is a j > i

such that ~! locally star-refines ~ (i.e.: for each A E ~. there 3 z 3

is a covering ~kE k, k=k(A), and a set BE ~ll such that star (A,I~)cB).

So, for every x E X, the sets Wi(x) = st(x,~i), i < ~ , form a nested

neighbourhoodbase for a topology T on X compatible with

(all A E Ni' i < ~u' are open in T); by the same techniques as used

in w I for semiuniform structures, the set ~ Wi(x), i~ k< ~, is

open with respect to T. Moreover, for every i and x, there is a

j = j(i,x) such that Wj(x) OWj(y) ~ 0 implies Wj(y) cWi(x ). But these

are exactly the sufficient conditions which - by the above mentioned

539

generalization of Frinks theorem - assure the existence of a

uniform structure ~@, with a linearly ordered base ~*, which is

compatible with the topology ~, and which is finer than ~. Finally

apply the theorem of Stevenson and Thron (see our theorem 1.1.).

(For the contable case, i.e. w =~o' compare A.K. Steiner and E.F.

Steiner [25], theorem 4.10. - For w >w note that the proof of H o'

the generalization of Frinks theorem is essentially different to

the countable case, for details see the above mentioned paper of @

Nyikos and Reichel [21~). - Now, if there is no countable base

for the nearness-structure u , we know from w I (e.g. from theorem 1.2.)

that p* has a base consisting of partitions �9 of X (i~ I). p* is i

finer than ~, thus (X,~p) is zerodimensional: ind X= Ind X= dim X= 0,

even stronger: �9 (the topology induced by U) has a base of rank I

(it is a non-archimedean topological space). See [22J.

Ba combining theorem 3.7., the remark before theorem 3.7., and the

theorem of Stevenson and Thron (viz. our theorem 1.1.) we obtain:

CorollarE_~.8.: A topology �9 on a set X is metrizable by a totally-

ordered abelian group G (of character w ) if and only

if it is induced by a regular N-structure u on X

with a linearly ordered base (of cofinality w ).

From this result follows that a topology �9 induced

by a regular N-structure ~ with a linearly ordered

base is paracompact, therefore the N-structure !JT=~ T

induced by �9 is a uniform (and topological, hence

paracompaet) N-structure on X.-

540

Summarizing we obtain:

Corollar~y ~.8.a.: If u is a regular nearness-structure on a set X

with a linearly ordered base ~, then its topological

coreflection u T is a uniform (and hence paracompact)

nearness-structure on X. -

~ote that, of course, ~l T need not have a linearly

ordered base itself).

Topological N-spaces and metrizability:

Now let us consider topological nearness-spaces with linearly

ordered Bases.

A topological nearness-space is paracompact iff it is uniform

(Herrlich E7J). Sharpening our results we study now paracompact

N-spaces (X,W) with linearly ordered bases. - (X,~) is w -contigual

(E7J, 18.11.) iff for every ~ ~ there exists a subset ~ of ~ with

less than w elements such that $~ ~. - As usual, a topological

space (X,~) is called ~ -compact (or "initially ~-compact; *)) iff every

open cover has a subcover consisting of fewer than w sets. A topolo-

gical N-space is w -contigual iff it is �9 -compact~

~eoremm_~.~: Let ~,~) be a uniform nearness-space with a linearly

ordered base ~ of least power $ and let ~ be the to-

pology on X induced by U. Let X have no isolated points,

then ~ is topological (hence paracompact) iff (X,~) is

�9 -compact (equivalently, iff (X,~ T) i s w - con t ig lxa l ,

where T denotes the topological coreflection of ~).

*) For more details about ~ -compact spaces, see e.g. ER.Sikorski:Remarks

on spaces of high power; Fund.Math. 37(1950),125-136~ or F26 ~ or D.

Harris: Transfinite metrics, sequences and top. spaces Fund.M.73(1972) 137-142.

-~o-COmpact spaces are exactly the compact ones.-

541

Remark: In [7], H. Herrlich defined an N-space (X,{) to be a

Nagata space iff (X,{) is metrizable and topological.

Theorem 3.9. can be interpreted as a generalization of

proposition 19.5. of [7] where Herrlich characterized the

class of Nagata spaces by using results of N. Atsuji,

M. Katetov, J. Nagata and A.H. Stone.

Proof: Let (X,u) be a uniform N-space with a linearly ordered base

of least power ~L' then u either is metrizable by a real (R-)

valued metric d (iff 9 = ~o ) or ~ has a well-ordered base

consisting of partitions ~a of X, a<w , (see the proof of

theorem 1.2.). In this case, all sets BE~ a are closed and

open with respect to ~. - Suppose now, that (X,~) is �9 -compact

and let ~= {Oi/i~ k<w } be an arbitrary open cover of X. If

U is metrizable by an ~-valued) metric d, i.e. iff w =~ o'

chose a Lebesgue-number 6' = 28 % R of ~ such that the covering

of X consisting of all balls B6(x)={yld(x,y)<6}, xE X, is

finer than {. Thus we conclude that every open cover of X is

refined by some ~s U which implies that U is topological, hence

paracompact.

If w >~o' consider the base ~ of ~ described above and note that U

the system of all sets B~ ~, a<w is a clopen base for the to-

pology ~. Suppose now that there is no ~a ~ a refining a given open

cover ~, then, for every a, pick a B aE ~ such that B a~O i for every

i_< k. Moreover, for every a(wl, pick x a~ B a, and obtain a w~-sequence

{x a} in X. Since (X,~) is w j-compact, {x a} has a cluster point x, E263,

theorem 1.1. and xE 0 i for some i.

542

Since 0 i is open there is a ~<wu and a set BE ~ such that xE Bc 0 i.

But all ~ are clopen partitions of X, and x is a cluster point of

{xa) , se there must be an index a such that our B~ is a subset of B.

And this yields a contradiction to ~ 0 i for all a<~ and all 0 lEG.

Therefore, concluding similarly to the countable case, every open

cover of (X,~) is refined by some ~E ~; thus ~ is topological, hence

paracompact.

Conversely, let (X,H) be a paracompact topological nearness space

with a base ~ ={~a/a<w ~ well-ordered by refinement. Since (X,~)

is paracompact we can assume that all ~a are locally finite open

coverings of X; if moreover w >~o, ~a can be visualized as clopen

partitions of X (see w 1).-Suppose (X',~) is not w -compact then there u

is a set Z={za/a<w } such that, for every ~, there is an open neigh-

bourhood V(za) which does not contain any other point of Z. Moreover,

chose V(z~) "small enough" such that V(z~)~N B (zae B E ~a )' then

the system

(X\ Z)U {V(~)/a<w } is an open covering of X which cannot be re-

fined by any ~a' hence (X,L~) cannot be topological, which is a contra-

diction.

From this theorem we obtain a corollary which generalizes a theorem

of N. Atsuji about metric spaces:

Corollary 3.10.: Let (X,U) be a uniform space with a linearly ordered

base of least power ~U' and let (X,~) have no isolated

points, then the following is equivalent:

(i) every continuous mapping f from (X,U) into any

other uniform space Y is uniformly continuous.

(ii) (X,~) is ~ -compact. (~ is the topology induced

by U.

543

Proof: (ii) ~ (i): follows from the proceeding theorem:

for every uniform covering~ of Y, f-1(~) is an open

covering of X, hence refineable by a uniform covering

of (X,U). - To prove (i) ~ (ii), remember that (X,~) is

paracompact if U has a linearly ordered base (see e.g.

Hayes [6j or Juh~sz [12j).

Therefore the "fin@'uniformity ~ of X consists of all open coverings

of X since X is fully normal by A.H. Stones theorem. Thus id:

(X,U) * (X,~), as a continuous mapping, is uniformly contiuous by

the assumption of our theorem, and therefore, every open cover of X

is refineable by an uniform cover of (X,U). - The rest of the theorem

is a consequence of theorem 3.9.

Remark: The above mentioned result of N. Atsuji follows from this

corollary by letting w =~o. In a similar manner we could

generalize the other parts of Atsujis theorem on metric

spaces in [I~:[H.C. Reichel: "On a theorem of N. Atsuji";~

to appear. (This paper studies also the case, where (X,~)

has isolated points).

Finally, by the same paracompactness-argument as it was used in the

last proof, we derive another corollary from theorem 3.9.:

Corollar2_~.11.: If the fine uniformity U of an arbitrary completely

regular topological space (X,~) has a linearly ordered

base of least power ~U' and if X has no isolated points,

then X must be w -compact. W

544

-In [7~j, w 4, H. Herrlich remarks the completely symmetric relation

between uniform and topological structures viewed in the realm of

nearness-structures: a uniform space (X,p) is "topologizable" iff

there is a topological N-structure ~ on X such that its uniform

(bi-)reflection ~, on X is equivalent with U. -

~ b ~ : Are there any non-trivial sufficient conditions for to-

pologizability of a uniform space with a linearly ordered

base? -

- Die Grundidee zu dieser Arbeit fa~te ich w~hrend eines Aufent-

haltes bei Prof. Dr. Horst Herrlich, Bremen. Ihm, wie auch Herrn

Dr. W. Ruppert danke ich f~r wertvolle Hinweise und Diskussionen~

545

BIBLIOGRAPHY :

[I] Atsuji, M.: Uniform continuity of continuous functions

of metric spaces;

Pacific J. Math. 8 (1958) 11 -16.

v

Topological spaces; Z. Frolik and M. Kat~tov (eds.);

Prague 1966.

[3] C~asz&r, A.: Grundlagen der allgemeinen Topologie;

Akad$micei Kiado, Budapest 1963.

[4] De Marr, R. and Fleischer, I.: Metric spaces over partially

ordered semi-groups;

CMUC 7 (1966) 501 - 508.

[5] Engelking, R.: Outline of General Topalogy;

North-Holland, Amsterdam 1968

[5a] Fletcher, P. and Lindgren, W.F.: Transitive quasi-uniformities;

J. Math. Anal. Appl. 39 (1972) 397-405.

[6] Hayes, A.: Uniform spaces with linearly ordered bases are

paracompact;

Proc. Cambridge Phil. Soc. 74 (1973) 67- 68.

[7] Herrlich, H.: Topological Structures; P.C. Baayen (ed.);

Math. Centre Tracts 52 (1974) 59-122.

[8] Herrlich, H.: A concept of nearness;

J.Gen.ToD.Appl. 5 097@ 191- 212.

[9] Herrlich, H.: Some topological theorems which fail to be true;

preprint 1975.(Int. Conf.on Categorical Top.,Mannheim 1 975 )

[10] Hunsake~ W.N. and Sharma, P.L.: Nearness structures compatible

with a topological space;

preprint 1974.

546

[11] Isbell, J.R.: Uniform spaces;

Amer. Math. Soc. Math. Surveys 12 (1964).

[12] Juhasz, I.: Untersuchungen ~ber w -metrisierbare R~ume;

Ann. Univ. Sci. Sect. Math., Budapest, 8

(1965) 129- 145.

[13] Kalisch, G.K.: On uniform spaces and topological algebra;

Bull. Amer. Math. Soc. 52 (1946) 936-939.

[14] Kat~tov, M.: On continuity structures and spaces of mappings;

CMUC 6 (1956) 257- 278.

[15] Krull, W.: Allgemeine Berwertungstheorie;

J. Reine u. Angew. Math. 167 (1932) 160-196.

[16] Mammuzi~, Z.: Introduction to General Topology;

Noordhoff, Groningen 1963.

[17] Morita, K.: On the simple extension of a space with respect to

a uniformity I- IV;

Proc. Japan Acad. 27 (1951) 65- 72, 130-137,

166-171, 632- 636.

[18] Nagata, J.: On the uniform topology of bicompactifications;

J. Inst. Pol. Osaka City Univ. I (1950) 28- 38.

F18~ Nagata, J.: Modern General Topology;

North-Holland, Amsterdam 1968.

[!9] Naimpally, S.A.: Reflective functors via nearness; Fund.

Math. 85(1974), 245-255.

[20] Nyikos, P.: Some surprising base properties in topology;

Studies in Topology (Proe. Conf. Univ. North

Carolina, Charlotte N.C., 1974; 427-450;

Academic Press, New York 1974.

[20a] Nyikos, P.: On the product of suborderable spaces;

Preprint 1974.

547

[21] Nyikos, P. and Reichel, H.C.: On uniform spaces with linearly

ordered bases II; (ca. 16 pg)

to appear in Fund. Math.

[22] Reichel, H.C.: Some results on uniform spaces with linearly

ordered bases; (ca. 26 pg)

to appear in Fund. Math.

[22a] Ribeiro, H.: Sur les ~spaces a metrique faible;

Portugaliae Math. 4 (1943), 21 -40.

[23] Schilling, 0.F.G.: General theory of valuations;

Math. Surveys IV, Amer. Math. Soc. 1950.

[24] Sion M. and Zelner G.: On quasi-metrizability;

Canad. J. 19 (1967) 1243-1249.

[25] Steiner A.K. and Steiner E.F.: On semi-uniformities;

Fund. Math. 83 (1973) 47-58.

[26] Stevenson F.W. and Thron W.J.: Results on w -metric spaces; u Fund. Math. 65 (1969) 317-324.

[27] Stone A.H.: Universal spaces for some metrizable uniformities;

Quart. J. Math. 11 (1960) 105-115.

[28] Weil, A.: Sur les ~spaces ~ structure uniforme et sur la

topologie g@neral; Paris 1937.

[29] Wilson, W.A.: On quasi-metric spaces;

Amer. J. Math. 53 (1931) 675- 684.

[30]Reichel, H.C. and Ruppert, W.: ~ber Metrisierbarkeit dutch Distanzfunktionen mit Werten in angeordneten

Halbgruppen (to appear in "Monatshefte f. Math.").

[31]Reichel, H.C.: A characterization of metrizable spaces(to appear).

Address of the author: Mathematisches Institut der Universit~t Wien

A-I090 Wien, Strudlhofgasse 4; A u s t r i a.

REFLECTIVE SUBCATEGORIES AND CLOSURE OPERATORS

by Sergio Salbany

Introduction

The following familiar examples illustrate the

problem of associating a closure operator with a

reflective subcategory.

V

Example I The Stone-Cech compactification 8 X of a

Tychonoff space X is a compact Hausdorff extension

of X , in which X is dense and such that every

continuous map into a compact Hausdorff space C , has

a continuous extension to 8 x § C .

In other words, the category of compact Hausdorff

spaces is a reflective subcategory of the category of

Tychonoff spaces T y .

In fact~ it is a reflection in the category of

topological spaces, and, as such, it may be regarded

as the composite of

(i) Initial reflection -

assigns to each topological space

Top + C_R , which

X the weak

topology induced by its continuous maps into I = [0~I]

with its usual topology.

(it) Separating reflection - C_~R § Ty , which

identifies the points in a completely regular space

549

X which are indistinguishable in the context of the

discussion - points x and y whose images under

continuous maps f : X § I are the same. For a given

X the Tychonoff reflection is simply the image of X

in the product I C(x'I) , where C(X,I) denotes the

set of continuous maps from X to I .

v

(iii) The Stone-Cech reflection in Ty .

An alternative way of regarding 8X , which has

been the basis of many generalizations of compactness

(e.g. [3],[7] ) , is based on the fact that the product

I c(x'I) is compact Hausdorff and that closed subspaces

of compact spaces are compact. If e is the product

map X § I c(x'I) , then 8X = e[X] , where -- denotes

the closure in the product topology on I C(X,I) .

Example 2

replaces

reflection is the Hewitt-realcompactification

re-emphasize that

X : e[X] c ~ c(x,~)

where

If ~ : real line with usual topology

I in the above example, the resulting

~X .

is the closure in the product topology.

Example 3 If e : X + I c(x'I) and Q denotes the

Q-closure operator of Mrowka ( x 6 Q(A) ~-- every

G~-neighbourhood of x intersects A ) , then

~X = Q[e[X]] .

We

550

Example 4 Every topological space is initial with respect

to its continuous maps into the Sierpinski two point

space D = {0,1} with only one non-trivial open set

{0} Let us compare this situation with that of

example 1

(i)

reflector.

(ii)

(iii)

To

80 X

The Initial reflector is the identity

The Separating reflector is the T0-reflection. v

The analogue of the Stone-Cech reflector in

is very interesting. A tempting analogue is

= e[X] c D c(X'D) , where D has the topology

C(X,D) specified above, e is the product map e : X + D

and denotes closure in the product topology.

This procedure was claimed to yield a space 80X

with the extension property for continuous maps from

X to compact Hausdorff spaces by Nielsen and Sloyer

[13] , but their claim was disproved with a simple

observation by Salbany and Br~mmer [1~ .

When Baron characterized epimorphisms in To [I] ~

the proper closure to use became apparent. The

reflector X r~ b-closure-< e[X] ] c D C(X,D) was

studied by Skula ~6] , Nel ~2] The b-closure of a

set A in a topological space

x such that cl x N V N A

V of x ( cl x = closure of X

X consists of all points

, for all neighbourhoods

{x} ).

551

2. The closure associated with a reflector

2.1 The closure operator

If C is a reflective subcategory wi~h reflector

R and reflection map qx : X + R[X] , then any map into

f : X + C has a unique extension ~ : R[X] § C such

that Ton x = f Thus, for any two maps h,g : R[X] § C

such that hon x = goq x , h = g . Let K(h,g) denote

the coincidence set of h and g ,consisting of all

points x such that h(x) = g(x) . The uniqueness

requirement is expressed by saying that K(h,g) m nx[X]

K(h,g) = R[X]

Motivated by these considerations and by the account

of Lambek and Rattray of the Fakir construction ([1~)

that we shall discuss in a later section, we propose

the following definition.

C E C

Definition Let C be a class of spaces. Given a

space X and A c X , put

[A] = N{K(f,g)IK(f,g) m A, f,g : X § C, C s i}

Proposition (i) A c [A]

(it) A c B ~ [A] c [B]

(iii) [[A] ] = [A]

Proof (i) is clear, since each K(f,g)

definition of [A] contains the set A

in the

552

(ii) Each K(f,g) in the definition of

[B] contains B , hence contains A , hence is in

the class of coincidence sets which determine [A]

Hence [A] c [B]

prove

[[A]] c [A] , let K(f,g) be such that K(f,g) m [A]

Then K(f,g) m[[A]] . Hence, the sets K(f,g) in the

class determining [A] are in the class determining

[[A]]. Hence [[A]] c[A], as required.

Note similar considerations establish the formula

[A] U [B] c [AU B] From this formula it follows that

[ [A] u [B] ] = [AU B]

Although I believe that examples where

[AU B] * [A] U [B] exist in profusion, I have not been

able to find one.

Thus [ ] is an operator whose induced neigh-

bourhood struoture is a Neighbourhood space (Espaoe

Voisinages) in the sense of M. Fr~ehet ([5])

( V is defined to be a neighbourhood of x if

x~ [X- V] ). Such expansive, monotone and idempotent

operators have been rediscovered and extensively used

by P.C. Hammer ([ 6 ]). These generalized closures

should be contrasted with generalized closures in the

sense of ~ech ([ 2 ]), where the operator is no longer

required to be idempotent but is expansive and

553

preserves finite unions (and is, consequently,

monot one ).

We show that

Sierpinski dyad D

[ ] induced by mappings into the

is a Kuratowski closure.

Proposition The operator [ ] induced by mappings

into the Sierpinski dyad is a Kuratowski closure.

Proof

that

x E [B]

X§

By our remarks above, it is sufficient to prove

[AU B] m [A] U [B] Suppose xE [A] and

Then there are functions f1' f2' g1' g2"

such that K(fz'f2 ) mA , K(g1'g2 ) �9 and

f1(x) # f2(x) , gz(x) #g2(x) =f v Let h I ig I f2g2

v f g2 (where (fvg)(x) = sup{f(x) , g(x)} ). h2 = f2gl i

Then K(hl,h 2) mAU B and hi(x) #h2(x) The proof

is complete.

Note The choice of h I ,h 2 is based on the following

formal identities: K(fl,f 2) U K(gl,g 2) : Z(fz-f z) U

UZ(gl-g2)= Z(fl-f2)(gl-g 2) = Z[(flg1+f2g2) - (flg2+f2gl)]

= K(flgl+f2g2,flg2+f2gz)

2.2 Comparing [ ]-cl'osure and closure

The following two propositions state sufficient

conditions for the neighbourhood structure T([ ]) to

be comparable with the topology T on a given set.

554

To state one of the relationships one has to

consider the initial category In(A) determined by

a class of spaces A This is a reflective sub-

category of Top whose objects are topological spaces

which are initial with respect to their mappings into

objects of A .

Proposition The following statements are equivalent.

(1) TcT([ ]) for all X in In(A)

(2) TcT([ ]) for all finite products of objects

in A .

Proof (1) ~(2) since all finite products of objects

in A are in In(A) (2) ~ (1) Suppose x ~ ,

then there is a finite product B = A I • 2 • ... • n

of objects As in ~ and a map f : X~B mld an

open set V in B such that f(x) C V and

f+[V] N A = % . By assumption (2) , there are functions

hl,h2 : B + A0 , A0 s A such that x ~ K(hl,h2) and

X-V c K(hl,h2) Then K(hlof,h2of) m ~ and

x ~ K(hlof,h2of) Hence ~ is T([ ])-closed. It

follows that T ~ T([ ])

A is closed under finite products Corollary Suppose _

and such that if X is in In(A) and A c X is a

closed set not containing x , then there are functions

fl,f2 : X + A0 , A0 s ~ , such that x ~ K(fl,f2) ,

K(fl,f2) m A Then T c_ T([ ])

555

Note The class of spaces considered in the example is

simply the analogue of the class of completely regular

spaces : points and disjoint closed sets can be separated

by a [0,1]-valued continuous function, 0 on the set

and 1 at the point.

Proposition

(I)

(2)

Proof

The following are equivalent :

T([ ]) c T for all X

consists only of Hausdorff spaces.

(1) ~ (2) Recall that (X,T) is a Hausdorff

space iff the diagonal

Now, let X be in A

are the projection maps from XxX

K(~1,z2) is T([ ])-closed, hence

X is a Hausdorff space.

(2) ~ (1) Note the following implications

T([ ]) c T ~ K(f,g) is T-closed ~ (fxg) [A]

is T-closed. Hence (2) ~ (1)

A is closed in (XxX,TxT) .

A = k(zl,z2) , where w1, ~2

to X Now

T-closed. Hence

Combining the above criteria, we have the following.

Proposition Suppose ~ is a class of spaces closed

under finite products and consisting only of Hausdorff

spaces. Then the following are equivalent

(1) T = T([ ]) for all objects in In(A)

(2) T = T([ ]) for all objects in A .

We now identify the

in the introduction.

556

[ ]-closure in the examples

Example i The class A consists of products of copies

of I , hence all the objects in A are Hausdorff

and completely regular. By the corollary to our

proposition and the comparison criterion, it follows

that T([ ]) = T .

Example 2 The class ~ consists of products of copies

of ~ As in example 1, T([ ]) = T .

Example 3 The class ~ consists of products of copies

of the Sierpinski dyad D By the corollary above,

T c T([ ]) ; and since 11 �9 _ a~ spaces involved in A are

not Hausdorff we have~ in fact, T ~ T([ ]) .

We have already shown that this neighbourhood

structure T([ ]) is in fact a topology. We prove

that this topology coincides with the b-topology

(It&I) (also called front topology by L.Nel ([,r

557

Proposition [ ]-closure = b-closure.

Suppose x ~ [A] , then there are f,g : Proof X D

such that A c K(f,g) , f(x) ~ g(x) . Assume f(x) = 0

and g(x) = 1 Then x 6 f§ N g [1] c X-A . Now

f§ is open, el {x} c g [1] , hence

x ~ b-closure (A) Similarly, if g(x) = 0 , f(x) = 1 .

Conversely, suppose x ~ b-closure (A) Then there

is V open, such that cl{x} N V NA = # Let f : X§

be f = 0 on V , f = 1 off V . Let g : X § be

g = 1 on cl{x} , g = 0 off cl{x} Then

Ac K(g,f.g) and x ~ K(g,f.g) Hence x ~ [A]

Proof is complete.

Note The above argument simply reproves Baron's

characterization of epimorphisms in m as ~4~g

maps onto b-dense subspaces ([1]) : Essentially

because T o spaces are subspaces of products of D ,

we have: A ~ B is epi in T O iff f[A] is

T([ ])-dense in B

We now discuss a related example involving

bitopological spaces (X,P,Q)

Example 5 A bitopological space (X,P,Q) is

pairwise completely regular if for every x and

disjoint P-closed set F , there is a bicontinuous

map f : (X,P,Q) § (l,u,l,) such that f(x) = 0

558

and f(x) = 1 on F , and for every x and disjoint

Q-closed set G , there is g : (X,P,Q) § (I,l,u)

such that g(x) = 0 and g(x) = 1 on G(D5])

It is shown in ([15]) that these are precisely the

bitopological spaces which are initial with respect to

their mappings into (l,u,l) (In �9 and I , the

non-trivial u-open sets are of the form (-~,a) and

the non-trivial 1-open sets of the form (b, ~) .)

There have been many proposals in the literature

for what a "compact object" ought to be in the

category of bitopological spaces. In ([1~) a case

has been made for calling a bitopological space (X,P,Q)

compact if the supremum topology PvQ is compact

(not necessarily Hausdorff). The present discussion

also shows how naturally these spaces arise : in

v

looking for an analogue of the Stone-Cech reflector

for bitopological spaces, one is led to consider the

embedding e : X + IC(X'l) where I = ([0,1],u,l)

and C(X,I) denotes the set of bicontinuous functions V

f : (X,P,Q) ~ (l,u,l) ; now, the Stone-Cech

bitopological compactification 82X "ought to be"

[e(X)] , where [ ] denotes T([ ])-closure as defined

previously via coincidence sets. We show that the

topology T([ ]) induced in (X,P,Q) is simply PvQ ,

confirming that it is natural to regard pairwise compact

spaces as PvQ-closed subspaces of products of copies

of (l,u,l) , in other words, as PvQ-eompact spaces.

559

Proposition Let [ ] be the operator induced by

mappings f : (X,P,Q) + (l,u,l) Then T([ ]) = PvQ .

Proof K(f,g) is PvQ-closed for any maps

f,g : (X,P,Q) + (l,u,l) , as f,g : (X,PvQ) § (I,uvl)

and u v i is the usual topology on I Thus

T([ ]) ~ PvQ Conversely, suppose that A is

PvQ-closed. To show that A is T([ ])-closed, we

assume x ~ A and exhibit hl,h2 : (X,P,Q) § (l,u,l)

such that x ~ K(hl,h2)

A is PvQ-closed and

P and a Q-open set Q

and A c K(hl,h2) Since

x ~ A , there is a P-open set

such that x s PNQ c X-A .

Let f : (X,P,Q) + (l,u,l) , g : (X,P,Q) + (l,u,l)

be such that f(x) = 0 , g(x) = 1 , f = 1 off P ,

g = 0 off Q Then x ~ K(g.f,g) , A c K(g.f,g) .

Note As remarked earlier in the discussion of example

4, this proposition essentially proves that epis in

the category of separated pairwise completely regular

spaces are maps onto PvQ-dense subspaces ([15]) .

The proof given above is a simplification of the

characterization of epis given in ~5]

Examples 4 and 5 above are intimately related.

Suppose (X,T) is a topological space. Let T be

the topology on X for which points x have minimal

neighbourhoods: T-closure {x} The bitopological

560

space (X,T,T*) is pairwise completely regular.

Moreover, it is pairwise separated if and only if X

is a T o - space. The correspondence (X,T) § (X,T,T *)

has been studied in [15 ]. It is interesting to note

that TvT * is simply the b-topology. Thus, the

characterization of epis in T o follows from the

characterization of epis in paimwise Tychonoff spaces.

However, in general, 82X #bX as a set of points,

where 82 is the bitopological Stone-~ech extension

reflector and b is the pc- reflector which takes

X to the b-closure of its image in the canonical

product of copies of D .

One further comment will finalize this diseussiom

Example 6 Let A be the category of uniform spaces

(not n ecessari!y separated). Let M be a metric space

and U the metrizable space of uniformly continuous

functions f : M+ I , I : [0,1] U is an injective in

A [ 9 ]. Lambek and Rattray [11] raise the question

of describing the associated reflective subeategory.

Based on the preceding work we can provide the first

step towards an answer: The reflective subcategory

consists of all uniform spaces which are isomorphic

to closed subspaces of products of copies of U

2.3 Characterizing reflections by coincidence kernels

This section raises a question which we have been

561

unable to settle even within situations which are fairly

algebraic, such as those involved with semi-continuous

functions (X,T) § (IR,u) or with bicontinuous functions

(X,P,Q) § (]R,u,l)

V

The Stone-Cech compactification can be

characterized as being a compact Hausdorff extension

C of X , in which X is densely embedded and such

that

Clc[Z(f) D Z(g)] -- ClcZ(f ) N ClcZ(g)

The problem is to find an analogous characteri-

zation which is more categorial. The obvious sub-

stitutes for the zero sets are the coincidence sets.

We conjecture that the proper substitute for cl e

ought to be the [ ]- operator discussed in this

note.

3. Lambek-Rattray localization

We conclude our discussion with a brief description

of the work by Lambek and Rattray in so far as it is

pertinent to this note.

3.1 Let A w

object in A

where (-,I)

composition

on A .

be a complete category and I a fixed

A m

The object I determines functors

(-~I) ~ EnsOP I(-) ~ A

is a left adjoint of I (') Thus, the

S=I (-'I) is a part of a triple, (S,n,Z)

562

The Fakir construction

another triple (Q,nl,Pl)

equalizer

Let

objects

[4] , associates with S

where Q : A + A is the

ns Q <~ s ,~ s 2

sn

FixQ denote the full sub-category of all

A such that hi(A) : A + QA is an isomorphism.

The question arises - when is Q a reflector ?

This is answered by the Theorem in the Lambek-Rattray

paper which we shall quote for completeness.

Theorem

(a) I

each A in

(b) Q

Q : A ~ Fix Q

(c)

The following statements are equivalent.

is injective with regard to <(A) , for

A

is idempotent, i.e. becomes a reflector

FixQ is the limit closure of I .

We add one more equivalent condition to the theorem

which we have found useful in discussing topological

examples.

Let us say that A--!+f B is C_-epi (where C_ is

a class of spaces) if A f~ B _S~ C -- A f, B ~ C

g = h , whenever C is in C

563

The equivalent condition that we mentioned is then

n,(A) (a') A , Q(A) is I-injective for every object

A in A .

3.2 Finally, the connection between Q and our

discussion of the closure associated with a reflection:

Q(A) = N{K(f,g) = N(A) If,g : S(A) + I} = [n(A)]

It was only after proving this that we realized

that this is the content of Lambek-Rattray's lemma 1,

that K(A) : QA § SA is the joint equalizer of all

pairs of mappings SA ~ I which coequalize

q(A) : A + SA .

I wish to thank the organisers of the Conference

for their invitation and financial support and also the

Council for Scientific and Industrial Research (C.S.I.R.)

and the University of Cape Town for their financial

support. I also wish to express my thanks, through Prof.

K.Hardie, to the T.R.G. for providing an opportunity for

the discussion of these ideas.

564

References

[1] S. Baron - Note on epi in T , Canadian Mathemati- O

eal Bull. 11, 503-504, 1968.

[2] E. Cech - Topological spaces. Academia Prague,

1966.

[3] R. Engelking and S. Mrowka - On E -compact spaces,

Bull. Acad. Polon. Sci., 6, 429-436,

(19581.

[4] S. Fakir - Monade idempotente associ~e ~ une monade~

C. R. Acad. Sci., Paris 270, A99-AI01, 1970.

[5] M. Fr~chet - Espaces Abstraits, Paris, 1928.

[6] P.C. Hammer - Extended Topology: Continuity I,

Portugaliae Math. 23, 79-93 (19641.

[7] H. Herrlich - @ - kompakte - R~ume, Math Z. 96,

228-255, (1967).

[8] H. Herrlich - Topologische Reflexionen und

Coreflexionen. Lecture Notes 78.

Berlin-Heidelberg - New York

Springer 1968.

[9] J.R. Isbell - Uniform Spaces, Amer. Math. Soc.

Surveys No.12, Providence, R.I., 1964

[i0] J.F. Kennison - Reflective functors in general

topology and elsewhere. Trans.

Amer. Math. Soc. 118, 303-315

( 1 9 6 5 ) .

[11] J. Lambek and B.A. Rattray - Localization at

injectives in complete categories,

Proc. Amer. Math. Soc. (41), 1-9, 1973.

565

[12] L.D. Nel and R.G. Wilson - Epi-reflections in the

category of T O spaces. Fund. Math.

75, 69-~4 (1972).

[13] R. Nielsen and C. Sloyer - On embedding in quasi-

cubes. Amer. Math. Monthly 75,

514-515 (1968).

[14] S. Salbany and G.C.L. Br~mmer - Pathology of

Upper Stone-cech compactifications,

Amer. Math. Monthly 78, 1971.

[15] S. Salbany - Bitopological spaces, compaetifiea-

tions and completions. Math. Mono-

graphs, University of Cape Town, 1974.

[16] L. Skula - On a reflective subcategory of the

category of topological spaces, Trans.

Amer. Math. Soc. 142, 137-141 (1969).

COMPACTNESS THEOREMS

by

M. Schroder

ABSTRACT: Theorems due to E. Binz, and to G.D. Richardson

and D.C. Kent, saying that in certain

categories of convergence spaces compact

spaces are topological, both have a common

extension. A convenient proof involves a

"convolution" of convergence structures.

INTRODUCTION

Lately several theorems have appeared, each saying that in a specified

category of convergence spaces, compact spaces are topological. To start

with, in 1968 E. Binz proved that the category of c-embedded convergence

spaces has this property. Then in 1972, as a by-product of their work on

compactifications, G.D. Richardson and D.C. Kent did the same for principal

T 3 spaces. A.C. Cochran and R.B. Trail [2] and C.H. Cook [3, Theorem 2]

obtained this result as well, using other methods.

Here it is shown first that the "principal" condition can be relaxed

to "pseudotopological". Together with [5], [7] or [9], which all characterise

c-embedded spaces internally, this yields an easier proof of Binz's theorem.

Second, the method used here allows one to weaken the "T3" condition as

well.

w TERMINOLOGY

Throughout this note, X is a non-empty set, ~ its power set and

F(X) the set of all proper filters on X. As usual, ~ is regarded as an

improper filter.

567

Mainly because the term "convergence space" has recently become

ambiguous, it is convenient to give the following definitions. A subset

H of F(X) is called a segment if for any @ in H every filter

finer than ~ also belongs to H, or a filter-ideal if in addition

~ X belongs to H whenever both ~ and X do. (Set notation is

used throughout, with exactly its usual set-theoretic meaning; for

example, if ~ and X are filters on X, then ~ is finer than X

iff ~ ~ X.)

Some authors use the term "convergence structure" for a map

y : X + F(X) under which each Yx is a filter-ideal containing ~, the

ultrafilter at x. Others call this a "limit structure", and use

"convergence structure" for those maps ~ in which each Yx is merely

a segment containing ~. Here, the latter idea is referred to simply as

a "structure", the term "convergence structure" being reserved for the

filter-ideal version. Similarly, the (convergence) space X is the set

X together with the (convergence) structure y.

A tool used often is the multiplication of structures described in

[i0]. Since it is not widely known, its definition and some of its

properties are sketched below.

As in [i0], a map ~ : X + F(X) is called a selection (on X); it

is called a y-selection if y is a structure and #x E X x for all x.

Any selection can be condensed onto a set or a filter as follows: if

is a selection, P a subset of X, and ~ a filter on X, then the

equations

~'P = n{r : x E P} and x

~.~ = U{~.P : p ( ~}

568

both define filters on X (except that ~-~ = X is improper). Now if

e is a selection as well, the filters e.~x clearly define a selection

0"9. It is easy to verify that the filters 0.(~.~) and (e.~).~ coincide,

and as a result, that multiplication of selections is associative.

Now let y and 6 be structures on X. By defining y.6 to be the x

set of all filters X such that X 2 ~'~, for some y-selection

and some ~ in 6x, one obtains a structure y.6, coarser than both

y and 8. Moreover, if both y and 6 are (principal) convergence

structures then so is y.6. The resulting multiplication or convolution

of structures is not very well understood. None-the-less, it is known

that

(i) the discrete topology is the identity element,

(ii) �9 is not always commutative,

(iii) for any structures y, 6, q on X,

y.(6.q) = (y'~)'q if either y or ~ is principal (however, I do not

know if the associative law holds in general), and

(iv) a convergence structure y is diagonal [6] iff

y = y.y. (O. Wyler's recent treatise [ii] provides a natural framework

for diagonal structures, and probably as well.)

Perhaps the most significant fact is that composition of adherence

operators reflects this multiplication; that is, ay. 6 = a6a Y for any

structures y and 6 on X. These facts and others dealt with in

[i0] were proved for convergence structures, but the proofs given there

still go through almost unchanged.

Using this fact, one can characterise those convergence structures

whose adherence operators are idempotent.

(i) ay is idempotent (meaning that ay = ayay) iff y'y is

finer than ~y, the principal modification of y.

P2"oo~ : Suppose first that

each subset A of X,

569

Y'T is finer than ~y. Then for

ay(A) c ay(ay(A))

= a (A), as noted above Y'T

a y(A) = ay(A).

Thus a = a a . Conversely, if a is idempotent, then Y Y Y Y

a =aa =a =a Y'Y T Y T ~T

Hence y.y is finer than ~y, since ~y is the coarsest structure

whose adherence operator is exactly a . # Y

This is probably a convenient place to point out that the processes of

solidification [9] and Choquet modification [3] are just different ways of

achieving the same end, as D.C. Kent told me in a letter. To be precise,

let H be a non-empty segment in F(X). A function T associating with

each ~ in H a member T(~) of ~ is called an H-cover. Now the set of

all H-covers defines a set oH of filters: X E oH iff for each H-cover

T, a finite subset {~i' .... ~k } of H can be found, with

T(~I) U . . . U T(~ k) E X �9

It is an easy exercise in set theory to prove

(2) X E oH iff every ultrafilter finer than X belongs to H

(that is, iff X belongs to the Choquet modification of H).#

Here too, ambiguous terminology has arisen. The difficulty is avoided

570

by referring solely to solidifications and solidity, while using lemma (2)

above to translate, when necessary.

The solidification oy of a structure y is obtained by using this

construction pointwise. Clearly oy is coarser than y, but finer than

Lemma (2) has the following corollary, whose proof is a similar exercise

in set theory.

(3) Let a be an adherence operator on X and H a segment,

and define a(H) to be the segment generated by the set

{a(~) : ~ E H}. Then

a(H) c a(oH) c oa(H) ,

and in particular, the same ultrafilters belong to all

three segments.#

Finally, w ends with a corollary of lemma (1) which turns out to be

surprisingly useful later.

(4) Suppose that y is a structure satisfying

(i) ~y = ~y and

(ii) y'y is finer than ~y.

Then oy is topological.

-P'~oo~: Any principal structure whose adherence operator is idempotent

is topological.#

571

w COMPACTNESS

Throughout, compactness means topological compactness, that is,

X is a compact space iff every ultrafilter on X belongs to some y . y x

In the proof of his result, C.H. Cook showed in effect that if an ultra-

filter belongs to Y'Yx then it already belongs to Yx" Equivalently,

y.y is finer than oy. Under the same conditions (namely that Xy is

compact and T3) a similar method shows that any ultrafilter in ~y ' x

belongs to Yx' or in other words, that ~y = o~. Applying lemma (4), one

sees that X is nearly topological, in the sense that its solidification Y

is topological. The main purpose of this section is to derive results

which can be similarly coupled to lemma (4).

From now on, let y be a fixed structure on X, and a its

adherence operator. First it is well known that if @ belongs to

Yx then ~ is finer than a~). But as a is the adherence operator

of ~y as well, this proves

(5) ~ E ~yx ~ i ~ a(~). #

(In fact, lemma (5) has a partial converse: namely, if ~ is an ultrafilter

such that i is finer than a(~), then it belongs to ~yx.)

Together with (4), the next two lemmas form a powerful tool for

studying a wide range of compact spaces.

(6) Yx N a(yy) = ~ ~ ~Yx n yy = ~ .

t~O0~: Suppose that ~ belongs to both ~Yx and yy By (5),

finer than a(~), and so ~ belongs to both a(yy) and Yx " #

is

572

(7) a(yy) = Yx N = ~ ~ Y'Yx N Yy 0 �9

Proof: Assume that Yx and a(yy) are disjoint, but that X belongs

to both Y'Yx and yy.. By definitio~ there is a y-selection ~ and a

filter ~ in Yx' with X ~ r The first assumption shows that there

can be no proper filter finer than both ~ and a(x). This means that

sets P in ~ and K in X can be found, such that P N a(K) = ~. Thus

each z ~ P lies outside a(K), and in particular, K' E ~z since

E y . In other words, z g

K' ~ ~.p c ~.~ c X

However, this completes the proof, by contradicting the fact that K

belongs to X.#

The first application is in the category of solid T 3 spaces, and

yields a theorem which could also be proved using the method of [8] almost

without change.

i~eorem i: Any compact solid T3 space is a compact Hausdorff

topological space.

Proof: Take such a space X . First, let ~ be an ultrafilter belonging Y

to y'y . By compactness, ~ E y for some y. However since X is x y Y

T~, if x # z then

Yx N a(yz) = Yx N Yz = ~ '

and so y'y N y = ~, by lemma (7). Thus x = y, showing that Y'Y is x z

573

finer than oy. Similarly, one can show using lemma (6) that ~y is finer

than ~y. Thus as X = X by assumption, X is topological by (4).# Y ~Y T

This theorem clearly extends those of [2], [3] and [8], since principal

spaces are solid. Moreover, it can be used to attack [i, Satz 9] as well:

this states that for any c-embedded space Xy, the following conditions are

equivalent -

(i) X is compact, Y

(ii) X is compact Hausdorff and topological, and Y

(iii) Cc(XT) is a Banach algebra.

Binz's proof of "(i) --~ (ii)" used the path "(i) ~ (iii) -----> (ii)";

a slightly more direct proof is now possible, as outlined below.

By [5, Theorem 2.4], [7, Satz 4] or [9, Theorem 3.6], every c-embedded

space is solid, Hausdorff and w-regular, and in particular, T 3. Thus

theorem i applies to any compact c-embedded space.

An alternative proof uses the fact that c-embedded spaces are solid

and functionally Hausdorff. Thus by [3, Theorem 8], if X is compact and Y

c-embedded, T can be identified with the compact Hausdorff weak topology

on X.

w RELAXATION

To extend theorem I to cope with spaces which are not necessarily solid,

Hausdorff or regular, one must unfortunately give a list of regularity

axioms (R) and symmetry axioms (S). To complete the picture and avoid

ambiguity, a few separation axioms (T) are given as well. Some of these

are filter versions of corresponding axioms in topology, while others are

suggested by lemmas (6) and (7). In this list, X is a space, with x and Y

y ranging over X.

574

R0 : Y ~ Yx y

- ~Yx x y

x y

RI. 2 : # ~ yx -----> Tx N a(yy) =

RI.5 : # ~ Yx ~ a(Yx) N a(yy) =

: a(Yx) = Yx' for all x in X

S O : ~ E Yx ~ Ty c Yx

Sl : Y ( ~x -------> Yy = 7x

T O : # E Yx and i ~ yy ~ x = y

T 1 : ~ E Yx ~ x = y

T2 : Yx n 7y = @ unless x = y

T 3 : T O and R 2

The following facts will be needed, some of which are well known,

and all of which are easily proved either directly, or with the help of

lemmas (3), (5) and (6).

575

(i) R 2 ~ RI. 5 ------> RI. 2 ------> RI. I ~ R I ~ R 0.

(ii) Even in the context of topological spaces, RI. 5 is strictly

weaker than R2; for example, it is well known that a functionally Hausdorff

topological space may fail to be R 2 , but it is RI. 5 �9

(iii) R 0 and T O ~ T I .

(iv) R 1 and T O ~ T 2 .

(v) R 0 and S O ~ S I .

(vi) The properties (R), (S) and (T) are all preserved under

solidification.

(vii) A space is RI. 5 if it is k-regular.

(The idea of k-regularity arose in [2], where it was shown that a space

is k-regular iff its principal modification is R2.)

Now by putting all these ideas together in the obvious way, one obtains

the following results, which significantly extend their earlier counterparts.

Theorem 2 : The solidification of any compact RI. 2 and S O space

is a completely regular (not necessarily Hausdorff)

topological space.

Proof: Let X be a compact space which is RI. 2 and S O . First, suppose Y

is an ultrafilter in ~y x. By compactness, ~ belongs to yy for some y.

576

Thus by (6), Xx and a(Xy) are not disjoint, and hence y 6 Yx (RI.2)"

or in other words, Consequently, ~ ~ X x (So). Thus ~Yx ~Yx'

~Y = ~Y. The second step is similar, with (7) showing that the other condition

of (4) is satisfied as well.

Thus X the

segments a(yy)

ultrafilter

and so # E Yx Xy is SI, ~ belongs to yy. It follows

that a(Oyy) ~ Oyy, as desired.

In short, X y is a compact R 2 topological space, and hence completely

regular [4, page 138 Theorem 5, and page ii0 Theorem 7]. #

is topological: more, it is R 2. By (3), for any y

and a(Oyy) share the same ultrafilters. So take an

in a(yy). Again by compactness, it belongs to some Yx'

(RI.2). Thus as

Corollary 1 : The solidification of any compact T 3 space is a compact

Hausdorff topological space.

Corollary 2 : Any compact solid space which is RI. 2 and S O is a

completely regular topological space.

Corollary 3 : The solidification of a compact k-regular S O space is

a completely regular topological space.

CoroIZ~ry 4 : Any compact solid k-regular T O space is a compact

Hausdorff topological space.

References:

[1] E. Binz "Kompakte L1mesraume und limitierte Funktionenalgebren" Comm. Math. Helv. 43 (1968), 195-203.

[2] A.C. Cochran and R.B. Trail "Regularity and complete regularity for convergence spaces" in Lecture Notes in Mathematics 375, 64-70 Springer (Berlin) 1974.

577

[B]

[4]

[5]

[6]

[7]

[8]

[9]

[io]

[ii]

C.H. Cook "Compact Pseudo-Convergences" Math. Ann. 202 (1973), 193-202.

S.A. Gaal "Point Set Topology" Academic Press (New York, London) 1964.

D.C. Kent, K. McKennon, G.D. Richardson and M. Schroder "Continuous convergence in C(X)" Pac. J. Math 52 (1974), 457-465.

H.J. Kowalsky "Limesr~ume und Komplettierung" Math. Nachr. ii (1954), 143-186.

�9 m ,

B. M~ller "L - und c-einbettbare Llmesraume' To appear. c

G.D. Richardson and D.C. Kent "Regular compactifications of convergence spaces" Proc. A.M.S. 31 (1972), 571-573.

M. Schroder "Solid Convergence Spaces" Bull. Austral. Math. Soc. 8 (1973), 443-459.

M. Schroder "Adherence operators and a way of multiplying convergence structures" Mathematics Preprint 29,(1975) University of Waikato.

O. Wyler "Filter space monads, regularity, completions" in Lecture Notes in Mathematics 378, 591-637 Springer (Berlin) 1974.

Mathematics Department, University of Waikato, Hamilton, New Zealand.

Differential Calculus and Cartesian Closedness

by

Ulrich Seip

Cartesian closed categories play an important r~le in many aspects

of mathematics. They appear in modern forms of Algebraic Geometry~ in

Logic~ in Topology. This leads quite naturally to the idea of trying

to use this notion also for Differential Calculus.

The first attempts in this direction were undertaken by A.Bastiani

[i], and then by A.Froelicher-W.Bucher [4]. They all used the notion

of limit spaces to generalize calculus in order to be able to form

function spaces of differentiable maps between limit vector spaces.

In retrospect one may say that A.Bastiani used a "good" definition of

continuous differentiability but the category she chose did not allow

the desired cartesian closedness property. A.Froelicher and W.Bucher

took cartesian closedness exactly as their goal. But because of their

"bad" definition of continuous differentiability, their special types

of limit vector spaces became increasingly complicated.

To me it seemed always desirable to establish differential calculus

in a pure topological setting. This because questions of continuity

and differentiability are local questions and topological spaces have

by definition a local structure. So the question was, whether or not

a differential calculus can be established in a topological setting

so as to obtain cartesian closedness in the infinitely often differen-

tiable case.

For a long time it even seemed impossible to establish anything

like cartesian closedness for continuous maps - not to speak of dif-

ferentiable ones. But in 1963 Gabriel and Zisman proved [6] that the

category ~ of compactly generated hausdorff spaces is cartesian

closed - and this is a full subcategory of topological spaces.

Starting from this category one is quite naturally led to the in-

vestigation of the category ~ of compactly generated vecto~ spaces

and continuous linear maps. Observing that the Hahn-Banach theorem is

the tool for proving the so-called mean value theorem of differential

calculusp it becomes clear that not all compactly generated vector

spaces can be used for a differential calculus if one wants the basic

well known theorems of calculus to hold (all of them are consequences

of the mean value theorem). But there is a nice complete and cocom-

plete full subcategory of ~D which is in a one-to-one correspondence

with a full subcategory of convex vector spaces. The objects of this

579

category ~ are suitable for a differential calculus, but they do

still not allow to prove such an important fact as the existence of

a primitive map for a given continuous map ~:~E. To obtain this,

one has to impose sequential completeness on the compactly generated

vector spaces under consideration, and this leads to the category ~y~

of convenient compactly generated vector spaces.

The objects of this category then form the base for our differen-

tial calculus. Our notion of (continuous) differentiability is the

simplest one possible: We say that a map ~:E~U-~F is (continuously)

di#ferentiable if it has a G~teaux derivative D~:U~L(E,F) which is

continuous (G~teaux derivative stands for directional derivative in

every direction). Since ~ is cartesian closed, the convenient vec-

tor spaces do not only behave well for continuous linear maps but al-

so for continuous maps. From this follows immediately that the no-

tions of "weak" and "strong" differentiability coincide in our set-

ting. And because banach and fr~chet vector spaces are also conven-

ient vector spaces, our calculus becomes a generalisation of the well

known fr6chet calculus.

The main theorem of our differential calculus for convenient vec-

tor spaces states that the category of convenient vector spaces and

smooth (infinitely differentiable) maps is cartesian closed.

The article is divided into 5 sections:

Section l: Consists mainly of the Gabriel-Zisman theorem for ~ and

the Kelley theorem which says that for X compactly generated and U

complete, the compact-uniform topology CU(X,U) for continuous maps

from X to the topological space underlying U is again complete.

Section 2: The general theory of compactly generated vector spaces

and continuous linear maps. The main theorem states that the catego-

ries ~ and ~ are complete, cocomplete, additive, have an inter-

nal functor L and a tensor product functor ~.

Section 3: Developes the differential calculus and gives the proofs

of the basic theorems like: Differentiability implies continuity,

functoriality of the tangent operator, differentiable maps into a

product and from a finite product, convergence theorem of differen-

tiable maps, symmetry of higher derivatives, existence of primitive

maps. Poincar@ lemma and Stoke theorem are left to the reader.

Section 4: Introduces convenient vector space structures for diffe-

rentiable and smooth function spaces. The highlight is theorem 4.g~

stating that the category of convenient real vector spaces and smooth

maps is cartesian closed.

Section 5: Contains other results and problems. Mainly an attempt is

made to extend theorem 4.9 to the case of smooth manifolds.

580

1. Topological Backqround We denote by ~ the category of hausdorff

spaces and by ~ the full subcategory of compactly generated spaces.

We remind the reader that a hausdorff space is called compactly gen-

erated if it carries the final topology with respect to the inclu-

sions of its compact subspaces. To each hausdorff space X we associ-

ate a compactly generated space CG(X) with the same points as X, by

requiring that CG(X) carries the final topology with respect to the

inclusions of the compact subspaces of X. Hence the identity function

l:CG(X)-->X becomes continuous and CG becomes a coreflector CG:~,

not changing the functions with respect to the underlying sets. Since

is complete and cocomplete, it follows that %~ is complete and co-

complete.

In order to avoid notational difficulties, we shall denote by the

symbol x the usual topological product (with respect to ~), whereas

the symbol m stands for the product with respect to %~ calculated as

n=CGox.

From elementary topology we recall that the compact-open topology

on function spaces of continuous maps defines an internal functor

CO:~~ Hence CGoCO=C:~~ is an internal functor for %~.

Proposition 1.1. Let X be compactly generated and S a subset of X.

If S is open or closed in X, the subspace topology on S is compactly

generated.

Proof. Clearly a hausdorff space is compactly generated iff it carries

the final topology with respect to the inclusions of its locally com-

pact subspaces. If S is open or closed in X, the intersection subspa~-

es SnK are locally compact for every compact subset K of X. The pro-

position follows.

Lemma 1.2. Let X be compactly generated, Y hausdorff. Then the evalu-

ation map ~:CGoCO(X,Y)nX~Y, defined by (~,x)~(• is continuous.

Proof. The domain of ~ is compactly generated. Hence it suffices to

prove continuity on compact subspaces. Every compact subset being con-

tained in the product of its projections, we may restrict ourselves

to prove the continuity of & on compact subspaces of type K• where

K is compact in CGoCO(X,Y) and L is compact in X. If (~,x)~K• and

~(x)~U with U open in Y9 we obtain in L~-lu a relative-open neighbor-

hood of x in L. L being compact, there exists a relative-open neigh-

borhood V of x in L such that the L-closure ~ is contained in Ln~-lu.

Hence (~,x)c[(~,U)nK]xV and ~ maps this relative-open neighborhood of

(~,x) into U.

Lemma 1~ Let X and Y be compactly generated. Then the map

~:X-~C(Y,XmY), defined by 7(x):y~-~(x,y), is continuous.

Proof. The domain of ~ is compactly generated and the category ~ is

581

coreflective in ~. Hence it suffices to show that ~:X~CO(Y,XmY) is

continuous on compact subspaces K of X. If (L,U) denotes a subbasis

open subset of CO(Y,XnY) with L compact in Y and U open in XnY, we

clearly have [~-l~(L,U)]nK=proJx[(K• whence continuity of

on K follows.

From the lemmas we obtain the theorems:

Theorem 1.4 (Gabriel-Zisman). The category ~ is complete, cocomplete,

and cartesian closed with C:~~ as internal functor. For each

object X the natural transformations ~X:I~--~C(X,-mX) and

x"C(X'-)mX~I~ are the unit and counit of the cartesian adjointness.

Hence a function ~:X--~C(Y,Z) is continuous iff the corresponding

function ~=Eo(~nl):XnY-~Z, defined by &(x,y)=~x(y), is continuous.

Theorem 1.5 (Steenrod). The following diagram commutes:

~OPx ~ CO~

~xCG ~CG

Theorem 1.4 is an immediate consequence of the lemmas. To prove

theorem 1.5, it suffices to prove commutativity on objects. Clearly

C(X,CG Y) and CGoCO(X,Y) have the same underlying sets. The identity

function l:CG Y-+Y being continuous, we obtain immediately the contin-

uity of the identity function l:C(X,CG Y)--~CGoCO(X,Y).- By lemma 1.2

the evaluation ~:CGoCO(X,Y)mX-~Y is continuous, whence by coreflective-

ness this is also true for the evaluation &:CGoCO(X,Y)mX--~CG Y. By

theorem 1.4 the continuity of I:CGoCO(X,Y)-->C(X,CG Y) follows.

We shall now exhibit relations between the category ~ and catego-

ries of uniform spaces. First we note that there is an adjoint rela-

tion between the complete and cocomplete categories ~ and ~ the

latter denoting the category of hausdorff uniform spaces and uniform-

ly continuous maps. This, because the usual topologizing functor

H : ~ clearly preserves limits and the solution set condition for

application of the adjoint functor theorem evidently can be satisfied.

We shall denote by H U : ~ the coadjoint to H.

Now observe that for a hausdorff space X and a hausdorff uniform

space U the function space CO(X,H U) is uniformizable as follows: If

K is compact in X and V an entourage of U, we define (K,V) by

(K,V)=~(~I,~2)~CO(X,H U)xCO(X,H U) I (~lX,~2x) ~V for all x~K]. These

sets (K,V) then form a subbase of a uniformity on the set of continu-

ous maps from X to H(U). The resulting uniform space will be denoted

by CU(X,U). Then CU becomes a functor CU:~~ the obvious way

and the following proposition follows:

582

Proposition 1.6. The category ~of hausdorff uniform spaces is com-

plete and cecomplete. The topologizing functor H:~L~ has a coadjoint

functor HU:~. Furthermore, the compact-uniform functor

CU:~~ the diagram ~~ CU >~%L commutative.

~lxH ~H ~op• ~ CO ~

The category ~Lcontains two important full reflective subcatego-

ries: The category ~of sequentially complete hausdorff uniform spa-

ces 9 and the category ~of complete hausdorff uniform spaces. The re-

flectors ~:~ı89 ~ : ~ c a n be constructed by using the adjoint

functor theorem. The fac~ that a dense subspace D of a hausdorff space 2card U

X implies card X~2 may be used to construct solution sets. The

explicit construction of the completion 0 of a hausdorff uniform space

U in terms of minimal cauchy filters can be found in [2]. Clearly U

may be considered a uniform subspace of U and 0 as the intersection of

all sequentially complete subspaces of U containing U.

Now again the category ~ enters the considerations. Kelley proved

that if X is compactly generated and U is complete uniform, then

CU(X,U) is again complete (whereas this is not true for arbitrary

hausdorff spaces X).

Theorem 1.7 (Kelley). The functor CU:~~215 by restriction

to ~~ ~, and by restriction to ~~ ~LL. Hence

the following diagram commutes:

cu d ~

r

op• CU ) ~

I•176 ICGoH c

Proof. Let X be compactly generated, U hausdorff uniform, and ~ be a

given cauchy filter on CU(X,U). Then there exists for any subbasis

entourage (K,V) of CU(X,U) a set F~ with F• If K=~xl, we ob-

tain F(x)xF(x)~V, whence for every x~X the image Ex(~) of ~ under

evaluation Ex:CU(X'U)-~U at x is a cauchy filter on U. To prove the

theorem we may therefore assume that the function • defined

by X:x~-~lim E (~), exists. Let us prove that this function is contin- x uous: Since X is compactly generated, it suffices to prove continuity

of X on compact subsets K of X. So let ~ be a convergent filter on K

with lim ~=x o. Then there exists to any continuous map ~:X~H(U) and

any entourage V of U a set G ~ such that (~• for all x~G~.

583

If V is any symmetric entourage of U, we choose first a set F~Wwith

FxF~(K,V), and then we select for a fixed ~EF a G~ with the property

mentioned before. Hence we have for all x~G~ that (X(x),~(Xo))~V S~ Be-

cause every entourage W contains a symmetric entourage V with V5~W,

the continuity of ~:X~H(U) follows.- To complete the proof of the

theorem, we observe that for a symmetric entourage V of U and a set

F~ with FxF~(K,V) we have (Xx,~x)~V 2 for all ~F and all x~K. This,

because by definition of ~ exists for every x~K a map ~ ~F such that X

(XX,~xX)~V. It follows F~(K,V)[X], whence X=lim ~.

We end this short discussion by exhibiting the adjoint relation-

ships between the various categories ~ ~, ~, ~L,~ in form of a

diagram.

Proposition 1.8. The categories ~,~, ~L, ~,~LLare all complete and

cocomplete. They are related to each other according to the following

diagram:

~ ~ CG ~ ~L

HU

In this diagram commute the inner and the outer triangles (we have

shortened the notation CG~H to CG). Moreover, every pair of functors

in opposite direction is an adjoint pair with the "outer" functor ad-

joint to its "inner" counterpart.

2. Compactly Generated Real or Complex Vector Spaces We begin by

reviewing the main properties of locally convex hausdorff topological

vector spaces (short: convex vector spaces) over the field F, with

either denoting the real numbers ~ or the complex numbers ~. The ca-

tegory of these spaces with the continuous linear maps as arrows will

be denoted by ~x~.

We observe the existence of the functor CO:~~ defined

as follows: If X is a hausdorff space and M a convex vector space,

the underlying vector space structure of CO(X,M) is obtained by

pointwise addition and scalar multiplication of continuous maps from

X to M and the convex topology on this vector space is the compact-

open topology. On arrows CO is defined by composition. Analogously

we obtain the functor LCO:~~ with LCO(M,N) the vector

space of continuous linear maps from M to N equipped with the com-

pact-open topology~ Finally we see that we have for any n~l~ corre-

sponding functors LnCO:(~)~ where LnCO(M1,..o,Mn;N ) is

584

n the vector space of n-linear continuous maps X:i~&Mi ~N equipped with

the compact-open topology.

There are two outstanding full reflective subcategoriee of ~:

~he first is the category ~ of sequentially complete, the second i8

the category ~of complete convex vector spaces. The reflectors

~ : ~ ~ and ~ : ~ ~ are constructed as for hausdorff uniform

spaces. Using Kelley's theorem 1.7, we obtain immediately:

Theorem 2.1. The f~nctor C0:~~ factors by restriction to

o ~ P• through ~. ~ Px~ through ~, and by restriction to ~o ~

Hence the following diagram commutes:

O A _ _

~op• ~s CO ~L's

1,

~op~ c0 ~]~

The main theorem for ~is the theorem of Hahn-Banach. Together

with the other relevant properties of @~ one has:

Theorem 2.2 (Hahn-Banach etc.). The categories ~ ~ , ~ are com-

plete, cocomplete, additive. The ground field ~ is a generator and a

cogenerator for each of these categories. More precise: If K is any

closed convex subset of a convex vector space M and if x~K~ then there

exists a continuous linear map X:M-->~ such that X(x)~X-K. Further: For

every n~N, the compact-open topology gives a functor

Lnc0:(~)~215 with LnC0(M1,...,Mn;N) the convex vector space

of n-linear continuous maps with the compact-open topology.

The proof of thi8 theorem is standard and can be found in any good

textbook on topological vector spaces [9].

What happens to a convex vector space M if we apply the functor CG

to its topological structure? Since CG preserves underlying sets and

products, we may - and shall - consider CG(M) as the same vector space

with addition and scalar multiplication now continuous with respect

to the compactly generated product. Since the ground field ~ is local-

ly compact 9 it is already itself a compactly generated vector space

(as it is a convex one).

More generally, we say that a vector space E over ~ and equipped

with a compactly generated topology, is a compactly generated vector

space if addition and scalar multiplication are continuous maps with

respect to the ~-product n. It is easy to see that En~=E• since the

latter already is compactly generated because F is locally compact.

We denote by ~ the category of compactly generated vector spaces

and continuous linear maps.

585

Evidently we have a functor C G : ~ ~ , defined on objects as de-

scribed before and not changing the linear maps underlying arrows of

~%~ We denote by ~ the full subcategory of ~ generated by all

compactly generated vector spaces of type CG(M) with M any convex

vector space. Further we denote by ~ the full subcategory of ~

generated by all compactly generated vector spaces of type CG(M) with

M any sequentially complete convex vector space. Clearly these defi-

nitions suggest to define ~ as the full subcategory with objects of

type CG(M) where M is complete. But we shall see that this category

is troublesome.

Now we define a functor L C : ~ 3 ~ as follows: If E is in ~%~

LC(E) has the same underlying vector space as E and carries the topo-

logy generated by the convex open subsets of E. One proves easily

that addition and scalar multiplication remain continuous for LC(E)

and the ordinary topological product. Since E=CG(M) for some convex

vector space M, the topology of LC(E), being finer than the topology

of M~ is hausdorff.

We prove now that the restriction of LC to ~ factors through

~: For this, let (Xn) be a cauchy sequence in LC(E), where E=CG(M)

with M sequentially complete. Since the topology of LC(E) is finer

than the topology of M, the sequence (Xn) is also a cauchy sequence

in M, hence convergent to some vector x ~ in M. The set ~Xnln~IN~W~Xo~

is then compact in M and hence compact in E=CG(M). Consequently (Xn)

also converges in E to x . Because the topology of E is finer than 0

the topology of LC(E), we finally see that the sequence (Xn) conver-

ges to x ~ in LC(E). A

If we try to do the same for ~ we clearly run into problems.

The ensuing difficulties can not simply be circumvented by applying

afterwards the completion functor a : ~ ~ , because this functor

generally enlarges the underlying vector spaces.

Lemma 2.3. The following diagram commutes:

~ , ~ CG ~ ~ LC ~ ~

Moreover~ the functor CG is left inverse and adjoint to the functor LC.

Proof. We are only left with the proof that CG is left inverse and

adjoint to LC. If E=CG(M), we have already seen that LC(E) has a topo-

logy finer than M. Hence CGoLC(E) has a topology finer than CG(M)=Eo

On the other hand~ the topology of E is certainly finer than the topo-

logy of LC(E). Hence also finer than the topology of CGoLC(E). It

follows that CG is left inverse to LC.- To prove adjointness~ we show

586

that ~(LC E,M)=~(E,CG M) for any E in ~ and M in ~: Let

Xe~q~LC E,M). Applying CG and observing that CG is left inverse to

LC we get X~(E,CG M). Conversely let XE~(E,CG M). Applying LC

we get • E,LCoCG M). Since the topology of LCoCG(M) is finer

than the topology of M, we obtain • E,M) as desired.

We now introduce an internal functor L : ~ ~ 2 1 5 by defining

L(EpF) as the vector space of continuous linear maps from E to F,

equipped with the subs,ace topology of C(E,F), where C(E,F) denotes

the vector space of continuous maps from E to F with the cartesian

closed compactly generated function space topology described in w 1.

Since the subs,ace L(E,F) is evidently closed in C(E,F), we know from

proposition 1.1 that the topology of L(E,F) is compactly generated.

From the cartesian closedness of ~ we deduce immediately that C(E,F)

is a compactly generated vector space. Hence L(E,F) is a compactly

generated vector space.

In an analogous way we get a functor C:~~

Lemma 2.4. The restriction of the internal functor L:~~215 ~

to~l~r~~ ~ factors through ~, and the restriction of L to

~o~• factors through ~. Moreover, the following diagram com-

mutes: ~ o px ~ L > ~

Lc~215 ~ o p• Q~ L C 0 > ~

Analogous statements hold in the mult i l inear cases, Proof. First we prove commutativity of the diagram: Since CG preserves

initial structures, we have L(E,F)=CGoLCO(E,F). Since CG~LC(F)=F, we

get from Steenrod's theorem 1.5 that CGoLCO(E,F)=CGoLCO(E,LC F). Since

CGoLC(E)=E, we get from lemma 2.3 that LCO(E,LC F)=LCO(LC E,LC F).

Hence CGoLCO(E,LC F)=CGoLCO(LC E,LC F). So the diagram commutes, and

this also shows that the restriction of L to ~ o p • factors

through ~.- If F belongs to ~, we know that CO(E,LC F) is sequen-

tially complete by theorem 2.1. Hence LCO(E,LC F) is sequentially com-

plete because it is a closed subs,ace of CO(E,LC F). Hence L(E,F) is

of the form CG(M) with M=LCO(E,LC F) sequentially complete.

This sets the stage for proving the main theorem for the categories

and ~:

Theorem 2.5. The categories ~ and $~ are complete, cocomplete,

additive. They are symmetric multiplicatively closed with the ground

field as unit. The internal closing functor is L and the multiplica-

tive functor | satisfies L(E| with a natural isomor-

phism. The ground field is a generator and cogenerator for these

categories.

587

Proof. Since CG is left inverse and adjoint to LC, completeness~ co-

completeness, additivity follow from theorem 2.2. The same argument

shows that the ground field is a generator and cogenerator. The func- n

tots Ln(El,~176 ;-) are limit preserving for every n~IN since C(i~iEi,-)

is limit preserving for ~ and the topology for limits is evidently

the one obtained in ~ after application of the forgetful functor.

Hence L2(E,F; -) has a coadjoint functor T(E,F ) by the special adjoint

funotor theorem and we obtain E~F as T(E F)(~). Clearly E|174

F~E~E~E| L(~,E)~E, L(E,L(F,G))~L(E~F,GI~L2(E,F;G) with natural iso-

morphisms in all variables.

We define the dual E* of a compactly generated vector space by

E*=L(E,~). The identity l:E*~E* gives us by adjointness the contin-

uous linear evaluation map ~:E*| and using the commutativity of

tensor products and then again adjointness we obtain the continuous

linear map ~:E~E** with ~x:X~X(x). If E is in ~ , the map ~ is

clearly injective because L(F,E)~E and ~ is a cogenerator for ~ .

We say that E is embedded in its double dual space if E carries the

initial compactly generated topology with respect to ~:E>-~E**.

Theorem 2.6 (Frblicher-Jarchow). Every compactly generated vector

space E of ~ is embedded in its double dual. Hence this holds also

for the category ~.

Proof. We have only to show that the ~-induced compactly generated

topology on the vector space underlying E is finer than the topology

of E. For convenience, denote by E' the vector space of continuous

linear maps from E to ~ and put Ev=LCO(LC E,F). By lemma 2.4 we have

E*=CG(EV). Now consider the injective linear function

~:LC(E)~-~LCO(LCoCG Ev,F) and denote by E~ the ~-induced convex vector

space structure on the vector space underlying LC(E). If U is any

closed circled convex zero neighborhood of LC(E), the polar U ~ of U

is defined as U~163 IXxl41 for all x~U I. Since U is closed,

circled and convex, we have U=~x~LC(E) I IXxl41 for all X~U~

=~-l(u~ Ifl~l). Hence U is a zero neighborhood in E~ if U ~ is compact

in LCoCG(E ~) Since U ~ obviously is an equicontinuous subset of E'

the topology of U ~ is the same whether one considers U ~ as a subspaoe

of E ~ or as a subspace of E' with the topology of pointwise conver-

gence. But as a subspace of the latter, U ~ clearly is compact. Hence

U ~ is compact in E v, whence compact in LCoCG(E~). Since the closed

circled convex zero neighborhoods of LC(E) form a zero neighborhood

base, it follows that the topology of E~ is finer than the topology

of LC(E). Sin~e the functor CG preserves initial topologies, CG(E~)

has the compactly generated topology induced by ~:E~E**, because

E**=CGoLCO(LCoCG E~F) by lemma 2.4. Hence this topology is finer

than the topology of CGoLC(E)=E.

588

As has been already mentioned, we have an evident functor

C:s176 Denoting by ~ont the category of compactly gen-

erated vector spaces with the continuous maps as arrows, we get also

evident functors C:~~ ont--~ ~ ~kc one" and C:~.~ ~ con~'• con~-->~a con~'" Replacing ~ by ~D or ~D, and replacing ~ ~ by s ~ or con~ d cone ~ont we of these functors the respective obtain restrictions all to

full subcategories and all the restricted functors factor through the

In case of i~Tt ~ and ~%'~^nt~ uu this follows corresponding subcategories. rx ~x

from Steenrod's theorem 1.5, and for ~ and ~cont this follows then

from Kelley's theorem 1.7. And because CGoLC(E)=E in these cases, we

have moreover that C(X,E)=CGoCO(X,LC E).

We obtain especially: ~x

Theorem 2.7. The category ~cont has arbitrary products which are ~x

calculated as in i%b ~'. It is cartesian closed with the internal funetor rx 0 rx

C:~ePntx~t~ont--~on t. The natural homeomorphism

~:C(E,C(F,G))>~>C(EnF,G) is linear�9 The ground field is a generator

and a cogenerator for ~%~cont"

3. Differential Calculus for Convenient Vector Spaces over the Reals

From now on we shall fix the ground field to be the real numbers ~.

We shall call a compactly generated real vector space E a convenient

vector space if it is of type CG(M) where M is a sequentially complete

convex real vector apace. Put differently: A compactly generated real

vector space is called convenient if it belongs to ~.

Definition 3�9 Let E and F be convenient vector spaces. Let U be

open in E and V open in F, and let ~:E~U-~VcF be a given function

(defined on U with values in V)�9 Then we call ~ differentiable (on U),

if there exists a continuous map ~:E~U-~L(E,F) such that for every

fixed (x,y)~UnE we have: lie ~(x+ty)-~(x)}=~x(y). O~t ~Q

Proposition 3.2�9 If ~:E~U -~VcF is differentiable, the map ~:U~L(E,F)

with ~x(y)= lie ~{~(x+ty)-~(x)~ is unique�9 OSt~O

This is clear since F is hausdorff.

Definition 3.3�9 Let ~:E~U~VcF be differentiable. Then we define: 1 (i) The derivative D~:U-~L(E,F) by D~x(y)= lim ~I~(x+ty)-~(x)l

O~t~O

(ii)lhe differential d~:UnE--~F by d~(x,y)= lim I 0 ~ o ~I ~(x+ty) -~(x) } (iii)The tangent T~:UnE~VnF by T~(x,y)=(~(x) lie ~(x+ty)-~(x)I ).

In each of these cases (x,y)~UnE is considered arbitrary but fixed,

so that the limit on the right side of the equations exists.

In case that the domain U of a differentiable map ~ is an open

subset of ~, we denote by ~':~U~F the continuous map defined by

~'(r)=D~r(!) We shall also use the notation d for ~'. �9 dr

We have:

589

Proposition 3.4. If ~:~U-~V~F is differentiable~ then ~ is continuous.

Proposition 3.5. If ~:IR=U~VcF is differentiable and X:F~ is a con-

tinuous linear map, then Xo~:~U-~is differentiable and (Xo~)'=XO~'o

These two propositions are simple consequences of the definitions.

Now we shall state and prove the central theorem for differential

calculusp often misleadingly called the mean value theorem. From this

theorem all the important theorems of differential calculus follow.

Lemma 3.6 (The fundamental lemma of differential calculus). Let I=EO,1]

be the closed unit interval in ~ let E be a convenient vector space,

and let K be a closed convex subset of LC(E)o If ~:~?I~E is a con-

tinuous map such that its restriction to the open interval (0,I) is

differentiable and if ~'(r)~K for all r~(O,1), then ~(1)-~(O)~K.

Proof. If E=IR there exists an ro~(O,1 ) such that ~'(ro)=~(1)-~(O) as

everyone knows. Hence the lemma holds far E=~- Assume that the lemma

does not hold for some ~:I~E, t.i. ~(1)-~(O)~K. By Hahn-Banach exists

a continuous linear map X:LC(E)-~I~ such that X(~(1)-~(O))~X--K. Since

CGoLC(E)=E, the linear map X is also continuous as a map X:E-~. By

proposition 3.5 we have (Xo~)'=Xo~'. Hence (N~)'(r)EX-K for all r~(O,l).

Since the lemma holds for i~ we must have Xo~(1)-Xo~(O)EXK.

Contradiction~

The fundamental lemma at work. First we introduce a useful notation

for differentiable maps: If ~:E~U--~V=F is differentiable, we define

eR~:EnE~I(x,y,t) I x~U and x+tycUl--~F by ~R~(x,y,O)=O and for t~O we

put eR~(x,y,t)={I~(x+ty)-~(x)l-d~(x,y). We observe that the domain of

~R~ is open in EnEn~K. We shall prove that ~R~ is continuous: First

observe that for any (x,y)~UnE and any real numbers s and t with

x+sty~U the formula ~R~(x,sy,t)=s~R~(x,y,st) holds. Hence the function

~R~(x,sy,t) is differentiable with respect to s for any fixed (x,y,t),

and the derivative is given by ~R~(x,sy,t)=d~(x+tsy,y)-d~(x,y)-~

=~(x,y,t,s). This function ~:EnEn~(x,y,t,s)Ix~U and x+tsy~UI~F

is continuous, has open domain, and satisfies ~(x,y,t,O)=O. Therefore:

If (Xo~Yo)~UnE is fixed and K is any closed convex zero neighborhood

in LC(F), there exist positive real numbers ~ and g and a neighborhood

N(xo,Yo) of (Xo,Yo) in UnE, such that ~(x,y,t,s)~K whenever ~ti<&,

Isl<g, (x,y)~N(xo,yo). From the form of ~ we infer that this implies

~(x,y,t,s)~K for Iti<6g, Isl~l, (x,y)EN(xo,yo). By proposition 3.4,

~R~(x,sy,t) is continuous in s for O~s~l for any fixed t with Itl<~E

and any fixed (x,y)~N(x~,Yo~... Applying the fundamental lemma we get

~R~(x,y,t)-~R~(x,O,t)=@R~(x,y,t)~K for all (x,y)EN(xo,yo~,, and Itl~E.

Since F=CG-LC(F) we see that GR~ is continuous at all points (x,y,O).-

Define Nyo=~y I (Xo,Y)~N(xo,Yo) ~ and Z=~ty I y~Ny~ and Itl<~E ]. Hence Z is a zero neighborhood in E and for any z~Z we have x +z~U and

0

590

@R~(Xo,Z,l)=~(Xo+Z)-~(Xo)-d~(Xo,Z)~K. This proves that

R~:UnE~(x,y) I x~U and x+y~U~--~F, defined by R~(x,y)=~(x+y)-~(x)-d~(x,y),

is continuous at all points (x,O). Since d~:UnE~F is continuous, we

obtain the continuity of ~:E~U-~VcF which in turn clearly implies the

continuity of 6R~ at all points (x,y,t) with t@O, x~U, x+ty~U.

Thus we have proved:

Theorem 3.7. If a map ~:E~U-~VcF is differentiable, then ~ is continuous.

Theorem 3.8. For a map ~:E~U--*VcF are equivalent:

(i) ~ is differentiable

(ii)there exists a continuous map ~:UnE~F which is linear in the

second variable, such that the map ~R~:E~E~(x,y,t)Ix~U ~ x§ ,

by 8R~(x,y,t)=~(x+ty)-~(x)I-~(x,y ) for t~O and 8Rr defined

is continuous.

Since we work in a cartesian closed setting for continuous maps,

it is clear that the continuity of the maps D~:U-->L(EtF), d~:UNE-~F,

T~:UnE--~VnF imply each other for any differentiable function ~:E~U--~VcF.

Thus, so-called "weak" and "strong" differentiability are the same

notions in our context. We shall make extensive use of this fact.

Next we prove compositions of differentiable maps to be differen-

tiable.

Proposition 3.g. Let ~:E~U~VcF and ~:F~V--~W~G be differentiable.

Then the composite map ~o~:E~U-*W~G is differentiable.

Proof. Consider the map 8R(~):E~E~o~(x,y,t) I x~U and x+ty~UI-->G , 1 defined by 8R(~.~)(x,y,t)=~(x+ty)-~o~(x)I-d~(~x,d~(x,y)) for t~O

and 8R(~o~)(x,y,O)=O. Then we have:

~R(l~o~)ix,y,t)=SR~(~x,~R~(x,y,t)~d~(x,y),t)+d~(~x,~R~(x,y,t)). By

theorem 3.8 this map 8R(~o~) is continuous. Oifferentiability of ~

follows since d~(~x,d~(x,y)) is linear in y.

Theorem 3.10 (The chain rules). Let ~:E~U-~VcF end ~:F~V-~W~G be dif-

ferentiable. Then the following formulae hold for the composite map:

( i ) D ( Fo~)=oomp ~{D~, D#o~ : E=U--->L(E,F )n L(F, G ) c~ L(E,G) ( i i ) d(Fo~)=dF,(~d~)o(&U~IE):EnE~UnE~U~UnE.--~WF--->G ( i i i )T (~ ,~)=T#oT~.

This is clear. The fact that only the tangent of differentiable

mappings behaves functorially is the main reason why this explicit

form of differentiation is in most cases more useful than the deri-

vative or the differential. Observe that T(1u)=lun E for the identity

map 1u:E~U-->U~E. We put TU=UnE and call it the tangent space of U~E.

In passing we note that constant maps, translations, continuous

linear and multilinear maps obviously are differentiable. Moreover~

pointwise sums and pointwise scalar multiplication with respect to

differentiable maps give differentiable maps whenever defined.

591

Theorem 3.11. Let ~:E~U--~VcF be differentiable. If U is connected in

E, the following statements are equivalent:

(i) ~ is a constant map

( i i ) o~=o

Proof. (i)==>(ii) is trivial. (ii)::>(i): Let XoeU. Then S:IxeOl~x:~Xol

is clearly closed in U. For any xeS choose a radial open neighborhood

~BR~(x,sy,t)=O for any yeR(x)-x, sel, teI, we see that R(x)cU. Since

~R~(x,y,t)eK for every t~[O,1] and any closed convex zero neighborhood

K in LC(F). Hence ~R~(x,y,1)=~(x+y)-~(x)cK for all closed convex zero

neighborhoods K in LC(F). Hence ~(x+y)=~(x), whence the set S is open

in U. The theorem follows.

Theorem 3.12 (Differentiable maps into a product). Let ~:EmU--~Vc ~IF L

be a map into a product space. Then the following statements are equi-

valent:

(i) ~ is differentiable

(ii) for every L~I the map pr o~:E=U--~V =pr~(V)cF is differentiable.

This is clear. To prove an analogous statement for maps from a

product, we have to restrict ourselves to the case of finite products.

Definition 3.13. Let ~:ElnE2mU--~V~F be a function, where U is open in

the convenient product vector space ElnE 2 and V open in the convenient

vector space F. We say that ~ is partially differentiable with respect

to the first variable, if there exists a continuous map ~I:U-*L(E1,F)

such that for any fixed (x,Yl)~UnE 1 we have:

lira ~(x+t(Yl,O))-~(x)~=~lX(Yl).~ ~ , - In an analogous way one defines ~ ~t--~o partial differentiability with respect to the second variable and ex-

tends the definition to the case of any finite number of convenient

factors E i for i>2. - Finally, we say that an ~:i~IEimU--~V~F is par-

tially differsntiable, if it is partially differentiable with respect

to all variables i with 143gn.

Proposition 3.14. If ~:i@IEi~U-~V~F is partially differentiable with

respect to the j-th variable, the corresponding ~j:U-~L(Ej,F) is

unique.

This is clear. Hence we define:

Definition 3.15. Let ~:i~IEi=U~V~F= be partially diffarentiable with

respect to the j-th variable. Then we define:

(i) the j-th partial derivative D.~:.~3~E.mU--~L(E.,F) by j 1:I i j 1 l~(x+t (O,. . . ,O,y j , 0, . . , , O))_~(x) 1 Dj~x(yj)= l i m

O~t ~0 . . R (ii) the j-th partial d~fferent~al d.~:( ~-~E )mE ~UnE.--*F by J i=~ ~ j J

dj~(x,yj):Dj~x(yj)

.~UnE.--~VnF~FnF by (iii) the j-th partial tangent Tj~:(i~IEi)nE J - J Tj~(x,yj):(~(x),dj~(x,yj))

The limits are always calculated by fixed (x,yj)s

592

Theorem 3.16 (Differentiable maps from a finite product). Let

~:i~=IEi~U-~VcF be a map out of a finite product space. Then the fol-

lowing statements are equivalent:

(i) ~ is differentiable

(ii) ~ is partially differentiable.

Proof. The implication (i)=~(ii) is trivial. To see that (ii)~(i) it

clearly euffices to consider the case of a product ElmE 2 of two fac-

tors. For fixed (x,Yl,Y2)~Un(El~E2) and t~O we consider the function

BR~(• ~(x+t(yl,y 2))-~(x) 1 -dl~(X,Yl)-d2~(x,Y2)=

-~(x+t(y!,Y2))-~(x+t(O,Y2))l-dl~(x+t(O,Y2),Yl)+

~(x+t(O ,y2))-~(x)I-d2~(x,Y2) + dl~(x+t(O,Y2),Yl)-dl~(X,Yl)=

~Rl~(x+t(O,Y2),Yl,t)+~R2~(x,Y2,t)+dl~(x+t(O,Y2),Yl)-dl~(X,Yl). So we are left to prove that lim ~R~(x+t(O,Y2),Yl,t)=O. To see this, con-

sider the map ~Rl~(x+t(~,s~l,t ) for t#O and ~Rl~(x,sYl,O)=O. This

map is differentiable with respect to s, and we obtain:

d~Rl~(x+t(O,Y2),sYl,t)=dl~(x+t(O,Y2)+st(Yl,O),Yl)-dl~(x+t(O,Y2),Yl) ~

=~(x,Yl~Y2~t~s ). The map ~ is a continuous map with open domain and

satisfying ~(X,Yl,Y2,t,O)=O. Hence: For any closed convex zero neigh-

borhood K of LC(F) exist positive real numbers ~ and E such that

~(x,Yl�Y2,t,s)~K whenever Itl<~ and Isl<E. From the form of ~ we infer

that this implies ~(x,Yl,Y2,t,s)~K whenever Itl<Min~&,~E I and Isl~l.

Application of the fundamental lemma gives BRl~(x+t(O,Y2),Yl,t)~K for

It1<MinI~,~E I. Hence lim @Rl~(x+t(O,Y2),Yl,t)=O. o , t ~ o

As a corollary we obtain that our notion of differentiability co-

incides in the finite dimensional case with the usual notion of con-

tinuous differentiability.

Next we prove the decisive lemma which will allow us later the

forming of convenient function spaces of differentiable mappings.

Lemma 3.17 (Convergence theorem for sequences of differentiable maps).

Let E and F be convenient vector spaces, and let U be open in E. Fur-

ther suppose that (~n) is a sequence of differentiable maps ~n:E~U~F.

Then the following holds: If (~n) is a cauchy sequence in CO(U,LC F),

and if (d~n) is a cauchy sequence in CO(UnE,LC F), then the limit

functions lim(~n)=~:U-->F and lim(d~n)=~:UnE-~F exist and are contin-

uous. Moreover, ~ is differentiable and dm=~.

Proof. Since CGoLC(F)=F, we obtain from Kelley's theorem 1.7 the

existence and the continuity of ~ and ~. Clearly, ~ is linear in the

second variable. - If (x,y)~UmE is fixed, there exists an E>O such

that x+ty~U for all Itl<E. Hence GR~n(X,y,t):~=(-~,E)--~F is defined

and continuous for all n~i~ Because ~R~ n is continuous, we get con-

tinuity of ~n:~n~-E,E)mI--~F with ~n(t,s)=BRmn(X,sy,t). We have

d~n(t,s)=d~n(X+tsy,y)-~n(X,y ). Since the d~ n form a cauchy sequence,

593

we see that the sequence (~n)~CO((-E,~)nI,LC F) is also oauchy.

Therefore, if C is compact in (-~,~) and K is any closed convex zero

neighborhood of LC(F), there exists a natural number N such that for d d

all m>N and n>N we have (~-~m-~-~n)(Cnl)cK. Application of the funda-

mental lemma yields (BR~m-~R~n)(X,y,t)~K for all t~C. This proves that

(GR~n(X,y,-)) is a cauchy sequence in CO((-E,s F). By Kelley's

theorem there exists a continuous ~:(-E,E)~LC(F) with ~ =lim(~R~n(X,y,-)) ,

and since CG~LC(F)=F we may consider ~ as a continuous map ~:(-E,~)-~F.

any fixed t~O we have ~(t)=limBR~n(t)=lim.~ ~-~~n(X+ty)-~n(X)l-d~n(X'Y)= For 1 =~(x+ty)-~(x)I-~(x,y ) and obviously we have ~(0)=0. Hence it follows:

lim ~(x+ty)-~(x)l=~(x,y ) which proves differentiability of ~ with

differential d~=~.

Higher orders of differentiability are introduced inductively with

respect to the derivative operator D or the tangent operator T. So we

have for example in the case of a 2-times differentiable map

~:EmU~V~F the first derivative D~:U--~L(E,F) and differentiating this

map we obtain the second derivative 02~:U--~L(E@E,F). These derivatives

are related to the second tangent T2~:UnEnE~E--~VnFnFnF by the formula

T2~(x,Y,Zl,Z2)=(~x,D~x(y),D~X(Zl),D2~x(Y~Zl)+D~x(Y2)). It is clear

that the notion of higher order differentiability is the same whether

defined inductively from D or from T. We have already defined the

tangent space of an open subset U of a convenient vector space E as

TU=UnE. We now define the tangent space of order n of U inductively

by TnU=T(Tn-lu) for n>l. With this notation we have for the i-th tan-

gent of an n-times differentiable map ~:U~V that Ti~:TiU~TiV where

l~i~n.

Theorem 3.18 (Functoriality of the tangent operator). Let ~:E~U--~V~F

and ~:F=V-~W~G be n-times differentiable. Then the composite map

~:U-~W is n-times differentiable, and we obtain for the i-th tan-

gent Ti()o~):TiU~TiW that Ti(~)=Ti~Ti~ for 14i~n.

We note in passing that constant maps, translations~ continuous

linear and multilinear maps are n-times differentiable for arbitrary

n~

The most important theorem on higher derivatives is:

Theorem 3.1g (Symmetry of higher derivatives). If ~:E~U~V~F is

n-times differentiable, then for each x~U the i-th derivative i D ~x:| ~s totally symmetric for every i~(1,...,n).

Proof. For i=l there is nothing to prove. Assume i=2: Fix

(x,Yl,Y2)~UnEmE and consider - whenever defined - the map

~(t,s)=@R~(x,sYl+Y2,t)-~R~(x,sYl,t)-~R~(x,sY2+Yl,t)+~R~(x,sY2,t).

Clearly we can find a 6>0 such that ~(t~s) is defined for O~s~l and

Itl<~. Oifferentiating with respect to s gives us:

594

~(t,s)=tlD2~-x(Y28Yl-Yl~Y2)+[6RD~(x,sYl+Y~,t)-eRD&(x,sYl ,t) ] (Yl)+

-[GRD~(x,sY2+Yl,t)-ORD~(x, sY2, t) ] (Y2) 1 =t [D &x (Y2~Yl-YleY2) +~(t, s) } where ~ is continuous and satisfies q(O,s)=O. Since [O~l] is compact,

there exists for any given closed convex zero neighborhood K of LC(F)

an E>O such that ~(t,s)~K for O~<s~<l and It~<E. Let us assume that

D2~x(Y2~Yl-Yl| Then select a closed convex zero neighborhood K

D2~.x(Yl@Y2-Y2~Yl)r and choose E>O such that ~(t,s)~K for such that

O~<s~<l and Itl<E. Applying the fundamental lemma we get that

O=~(t,1)-~(t,O)~t[D2~x(Y2~Yl-Yl| I for Itl<E. Contradiction. -

By the inductive definition of higher differentiability it is now

clear that D1~x is also totally symmetric for i>2.

Finally we prove the existence of a primitive function for a given

continuous map ~:~,~U-~E. Here a map ~:IR~U-+E is called a primitive

map for e. if ~'=~.

Lemma 3.20. Let E be a convenient vector space and o~:~=[O,1]=I-->E a

continuous map. Then there exists a continuous map ~:~I-~E which is

differentiable on (O,1) with i~'=~ on the open interval (O,1).

Proof. We subdivide I into 2 n parts of equal length and define

~n'I-~E by ~n(t)=~(2"ni)+(2nt-i)I~(2-~i+l))-~(2-ni)I for i~<2ntx<i+l 2 n-1 } and i~O,l,..., . Further we define ~n'l-~E by

(t)=2-1(t-2-ni)I2~(2-ni)+(2nt-i){&(2-~i+l))-~(2-ni)}} + n . L-i +2 -n-~ ~- I~(2-nj)+~(2-n(j+l~)l for i~<2nt~<i+l and i~O,1,...,2n-1}.

Obviously ~n is differentiable on (O,1) with ~n=~n . Moreover ~n(O)=O.

Let us consider now the sequences (~n) and (~n) in CO(I,LC E). Since

I is compact and ~ is continuous, we can find a natural number N to

any closed convex zero neighborhood K of LC(E) such that ~(t)-~(t')~K

whenever It-t'l~<2 -N. Hence ~n(t)-~(t)~IK+K whenever n>N. Consequently

the sequence (&n) is a cauchy sequence in CO(I,LC E) convergent to ~.

Now consider the maps ~m,n(t,s)=i~m(tS)-i~n(tS ). These continuous maps

satisfy ~m,n(t,O)=O and are differentiable for any fixed t with re-

d ,s)=tI~m(tS)-~n(tS)I Since the (~n) form spect to s. We have ~s~m,n(t

a cauchy sequence there exists for any closed convex zero neighborhood

K a natural number N such that (~m-~n)(1)mK whenever m>N and n>N. Then

we obtain d~ m n(t,s)s whence application of the fundamental lemma

yields ~Om,n(t,1)=(~m-~n)(t)~tK whenever m~N and n>N. From this we con-

clude that the sequence (~n) is a cauchy sequence in CO(I,LC E).

Lemma 3.17 now tells us that the continuous map ~=lim(~n):I-~E is dif-

ferentiable on (O,1) with ~'=~.

Theorem 3.31 (Existence of primitive maps). Let E be a convenient

vector space, let U be open in ~, and let ~:~=U-~E be a continuous

map. Then there exists a primitive map ~:~=U-~E for ~. If U is con-

nected (t.i. an open interval), the difference of any two primitive

maps for ~ is constant.

595

Proof. The existence of a primitive map follows directly from

lemma 3.30. The difference of any two such maps in the connected case

must be constant by theorem 3.11.

4. Convenient Function Spaces for Differentiable and Smooth Maps

Let E and F be convenient vector spaces and let U be open in E. If

and ~ are n-times differentiable maps from U to F 9 and if r is any

real number, the pointwise sum ~+~ and the pointwise scalar product

r~ are obviously again n-times differentiable maps from U to F and

we have Dn(r~+~)=rDn~+Dn~ and Tn(r~+~)=rTn~+Tn~. Hence the n-times

differentiable maps from U to F form a vector space denoted by

Dn(U,F). We extend this notion to the case n=O by defining D~ as

the vector space of continuous maps from U to Fo Since differentia-

bility implies continuity, and higher differentiability is defined

inductively, we have linear inclusions i :Dn(U,F)~Dm(U,F) whenever n,m

O~m~n. Hence the intersection ~ Dn(U,F) is a vector space, denoted n=o

by D ~ (UtF) and called the vector space of smooth maps from U to F.

Clearly D ~ (U,F)=lim Dn(U,F) in the category of real vector spaces.

Hence we have for every n~IN linear inclusions in:D ~ (U,F)~-,Dn(U,F)

satisfying the universal property associated with a limit.

We shall now turn the vector spaces Dn(U,F) and D ~ (U,F) into con-

venient vector spaces as follows: For n=O we provide D~ with the

convenient structure given by the functor C:~~

described at the end of section 2. The resulting convenient vector

space will be henceforth denoted by C~ For n=l we consider the

linear injective function T~:DI(u,F)~-*C~ defined by ~-~T~.

Now we induce on Dl(u,F) the initial compactly generated topology

with respect to T#. This topology turns Dl(u,F) into a compactly

generated vector space. Let us prove that this is even a convenient

vector space: We know that C~ is a convenient vector space. If

(~n)eDl(u,F) is a sequence such that the sequence (T~n)~CO(TU,TF) is

cauchy, we have by lemma 3.17 that the sequence (~n) converges to a

differentiable map ~ with T~=lim(Tmn)~CO(TU,TF ). It follows that

Dl(u,F) with the initial compactly generated topology induced by T~

is a convenient vector space. This vector space will be denoted by

cl(u,F). Now we proceed inductively by setting cn(U,F)=cl(cn-l(u,F))

for n>l. Since the linear inclusion il,o:CI(u,F)~-~C~ ) is contin-

uous, we see that all linear inclusions i :cn(u,F)~-,cm(u,F) are n,m

continuous whenever O~m~n.

Consequently~ since ~ is complete, the limit for the diagram

~in,m:Cn(U,F)~+cm(u,F) I n,mel~and O~m~n I exists in ~, and we may

and shall select D(U,F) equipped with the corresponding compactly

596

generated limit topology to define the convenient vector space S(U~F)

of smooth maps from U to F. We observe that by construction the linear

inclusions in:S(U,F)~-~cn(u,F ) are continuous. Since we may identify

S(U,F) as a subspaoe of the product space ~ cn(U,F) in ~, we get n=o

from theorem 3.12 that a map ~:G~W-~S(U~F) is differentiable iff the

maps ino~:W-->cn(u~F ) are differentiable for all n~q. Since theorem

3.12 clearly extends to n-times differentiable maps into a product~

the construction of S(U,F) shows as well that a map ~:G=W--~S(UtF) is

smooth iff the maps ino~:W--~cn(u,F ) are smooth for all n~l~

We have thus proved the theorem:

Theorem 4.1. Let E t F 9 G be convenient vector spaces~ let U be open

in E and W open in G. Then we have for each n~IN a convenient vector

space cn(U~F) with elements the n-times differentiable maps from U to

F such that

(i) C~ in the notation of section 2

(ii) cn(U~F) has the initial compactly generated topology induced by

n:cn(u,F)~-~C~ which is the continuous linear injective map T~

defined by T~(~)=Tn~.

Hence for all O~j~n the linear injections T~:cn(U,F)~cn-J(TJu,TJF)--

are continuous and cn(U,F) has the initial compactly generated topo-

logy with respect to all of these maps.

Further the linear inclusions i :cn(u,F)~-~cm(u,F) are continuous n~m

whenever O~m~ n.

Since the functor C(U,-): ~cont--~ ~1~cont is compatible with arbitrary

products and by the validity of theorem 3.12 the obvious linear map

L:cn(u, n F )~--~cn(u,F ) is a homeomorphism. t~ L n L~

A map ~:G=W-~C (U~F) is j-times differentiable iff the map

no~:G=W--~C~ is j-times differentiable. T~

The convenient vector space S(U~F) of smooth maps from U to F is the

intersection of the convenient vector spaces cn(U~F) for all n~i~ and

is equipped with the corresponding compactly generated limit topology.

Therefore we have continuous linear inclusions in:S(U,F)~-~cn(u,F ) for n S(U,F)~_~S(TnU,TnF). every n~N and continuous linear injective maps T~:

A map ~:G=W--~S(U,F) is n-times differentiable (or smooth) iff for

every n El~ the map inO~=~:G~W~Cn(U,F ) is n-times differentiable (or

smooth).

Finally we have for any product ~ F. the obvious linear homeomorphism

F, ) ~-~S (U, FL ). B:S(U~ L~Z End of theorem 4.1.

We note especially that constant maps, translations~ continuous

linear and multilinear maps are smooth.

597

Lemma 4.2. Let E, F, G be convenient vector spaces, let U be open in

E, and let ~:F~G be an n-times differentiable map where n~l. Then

induces a differentiable map ~.:cn(u,F)~Cn-I(u,G) defined by ~.(~)=~~

Proof. First we show continuity of ~.:This is immediate from the

commutativity of the diagram

cn(U,F) T~ ~cO(TnU,TnF)

I ~ * n ~T~ 13)* ~. cn(U,G) T~ > C~ G)

cn-l( Next we define T~.:TCn(U,F)-~TCn-I(u,G) by commutativity of the upper

side of the diagram

Tcn(u,F)=Cn(U,F)nCn(U,F) ~I ~- i

cn(j, TF) in, _l cn'I(u,TF

~ Tn-1 C~

T~. ~ Tcn_I~u,G)=Cn_I(u,G)nCn_I(u,G)

(Tp). ~ cn_I(~,TG)

T -i

(Tn, 5)* > C~

The lower side of this diagram clearly commutes. Hence T~. - so de-

fined - is continuous. The differential d~.:TCn(U,F)--~cn-I(u,G),

corresponding to T~., is then given by d~.(~,~)(x)=d~(~x,~x), whence

linear in the second variable. So we are left to prove that for fixed (~,~)eTCn(U,F) the map @R~.(~,~,-):~-~cn'I(u,G), defined by

1 ~R~.(~,~,t)=~I~.(~+t~)-I~.(~)l-dI~.(~,~ ) for t~O and @R~.(~,~,O)=O is

continuous for t=O. This is by definition of cn'I(u,G) equivalent to

continuity of ~n-lo@R~.(~,~9- ) and this is by cartesian closedness of

C ~ with respect to continuous maps equivalent to continuity of the associated explicit map ~:Tn-lun~--~Tn-lG with

~,t)=[T~-loBR~.(~,~,-)](~,t). Since T n-1 defines and is defined by

all the D l for O~i~n-1, we have to show that for fixed tG~ the map

~R~.(~,~,t):E~U--~G is (n-1)-times differentiable and the maps

DZ@R~.(~,~,t):U-~L(&E,G) are continuous in (x,t) with Di~R~.(~,~,O)=O

for Or An easy computation gives us Di~R~.('~,~,t)(x)(xl| |

=@RDi~(~x,~x,t)(d[~+t~](X,Xl)~... | i+l +tD ~(~x) (d[~+t~] (x, x.)| ,,, | ~ i 1 • [~+t ~] (x ' xi-1)| xi)~(x) ) +

K~= ~RD - i3(~x,~x,t)(d[~+t~](X,Xl)~...ed[~+t~](X,Xk_l)~d2[~+t~](X,Xk~Xi)|

~d[~+t~] (X,Xk+l)| (x,xi))=

=~RDi~(~x,~x,t)(d[~+t~](X,Xl)e.,. ed[~+t~](x,x i))+Fi((x,t),xl,,,,,xi). For i=O we have ~R~.(~,~,t)(x)=BR~(~x,~x,t)~ Hence Fo=O. By induction

we see from our formula that ~RI~.(~,~,t):U--~G is (n-1)-times differen-

598

tiable. By another induction we see that the F. are (n-l-i)-times l

differentiable, are continuous in ((x,t),xl,...,xi) and satisfy

Fi((x,O),Xl,...,xi)=O. This proves the assertion. Corollary 4.3. Let E, F, G be convenient vector spaces, let U be open

in E and ~:F-~G an n-times differentiable map. Then ~ induces i-times

differentiable maps B.:cn(u,F)-~cn-i(U,G) for all i~(O,1,...,n) which

are defined by ~.(~)=~o~.

Proof. For i=O this is clear by commutativity of the diagram

cn(U,F) T~ ~ co(TnU,TnF)

1 ~* n I (Tn~)* cn(U,G) T# ~ co(TnU,TnG)

For i=l this has been proved in lemma 4.2. Suppose the corollary true

for j-1 with l~j-l<n and let ~:F-~G be j-times differentiable. Then

~.:cn(u,F)~cn-j+I(u,G) is (j-1)-times differentiable by hypothesis,

whence ~.=in.j+I,n_j~.:cn(u,F)-~cn-J(u,G) is (j-1)-times differen-

tiable. The commutative diagram c ~ i

TJ-lcn(u,F)> ~ bcn(u,TJ-1F) n~ > cn-j+I(u,TJ-IF)

~j-l~. ~ $(TJ-I~). !$~!-~).

TJ-l~. TJ-Icn-j+I(u,G)~I~cn-j+I(u,TJ-IG) in-j+l,n-J~cn-J(u 1 )

Wi . . .~ $(TJ-I~). �9 - ~ n-j+i.n-3 . . w

TJ-• ~ Bcn-J(U,TJ-IG)

shows TJ-l~. differentiable, since by lemma 4.2 the far right map (TJ-I~).:cn'j+I(u,TJ-1F)->cn-J(u,TJ-IG) is differentiable.

Lemma 4.4. Let E, F, G be convenient vector spaces, let U be open in

E and V open in F, and let ~:E~U-~VcF be an n-times differentiable

map. Then ~ induces a continuous linear map ~*:cn(V,G)--~cn(u,G), de-

fined by ~*(~)=~o~.

Proof. Clearly ~* is linear. The continuity follows from the commuta-

tive diagram

n C ~ cn(V,G) T~ ~ (TnV,TnG)

cn(U,G) T~ ~ co(TnU,TnG)

Lemma 4.5. Let E, F be convenient vector spaces and let U be open in

E. If n~l, the evaluation map 6:cn(u,F)nU~F, defined by E(~,x)=~x,

is differentiable.

Proof. Continuity of evaluation is clear from the commutative diagram i omlu

cn(U,F)oU n, ~ C~

F F

and cartesian closedness with respect to C ~ . Next we define maps

599

l i s o : T ( C n ( U , F ) n U ) - - > T C n ( U , F ) m T U by l i s o ( ~ , x , ~ , y ) = ( ( ~ , ~ ) , ( x , y ) ) ,

p r j : T C n ( U , F ) - ~ c n ( u , F ) by P r l ( ~ , ~ ) = ~ and P r 2 ( ~ , ~ ) = ~ , ~.:TF~TFj by ~l(y,z)=(ytz ) and ~2(ytz)=(O~y ). Obviously all these

maps are continuous linear maps. From these maps we obtain by compo-

sition for j=l,2 the maps ~j=~oEo(T~nlTu)o(pr mlT.)olison 1 J .u from T(cn(u,F)nU) to TF. Since we know that s - (TU,TF)nTU-~TF is con-

tinuous, we deduce that ~1+~2 is continuous. We shall prove that

T~=~l+~2: To see this, we observe that the corresponding differential

ds satisfies ds whence d& is linear in the

second variable. For fixed ((~,x),(~,y))~T(Cn(U,F)nU) and t~O we have

~R~((~,x),(~,y),t):~IE(~+t~,x+ty)-E(~,x)~-dE((~,x),(~,y))=

=s Hence lira eRE((~,x),(~,y),t)=O. O~t-->O

C o r o l l a r y 4 . 6 . Let E, F be c o n v e n i e n t v e c t o r spaces and l e t U be open i n E. Then the e v a l u a t i o n map s i s n - t i m e s d i f f e r e n t i a b l e . P r o o f . For n=O we have c o n t i n u i t y o f & by c a r t e s i a n c l o s e d n e s s w i t h r e s p e c t to c o n t i n u o u s maps. For n= l we have d i f f e r e n t i a b i l i t y by lemma 4 . 5 . Suppose the c o r o l l a r y t r u e f o r n = k ~ l . Then we have f o r n=k+ l d i f - f e r e n t i a b i l i t y by lemma 4 .5 w i t h d e r i v a t i v e TE=~I+~2. The d e f i n i t i o n of the ~ j i n v o l v e d - a p a r t f rom E:ck(Tu,TF)mTU-->TF - o n l y c o n t i n u o u s l i n e a r maps. By h y p o t h e s i s the e v a l u a t i o n map E:ck(TU,TF)mTU-~TF i s k - t i m e s d i f f e r e n t i a b l e . Hence T& i s k - t i m e s d i f f e r e n t i a b l e . Lemma 4 . 7 . Let E, F be c o n v e n i e n t v e c t o r spaces , l e t U be open i n E and V open i n F. Then the i n s e r t i o n map ~ :E=U~Cn(F~V ,EmF) , d e f i n e d by ~ x : y ~ ( x , y ) , i s i - t i m e s d i f f e r e n t i a b l e f o r eve r y iEIN. P r o o f . We have ~=~+X where ~ :U- ->cn(v ,EnF) i s d e f i n e d by ~ x : y ~ - ~ ( O , y ) , and where X :U-~Cn(V ,EnF) i s d e f i n e d by X x : y ~ - ~ ( x , O ) . Hence ~ i s the sum of the c o n s t a n t map ~ and the l i n e a r map X. We are l e f t to p rove the c o n t i n u i t y o f X wh ich i s e q u i v a l e n t to the c o n t i n u i t y of

n oX(x) ( y l , . . . , y ) = ( ( x , O ) ( 0 , 0 ) ( 0 , 0 ) ) T~X:U-->C~ Since T~ 2 n , ,..o,

this is the case.

Definition 4.8. We denote by ~mooth the category with objects the

convenient vector spaces and arrows the smooth maps.

We have the following fundamental theorem:

Theorem 4.9 (The fundamental theorem for smooth maps). The category

~smooth contains the category ~ and is contained in the category

~%~cont" All these categories have the same objects and the same (ar-

bitrary) products. The category ~ is cartesian closed with smoo~n ~op • ~ __~ S: ~ smooth ~ smooth ~mooth as internal functor. This funotor S is defined on objects as the limit of the diagram

~Cn(E,F) in, n-1 ) cn-l(E,F) I ns 1 in ~, where C ~ is the cartesian

internal functor for ~ont and Cn(E,F) has the initial compactly

generated topology induced by the linear injective map

600

Tn:cn(E,F)~C~ For arrows (t.i. smooth maps) the functor S

is defined by composition.

The unit for cartesian closedness is given by the smooth insertion

maps ~:E--~S(F,EnF) with ~x:y~-~(x,y). The counit for cartesian

closedness is given by the smooth evaluation maps &:S(E,F)mE~F with

E(~,x)=~x. A map ~:E~S(F,G) is smooth iff the corresponding map

&=Eo(mnl):EnF-~G, defined by &(x,y)=~x(y), is smooth. The natural

diffeomorphism ~:S(E,S(F,G))~>S(EnF,G) is linear.

The ground field lq is a generator and a cogenerater for ~mooth"

The tangent functor T:~mooth~mooth with T:E~-~TE=EnE and

T:m~T~ with T&(x,y)=(~x,d~(x,y)) is linear.

Finally the diagram

~op x "~ ~mooth ~smooth l~TLSmooth S ~ ~g ! ~q~smooth ~mooth S ~mooth

commutes up to a smooth natural isomorphism.

Proof. 1. Functoriality of S: Let ~:E-~F be smooth. Then the following

diagram commutes for all n~IW: i

S(F,G) n , c n ( r , G )

S(E,G) n , Cn(E,G)

From lemma 4.4 we know that ~*:Cn(F,G)-->Cn(E,G) is smooth for all nEIM.

Hence by the limit definition of S we see that m*:S(F,G)--~S(E,G) is

smooth. - Let ~:F +G be smooth. Then the following diagram commutes

for all nelN and k&Ikl:

S(E,F) in+k ~ cn+k(E,F)

S ( E ' ~ ) = F * 1 i n 1 ~* S(E,G) w Cn(E,G)

The map ~ . : c n + k ( E , F ) - - ~ c n ( E , G ) i s k - t i m e s d i f f e r e n t i a b l e by c o r o l l a r y 4 . 3 . Hence by the l i m i t d e f i n i t i o n of S we see t h a t G . : S ( E , F ) ~ S ( E j G ) i s k - t i m e s d i f f e r e n t i a b l e . S ince k i s a r b i t r a r y we see t h a t 5.:S(E,F)-->S(E,G) is smooth..

2. Smoothness of the unit and counit: The insertions 7:E--~Cn(F,EmF)

are smooth for all neN by lemma 4.7. Hence ~:E-->S(F,EnF) is smooth by

the limit definition of S. - The evaluations s are

n-times differentiable for every nelkL by corollary 4.6. Since the dia-

gram S(E,F)mE ~ ~ F

Cn(E,F)nE E_+ F

commutes, we see that E:S(E~F)nE-~F is smooth.

601

This proves that ~smooth is cartesian closed with S as internal

functor. All other statements are evident.

We close with the following important remark: If we define the

category open~mooth as the category of open subsets of convenient

vector spaces and smooth maps, we obtain the more general functor op ~

open ~ m --~ . We see as before that insertion S: ~s ooth • ~mooth ~ smooth" ~:E~U~S(F~V,EmF) and evaluation E:S(E~U,F)nU-~F are smooth maps,

that a map ~:E~U--~S(F~V,G) is smooth iff the corresponding map

~=O(~nl):EnF=UnV-~G is smooth, and that we have a smooth linear dif-

feomorphism ~:S(E~U,S(FmV,G))~-~>S(EnF~UnV,G).

5. Other Results and Problems Arisinq in this Context It is fairly

simple to establish the main theorems of differential forms in this

setting. The Poincar@ lemma and the Stoke theorem are easily seen to

hold. - The most troublesome question coming up concerns the inverse

function theorem. But this is unavoidable as we see from the follow-

ing example: Let exp:S(~,~)-~S(~,~) denote the exponential map defi-

ned by exp(~)(t)=e ~t). Clearly Dexp(O)=IS(~,~... Hence Dexp(O) is in- t ~

vertible, and the usual inverse function theorem would imply the in-

vertibility of exp in a neighborhood of the zero map. But because on

the one side exp(~):~ only takes positive values (whatever ~ : ~

one starts with), and since on the other side there exist smooth maps

also assuming negative values in every neighborhood of exp(O):~-->~,

we see that even the invertibility of exp near the zero map is impos-

sible. Therefore one has to impose additional conditions then are

necessary in the banach space case, in order to obtain an inverse

function theorem in our generalized setting. - Another interesting

point concerns differential equations in the new setting. By proving

suitable fixed point theorems for convenient vector spaces one ob-

tains the necessary foundation for existence and uniqueness theorems

for ordinary differential equations. These fixed point theorems

clearly involve bounded subsets. More details can be found in [lO].

Until now I have not established a Frobenius theorem for partial dif-

ferential equations.

Finally I shall touch upon some questions concerning the general

theory of differentiable manifolds. Instead of modelling differenti-

able manifolds on hilbert or banach spaces [8 ], we shall now consi-

der differentiable manifolds modelled on convenient vector spaces,

where atlases consist of differentiably compatible charts in the

sense of our generalized differential calculus. We shall then speak

of convenient differentiable manifolds. In the same spirit we define

convenient differentiable vector bundles. What one would like to

602

obtain is the cartesian closedness of the categorY~smooth of conven-

ient smooth manifolds and smooth maps. There are several obstructions:

The first consists in defining charts for the set S(M,N) of smooth

maps from one convenient smooth manifold to another. It is clear how-

ever, on what convenient vector spaces the "manifold" S(M,N) has to

be modelled: Near a smooth map ~:M~N the corresponding vector space

has to be the vector space S~(~N) of smooth sections over ~ in the

tangent bundle ~N:TN~N. More formally: S~(~N)=~:M--~TNI~N~=~, ~ smooth~

This fact was noticed by Eells [3]. The convenient vector space topo-

logy of S~(~N) is now obtained as follows: For every iEIN we have i S~(~N)>-~C(TiM,Ti+lN) and LCoC(TiM,Ti+IN) clearly is a sequentially T~:

complete hausdor~f convex vector space. Hence CGoLCoC(TiM,Ti+IN) is a

convenient vector space and we take for S~(~N) the convenient compact-

ly generated initial topology with respect to the linear injective map

~ - - -~Ti~:s~(~N)~-~i~oCGOLCoC(TiM,Ti+IN). So we know where the charts have

to be situated, but we lack the corresponding chart maps.

To obtain chart maps I recall the following construction, possible

for every smooth hilbert manifold N [ 8]: There exists the so-called

exponential map exp:TN~U~N, defined on a neighborhood U of the zero

section O:N~TN, which together with the projection ~N:TN-->N gives a

smooth diffeomorphism ~NIU, expI:U>~V~NxN with an open neighborhood

V of the diagonal in NxN. Applying now lemma II, 7.4 in [7] we can

select a smaller neighborhood U of the zero section in TN which is

fiberwise smoothly diffeomorphic to TN itself. These two results to-

gether imply in the case of smooth hilbert manifolds the existence

of a smooth map E:TN-~N satisfying the following two conditions:

(i) the composition EoO of ~ with the zero section is the identity

map on N, and (ii) the smooth map ~%N,~l:TN~NxN establishes a smooth

diffeomorphism between TN and its image in NxN where this image is an

open neighborhood of the diagonal in N•

In the general case we take now this situation as the definition

of what we call a smooth addition for a convenient smooth manifold.

Definition 5.1. Let N be a convenient smooth manifold. Then we say

that a smooth map ~N:TN~N is a smooth addition for N if the follow-

ing holds: (i) If ON:N--~TN is the zero section, then ~NOON=IN:N~Np

and (ii) the map ~%N,~NI:TN-->N• is a smooth diffeomorphism between

TN and its image in NxN with this image open in NxN.

Assume that N is a convenient smooth manifold with a smooth addi-

tion ~N:TN~N. Then we obtain for every x~N by restriction of E N to

~l(x) over x a chart Ex=ENl%-l(x):~-l(x)>---~Ux~N near x, the fiber

and these charts form a smooth atlas which defines the (original)

smooth structure for N.

603

Now we consider S(M,N) where N is a convenient smooth manifold

with a smooth addition EN:TN-~N and M is any convenient smooth mani-

fold. Then we obtain for every smooth ~S(MtN ) by composition with

~N a map ~:S~(~N)~S(M,N ) defined by E~(~)=Eo~. Clearly these maps

E~ are one-to-one for every ~, whence they are the natural candidates

for defining (induced) smooth charts for S(M~N). The only problem is

their smooth compatibility: In general they are only compatible if M

is a compact smooth manifold. But in this case the smooth structure

of S(M,N) does obviously not depend on the choice of the smooth addi-

tion ~N for N.

Denoting by comp~smooth the category of compact smooth manifolds

and smooth maps (which coincides with the category of compact conven-

ient smooth manifolds) and denoting by ad~smooth the category of

convenient smooth manifolds with a smooth additionp we obtain the

following theorem: op

Theorem 5.2. There exists a functor S:com~moothXadd~smooth-->ad~mooth

given on objects by S(M,N) with smooth charts as above, and defined

by composition for smooth maps. This functor has the property that

whenever M end N are compact smooth manifolds and P is a convenient

smooth manifold with a smooth addition~ then the smooth manifolds

S(M,S(N,P)) and S(MnN,P) are naturally diffeomorphic by the usual

correspondence of ~:M--~S(N,P) with ~:MnN--~P where &(x,y)=~x(y).

The proof of theorem 5.2 is straight forward, using of course the

generalisation of theorem 4.9 at the end of section 4 to show the

smoothness of induced maps. To obtain a smooth addition for S(M~N)

one establishes first a natural smooth diffeomorphism

~:TS(M,N)>~S(M,TN) analogous to the one in theorem 4.g. The smooth

addition EN:TN~N then induces the smooth map (~N).:S(M,TN)-~S(M,N)

and one gets in (EN).O~:TS(M,N)~S(M,N) the desired smooth addition

for S(M,N).

It rests to state that we did not exactly obtain what we wanted:

Namely a cartesian closed category of convenient smooth manifolds.

The compactness condition on M for forming S(M,N) as a convenient

smooth manifold should be removed. But for doing this 9 the topology

on the section spaces seems to be toocoarse. May be that there

exists another generalisation of differential calculus Which works

as good in the manifold case as the one developed here works in the

vector space case.

604

References

[7]

[B] [9]

[i0] U.Seip

[1] A.Bastiani: Applications diff@rentiables et vari@t@s diff6renti-

ables de dimension infinie, J~ Math.13 (1964),

1-114

[2] N.Bourbaki: Topologie g@n@rale, Hermann, Paris

[3] J.Eells : A setting for global analysis, Bull. Am. Math. Soc.72

(1966), 751-807

[4] A.FrBlicher-W.Bucher: Calculus in vector spaces without norm,

SLN 30, Springer, Berlin (1966)

[5] A.Fr~licher-H.Jarchow: Zur Dualit~tstheorie kompakt erzeugter

und lokal konvexer Vektorr~ume, Comm~ Helv.47

(1972), 289-31D [6] P.Gabriel-M.Zisman: Fondement de la topologie simpliciale,

S@minaire homotopique, Universit@ de Strasbourg

(1963/64)

M.Golubitsky-V.Guillemin: Stable mappings and their singularitiee~

GTM 14, Springer, New York-Heidelberg-Berlin (1973)

S.Lang : Differential manifolds, Addison-Wesley, Reading (1972)

H.H.Sch~fer:Topological vector spaces, GTM 3, Springer,

New York-Heidelberg-Berlin (1971)

: Kompakt erzeugte Vektorr~ume und Analysis, SLN 273~

Springer, Berlin-Heidelberg-New York (1972)

Ulrich Seip

Fachbareich Mathematik

Universit~t Konstanz

Postfach 7733

D 775 KONSTANZ

Deutschland

Ulrich Seip

Instituto de Matem~tica e Estat~stica

Universidade de S~o Paulo

Cx. Postal 20.570 (Ago Iguatemi)

BR 01451 SAO PAULO

Brasil

Abstract:

PERFECT SOURCES

by

G. E. Strecker

A survey is given of various approaches to suitable categorical

analogues of perfect maps. The notions of ~-perfect source and

-strongly perfect source are defined, and are shown to be ideal ly

suited to fac tor iza t ion theory and theorems demonstrating the existence

and construction of ep i re f lec t i ve hulls in a quite general set t ing. A

characterizat ion is given of when the two types of perfectness coincide,

and suggestions for fur ther study are provided.

51. Introduction

I t is not surprising that soon af ter f inding appropriate categorical

analogues for the important topological en t i t i es of homeomorphisms,

in jec t i ve and sur ject ive mappings, embeddings, closed embeddings, dense

maps, and quotient maps, topologists would focus at tent ion on the problem

of obtaining a suitable analogue for the important class of perfect

mappings.

The signi f icance of these mappings stems from the fact that even

though they need be neither in jec t i ve nor sur jec t ive , they come close to

being both structure preserving as well as structure re f lec t ing , and are very

closely related to the important notion of compactness. Consequently, as

a class, perfect maps have qui te nice character is t ics and many topological

properties are preserved or inversely preserved by them.

A topological map f : X--~ Y is said to be perfect i f and only i f

i t is continuous, closed, and has compact point- inverses. Bourbaki [B~ has

shown that th is is equivalent to:

606

(*) fo r each space Z, f x i z : X x Z--+ Y x Z is closed.

In [HI] Henriksen and Isbel l showed that for a mapping f : X--+ Y

between completely regular spaces:

(**) f is perfect i f and only i f ~f[~X \ X] C ~Y \ Y ,

Each of the above characterizations is subject to a categorical

general izat ion. For the f i r s t , however, one needs the existence of f i n i t e

products and, more importantly, the notion of "closed mapping" and so

a f o r t i o r i the notion of "closed sets" in objects of the category in

question. Such a general ization has been obtained by Manes IMp. The

assumption is made that one is dealing with a category of structured sets

which reasonably creates f i n i t e products and for which each object is

assigned a family of "closed" subsets of i t s underlying set that behave

appropriately (e.g. they are f i n i t e l y in tersect ive and each morphism

inversely preserves them). He defines a perfect map as the obvious

analogue of ( * ) , cal ls an object compact i f f the terminal map is perfect

and cal ls i t Hausdorff i f f i t s diagonal is closed. Some quite general

proofs of standard topological resul ts are obtained; e.g.

(1) perfect maps form a subcategory.

(2) an object is compact i f f each project ion paral lel to i t

is closed.

(3) compactness i f f i n i t e l y productive, closed hereditary,

preserved by sur ject ive maps and inversely preserved

by perfect maps.

(4) an object is Hausdorff i f f the graph of any map with

i t as codomain is closed.

(5) any map with compact domain and Hausdorff codomain is

perfect.

607

The obvious analogue of (**) would be in the sett ing of a category

~ having a ep i re f lec t ive subcategory, ~ ( a n d maps given fu l l re f lect ion

rA:A + rA).

A morphism f:A § B is called ~ - strongly perfect i f f

r A A ~ rA

f [ i r f

B- �9 rB r B

is a pullback square.

Franklin [F3~. , Hager [H1], Tsai IT], and B~'aszczyk and Mioduszewski

IBM] have each used the above idea to generalize or obtain analogues of

perfect mappings in various rest r ic ted sett ings. In par t icu lar , Franklin

and Hager deal only with the categories of Tychonoff spaces and uniform

spaces and the i r re f lect ion maps are assumed to be embeddings, Tsai only

considers subcategories of Hausdorff spaces, and Blaszczyk and Mioduszewski's

sett ing is within the category of Hausdorff spaces and mappings which can

be extended to the Kat~tov H-closed extensions [KI] of the i r domains and

codomains. Whereas in IT] and IBM] the main thrust is to obtain topological

( in ternal ) characterizations of the ~ - s t r ong l y perfect morphisms and the i r

relat ionship to the question of ex tend ib i l i t y , in IF3] and [H1] some easi ly

abstracted, quite general proofs of character ist ics of ~-~-strongly perfect

maps are obtained; namely i t is shown that ~ - s t r o n g l y perfect maps:

(1) form a subcategory

(2) are a r b i t r a r i l y productive

(3) are closed under the formation of pullbacks;

e.g. , projections paral le l to ~-- factors.

In th is sett ing Franklin and Hager have also obtained generalizations

of Frankl in 's

608

IF2] and Herr l ich and van der S lo t ' s [HS~ theorems deal ing wi th l e f t -

f i t t i n g hu l l s of various topological proper t ies. In th i s connection we

should also mention a very nice genera l iza t ion of F rank l in ' s r esu l t due to

Nel [N~. Also see [HS 3, 37C].

Since the character izat ions (*) nad (**) are the same for maps between

completely regular spaces i t seems natural to inqu i re as to the d i s p a r i t y

of t h e i r genera l iza t ions. Herr l ich [H~ has shown that they are d i f f e r e n t

even in the category { of Hausdorff spaces, where ~ is the subcategory

of compact spaces. Namely, there are closed embeddings that are not

s t rong ly perfect. In [H2] Her r l i ch has suggested and la te r ([H37) more

f u l l y analyzed a t h i r d general approach to perfect maps that agrees with

the analogues to (*) and (**) for completely regular spaces and l i es

s t r i c t l y between them when applied to Hausdorff spaces. Nakagawa [N~

has also invest igated an approach that for su i tab ly nice categories y ie lds

the same classes of perfect maps. Her r l i ch ' s approach, which is c lose ly

re lated to the r i g h t b icategor ica l s t ructures of Kennison

described as fo l lows:

For any class of objects w ~ , ca l l

m-extendible epimorphsim i f f e

and f :A § K, there ex is ts some

is cal led m-perfect i f f whenever

[K~ , can be

in a category e:A + B an

is an epimorphism and whenever K E m

h such that f = hoe . A morphism f

rl Is

609

is a commutative square wi th e an m-extendible epimorphism, there

ex is ts some d such that r = d ~e and s = f ~d ,

In [H2] the classes of epimorphisms that are the m-extendible

epimorphisms fo r some m are character ized, the m-perfect morphisms

are shown to be:

(1) closed under composition,

(2) closed under (mul t ip le ) pul lbacks,

(3) closed under products,

(4) l e f t cance l la t i ve , and

(5) a superclass of the class of a l l strong monomorphisms.

The re la t i onsh ip between the m-perfect morphisms~ e p i r e f l e c t i v e h u l l s , and

fac to r i za t ions is invest igated and a character izat ion for those classes of

morphisms that are the m-perfect morphisms for some m is cal led for .

Suitable such character izat ions of perfect morphisms have since been

given in [S~] and [$3]. In these papers and in [H3], [N2~, and IN3]

numerous re lated resu l t s and refinements and improvements in the theory

have been obtained and many examples given.

In the next sect ion we shal l make some fu r the r refinements, a major

one being to extend the theory to perfect sources, ra ther than perfect

morphisms. I t has been brought to our a t ten t ion that an approach to source d iagonal izat ions and fac to r i za t ions s im i la r to that given below w i l l appear in [P].

w Def in i t i ons

We w i l l assume throughout tha t we have a given category ~ and that

a l l e p i r e f l e c t i v e subcategories are both f u l l and isomorphism-closed.

A source is a pa i r (A,F) where A is a ~ -ob jec t and F is a class

610

of morphisms each wi th domain A . Such a source w i l l sometimes simply be

denoted by F . I f (A,F) is a source and fo r each f : A § Bf in

F) (Bf, Gf) is a source, then (Gf) oF w i l l denote the source (A,H)

where H = {g ~ f l f ~ F and g ~ Gf} In the special case where (A,F)

is a source and g : B § A is a morphism, F : g is the source

(B, ( f ~ g ) f ~ F ) .

I f B is a fami l y of sources, then B 1 denotes the fami l y of a l l

s ing le ton sources in B �9 Each such source (B, { f } ) w i l l be i d e n t i f i e d w i th i t s s ing le morphism f .

A fami l y of sources, B , is ca l led basic provided tha t ;

(B1) B1 contains a l l isomorphisms in

(B2) B is closed under composi t ion; i . e . , whenever F

is a source in B and (Gf) is a fami ly of sources

in B such tha t (Gf) ~ F e x i s t s , (Gf) o F must be

in B .

(B3) B is closed under the format ion of j o i n t pu l lbacks ;

i . e . , i f (L, K U { h } ) is the l i m i t of the diagram

L -" ~ (#f )

h~ ~ ( r f )

A r F > (Bf)

where (A,F) ~ ~ , then (L,K) c B .

(B4) B is closed under the format ion of m u l t i p l e pu l lbacks ;

i . e . , whenever (F i ) I i s a fami ly of sources belonging

to B wi th the proper ty tha t f o r each i , j ~ I the

611

class of a l l codomains of F. is the same as the 1

class of a l l codomains of Fj , then i f (L,K) is

t h e i r l i m i t , i t fo l lows that (F i ) oK belongs to 8,

A class of morphisms is ca l led cobasic provided that i t , considered

as a class of s ing le ton s inks, is a cobasic class o f s inks; i . e . , s a t i s f i e s

the cond i t ions dual to (BI ) - (B4) above,

Let a be any class of morphisms. A source F is ca l led a- lower

d iagona l izab le provided tha t whenever e and r are morphisms wi th

e ~ a and G is a source such tha t G~e = F~ r , there ex i s t s a

morphism d such that the diagram

e o > o

F

commutes. The class of a l l a- lower d iagona l i zab le sources w i l l be denoted

by A(a) S im i l a r l y i f 6 is a fami ly of sources, a morphism e is

ca l led 6-upper d iagona l izab le , and we w r i t e e E ~!P(6), i f f whenever r

is a morhpism and F and G are sources wi th F ~ 6 such that

G o e = F e r there ex i s t s some d such tha t the above diagram commutes.

I f a is a class o f epimorphisms, then a member of A(a) is ca l led an

a -pe r fec t source and a member of A(a) I is ca l led an ~ -pe r fec t morphism.

6 is ca l led a class o f per fec t sources i f 6 = A(a)

epimorphisms a.

I f ~ is any class of ob jec ts , then we say tha t f : A + B is

m-extendible provided tha t f o r each D ~ m and

fo r some class of

612

g ; A § D there exists some h : B § D such that g = h o f . The class of m-extendible epimorphisms wi l l be denoted by .x(m)

I f 6 is a family of sources, then A(6) w i l l denote the class

of al l objects A having the property that each source with domain A

belongs to 6 .

A source F is called a mono-source i f f whenever F ~ f = F ~ g i t

follows that f = g . The class of al l mono-sources that are also

epi-perfect is called the class of al l strong_ mono-sources (cf. [K3])

A family of sources is said to be:

le f t -cance l la t ive

iffwhenever F is a source and (Gf) is a family of

sources such that (Gf) ~F exists and belongs to ~ ,

then F must belong to ~ .

fundamental i f f 6 :

(F1) is basic,

(F2) contains al l strong mono-sources, and

(F3) is le f t -cance l la t ive .

closed under the formation of products

iffwhenever for each i ~ I , (Ai,F i ) is a source in

B for which the induced map KF i > from A i to

the product of the codomains of F i exists, and such that

(HAi,~ i) exists, then the source (~A i , (<F i > ) ~ (~ i ) ) belongs

to 6 .

For a given class, ~, of epimorphisms, ~ is said to be:

~-compatible

i f f eve ry ~-source in ~ has a cointersection, and each

two-element source in

613

having at l eas t one member in ~ has a pushout.

an ~-Eer fect category

provided tha t each source F in # has an e s s e n t i a l l y

unique f a c t o r i z a t i o n F = G ' e , w i th e ~ ~ and G

s -pe r fec t ,

I f %~ is an e p i r e f l e c t i v e subcategory of ~ w i th r e f l e c t i o n

morphisms r A ; A - - . rA , then a source (A,F) is ca l led an ~{~-strongly

per fec t s o u r c e i f f t h e source (A, F U { r A } ) is the l i m i t of the diagram

Bf rB

where fo r each f ~ F , f : A § Bf .

w Results

rA

I f )~rBf

Theorem 1

For any c lass of sources,

(1)

(2)

(3)

(4)

B, the fo l l ow ing are equ iva len t :

B is a c lass of per fec t sources.

= A ( ~ B N ~ ) , fo r some c lass of epimorphisms, ~.

B = A ( ~ B n e p i ) , where epi i s the c lass of a l l epimorphisms

of ~o

6 = A(~) fo r some cobasic fami l y of epimorphisms, a

614

Theorem 2

For any class m of morphisms:

( i ) m~A(m) 1 is contained in the class of a l l isomorphisms.

(2) A(m) is basic and closed under the formation of products.

Theorem 3

For any class m of epimorphisms;

(1) A(m) is fundamental, and A(m) 1 is closed under the formation

of inverse l i m i t s of inverse spectra; i . e . , i f each spectrum

map is in A(m) I , then each member of the inverse l i m i t source

w i l l be in A(m)1

(2) i f ~ is s -per fec t , then every class ~ of objects in

has an s - r e f l e c t i v e h u l l , Ra(~ ). Furthermore, for each

object A the Rm(m)-reflection morphism is the f i r s t factor

of the (m, A(m)) f ac to r i za t i on of the source

cons is t ing of a l l morphisms with domain A and codomain

belonging to ~.

(3) i f ~ is s-compatible, then the fo l lowing are equivalent :

(a) m is cobasic,

(b) ~ is u-per fect and a is closed under composition with isomorphisms,

( c ) a =~A(~);

i f , in add i t ion , m has the property that whenever f o g ~ m

and g is an epimorphism, i t fo l lows that g E m , then

(a), (b) and (c) are equivalent to:

(d) m = x(w) for some class of objects

(e) ~ = x A A(m).

(4) is any class of objects for which (x(~) / -~a) i f m is cobasic

and ~ is (• m)-compatible, then ~ has an u - r e f l e c t i v e hu l l

given by: Rm(~ ) = AA(X(~)/')m) . in

615

Thus under these conditions m = AA(x(m)O~) i f and

only i f m is the object class of a ful l ~-reflective

subcategory of ~ ,

Theorem 4

and

be any class of objects, ~ be any class of epimorphisms, Let

be s-ref lect ive in ~ Then

(1) each ~(~-strongly perfect source is (x(l~)/~ ~)-perfect.

(2) the following are equivalent:

(a) The ~(P -strongly perfect sources and the (x(~)(~ ~)-perfect

sources coincide and ~ is an (x(~)/') ~)-perfect

category.

(b) For each source (A, ( f i ) ) in ~ and each diagram

A. rA ~rA

( f i I ~ ( r f i )

( i ) > (rB i) (rB.)

l

the lower corners have a l im i t (L, (k i ) , k A)

induced morphism, h, belongs to ~ .

A

( f i )

r A

~k L

and the

> rA

( ) ~ (rB i ) (rB.)

1

~4 Consequences and Applications

Since singleton perfect sources are perfect morphisms, Theorems 1,

616

2, and 3 c lear ly generalize many of the main results of [H~, [H~,

ENd, Es2] and [S~. Note that we have defined perfectness re la t i ve to

classes of epimorphisms, rather than to classes of objects as was done in

the papers ci ted above. In par t icu lar for any class of objects, m, our

A(• 1 is precisely the m-perfect morphisms of [H~, [H3], [N~,

IS2 and [$3]. This a l tera t ion has allowed us to obtain as special cases

of our theorems many of the results of Nel IN3], who defined perfect

morphisms re la t ive to development classes of morphisms, thereby broadening

the scope of the theory. The general resul ts of w also shed new l i gh t

on factor izat ions of the "monotone quot ien t - l i gh t " var ie ty and the i r

attendant " t o t a l l y disconnected" ref lect ions (cf . [$4]) . Note that

in Theorem 2 there is no res t r i c t i on on the class of morphisms ~; in

par t icu lar members of m need not be epimorphisms. Also there is no

assumption of uniqueness of the diagonal mcrphism in the de f in i t i on of A.

Thus Theorem 2 y ie lds results e .g. , for closed embeddings since in the

category of topological spaces they are AI(~) where m = al l dense maps;

and for (amnestic) topological functors since, by the recently announced

resu l t of Br~mmer and Hoffman, in the category of al l categories and

fa i t h fu l functors, they are A(m), where e = al l f u l l f a i th fu l functors.

617

Besides special izat ions to the general theories mentioned above,

there are also in terest ing new special izat ions al l the way down to classical

topological resul ts. For example Theorem 3(1), for the case where ~ is

the completely regular spaces and ~ is the compact extendible epimorphisms

y ie lds Mori ta 's resu l t ~2 ] that every inverse l i m i t of an inverse spectrum

of metrizable spaces and perfect bonding maps is a paracompact M-space.

I t should be mentioned that a special case of our general notion of

factor ing sources so that the second factor is a perfect source has in

essence been considered by Whyburn ~ i ] , ~2 ] , Cain ~ i ] , ~2 ] , and others.

In the Whyburn-Cain set t ing one has a continuous function (between Hausdorff

spaces) f :A § B and a compactif ication ~ of A. An attempt is then

made to obtain a perfect mapping f* :A* § B with A a dense subset of

A*, f * IA = f and a mapping h:A*~ A whose res t r i c t i on to A is the

inclusion of A in to A. Looking at th is from a "source" standpoint

one has the given source (A, ( f , e ) ) , where e:A § I f one then takes

618

the (epi, strong monosource)-factorization in the category of Hausdorff

spaces, (A*, ( f * ,h ) ) is immediately obtained as the second factor .

Using th is approach many of the theorems involving "mapping compact i f icat ions"

become qui te easy to prove. For example the two main resul ts of [C2]

reduce to t r i v i a l i t i e s . Also using the " fac to r iza t ion of two-element

sources" approach, the resu l t of Nagata [N~ that a space of weight m

is a paracompact M-space i f and only i f i t is homeomorphic with a closed

subspace of the product of a metrizable space and [0,1] m , is easi ly

shown and the construct ion of Mor i ta 's paracompactif ication uX of a

given M-space [M~ and the Kat~towVigl ino absolute closure of a Hausdorff

mapping IV] are read i ly obtained, For more deta i ls see [DS].

Theorem 4 above has some s igni f icance in that i t characterizes the

s i tua t ion when two of the major categorical approaches to perfectness

coincide, and thus gives perhaps some more ins ight as to why they y ie ld

the same classes of "compact-perfect" mappings (resp. sources) in the

category of completely regular spaces, but d i f fe ren t classes in the

category of Hausdorff spaces.

w Areas for Further Study

The theory of perfect maps of Manes [M 1] which was described at the

beginning is the most recent of the three "perfect map" theories and is

one which i t seems deserves fur ther exp lo i ta t ion. An ear l i e r paper of

Brown [B3] essent ia l l y shows that the Manes approach "works" for the

category of topological spaces and sequent ia l ly continuous funct ions.

Here "closed" means "sequent ia l ly closed" IF4] , "compact" t ranslates to

"sequent ia l ly compact" and "Hausdorff" becomes "space with unique

619

sequential l im i ts . " Since sequential compactness is f i n i t e l y productive,

but not a rb i t r a r i l y so, th is shows that the Manes approach is appropriate

in situations where the ~-perfect or ~ - s t r o n g l y perfect approaches are

not. Also i t is apparent that with some s l ight t ightening of the Manes

axioms (e.g., making closed sets a rb i t r a r i l y intersective~ and having the

canonical inclusion of a closed subset of a structured set be not only

perfect but also an optimal l i f t ) one could obtain appropriate general

versions of many more classical results relat ing perfectness, compactness,

Hausdorff~and various factorizat ions. Thus~ in the fu l l subcategory of

Hausdorff objects, each dense morphism would be an epimorphism, each

regular monomorphism would be a closed embedding, and unique (dense,

closed embedding dyad)-factorizations would occur, (cf. ~HS~ and ~DS])~

Not long af ter Gleason's [G] fundamental discovery concerning onto-

projectives and projective covers in the category of compact Hausdorff

spaces, i t became clear that to extend the results to wider topological

categories the appropriate mappings for pro jec t iv i ty to be relat ive to,

should be the perfect onto maps (see e.g., IF1] and [S~) . Banaschewski

[B] has obtained quite nice general results that extend many ear l ier ones

in the area. Here, again, pro ject iv i ty is considered relat ive to the

perfect onto mappings between Hausdorff spaces. I t seems as though a

f ru i t f u l area to investigate would be that of projectives in s t i l l more

general settings using the appropriate categorical analogues of perfect-

ness given above.

We should also mention that since the approaches to perfectness

mentioned in this paper are categorical, they are al l subject to dualization;

i . e . , coperfect morphisms, coperfect sinks, monocoreflective hulls~ etc,

620

Although some in terest ing examples and applications in th is dual realm

are presented in [H3] and [$3], i t appears to be an area which h i s t o r i ca l l y

has been neglected and which deserves fur ther study and development, In

th is connection i t is in terest ing to observe that sequential spaces

y ie ld a nice example for Manes-perfectness and at the same time a funda-

mental example of a monocoreflective hul l .

F ina l ly , i t appears that many useful resul ts could be obtained by

a determination of internal characterizat ions of various categorical

perfect sources and morphisms in the recently developed, quite important

and convenient categories of nearness spaces, seminearness spaces and g r i l l s

[BHR], [H43, [HS], [H63, and [K2]

621

REFERENCES

B. Banaschewski, Projective covers in categories of topological spaces and topological algebras. General Topology and It5 Relations to Modern Analysis and Algebra, I I I (Proc. Conf. Kanpur, 1968), pp. 63-91. Academia, Prague, 1971,

[B2] N. Bourbaki, General Topology, Part 1~ Addison-Wesley, Reading, Mass., 1966.

[B 3] R. Brown, On sequentially proper maps and a sequential compactification, J, London Math, Soc. (2), 7 (1973), 515-522.

[BHR] H. L. Bentley, H. Herrlich, and W, A. Robertson, Convenient categories for topologists, preprint,

EBM]

CCl]

A. B?aszczyk and J, Mioduzewski, On factorization of maps through ~X, Colloq. Math. 23 (1971), 45-52.

G. L. Cain, Jr . , Extensions and compactifications of mappings~ Math. Ann. 191 (1971), 333-336.

, Metrizable mapping compactifications, General Topology and Appl.,2 (1972), 271-275.

F. A. Delahan and G. E. Strecker, Graphic extensions of mappings, preprint.

J. Flachsmeyer, Topologische Projektivraume, Math. Nachr. 26 (1963) 57-66~

I-F 3] S. P. Franklin, On epi-ref lect ive hulls, General Topology and Appl. i (1971), 29~31.

, On epi-ref lect ive hulls I I , Notes for Meerut Univ. Summer Inst. on Topology, 1971.

, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115.

A. M, Gleason, Projective topological spaces, I l l i no is J. Math. 2 (1958), 482-489.

622

[H2]

[H3]

A. W. Hager, Perfect maps and epiref lect ive hul ls, Canad. J. Math. 27 (1974), 11-24.

H. Herrl ich, A generalization of perfect maps, General Topology and I ts Relations to Modern Analysis and Algebra, I I I (Proc. Third Prague Topological Sympos. (1971), pp, 187-191. Academia, Prague, 1972.

, Perfect subcategories and factor izat ions, Topics in Topology (Proc. Colloq. Kaszthely, 1972), pp. 387-403. Colloq. Math. Soc. J~nos Bolyai, Vol 8, North Holland, Amsterdam, 1974.

[H4] , A concept of nearness, General Topology and Appl, 5 (1974), 191-212.

[H5] , Topological structures, Math, Centre Tract 52 (1974), 59-122.

[H63 , Some topological theorems which fa i l to be true, preprint.

[HI]

[HS I]

M. Henriksen and J. R. Isbel l , Some properties of compactifications, Duke Math. J., 25 (1958), 83-106.

H. Herrlich and J. van der Slot, Properties which are closely related to compactness, Indag. Math. 29 (1967), 524-529.

[HS 2] H. Herrlich and G. E. Strecker, Coreflective subcategories in general topology, Fund. Math. 73 (1972), 199-218.

EHS3] Bacon, Boston, 1973.

, Category Theory, Allyn and

[K1] M. Kat~tov, Uber H-abgeschlossene und bikompakte R~ume ~asopis P~st, Mat. Fys. 69 (1940), 36-49.

[K2] , On continuity structures and spaces of mappings, Comment. Math. Univ. Carolinae 6 (1965), 257-278.

G. M. Kelley, Monomorphisms, epimorphisms, and pull-backs, J. Austral. Math. Soc. 9 (1969), 124-142.

623

J. F, Kennison, Full reflective subcategories and generalized covering spaces, I l l ino is ,1, Math. 1_22 (1968), 353-365.

[M 1]

["1]

IN 2 ]

m31

[P] [Si]

Is2]

IT]

[V]

E. G. Manes, Compact Hausdorff objects, General Topology and Appl. 4 (1974), 341-360.

K. Morita, Topological completions and M-spaces, Sci, Rep. Tokyo Kyoiku Daigaku 10 , No. 271 (1970), 271-288.

J. Nagata, A note on M-spaces and topologically complete spaces, Proc. Japan Acad. 45 (1969), 541~543.

R. Nakagawa, Classes of morphisms and reflections, preprint.

L. D. Nel, Development classes: an approach to perfectness, reflectivesness and extension problems~ TOPO 72-General Topology and i ts Applications (Proc, Second Pittsburgh Internat, Conf.) pp. 322-340. Lecture Notes in Math., Vol. 378, Springer, Berlin 1974. D. Pumpl~n, Kategorien, to appear. D. P. Strauss, Extremally disconnected spaces, Proc. Amer. Math. Soc. 18 (1967), 305-309.

G, E. Strecker, Epireflection operators vs. perfect morphisms and closed classes of epimorphisms~ Bull, Austral.Math. Soc. 7 (1972), 359-366.

, On characterizations of perfect morphisms and epireflective hulls, TOPO 72-General Topology and i ts Applications (Proc. Second Pittsburgh Internat. Conf.) pp. 468-500. Lecture Notes in Math., Vol, 378, Springer, Berlin 1974,

, Component properties and factorizations, Topological Structures, Math. Centre Tract 52 (1974), 123-140.

J. H. Tsai, On a generalization of perfect maps~ Pacific J, Math, 46 (1973}, 275-282.

G. Viglino, Extensions of functions and spaces, Trans. Amer. Math, Soc. 179 (1973), 61-69.

624

G. T. Whyburn, Compactifications of mappings, Math. Ann. 166 (1966), 168-174,

Dynamic topology, Amer. Math. Monthly 77 (1970), 556-570.

Department of Mathematics Kansas State University Manhattan, Kansas 66506

Espaces fonct ionnels et s t ructures syntopog6nes.

par Daniel Tanr~

Les st ructures syntopog~nes ont ~t~ i n t rodu i tes par Csaszar (4) a f in de pouvoir consid~rer les appl icat ions continues entre espaces topologiques, un i fo r - mes entre espaces uniformes, les morphismes usuels entre espaces de proximit~ comme des cas p a r t i c u l i e r s d 'app l i ca t ions T-continues entre espaces syntopog~nes. Cependant, la cat~gorie des st ructures syntopog~nes d~f in ies dans (4) n '~ tant pas proprement f ib r~e (8), ce que nous appellerons i c i s t ruc ture syntopog~ne est une classe d'~quivalence de st ructures syntopog~nes au sens de (4). Ceci coTn- cide d ' a i l l e u r s avec la d ~ f i n i t i o n de T-espaces pr~idempotents donn~e par Hacque (7) La d i f fe rence de presentat ion entre les deux notions est la suivante: A < B dans (4) s i g n i f i e B e p(A) dans (7). Nous notons V l a cat~gorie des T-espaces et S }a sous-cat~gorie pleine de g form~e des espaces syntopog~nes. Nous montrons que V e s t une cat~gorie topologique proprement f ib r~e (8) et que S est s table pour la formation de st ructures i n i t i a l e s dans V.

Nous abordons ensuite l ' e s s e n t i e l de notre ~tude: les espaces fonct ionnels de st ructures syntopog~nes. Si ~ e s t un espace syntopog~ne sur Y e t X un ensemble, nous mettons sur l 'ensemble M(Y,X) des appl icat ions de X vers Y une s t ruc ture d'espace syntopog~ne qui se r~du i t ~ la s t ruc ture de la convergence uniforme si Y est un espace uniforme.

Si ~ e t ~ sont des espaces syntopog~nes sur X et Y respectivement, nous d~f inissons les par t ies T-~quicontinues de H(Y,X), de sorte que, si ~ e t ~ sont des espaces topologiques (resp. des espaces uniformes, resp. si ~ e s t un espace topologique et ~un espace uniforme) nous retrouvions la d ~ f i n i t i o n de l ' ~ga le cont inu i t~ (/0) de Kel ley (resp, ~quicont inu i t~ uniforme (2), resp. ~quicont inu i - t@). Enf in, nous re l ions ces deux notions en les appl iquant dans le cas par t i cu - l i e r d 'app l i ca t ions p a r t i e l l e s d~duites d'une app l ica t ion f de XxK vers Y, o~ K est un ensemble.

Dans ce contexte, on ne peut que reg re t t e r la lourdeur des notat ions et la techn ic i t~ des d~monstrations. Aussi, pour terminer , nous d~f in issons un fonc- teur Hom interne ~ la cat~gorie des N-espaces (~) , foncteur qui coTncide avec la convergence simple. I c i , les d~monstrations se d~duisent, dans leur grande major i tY , des r~su l ta ts cat~goriques obtenus sur les N-espaces.

I - Les categories des T-espaces et des st ructures syntopog~nes:

Soit E un ensemble, P(E) l 'ensemble des par t ies de E et F(E) l 'ensemble

des f i l t r e s sur E. Nous noterons ~ l 'ensemble vide.

626

De f i n i t i on 1-1: Une appl icat ion p de P(E) dans p2(E) d 6 f i n i t un ordre s ~ i - t o - pog~ne (4) sur E si les axiomes suivants sont v ~ r i f i ~ s :

(STI) @ c o(@), E ~ o(E); (ST2) B c B' , A c A' et B c p(A) ent ra inent B' ~ o (A ' ) ; (ST3) B E o(A) entra~ne A c B.

De f i n i t i on 1-2: Une app l ica t ion p de P(E) dans p2(E) d e f i n i t un ord~e topog~ne (4) sur E (ou (E,p) est un T-c~pace simple (7}) s i p est un ordre semi-topog@ne v e r i f i a n t l 'axiome supplementaire:

(Q) B e o(A) et B' ~ p(A') ent ra inent B n B' c p(A n A') et B u B' e p(A u A ' ) .

Nous pouvons exprimer la d e f i n i t i o n d 'ordre topog6ne et de T-espace sim- ple sous la forme plus leg#re suivante ( 7 ) ' ( / / J : p e s t une app l ica t ion de P(E) dans F(E) v e r i f i a n t :

(T I ) ~ ~ p(@); (T2) B ~ p(A) entra~ne A c B; (T3) p(A u B) : p(A) n p(B).

L'ensemble des ordres topogenes sur E est ordonne par la re la t i on

P l c P2' P2 est plus f ine que Pl" Si Pl et P2 sont deux ordres topogenes, on d~- f i n i t un nouvel ordre topog6ne p = Pl o P2 en posant, pour tout A c E,

p(A) = Bep~(A) Pl (B)"

A un ordre topog~ne p correspond canoniquement une re l a t i on d'eloignement caract#r isee par A T B ssi i l ex is te C e p(B) te l que C n A = 9.

Un ordre topog@ne p sur E est d i t :

ponctuel si p(A) = x~A p (x ) , pour tout A E P(E); p#J'cfaZ~ s ' i l est ponctuel et si les f i l t r e s p(A) sont pr inc ipaux; sgm~t~que si T e s t symetrique;

2 idempotent s i p = p.

Remarque: La terminologie ci-dessus est ce l le de (7); darts (4) ponctuel est rem- plac~ par p a r f a i t et p a r f a i t par b i p a r f a i t . De plus, dans (4), la r e l a t i on p e s t d i t e sym@trique si B ~ p(A) implique E-A c p(E-B). On constate imm#diatement que les deux d e f i n i t i o n s d 'ordre topog~ne symetrique coTncident.

De f i n i t i on 1-3: Soit (E,p) et ( E ' , p ' ) deux T-espaces simples. Une app l ica t ion f de E vers E' d e f i n i t une application T-continue (7} de (E,p) vers (E ' , p ' ) s i , pour tout B E P(E), l ' image r@ciproque par f de tout p ' -vo is inage de f(B) est un p-vois inage de B, i . e . f - l p ' ( f ( B ) ) c p(B).

Introduisons maintenant les st ructures etudiees dans (4) et (7): les T- espaces et les s t ructures syntopog~nes.

627

D@finit ion 1-4: Une T-b~e B sur un ensemble E est un ensemble f i l t r a n t ~ d ro i te d'ordres topog~nes sur E, i . e .

(SI) pour ~ u t (pl,P2) ~ B• i l ex is te ~ ~ B te l que P l c ~ et p2 c ~.

L'ensemble des T-bases sur E est pr~ordonne par la r e l a t i on suivante: B < B' ss i , pour tout p ~ B, i l ex is te p' ~ B' te l que ~ ~ ~ ' . Cette r e l a t i on de preordre d~termine une r e l a t i o n d'@quivalence et une r e l a t i o n d 'ordre sur l 'ensemble quot ient . D@finit ion 1-5: Un T-~pace~E = (E, [~ ] ) est un ensemble E muni d'une classe d'@- quivalence [~] de T-bases sur E. Par abus de nota t ion, une classe d'~quivalence sera souvent d@sign@e par l 'un de ses elements B, les ordres ~i de B ~tant appel~s ordres g~n~riques delE. Un T-espace a une propr i~t~ (par exemple sym~trique) si les ordres g~n~riques d'une de ses T-bases la possedent.

(4) Def i n i t i on 1-6: Un es~ace syntopog~ne T-espacelE = (EJ~ ] ) v ~ r i f i a n t :

(S2) pour tout Pi ~ [ ~ ] ' i l ex is te ~

(ou T-~space pr~idempotent (~)) est un

[~] te l que Pi c P~"

i Def i n i t i on 1-7: Soi t lE = (E, [~ ] ) e r iE ' = ~ ' , [ ~ ] ) deux T-espaces. Une app l ica t ion f deIE danslE' est T-continue s i , pour tout ordre g~n~rique ~ de lE ' , i l ex is te un ordre g~n~rique ~i deIE te l que l ' a p p l i c a t i o n f s o i t T-continue du T-espace

simple (E~) dans le T-espace simple ( E ' , p i ) . (3) Parmi les T-espaces, on d is t ingue les espaces pr~topologiques , topolo-

giques, de proximit~ (5), uniformeso Pour plus de d~ ta i l s sur ces r e s u l t a t s , le

lec teur consultera (4) et (7).

Nous al lons maintenant ~ t a b l i r l ' ex i s tence de st ructures i n i t i a l e s dans V. Cette construct ion n 'es t f a i t e ni dans (4) ni dans (7), on peut cependant d~- duire cette existence de r~su l ta ts de (4) pour la cat~gorie S (S est ~ produi ts et ~ st ructures images r~ciproques) des espaces syntopog~nes. Nous noterons V la cat~gorie des app l ica t ions T-continues entre T-espaces dont les ensembles sous-jacents appart iennent ~ un univers U; S est une sous-categorie pleine de V. Proposit ion 1-8: La cat~gorie V ~ t ~ structures i n i t ~ .

Soi t , pour tout ~ c j , o0 J appar t ient ~ l ' u n i v e r s U, un T-espace ~ = (Y , ( n B ) ~ l ). Soit E un ensemble, ~l~ment de U, et f une app l ica t ion de E dans Y , pour to~t ~ ~ J. Soi t J' une par t ie f i n i e de J; pour tout ~ ~ J ' , on cho i s i t y(~) ~ I~. Notons y ce choix. On pose, pour tout B c E"

C~ pJ "u ss i , pour tout ~ J' i l ex is te C ~ n ~ , ~ y~ ~(f (B)) ~ ) t e l s que

628

c m j , f : l ( c a ) c C.

pJ' J ' ,y L'ensemble 'Y(B) ~tant satur# par induct ion et f i l t r a n t a gauche, p est une app l ica t ion de P(E) dans F(E). L'axiome (TI) est aussi t r i v ia lement v ~ r i f i ~

�9 J ' par pJ' 'Y Pour (T2), so i t C ~ p 'Y(B), i l ex is te C a ~ n~(a) ( fa(B) ) te ls que ~ j , f : l ( c a ) c C. De fa(B) c Ca, on d#duit B c c ~ j , f : l(Ca) c C. Pour l 'axiome

j ~ (T3), nous remarquons d'abord que B c B' entra~ne pJ 'Y(B' ) c p 'Y(B). On a a ins i , pour deux par t ies B e t B' quelconques de E: J ' 'Y(B ' J ' J ' pJ' J ' ' Y ' p u B ) c p 'Y(B) n p 'Y (B ' ) . Soi t C �9 'u n p ( B ) , i l ex is te

�9 n a C' n a I ( C ) C y (~) ( fa(B)) et ~ y ( ~ ) ( f ( B ) ) te l s que ~ j , f : �9 C et

: l ( c u C ' E n a ' :l(Cc~ ' o 2 j , f ~) c C. Or C a y (a ) ( fa (B u B )) et a ~ j , f u C ) c C entra~- J' ,y( nent C�9 p B u B').

On a ainsi montr~ que pJ' 'u est un ordre topog~ne sur E. Faisons parcourir ~ J' l'ensemble des parties finies de J e t ~ y l'ensemble

J' ,y]) des choix possib les; IE = (E,[p est un T-espace, c ' e s t - ~ - d i r e v ~ r i f i e l axiome j , ,y, j ,, ,y"

(S l ) . En e f f e t , so i t p et p deux ordres topog~nes g#n~riques de E; on pose K = J ' u J" et on d ~ f i n i t un choix y de la fagon suivante:

a C% . s im e J ' e t a El J " , on prend qy(a) = q y ' ( a ) ' si a (~ J' et m �9 J" , on prend a c~ ny(a) = qy " (a ) ; s im �9 J' e t a E J" , on sa i t q u ' i l ex is te n m te l que

cz C qe~ .,~ cz ny,(~) y(~) et q "(a) �9 n (~). On a a lors :

pJ''Y' c pK'y et pJ"'Y" �9 pK,y.

II reste ~ montrer queiE est la structure in i t ia le pour la famille (Y/ ,fa)~e J. Soit Z! = (Z,[~])un T-espace et g une application de Z dans E; on pose gm = faog. L'application f~ est T-continue delE vers Yrm. En effet, soit n~ un ordre g~n~ri-

que de Yia, on pose J' = {m} et y(~) = ~; si C �9 n~(fa(B)), alors f-1(C)~ appartient

pJ"Y(B), par d~finition. La T-continuit# de l'application g de Z!verslE entra~- ne donc celle de gm de Y( vers Z/.

R~ciproquement, supposons que g~ soit T-continue de Y( vers Zz. Soit pJ"Y un or- a a

dre g@n@rique delE et so i t (ny(~))~ej , la fam i l l e d~ f in issan t pJ 'Y Par hypo-

th~se, pour tout a E J ' , i l ex is te un ordre g~n#rique z ~,j de Z/ te l que gm so i t T-cont inue de z ~. ~ j vers y (~) . D'apr~s (Sl) , l 'ensemble J ' ~tant f i n i , i l ex is te

z J'j te l que z ~.j c z~ , pour tout a E J ' . On v ~ r i f i e ais~ment que g est T-cont inue

de (Z,z ) vers (E,p J ' ' Y ) , § J

La cat~gor ie V e s t donc une cat6gor ie topologique proprement f i b r~e au sens de (8).

629

Proposition 1-9: Si t o w les o r d r ~ topog~n~ consid~r~s dans la d~monst~ation pr~c~dente sont sym~triques, les o r d r ~ topog~nes pJ' 'u c o ~ t r u i t ~ sont a ~ s i s ym~tr iqu~.

J' ~ pJ' § Soit p 'Y d#f in i par la fami l le (ny(m))m~j,. Soit C e 'Y(B), i l ex is te

C~ c n(%y ((%) (f(% (B)) que c2 j , f~ pour (% ' te ls I(C(%) c C. On obt ient a lors , tout c j :

Y e n(% a' 'Y ~1(C )) ' (%-f(%(B) y((%)(Y -C ) et E-Be p (E-f . L'ensemble J ~tant f i n i , j , = pJ' 'Y(E- I(C(%) E-B appart ient ~ (%~a,O 'Y(E-f~I(c(%)) menj,f~ ). On en d@duit:

E-B e pJ"Y(E-C) . §

Coro l la i re 1-10: La sou~-ccut#go~uLe ple/ine V s de V, doFut le~ o b j ~ sont l ~ T- espaces s y m ~ i q u e s , e s t ferm~e pour la cons truct ion de s t ruc tures i n i t i a l ~ dans V. Proposition 1-11: La sous-cat~gorie p le ine S des ~paces syntopog~nes ~ t f~m~e pour la construct ion de s t r u c t u ~ i ~ t i a l e s da~ V.

J ' , y ' § On suppose donc t o u s l e s Y~ syntopog~nes. Soit p d~f in i par la fami l le (%

(n~, 2m ((%))c~ej" Pour tout (% ~ J ' , i l ex is te ny(~)~ te l que n(%y,((%) c ny((%). Soit J ' , y p l ' o rd re topog@ne sur E associ~ ~ (ny((%))~j , et montrons: J',~' 2 J ' p c '~

j ' ,u ~ 1(C(% Soit C E p (B), i l ex is te C~ ~ n , ( e ) ( f (B)) te l que ~ j , f ~ ) c C. De

C ~2(% (% ny(m)(f(%(B)), on d@duit l ' ex is tence d'un D te l que C ~ ~ ny((%)(D ) et

D(% @

ny(~)(f(%(B)).j,f~l(D~ ) J ' 'Y (B) par construct ion et on montre, en Posons D = c~ , on a D ~ p a' 2 u t i l i s a n t la cont inui t~ de f : C ~ p 'Y(D). II en r~sul te C e J ' 'Y (B ) . +

(%

Remarques: I) Si l 'ensemble J e s t i n f i n i , la s t ructure i n i t i a l e d'une fami l le de structures simples n 'es t pas en g#n~ral simple. I I s 'ensu i t en p a r t i c u l i e r que la cat~gorie T des topologies n 'es t pas ferm#e pour la construct ion de pro- dui ts dans V.

2) La s t ructure i n i t i a l e d'une fami l le de structures ponctuelles n 'es t g#n~- ralement pas ponctuel le. Cependant, la ponctual i t# est conserv~e par s t ructure image r~ciproque. On en d~duit (#nonc# darts (4) pour S): la s t ructure image r~- ciproque dans V d'un espace topologique est un espace topologique.

3) Ce qui precede et (8) nous donnent l ' ex is tence de structures f ina les et de l imi tes dans V e t dans S.

4) Pour terminer,remarquons que la r@union des T-bases d'une m~me classe d '~- quivalence pour < est encore une T-base de cette classe. C'est cet ~l~ment maxi- mal que nous prenons comme repr~sentant, dans la p lupart des d6monstrations.

630

2- Espaces fonc t ionne ls : s t ruc tu re de la convergence t o ta l e : Nous nous placerons i c i dans la cat@gorie S des espaces syntopog@nes. Soi t ~ = (Y , [~ ] ) un espace syntopog6ne et X un ensemble. Notons M(Y,X) l ' e n -

semble des app l ica t ions de X darts Y. Soi t ~ et n' des ordres g@n~riques de te l s que n c ~ ' ; nous d~f in issons B sur P(M(Y,X)) par: B e B(A) ss i , pour tout C c Y, i l ex is te U C �9 4(C) et i l ex is te V c �9 n'(C) te ls

que U c �9 n' (Vc) et C~Y {g I t �9 A-I(Vc ) ~ g( t ) �9 U C} c B.

�9 A-I(Vc) s i g n i f i e f ( t ) �9 V c, pour tout f �9 A. Si A = @, on L 'expression t pose

�9 n(~) . Proposi t ion 2-1: Po~ tout couple (n ,n ' ) t ~ que n c ~' , l ' app l i c~ t ion B d~fin~t un ordre semi- topog~ne sur M(Y,X).

§ (ST1) et (ST3) sont bien entendus v ~ r i f i ~ s . Pour (ST2), remarquons que, par d ~ f i n i t i o n , si B e s t inclus dans B' et si B ap- pa r t i en t ~ ~(A), on a: B' �9 ~(A). Soi t A c A' et B �9 B(A ' ) , a lo rs , pour tout C c y, i l ex is te U c �9 n(C) et ~C �9 n'(C) te l s que:

U C �9 n'(Vc) et C~Y {g I t ~ ArZ(Vc) ~ g( t ) �9 U C} �9 B.

Notons: H = C~Y {g I t �9 A-I(Vc ) = g( t ) �9 UC},

H' = C~Y {g I t �9 A' - I (Vc ) ~ g( t ) �9 UC); on a: H c H ' , d'o~: B �9 @(A) et B(A') c @(A). §

Notons B q l ' o r d r e topog~ne associ~ (4) ~ B. Proposi t ion 2-2: En faisant varier dam [~ ] • couple (n ,n ' ) t e l que n c 2,,

l ~ app l ica t io~ B q ~ s o c i ~ d~finissent une structure sy~opog~ne s~r M(Y,X), notre M(Yf, X).

§ Etabl issons d'abord (Sl ) . Soi t B associ@ a (n ,n ' ) et B I associ~ ~ (n l ,n~) ; I I I . i l ex is te n 2 te l que n c n2 et n I c n2 ; i l ex is te n~ te l que n 'c n 2 et n I c n2 ,

I i l ex is te n" te l que n 2 c 2,, et i l ex is te n 3 te l que q2 c n3 et ~" c n3. Notons ~2 l ' a p p l i c a t i o n associ~e ~ (n,n3) ; on a B c @2 et BI c B2.0n d@duit de ceci (4)"

Bq c B~ et B q c B~.

Montrons maintenant l 'ax iome de pr~idempotence ($2). Soi t B l ' o r d r e semi-topog~ne associ@ ~ ( n , n ' ) ; i l ex is te n" te l que ~' c 2,,. Notons B' l ' o r d r e semi-topog6ne associ# ~ ( 2 " , n " ) ; on a: n c 2, c n' c 2" c n".

Soi t B e B(A), alors pour tout C c y, i l ex is te U C ~ n(C) et i l ex is te V C e n'(C) t e l s que U c c n ' (Vc) et C~Y {g I t ~ A-I(Vc) ~ g( t ) �9 U C} �9 B.

De n' c 2,, et U C �9 q ' (Vc ) , on d#dui t l ' e x i s t ence d'un V~ te l que U c �9 n"(V~) et �9

On a obtenu: pour tou t C c y, i l ex is te V~ �9 2"(C) et V c �9 B"(C) te l s que

631

V~ E n"(Vc). Posons B ' = C~Y {g I t A-I(Vc) : g ( t ) EVE} . On a B' c B e t B' E B' (A) . Posons maintenant: B" = C~Y {g l t E B ' - I (vE) ~ g( t ) E Uc}. Par d ~ f i n i t i o n , B" appar t ien t ~ ~ ' ( B ' ) .

go et t g ( t ) Vc ~I Montrons B" c B. Soi t E B" E A- I (Vc) . Par d@f in i t ion de B ' , E pour tou t g E B ' , donc t E B ' - I ( vE ) . I I s ' ensu i t go( t ) E U c car go E B". Ceci est vrai pour tou t C c y, doric go E B e t B E ~ ' ( B ' ) . En r~sum~, nous avons montr@: so i t Bun ordre semi-topog~ne d~dui t de ( n , n ' ) , i l ex is te ~' te l que: pour tout A c M(Y,X), tout B E ~(A), i l ex is te B' c M(Y,X) te l que B E ~ ' (B ' ) et B' E ~ ' (A ) , d'o~ ~ c ~ ' . La d~monstration d'une propr i~ t~ semblable pour le topog~nis6 ~q de B se f a i t simplement en u t i l i s a n t une carac t~ r i sa t ion du topog~nis~ mise en @vidence dans (4). On ob t ien t a ins i Bq c ~ 'q. +

On peut a ins i d ~ f i n i r un foncteur M(~,-) de M ~ dans S, o0 M * est la duale de la cat~gor ie des app l i ca t ions .

Si ~ e s t ponctuel , l 'espace fonct ionnel M(~,X) n 'es t pas n@cessairement ponc- t ue l ; mais l ' i nconv~n ien t est mineur car on peut consid~rer l 'espace ponctuel M(~,X) p associ~ pour avoir des espaces fonct ionne ls dans la sous-cat~gor ie p l e i - ne Sp des espaces syntopog~nes ponctuels. Par cont r~ si ~ est d~ f in i par un seul ordre n, l 'espace fonct ionnel cons t ru i t ci-dessus est simple. Si ~ est topo log i - que, i l s u f f i t de prendre M(~,X) p pour obten i r un espace topologique'. Nous al lons maintenant s i m p l i f i e r la const ruct ion de l 'espace fonct ionnel ponctuel M(~,X) p. Pour cela, supposons les ordres n i de [~] tous ponctuels et so i t (~ ,n ' ) t e l s que n c ~,. Pour tout f c M(Y,X), notons y ( f ) le f i l t r e engendr~ par les in te rsec t ions f i n i e s des par t ies B t e l l e s que:

Uy E n ' ( y ) te l s que Uy E n'(Vy) et pour tout y c y, i l ex is te c n(y) et Vy

B = y~_V {g ] f ( t ) c Vy g( t ) E UW}. On pose y(a)= i~ A y ( f ) .

Proposi t ion 2-3: Avec l ~ n o t i o n s p receden ts , on a M(~,X) p = (M(Y,X) , [~ ] ) . § Par const ruc t ion , y est un ordre topog~ne ponctuel plus f i n que ~, on en

d~du i t (4 ) : BqP c y. En e x p l i c i t a n t les d ~ f i n i t i o n s de y e t B , on ob t ien t : B c y ( f ) implique B E Bq( f ) . I I s ' ensu i t : y c ~qP. +

Proposition 2-4: S i V e s t un espace uniforme e t X un ensemble, M(~,X) p e s t /a s t r u c t u r e de la convergence uniforme.

§ Soi t (~ ,n ' ) des ordres g~n~riques de ~ te l s que ~ E ~ ' . Par hypoth~se, n (resp. n ' ) est canoniquement associ~ a un entourage W (resp.W') de Y, te l s que ~, c W. Notons u l ' a p p l i c a t i o n associ~e ~ ( n , n ' ) . Soi t <W(f)> = {g ] ( f ( x ) , g ( x ) ) ~ W, pour tout x ~ X}.

632

= U " W' U' U' Soit B ~ y ( f ) d~ f in i par Uy W(y) et y , on a: (y) c y , W ' ( y ) c W(y) et

B= y{gl f(t) EU' y g( t ) E W(y)}. Si on chois i t y = f ( x ) , de f(x) E U~(x~ , on d~duit g(x) E W(f(x)), ceci pour

tout x E X et tout g E B, d'o~ B c <W(f)>.

Soit maintenant g E <W'(f)> et t tel que f ( t ) E U'. De ( g ( t ) , f ( t ) ) E W', on d~- Y

du i t : g ( t ) E W'(U~) et g ( t ) E W(y), dIoQ: <W'(f)> c B. Notons ~ (resp. ~ ' ) l ' o rd re topog@ne engendr@ par ~ (resp. ~ ' ) sur M(Y,X), i . e . : 6 ( f ) est le f i l t r e des surensembles de <W(f)>. On daduit de ce qui pr@c~de: c y c f i ' .

En notant [~] la structure syntopog~ne de la convergence uniforme et [T] celle

d~f in ie pr@c~demment, on a [~] < [y] et [y] < [ ~ ] , pour la r e l a t i on < d~ f in ie dans le premier paragraphe. A ins i , /y ] est la s t ruc ture de la convergence uni- forme. §

3- T-6quicont inu i t~ :

Soit ~ = (X, [~ ] ) et ~= (Y,[~]) deux espaces syntopog@nes; notons H(Y,X) l 'ensemble des app l ica t ions de X dans Y. D# f i n i t i on 3 - i : Une par t ie H de M(Y,X) est T-~quW_comtinue s i , pour tout couple (n ,n ' ) d'ordres g#n#riques de ~ te l s que n c ~ ' , i l ex is te un ordre g~n#rique

de ~ te l que: pour tout Cc Y, tout Ac X, tout UE n(C), i l ex is te VCn'(C) et U' E p(A) te l s que U~ q ' (V) et: f ,e H et f (A) c V impl iquent f (U ' ) c U.

Proposi t ion 3-2: S i ~ ~t Yf son t des e s p a c ~ t o p o l o g i q u ~ , l a d ~ f i n i t i o n pr~c~- dente co lnc ide avec c ~ l e d ' ~ g ~ e c o n t i n ~ de Ke l l ey (I0) .

Proposi t ion 3-3: I) S i ~ e t Yf son t d ~ ~ p a c ~ uni formes, l a d ~ f i n i t i o n 3-I e s t c ~ e d' ~quicont in~i t~ uniforme (2)

21 S i ~ e s t un espace topo log ique e t Y( un ~ p a c e u~iforme, l~ T -~qu icon t inu i - t~ e s t l ' ~ q ~ c o n t i n ~ ( 2 )

§ i ) Soi t >% et Yf deux espaces uniformes. Soit H une par t ie de M(Y,X) v ~ r i f i a n t la d@fin i t ion 3-1. Montrons que H est uniform~ment ~quicontinue. Soi t W un entou- rage de Y, i l ex is te W' te l que ~' c W. Par hypoth@se, i l ex is te un entourage

V' de X te l que: pour tout y = f (Xo) , pour tout XoE X, pour tout U = W(y), i l ex is te V te l que W'(f(Xo) ) c V e t W'(V) c W(y), et i l ex is te U' te l que V'(Xo) c U' , v ~ r i f i a n t :

f E H et f(Xo) E V impl iquent f (U ' ) c U. Or f(Xo) E W'(f(Xo) ) c V; donc, pour tout f ~ H, on a f (U ' ) c U et par su i te

f (V ' (Xo) ) c U, i . e . : (X,Xo) ~ V' implique ( f ( x ) , f ( X o ) ) c W. On a a ins i montr6: pour tout entourage W, i l ex is te un entourage V' te l que

633

( x ' , x ) ~ V entra~ne ( f ( x ' ) , f ( x ) ) ~ W. R~ciproquement, so i t H une par t ie uniform~ment @quicontinue de M(Y,X); montrons qu ' e l l e est T-~quicontinue. Soit (W,W') un couple d'entourages de Y te l s que ~' c W. Soi t V l 'entourage de X correspondant ~ W', c ' e s t - ~ - d i r e v ~ r i f i a n t :

( x ' , x ) ~ V implique ( f ( x ' ) , f ( x ) ) ~ W', pour tout f ~ H. Notons n ( resp .n ' , resp.p) les ordres topog~nes associ~s canoniquement ~ W (resp.W', resp.V). Les ordres topog~nes consid~r~s sont ponctuels, nous ra ison-

nerons donc sur les ~l~ments de Y e t de X. Soit y ~ Y e t x ~ X, s o i t U ~ n (y ) , i . e . W(y) c U; choisissons U' = V(x)~ p(x) et V = W'(y) c ~ ' ( y ) . On a: W'(V) c W(y) c U. Soi t f ~ H te l que f ( x ) ~ W'(y). S i x ' e V(x) , par hypoth~se, ( f ( x ' ) , f ( x ) ) est un @l~ment de W'. On obt ien t a ins i :

( f ( x ' ) , y ) = ( f ( x ' ) , f ( x ) ) o ( f ( x ) , y ) c W' o W' c W. I I s 'ensu i t : f (V (x ) ) c W(y) c U.

2) La d~monstration est semblable ~ la pr~c~dente. +

Nous a l lons maintenant r e l i e r les deux d~ f i n i t i ons pos~es: ce l le d'espaces

fonct ionnels et ce l l e de T-~quicont inu i t~. Soit K un ensemble, ~ e t ~deux espaces syntopog~nes. Soit f une app l ica t ion

de KxX dans Y; on note H l 'ensemble des app l ica t ions p a r t i e l l e s f ( t , - ) de X dans

Y, o0 t parcourt K. Proposi t ion 3-4: L'applicat~on ~: x ~ f ( - , x ) de X dans M(Y,K) es t T - c o ~ n u e de ~ vea~ M(~,K) ss i H ~ t une pa~2ie T-~quicontinue de M(Y,X).

§ Exp l ic i tons chaque d ~ f i n i t i o n . H est T-~quicontinue: pour tout couple (n ,n ' ) te l que n c ~ ' , i l ex is te o, ordre g~n@rique de ~, te l que: pour tout C c y, tout A c X, tout U ~ n(C), i l ex is te V ~ n'(C) avec U c n'(V) et i l ex is te U' ~ ~(A) v ~ r i f i a n t : f ( t , A ) c V implique f ( t , U ' ) c U.

est T-continue de ~ v e r s M(~,K): pour tout ordre g@n~rique 8 q de M(~,X), i l ex is te un ordre g~n~rique p de ~ t e l que ~ so i t continue de p vers 8 q. L 'ordre

p ~tant topog6ne, la T-cont inu i t~ de @, de p vers @q, est ~quivalente ~ la T- continuit@ de p vers 8. Par d@fin i t ion de 8, l 'express ion pr~c~dente devient : pour tout couple (n ,n ' ) te l que n c ~ ' , i l ex is te p, ordre g~n~rique de ~, te l que: pour tout B ~ 8(r i l ex is te U' ~ ~(A) te l que @(U') c B. En e x p l i c i t a n t la d ~ f i n i t i o n de 8, on constate que les deux assert ions sont @quivalentes. +

Appendice: Convergence simple dans les N-espaces. Nous al lons mettre une s t ruc ture de foncteur Hom interne sur la cat~gorie des

N-espaces {8) (que nous traduirons par espaces de rapprochement), s t ruc ture pr~- sentant une comparaison avec l ' ana lyse usuel le. Nous ne rappel lerons pas i c i les d~ f i n i t i ons et propri~t~s des N-espaces, le lec teur se reportera ~ (8}.

634

Soit E un ensemble et (E' ,~) un espace de pr@rapprochement. Soit A c P(M(E' ,E)) , on pose: A~ ~ s s i , pour tout x ~ E, {A(x) I A~ A} ~ ~. Proposi t ion: (M(E',E),~) ~ t un ~pace de pr~rapprochement. C'est l 'espace produi t de E copies de (E ' ,~ ) .

D@sormais, nous supposerons que ~ est un N-espace et nous noterons qq l 'espa- ce Q-proche associ~ ~ n (8)

Proposi t ion: (M(E',E),nq) es t un N-espace.

Soit (E ' ,~) un N-espace uniforme ( i . e . provenant d'une uni formi t~ W sur E ' ) . Soi t n la s t ruc ture d~ f in ie pr#c#demment. Proposi t ion: (M(E',E),~) ~ t le N-espace uniforme de la convergence simple.

Soit ~ une s t ruc ture de N-espace sur E, nous noterons ~' la s t ruc ture indu i te par n sur l 'ensemble Hom((E' ,~),(E,~)) des N-appl icat ions de (E,~) vers (E ' ,~ ) . Soit (E",6) un N-espace et f une N-appl icat ion de (E' ,~) vers (E",B); notons n" la s t ruc ture de N-espace sur Hom((E",B),(E,m)).

A lors , Hom(f,(E,~)) est une N-appl icat ion de (Hom((E ' ,~ ) , (E ,~) ) ,n ' ) vers (Hom((E",B),(E,m)),n") qui ~ g associe f o g. De m#me, Hom((E,m),f) est une N-appl icat ion de (Hom((E,m),(E",B)),n") vers (Hom((E,m) , (E ' ,~) ) ,n ' ) associant h o f ~ f .

A ins i , la s t ruc ture de convergence simple d # f i n i t un foncteur Hom interne a la cat#gorie des N-espaces.

Remarque: Dans ce paragraphe, les d~monstrations se d~duisent dans leur presque t o t a l i t ~ des propri~t~s cat~goriques ~tab l ies dans (8).

Cette ~tude nous conduit aux question suivantes: d ~ f i n i t i o n d'une a-conver- gence et existence d'un th~or~me d 'Ascol i dans le cadre des structures synto- pog6nes et des N-espaces. Nous aborderons ce l l es - c i dans un prochain t r a v a i l .

_ - _ o _ - _ , _ . _ . _ , _ . _

(I) A .B~t ian i : Topologie g~n~rale, Cours m~Ctigraphi~, Amiens 1973. (2) N.Bourbaki: Topologie g~n~r~e, Hermann, Par~. (3) G.Choqu~. Convergenc~, Ann. Univ. Grenoble, nouv~le s ~ e , 23, 1947. (4} A.Csaszar: Foundations of g e n i a l topology, Macmillan, New-York 1963. (5) V.A.Efremovic. Geometry of proximity, Mat. Sbornik, 31 (73), 1952. (6) C.Ehresmann. C a t ~ g o ~ ~t s t ~ u ~ e s , Dunod, Par~ 1965. (7) M.Hacque: L~ T -~pac~ et l e ~ applications, Cahi~%s de Top. ~t G~om. Di f f . , IX-3, 1968. (8) H.Her~ch. Topological structures, Hathematic~ Centre T r a ~ , 52, 1974. (9) J.R. Isbell: Uniform spaces, Am~. Math. Soc. Math. S~veys, 12, 1964. (10) J.L.Kelley: Gener~ Topology, Van Nostrand, 1955. (11) D.Tanr~: Sur l ~ T -~pac~ s impl~ , Esqu~ses math. 18, P ~ , 1972.

FILTERS AND UNIFORMITIES IN GENERAL CATEGORIES

S.J.R. VORSTER t

w 1 INTRODUCTION

In [ i] B. Eckmann and P.J. Hilton translated the group axioms into

categorical language and studied the resulting concept of a group-

object in the context of general categories. In this paper a

similar approach will be followed by using uniform spaces instead

of groups.

Arbitrary sets equipped with uniformities are replaced by arbitrary

objects supplied with "uni-structures", which yields the concept of

uni-objects in general categories. Uniformly continuous maps

translate into categorical language without any difficulty.

Furthermore, .the ideas of filters on sets, groups, etc. are unified

by introducing objectfilters on objects of categories. Analogues

of some basic convergence properties of filters on uniform spaces

are proved in a more general context, viz. for objectfilters on

uni-objects in categories. The theory of uniform spaces will

thus be made available and may possibly become useful in areas of

mathematics other than general topology.

w 2 PRELIMINARIES

Any notations and concepts which are used but not defined in this

paper, will have the meaning assigned to them by H. Herrlich and

G.E. Strecker [4].

t This research was made possible by a grant from the South African C.S.I.R.

636

Let C be a well-powered category with A �9 0b(C) an arbitrary

object of ~ and 4: A ~ B any morphism of ~. The following

notations will be used (where all limits mentioned are assumed to

exist) :

E for a terminal object in C; E for the class of all epimorphisms

(or extremal epimorphisms) in C; M for the class of all monomor-

phisms (or extremal monomorphisms) in C; Sub(A)(resp. M(A)) for

the set of all subobjects (resp. all M-subobjects) of A;

(X,f) ~< (Y,g) if there exists a morphism h: X ~ Y such that f = gh,

and (X,f) ~ (Y,g) if both (X,f) ~< (Y,g) and (Y,g) ~< (X,f) hold, n

where (X,f), (Y,g) �9 Sub(A); N (Xi,f i) for the intersection of any i=l

finite family of subobjects (Xi,fi) �9 Sub(A), i = 1,2,...,n.

If

Diagram 1

A ~ >B

h

with (X,f) E Sub(A) and (Y,g) E Sub(B), is a pullback square

(resp. (E,M)-factorization of 4f, i.e. 4f = gh with h E E and

g �9 M) then we shall write ~,i (y,g) = (X,f) (resp. 4 (X,f) = (Y,g)).

Let I be an arbitrary set (resp. I = {1,2}). The product of a

family (Ai)iEi, A i �9 0b(C) will be denoted by (KAi,Pi) (resp-

(A, x A 2 ,Pl ,P2 ) ) or simply by KA i (resp. A I • A 2 ) and, if

(4i: B ~ Ai)i�9 I is given <~i} : B ~ ~A i (resp. <41 ,~2 } :B~AI k A2)

will denote the unique morphism such that pi < 4i> = 4i for each

i �9 I (resp. PI(4, ,~2 ) = 4, and p2<4, ,42) = 42). For a given

B i) we shall write family (4i: A i i�9 I

637

K~i = (~iPi } : ~Ai ~ KBi (resp. ~i x ~2= (~Ipl ,~2P2> : A1 • A 2 ~ BI • B2).

The following results will frequently be used:

(2.1) Proposition If C is a category with finite products, m

then

(i) (~i ,~2)~=(~i~,~2~ (ii) (~i ~ ~2)(~1 ,~2)= (~i~i ,~2~2 ) ,

(iii) (~I • ~2 ) (~1 ~ ~2 ) = ~I ~I • ~2 ~2 '

hold whenever both sides of each equality make sense.

The commutator morphism k = kAB: A XB ~ Bx A defined by k = <p2 ,P,)

will play an important role. Clearly, k is an isomorphism with

inverse k -I = k. One easily verifies

(2.2) Proposition If C is a category with finite products, then

(i) k( ~, ,~2 } = ( ~2 ,~,) , and (ii) k(~ x r = (4• ~)k

hold whenever both sides of each equality make sense.

(2.3) Proposition If C is a well-powered category with finite

products and if A e 0b(~) and (X,f),(Y,g) 6 Sub(A• then

(i) (X,f) ~ (Y,g) if and only if (X,kf) < (Y,kg),

(ii) (X,kf) ~ (Y,g) if and only if (X,f) ~ (Y,kg).

w 3 OBJECTFILTERS IN CATEGORIES

In this section the concept of filters on objects of general

categories will be introduced, which will unify the concepts of

filters on sets, groups and objects of other categories.

(3.1) Definition Let C be any category with a terminal object E.

(i) An A 6 0b(~) is a non-empty object if and only if there exists

an m: E ~ A (which is necessarily a monomorphism).

(ii) A subobject (X,f) of an A E 0b(~) is a non-empty subobject of

638

A if and only if there exists an m: E ~ A such that (E,m) < (X,f).

(3.2) Examples In all concrete categories having the "single-

tons" as terminal objects (for example, Set, Top, Grp, etc.) non-

empty objects and subobjects correspond precisely to objects of

these categories having non-empty underlying sets.

(3.3) Definition Let ~ be a well-powered (~,~) category with

finite intersections and a terminal object. A non-empty subset

of M(A), A E 0b(~), is an objectfilter on A if and only if

(OF.l) (X,f),(Y,g) E [ implies (X,f) N (y,g) @ [,

(OF.2) (X,f) e [ and (X,f) ~ (Y,g) imply (y,g) E [,

(OF.3) (X,f) 6 F implies (X,f) is a non-empty subobject of A.

If F and G are objectfilters on A such that (X,f) E F implies

(X,f) E G then F is said to be contained in G and we write F C G.

(3.4) Examples (i) In Set objectfilters correspond precisely

to filters on sets.

(ii) In Gr~ (resp. Ab) an objectfilter on a group (resp. abelian

group) G is a non-empty family of subgroups of G such that [ is

closed under finite intersections and any subgroup of G which

contains an element of F also belongs to F. (It is known that m

an objectfilter on an abelian group gives rise to a topology on G

such that G becomes a topological group.)

(iii) In Top an objectfilter on a topological space X is a non-

empty family [ of non-empty subspaces of X which is closed under

finite intersections and such that any subspace of X which con-

tains an element of F also belongs to F.

(3.5) Definition Let ~ be a well-powered (~,~) category with

finite intersections and a terminal object E. A non-empty sub-

set B of M(A), A E 0b(~), is a basis for an objectfilter on A if

639

and only if

(X,f) 6 B implies that (X,f) is a non-empty subobject of (OFB. 1 )

A.

(0FB. 2 ) (X,f),(Y,g) 6 B implies that there exists a (Z,h) E B

such that (Z,h) ~< (X,f) N (y,g).

(3.6) Proposition Let C be a well-powered (E,M) category with

finite intersections and a terminal object. If A is a non-empty

object of C and if a subset B of M(A) is a basis for an object-

filter on A, then

(B) = {(X,f)e Sub(A) / (Y,g) ~< (X,f) for some (y,g) e B}

is an objectfilter on A. (B) is said to be the objectfilter

generated by B.

(3.7) Definition If C is a well-powered category, a morphism t

: A ~ B of C is said to preserve terminality if, for any

(E,m) E Sub(A), E a terminal object in C, it holds that

~(E,m) = (E',m') E Sub(B), where E' is also a terminal object in

C.

(3.8) Theorem Let C be a well-powered (E,M) category with finite

intersections and a terminal object E. Suppose every ~: A ~ B

of C preserves terminality. For any objectfilter F on A the set

B = {~(X,f)/(X,f) @ F} forms a basis for an objectfilter (denoted

by ~(F)) on B.

Proof We have to verify (OFB.I) and (OFB.2). Firstly, let

(Y,g) e B, i.e. (Y,g) = ~(X,f) for some (X,f) e F. Now, since

(X,f) is a non-empty subobject of A there exists an m: E ~ A such

that (E,m) ~< (X,f), i.e. there is an f': E ~ X satisfying

ff' = m. Let ~(E,m) = (E',m') and consider

640

Diagram 2

E

A ~ > B

-~X 9, > Y

where g, ,m I E ~ and g,m' E M. Then we have

m'm I = ~m = ~ff' = ggl f' and hence it follows from the (E,M)-

diagonalization property ([4], 33.2 and 33.3) that there exists an

m": E' ~ Y such that m' gm" = , i.e. (E',m') < (Y,g) and since

preserves terminality (Y,g) is a non-empty subobject of B. Thus

(OFB.I) is satisfied.

Secondly, one easily verifies that B also satisfies (OFB.2), by

making use of the (E,~)-diagonalization property ([4] ,33.2) and

the definition of intersection.

(3.9) Theorem Let ~ be a well-powered (~,M) category with

finite intersections and products. Let F be an objectfilter on --1

A. E 0b(C) for each i E I, I an arbitrary set. For any family 1

(Xi'fi)i E I' (Xi,fi) E[i for each i E I, it holds that

pj(HXi,Hf i) ~ (Xj,fj) for every j E I, where pj: KA i ~ Aj denotes

the j-th projection morphism.

Proof Clearly, C is finitely complete (by [4] ,23.7) and hence

has a terminal object E. Consider any fixed j E I. It will be

shown that p j ~ _E" Now, s i n c e ( X i , f i ) E --1F' i s a n o n - e m p t y s u b -

object of A. for each i E I, there exists an m : E ~ A. such that 1 1 1

__(E,m i) ~ _.(Xi,fi) , i.e. there is an m~:l E ~ X.1 such that

641

Diagram 3

X. ~ >A~

E

commutes. For each i E I, i ~ j, let

e. m[ hl: X. ~ X = X. ~3 E ~3 X where Hom(X ,E) = {ej} and let

�9 3 i 3 i j

h = i: X ~ X.. Then, by the definition of product there exists 3 3 3

~ ~X such that pj(h i) = I, i.e. pj is a retraction and (hi) : X 3 1

therefore also an epimorphism. In addition, pj is an extremal

epimorphism since pj = mf, m a monomorphism, implies that

1 = pj(h i) = mf(h i) so that m is also a retraction (See [4],6.7).

Hence pj E ~.

Now, let pj(KXi,Zfi) = (Yj,gj), then in the

Diagram 4

TrA Aj

! ! ! ! PjKfi = gjgj = fjPj' where gj,pj e ~ and gj,fj E ~, and hence it

follows from the (~,M)-diagonalization property that

pj(~Xi,Kf i) = (Yj,gj) ~ (Xj,fj).

(3.10) Theorem Let C be a well-powered, finitely complete

(~,~) category with products. Let (Ai) iE I be an arbitrary

family of non-empty objects of C and let F. be an objectfilter on

A. for each i ~ I. Then 1

642

(i) ~ = {(KXi'Hfi)/(Xi'fi ) �9 [i for each i �9 I and there exists

a finite set J such that (Xi,fi) ~ (Ai,l) for all i �9 I-J} is a

basis for an objectfilter on A = KA . 1

(ii) pj([) = [j for each j �9 I, where [ = (~) is the objectfilter

generated by B. m

Proof (i) One easily verifies that B satisfies (OFB.I) and (OFB.2).

(ii) Firstly, let (X,f) E pj([), i.e. there exists

(y,g) = (~yi,~gi) e B such that pj(Y,g) < (X,f). It follows

from (3.9) that (Yj,gj) < pj(Y,g) < (X,f) where (Yj,gj) �9 [j so

that (X,f) �9 [j.

Conversely, suppose (X,f) E F and let (Y,g) = (~Xi,Kfi) , where --3

(Xi,f i) = (Ai,l) for each i ~ j and (Xj,fj) = (X,f). Then, by

(3.9) pj(Y,g) < (X,f) and since (Y,g) e [ = (~) (because

(Ai,l) E --1F for each i ~ j) it follows that (X,f) E pj(F)._ Thus

pj ([) = [j, where [ = (~) .

w 4 UNI-OBJECTS AND UNI-MORPHISMS

In order to generalize the definition of uniform spaces to abstract

categories, the idea of the "composition of relations" will be re-

placed by a more general concept.

(4.1) Definition Let C be a well-powered category with finite m

products and consider an A C 0b(C) with product (A • A,p I ,P2 ).

The

Diagram 5

Z ~ > X

"7' > Ay, A

643

with (X,f) ,(Y,g) ,(Z,h) E Sub(AxA) is said to (Pl ,P2 )lcOmmute

(or alternatively, (h,u,v) (p~ ,P2)-commutes with (f,g)) iff

plh = p, fu, p2h = p2gv and p2fu = p, gv.

(4.2) Definition Under the conditions of (4.1), diagram 5 is

called a (Pl 'P2)-pullback iff it (Pl 'P2)-c~ and for any other

(Pl ,P2 )-c~ square

Diagram 6 Z' u' ~ X

Y >A/A

with (Z~h') 6 Sub(AxA), there exists an h": Z' ~ Z such that

h' = hh".

Diagram 7

r

y >AxA

If no confusion is possible we shall simply call (Z,h) a (p, ,P2 )-

pullback of (X,f) and (Y,g) (resp., of (X,f) if (X,f) = (Y,g)) and

write (Z,h) = (X,f) ~ (Y,g).

(4.3) Examples (i) Let C be any well-powered category with

finite products. For any A 6 0b(C) and each (X,f) e Sub(A), with

I products (A xA,p, ,P2 ) and (X xX,p I ,p~) respectively.

644

Diagram 8

X • " J > X .X

Xx X -I:>~-F > AxA

is a (P, ,P2)-pullback square, for suppose

Diagram 9

X u > XxX

X~ X > AxA

(Pl 'P2 )-cOmmutes, where (y,g) E Sub(Ax A), then by using (2.1)

h ( ' 'v) Y ~ Xx X satisfies = pl u,p2 :

! I (f xf)h = (fplu,fp2v) = <p, (f x f)u,p 2 (f x f)v> = <plg,p2g) = g.

(ii) Let C = Set and consider any set A with Cartesian product

(A x A,p I ,P2) and arbitrary subsets X and Y of A x A with inclusion

maps ix: X ~ Ax A and iy: Y ~ A• A respectively. If

Z = Y ~ X = {(x,y) E A • there exists a z E A such that (x,z) E X

and (z,y) E y} with inclusion iz: Z ~ AX A, then one easily verifies

that

Diagram i0

Z .... ~ >

/.y 7" > A•

is a (Pl ,P2 )-pullback square. The maps u and v are defined by

u(x,y) = (X,Zxy) and v(x,y) = (Zxy,Y) for every (x,y) E Z, where

645

for each (x,y) �9 Z a z �9 {z �9 A/(x,z) �9 X,(z,y) �9 Y} has been xy

selected by the Axiom of Choice.

(iii) (Due to R.J. Wille) Let C = Vect R be the category of all

vector spaces over the field R of real numbers. For any vector

space A with subspace X of A x A, we consider the subspace

X ~ = {(x,y) I ~z �9 A such that (x,z),(z,y) �9 X }

of A x A. We shall show that there exist linear maps u,v such that

Xo x u > X

X c > A•

is a (Pl ,P2)-pullback diagram.

By using the Axiom of Choice, with each (x,y) E X o X we associate

a z(x,y) E Z(x,y) = {z E A I (x,z) E X, (z,y) e X} in the following

way: With (x,y) 0 = (0,0) E X o X we associate z(O,O) = 0 E Z(O,O).

Since X' = Xo X - {(0,0)} is well-ordered it has a least element

(x,y) I for which we choose a z(x,y) I E Z(x,y)1. With r(x,y) I ,

r E R, we associate rz(x,y) I . For the least element (x,y) 2 of

X' - {r(x,y) I r e R} select a z(x,y)2 E Z(x,y) 2 and with s(x,y)2,

s E R, associate sz(x,y) 2 . With r(x,y), + s(x,y) 2 , r,s E R,

associate rz(x,y), + sz(x,y) 2 . For the least element (x,y) 3 of

' E Z (x,y) X - {r(x,Y)1 + s(x,Y)2 I r,s �9 R} choose a z(x,y) 3 3"

Associate tz(x,y)3 with t(x,y) 3 , t �9 R. For all linear

combinations of (x,Y)i, i = 1,2,3, the associated

z(x,Y)i �9 Z (x,Y)i is determined. Continue this process

646

indefinitely.

One easily checks that the above diagram with u,v defined by

u(x,y) = (x,z(x,y)) and v(x,y) = (z(x,y),y) for all (x,y) �9 Xo X,

is a (Pl ,P2 )-pullback.

If C is a well-powered (E,M) category with finite products and

A E 0b(C) we shall write

M~(A xA) = {(X,f) E M(A• * (X,f) exists}, which is non-

empty since (A• A,I) E M * (A x A) .

(4.4) Proposition Let C be a well-powered (E,M) category with

finite products. Consider any A 6 0b(~) and

(X,f) , (Y,g) , (Z,h) E Sub(A xA) such that (X,f) ~< (Y,g) .

(i) If (h,u,v) (Pl ,P2) -cOmmutes with (f,f) then it also (p, ,p2)-

commutes with (g,g).

, M ~ (ii) If (X,f) (Y,g) E (A• A) then (X,f) ~ (X,f) ~< (Y,g) �9 (Y,g).

(iii) If (X,f) , (Y,g) E M ~ (A x A) and (X,f) ~ (Y,g) then

(X,f) ~ (X,f) ~ (Y,g) * (Y,g).

(4.5) Proposition Let ~ be a well-powered (~,~) category with

finite products and let A E 0b(~).

(i) (X,f) E M ~ (A x A) iff (X,kf) E M ~ (Ax A) .

(ii) If (X,f) E M~(A • and (X,f) ~ (Y,g) then (X,f) ~ (Y,g)

exists and is equivalent to (X,f) * (X,f).

It is now possible to generalize the definition of uniform spaces.

(4.6) Definition Let C be a well-powered, finitely complete D

(~,M) category. A pair (A,~),A e 0b(~) and ~ a subset of M(AX A),

is a uni-object in C iff N

(U0.1) (X,f) E ~ implies (A,A) ~< (X,f) , where A = (i,i) : A ~ A • A,

(U0.2) (X,f) E ~ implies (X,kf) E ~,

647

(X,f) �9 ~ implies that there exists a (Y,g) �9 M ~ (A • A) (U0.3)

such that (y,g) 6 ~ and (Y,g) �9 (Y,g) ~< (X,f),

(U0.4) (X,f), (Y,g) E ~ implies (X,f) N (y,g) �9 ~,

(U.05) (X,f) �9 ~ and (X,f) ~< (Y,g) imply (Y,g)�9 a.

If (A,~) is a uni-object, ~ will be called a uni-structure on A.

(4.7) Examples (i) If C = Se__~t and if (Pl 'P2 )-pullbacks are

interpreted as in example (4.3) (ii), uni-objects correspond pre-

cisely to uniform spaces.

(ii) Let C = Vect R and consider any real vector space A. Clear-

ly, any family e of subspaces of A• A satisfying the following

properties is a uni-structure on A: (a) A C X for any X �9 ~;

(b) X �9 ~ implies X' = { (y,x) I (x,y) E X} �9 ~;

(c) X,Y �9 ~ implies X n y �9 ~;

(d) x �9 e , x ~< Y imply Y �9 e;

(e) For any X �9 ~ there exist a Y �9 ~ such that y o Y ~< X

(see Example 4.3 (ii)).

The reader is urged to find some more examples by interpreting

the definitions of (P, ,P2 )-pullbacks and uni-objects in other

categories.

(4.8) Proposition Let C be a well-powered, finitely complete m

(E,M) category and consider any uni-object (A,~) in C. For any

(X,f) �9 ~ there exists 6u (Y,g) �9 M ~ (A • A) such that

(Y,g) �9 ~, (Y,g) ~ (Y,kg) and (Y,g) * (Y,g) ~< (X,f).

The definition of uniformly continuous maps between uniform spaces

can now also be generalized without difficulty.

(4.9) Definition Let C be a well-powered, finitely complete

(E,M) category and let (A,~) and (B,~) be uni-objects in C.

648

A morphism ~: A ~ B of ~ is a uni-morphism iff

(~ x ~)-i (X,f) E e for each (X,f) E 6.

(4.10) Proposition Let ~ be a well-powered, finitely complete

(~,~) category. The class of all uni-objects in C together with

all uni-morphisms in C form a subcategory Uni of C.

w 5 OBJECTFILTERS ON UNI-OBJECTS

In this section it will be shown that analogues of the results that,

in a uniform space every convergent filter is a Cauchy filter and

every Cauchy filter converges to each of its adherence points,

also hold in general categories.

(5.1) Definition Let ~ be a well-powered, finitely complete

([,~) category (with terminal object E). Let (A,~) be a uni-

object in C and F an objectfilter on A. m

(i) The objectfilter F converges to an m: E ~ A on (A,~) (and m

is called a limitmorphism of F) iff for any (X,f) @ e there exists

a (Y,g) E F such that (Y x E,g • m) ~< (X,f) .

(ii) [ is a Cauchy objectfilter on (A,e) iff for any (X,f) E

there exists a (Y,g) E ~ such that (y • y,g x g) < (X,f).

(iii) An m: E ~ A is adherent to an objectfilter F on (A,e) iff m

there exists an objectfilter G on A such that F C G and G con-

verges to m: E ~ A on (A,e).

(5.2) Remarks (i) Note that in a uniform space (X,~) ~5.1) (i)

means that a filter F converges to x E X iff given u E U there

exists an F E ~ such that F• ~ u. This is a generalization of

sequential convergence in metric spaces for, it is easily seen that

the Frechet filter associated with a sequence {x } converges to n

x iff {x n} converges to x.

(ii) Definitions (5.1) (ii) and (iii) are obvious generalizations

649

of the usual definition of Cauchy filters on uniform spaces and

a well-known result respectively.

( 5.3 ) Theorem Let C be a well-powered, finitely complete

(~,M) category. If F is an objectfilter on A which converges to

m: E ~ A on a uni-object (A,e) then F is a Cauchy objectfilter

on (A,e).

Proof Consider any (X,f) E e. By (4.8) there exists a

(y,g) E M ~(A xA) such that (Y,g) E ~, (Y,g) ~ (Y,kg) and m

(Y,g) ~ (Y,g) < (X,f). Hence (Y,g) ~ (Y,kg) ~ (Y,g)~ (Y,g) by

(4.5) (ii). Now, since F converges to m: E ~ A on (A,e), given

(y,g) E e there exists a (Z,h) E F such that (Z • E,h x m) ~ (Y,g),

i.e. there is an h': Z xE ~ Y such that h'g = h x m.

! ! Now, consider the products (A • ,P2 ) and (Zx Z,p, ,P2 ) and let

Hom(Z,E) = {e}. It will be shown that the

Diagram ii Z x Z

V

7

! !

(Pl ,P2)-c~ where u = h'(l,e) p, and v = h'( l,e~ P2 �9

650

Z,xZ /~P~ / >, Z <J~> h ~ ' ' ' ' " ~ ~ Z x E ", "7'

Z <e,1> ~ ExZ ~h >AxA

x E ~ x ~ > Ax A

7 Diagram 12

By using (2.1) and (2.2) , we have

= , = , ( ' ,mep I' ) gu gh'( l,e>p1' = (hx m) (l,e) p, (h,me) pl = hpl

and

kgv = kgh'(l,e) P2' = k(h x m) ( l,e} p~ = (mx h)k< l,e) p~

= (m• = (me,h) p~ = (mep~,hp2') .

! !

Hence, p, gu = hpl = Pl (hxh) and P2kgv = hp2 = P2 (hxh). Further-

more, since E is terminal it holds that ep~ = e' = ep2', where we

have put Hom(Z • = {e'}, so that p2gu = mep,' = me' =

I

= mep2 = p, kgv.

Thus diagram Ii (Pl ,P2 )-cOmmutes, so that

(Z x Z,h xh) <~ (y,g) �9 (Y,kg) ~ (Y,g) ~ (Y,g) ~< (X,f).

Hence, given any (X,f) E ~ there exists a (Z,h) E F such that n

651

(Z • Z,h x h) ~ (X,f), i.e. [ is a Cauchy objectfilter on (A,~).

(5.4) Theorem Let ~ be a well-powered, finitely complete

(E,M) category. If F is a Cauchy objectfilter on a uni-object

(A,~) in ~ and if m: E ~ A is adherent to [, then [ converges to

m: E~A.

Proof Consider any (X,f) E ~. By (U0.4) there exists an

(X~f')eM~(A• that (X~f') E ~ and (X~f') ~ (X~f')~ (X,f).

Since F is a Cauchy objectfilter, given (X~f') E e there exists a

(Y,g) 6 _F such that (Y x y,g • g) < (X',f') , i.e. there is a

! !

g :Y • Y ~ X' such that f'g = g • g. Furthermore, since m : E ~ A

is adherent to F there exists an objectfilter G on A such that

[ C G and ~ converges to m: E ~ A on (A,~). Hence, given

(X' ') ,f E ~ there exists a (Z,h) E G such that (Zx E,h x m) ~ (X~f'),

i.e. there is a g" : Z • ~ X' such that f'g" = h• m. Since F C G

it holds that (Y,g) E _G and therefore (Z',h') = (Y,g) N (Z,h) E _G,

which implies that there exist g, : Z' ~ Y and h, : Z' ~ Z such

that the

Diagram 13

Z < ~' Z ' ~' > "7'

! !

commutes. Since (Z,h) E G it follows from (OF.3) that there

exists an m': E ~ A such that (E,m') ~< (Z',h') , i.e. there is an

h 2 : E ~ Z' such that h'h 2 = m'.

It wiil now be shown that

652

Diagram 14 Y x E u > X'

X / ~ > AxA

w g,, (pl ,p2)-commutes, where u = g (i xg, h 2 ) and v = (hlh2e x i).

Diagram 15

7 ; < - > "/ ~ 7 > X

>A~A

We have

f'u = f'g' (i x gth 2) = (gx g) (i x glh 2) = gx gglh 2

= gxh'h 2 = g xm'

and

f'v = f'g"(hlh2e x i) = (hx m) (h, h2ex i) = hh,

= h'h2e xm = m'e xm.

Hence, p,f'u = Pl (g xm') = gPl = p~ (gxm), p2f'

= rap2

= g x h'h 2

h2exm =

v = P2 (m'ex m)

! I ! ! ! ! = P2 (g xm) and p2f u = m P2 = m epl = Pl f v, where

653

! I Pl "~ Y xE ~ Y and p~ : Y x E ~ E are projection morphisms.

Thus diagram 14 (Pl ,P2 )-commutes, so that

(y x E,g x m) <~ (X,f') + (X'f') ~< (X,f) . Hence, given (X,f) E

there exists a (y,g) E F such that (yx E,g• m) ~< (X,f), i.e.

F converges to m: E ~ A on (A,~).

[1 ]

[2]

[3]

[4]

[5]

[6]

REFERENCES

B. ECKMANN and P.J. HILTON. Group-like structures in general categories I. Math. Ann. 145 (1962), 227 - 255

M. FRECHET. Sur quelques points du calcul fonctionnel. Rend. Palermo, XXII (1906), 1 - 74

M. FRECHET. Les ensembles abstraits et le calcul fonctionnel. Rend. Palermo. XXX (1910), 1 - 26

H. HERRLICH and G.E. STRECKER. Category Theory. Allyn and Bacon, Inc., 1973

G. PREUSS. Allgemeine Topologie. Springer Verlag, 1972

A. WEIL. Sur les espaces a structure uniforme et sur la topologie g~n~rale. Hermann, 1938

CATEGORIES OF TOPOLOGICAL

TRANSFORMATION GROUPS

J. de Vries

I . INTRODUCTION

The theory of topological transformation groups (ttg's) forms a fasci-

nating and comprehensive realm in the world of mathematics, bordering on

the domains of abstract harmonic analysis, ergodic theory, geometry, dif-

ferential equations and topology. In this talk I cannot give you even a

flavour of the subject. Instead, I would like to discuss certain categories of ttg's. I shall use category theory in a rather "naive" way. Some catego-

ries of ttg's will be defined and investigated. In this context, the catego-

ries TOPGRP (all topological groups and continuous homomorphisms) and T0P

(all topological spaces and continuous functions) will be regarded as

"known", and ,most questions will be reduced to questions about these cate-

gories. In the attempt to do so, some "classical" problems and techniques

appear in a natural way. This shows that these problems are interesting,

not only because they turned out to be so in the development of the subject

(by "accident"), but also from the more "intrinsic" point of view of cate-

gory theory. However, the solutions of these problems have been given inde-

pendently of category theory. On the other hand, attempts to place certain

problems and their solutions into a categorical setting can be very illumi-

nating, and may require the definition of interesting new ~ategories.

This will be illustrated when we consider the problem of embedding arbitrary

ttg's in linear ttg's.

Let me first recall some definitions. A topological transformation group (ttg) is a system <G,X,~> in which G is a topological group (the

phase group), X is a topological space (the phase space) and ~ (the action of G on X) is a continuous function, ~: G x X -~ X, such that

~(e,x) = x; ~(s,~(t,x)) = ~(st,x)

655

for every x �9 X and s, t �9 G (e denotes the unit of any group under con-

sideration). A ttg with phase group G is often called a G-space. If ~ is

an action of G on X, continuous mappings t: X § X and ~ : G + X can be x

defined by

t x: = ~(t,x) =: ~ t

x

for t ~ G and x c X. Plainly, each t is an autohomeomorphism of X, and

the mapping ~: t~+ t is a homomorphism of the underlying group of G into

the full homeomorphism group H(X) of X. The closure of the group ~[G] in

X X is a semigroup (with composition of mappings as multiplication), called

the enveloping semigroup of <G,X,~>. For x E X, the subset ~ [G] of X is x

called the orbit of x under G. The orbits form a partition of X. The cor-

responding quotient space and quotient mapping are denoted by X/C and

c~: X + X/C , respectively.

We give here a few examples of ttg's . In all cases, G denotes a topo-

logical group, and %: G • G § G its multiplication.

(iii)

G (i) For every topological space X, define ~X := % x IX: G x G • X + G x X.

Then <G,GxX,~> is a ttg.

(ii) If H is a subgroup of G then G acts on the space G/H of left cosets

by means of an action ~, defined by ~t(sH) := tsH, s,teG .

If Y is a topological space then a mapping ~: G • Cc(G,Y) + Cc(G,Y)

can be defined by ~tf(s) := f(st), s,teG, f~ C(G,Y) . If G is local-

ly compact, then ~ is continuous, and <G,Cc(G,Y),~> is a ttg.

(v) Similarly, a ttg <G,LP(G),~> can be defined if G is locally compact

and ] ~ p < ~. Here LP(G) is the usual space of measurable functions

whose p-th power is integrable with respect to the right Haar measure

on G.

(vi) If f is a sufficiently nice ~n -valued function on an open domain

in A n, then the autonomous differential equation ~ = f(x) defines

an action of ~ on fl such that the orbits are just the solution

curves. (Due to this example, actions of R on arbitrary spaces are

often called flows.)

656

We cannot go into details here about the application of ttg's. Origina-

ting from differential geometry (the work of S. LIE on "continuous groups")

there is the theory of Lie groups and their actions, including work of

HILBERT, BROUWER, CARTAN and WEYL, to mention only a few names from the

classical period. Introductions into these areas can be found in [29] or

[9]. For applications in harmonic analysis, see for instance [40]. Related

is the theory of fibre bundles.

Ttg's are also studied in Topological Dynamics. This field of research

grew from classical dynamics and the qualitative theory of differential

equations in an attempt to prove theorems about stability, recurrence,

asymptoticity, etc. by purely topological means, whenever possible. The

most notable early work was by H. POINCARE and G.D. BIRKHOFF. A large body

of results for flows which are of interest for classical dynamics has been

developed since that time, without reference to the fact that the flows

arise from differential equations. Later, results were extended to general

ttg's. A landmark in this development towards abstraction is the book [19];

a more recent introduction is [16]. Here the link between ttg's and dynam-

ics is not so clear for a non-specialist. More closely related to differ-

ential equations are books like [8] or [21]. Also in some books on differ-

ential equations one can find results on flows. See for instance [31] or

[25] *). These theories are "local", in the sense that questions are asked

like "what does the w-limit set look llke?"; "what happens in the neighbour-

hood of a fixed point?"; etc. Related are the "global" theories of SMALE

and others, where the object of study is vector fields on manifolds. For

an introduction, see [2], and for applications, [I].

In the development of the theory of ttg's an important role has been

played by the quotient mapping c~: X + X/C for a ttg <G,X,~>. Let me for-

mulate here two related problems:

(i) If G is compact, then c is a perfect mapping, and there exists a

nice relationship between the topological properties of X and X/C .

In this context, also the normalized Haar measure of G can be used.

.) For flows in the plane, see also [7].

657

For which ttg's with a non-compact phase group does there exist such

a nice relationship? Paracompactness of X/C turns out to be of partic-

ular interest.

(ii) A (global) continuous cross-section of a ttg <G,X,~> is a pair (S,T)

with S S X and ~: X + G a continuous function such that, for every

x e X, T(x) is the unique element of G for which ~(T(x),x) E S. It is

easily seen that <G,X,~> has a continuous cross-section iff it is

isomorphic as a G-space with <G,GxY,~> for some space Y. In that

case, Y, S and X/C are homeomorphic. The question of which ttg's

have such a global continuous cross-section is important, not only

in abstract theories, but also in the study of flows ("which flows

are parallelizable?").

These two problems are also related to general questions in Topological

Dynamics, dealing with the structure of orbit closures. See e.g. [22].

Concerning the relationship between (i) and (ii), we confine ourselves to

the remark that in order to prove that certain ttg's have a global contin-

uous cross-section one usually shows first the existence of local contin-

uous cross-sections; then, using nice properties of X/C and G, these are

pasted together to a global one. Cf. for instance [34], [27], [22] and the

references given there.

A number of solutions of the following problem use also the existence

of local cross-sections for certain ttg's. The problem is

(iii) Which ttg's <G,X,~> can be embedded in a topological vector space V,

or even in a Hilbert space, in such a way that the ~t's become re- t

strictions of invertible linear operators p , t e G, such that

<G,V,p> is a ttg.

For flows, see for example (the proof of) BEBUTOV's theorem and generaliza-

tions thereof in [30], [26] or [23]. For actions of Lie groups G and embed-

dings in Hilbert G-spaces using the method of local cross-sections (or,

more general, of slices), see [33] and the references given there. We shall

return to problem (iii) in the last part of this paper. First, I shall in-

dicate why problems (i) and (ii) are also interesting from a categorical

point of view.

658 2. THE CATEGORY TTG

2.1. A mo~hism of rig's from <G,X,~> to <H,Y,o> is a morphism

(~,f): (G,X) + (H,Y) in the category TOPGRP • T0P for which the diagram

GxX

~• 1 HxY

> X

Y O

.) commutes. Notation: <~,f>: <G,X,~> § <H,Y,o>. Here, f will be called a

~-equivari~t mapping; an ]G-equivariant mapping will just be called equiv-

ariant.

Let TTG denote the category having the class of all ttg's as its ob-

ject class (also ttg's with an empty phase space are admitted). The mor-

phisms in TTG are the above defined morphisms of ttg's, with coordinate-

wise composition.

2.2. Important for the investigation of the category TTG are the following

forgetful functors, whose obvious definitions we leave to the reader:

K: TTG --+TOPGRP x TOP;

G: TTG ~ TOPGRP;

S: TTG --~ TOP.

These functors forget all about actions, so they cannot be expected to

reveal much about the "internal" structure of ttg's. In this respect, the

following functor may be expected to be more useful:

$I: TTG ~ TOP.

It is defined in the following way. For an object <G,X,~> in TTG, set

SI<G,X,~>: = X/C , the orbit space of <G,X,~>. If <~,f>: <G,X,~> § <H,Y,o>

*) The products here are ordinary cartesian products, i.e. products in the category T0P. In this context, we shall consider TOPGRP just as a sub- category of TOP, and we shall always suppress the corresponding inclusion functor.

659

is a morphism in TTG, then f maps each orbit of X into an orbit of Y, hence

there is a unique continuous function f': X/C + Y/C ~ such that f' o c~ =

c o o f. Now set Sl<$,f>: = f'

2.3. THEOREM. The functor K: TTG § TOPGRP x TOP is monadic. Consequently,

TTG is complete, and K preserves and reflects all limits and all monomor-

phisms.

PROOF. Let C: = TOPGRP x TOP, and define a functor H: C § C by means of

the assignments

H: S(G,X)~ ~-+ (G,G• on objects;

L($,f) 6-+ (~,$xf) on morphisms.

Some straightforward arguments show that by

G n(G,X): = (IG,q ~) and ~(G,X): = (IG'~X)'

(G,X) any object in C, two natural transformations

q: I c § H and ~: H 2 + H

~ G are defined. Here q (x): = (e,x) and ~x(S,(t,x)): = (st,x) for s, t E G

and x ~ X. It is easily verified that the triple (N,q,~) satisfies the def-

inition of a monad (cf.[28], Chap.Vl). The algebras over this monad are

easily seen to be the systems ((G,X), ($,~)) with (G,X) an object in C,

= IG, and 7: G x X § X a morphism in TOP making the diagrams

X

G qX 1 x ~

> G • X G x (G • X) G > G • X

X G x X - - > X

cormnutative, i.e. ~ is an action of G on X. So the algebras over (H,q,~)

can unambiguously he identified with objects in TTG. In doing so, the mor-

660

phisms between such algebras become morphisms in TTG, and the category of

all algebras over (H,n,~) turns out to be isomorphic (can be identified

with) TTG.In making this identification, K corresponds to the forgetful

functor of this category of algebras to C; this is equivalent to saying

that K is monadic.

Now the remaining statements in the theorem are a direct consequence

of the general theory of monads (cf.[28], Chap.Vl). D

2.4. COROLLARY. The functor K: TTG § TOPGRP x TOP has a left adjoint F, defined by the rules

~(G,X) ~-+ <G,GxX,~> on objects;

F: ((~,f) i-+ <~,~• on morphisms.

PROOF. Either by the theory of monads, using the identification of TTG with

the category of algebras over (H,n,~) as indicated in the proof of 2.3, or

by a direct argument, showing that for every object (G,X) in T0PGRP • T0P

the arrow

n(G,X ) : (G,X) > (G,G•

has the desired universal property.

2.5. The unit of the adjunction of F and K is the natural transformation n

(cf. also the proof of 2.4); the counit is given by the arrows

<]G,~ > ~<G,X,~>: <G,GxX,~> - > <G,X~>

Therefore, we may call the objects <G,G• in TTG free ttg's in TTG. (compare [24], p.231). This terminology can cause some confusion9 because

usually a ttg <G,X,~> is called free if ~tx = x for some x �9 X implies

t = e; we shall use the term strongly effective for this notion. It is ob-

vious that a free ttg is strongly effective, but the converse is not gen-

erally true; a well-known class of counterexamples is provided by groups G

which are subgroups of topological groups X such that the quotient mapping

of H onto the space of right cosets does not admit a continuous section

661

(let G act on X by left translations). The free ttg's are plainly just the

ttg's which have a continuous global cross-section.

There is yet another way in which we arrive at the need of character-

izing ttg's with continuous global cross-sections. Indeed, as in any ad-

joint situation, we have not only a monad, but also a comonad which is de-

fined by the adjunction (F,K,q,$). Thus, in TTG, we have the comonad

(FK,~,FnK).

2.6. THEOREM. The coalgebras for the comonad (FK,$,FqK) in ITG are the sys-

tems (<G,X,~>,(S,u)) with (S,u) a continuous cross-section of <G,X,~>. The

morphisms of coalgebras from (<G,X,~>,(S,u)) to (<H,Y,o>,(T,v)) are the

morphisms <~,f>: <G,X,~> § <H,Y,o> in TTG with f[S] E T.

PROOF. Straightforward. For details, cf. [39], p. 92. D

2.7. As is well-known, eomonads give rise to a cohomology theory (see e.g.

[28], Chap.VII,w It would be interesting to investigate how this can be

used with respect to the above mentioned comonad, (if not in TTG, then re-

stricted to a suitable subcategory).

2.8. We shall show now that the category TTG is cocomplete, but that the

functor K does not have n~ce preservation properties for colimits. In view

of Beck's theorem (cf.[28], p.147) and theorem 2.3 above, certain coequal-

izers are preserved by K. An example where K does not preserve the coequal-

izer will be given in 2.13 below. The bad behaviour of K with respect to

colimits is due to the functor S. Indeed, we have the following results:

2.9. LEMMA. The functor G: IIG § IOPGRP has a right adjoint. Hence it pre-

serves all colimits and all epimorphisms. D

2 . 1 0 . LEMMA. The functor SI: TIG § TOP has a right adjoint. Hence it pre-

serves all colimits and all epimorphisms.

2 . ] 1 . COROLLARY. The funotor K: TTG § TOPGRP x TOP preserves and re f l ec t s all epimorphisms.

662

PROOF. Reflection: K is faithful.

Preservation: If <~,f>: <G,X,~> § <H,Y,o> is epic in TTG, then by 2.9, ~ is

a surjection. Hence f maps each orbit of X onto some orbit in Y. By 2.10,

S]f maps X/C onto Y/C ~ . Combining these results it follows easily that f

is a surjection. D

2.12. The category TOPGRP • TOP is well-powered and co-(well-powered).

Since the functor K preserves all monomorphisms and all epimorphisms, the

category %7G is well-powered and co-(well-powered) as well. In addition,

IIG is complete. Hence theorem 23.11 in [24] implies that the category I%G

has all coequalizers.

2.13. EXAMPLE. Let <G,Y,o> be a ttg. Call an equivalence relation X in Y

invariant if, considered as a subset of Y • Y, it is invariant under the

coordinate-wise action ~ of G on Y • Y. In order to obtain an example which

shows that the functor K (or rather, the functor S) does not preserve co-

equalizers, it is sufficient to construct a ttg <G,Y,o> and an invariant

equivalence relation X in Y such that there exists no continuous action of

G on Y/X making the quotient mapping g: Y * Y/X equivariant.

To this end, take G: = Q, the additive group of the rationals,

Y: = ~ • ([0,1] • ~), and X: = AQ • R, where AQ is the diagonal in Q • Q

and R is the equivalence relation in [0,1] • ~ obtained by identifying

all points (O,n), n E ~, with each other. If we consider the action

of ~ on Y = ~ • ([0,1] x ~), then the equivalence relation [0,l]x~

X in Y is invariant, and there is only one candidate ~ for an action of Q

on Y/X which makes the quotient mapping g: Y § Y/X equivariant. Now conti-

nuity of ~: Q x (Y/X) § Y/X can easily be seen to imply the equality of the

following two topologies on Y/X: (i) the quotient topology induced by g and

(ii) the product topology obtained by identifying Y/X with ~ • • ~)/R),

where ([0,1]• has its usual quotient topology. It can be seen, however,

that these two topologies do not coincide. For details, cf. [39], 3.4.4.

2.14. The easiest way to prove that TTG is cocomplete is by invoking theorem

23.13 in [24]. First, we recall some definitions. An epi-sink in a category X

is a family {fi: Xi § X}icI of morphisms in X such that for every pair of

663

morphisms g, h: X § Y in X the condition gfi = hfi for all i ~ I implies

g = h. The category X is called strongly co-(well-powered) if for every

set-indexed family {X. I i e I} of objects there is at most a set of objects 1

X in X for which there exists an epi-sink {f.: X. § X} The theorem re- l z icl"

ferred to above reads as follows:

If the category X is complete and well-powered, then the following

are equivalent:

(i) X is strongly co-(well-powered);

(ii) X is cocomplete and co-(well-powered).

Observe that the categories TOPGRP and T0P are complete and well-pow-

ered, and that they satisfy condition (ii). Hence these categories are

strongly co-(well-powered). We shall use this in proving the following

2.]5. LEMMA. The category TTG is strongly co-(well-powered).

PROOF. Let {<~i,fi>: <Gi,Xi,~i > § <G,X,~>}ic I be a set-indexed epi-sink in

TTG. Since left adjoints preserve epi-sinks, if follows from 2.9 that

{@i: Gi § G}iel is an epi-sink in T0PGRP. This allows G only to be taken

from a set of possible topological groups. Similarly, 2.]0 implies that

{S]fi: Xi/C + X/C }ie I is an epi-sink in TOP, leaving for X/C only a

set of possibilities. Plainly, card(X) ~ card(G).card (X/C), hence there

is at most a set of possibilities for X. Finally, for each G and each X

there is only a set of actions of G on X. So there is at most a set of ob-

jects in TTG from which <G,X,~> can be taken.

2.]6. THEOREM. The category TTG is cocomplete.

PROOF. Clear from 2.14 and 2.]5.

2.]7. In [39], a different proof of the cocompleteness of TTG is given,

using a technique which is a generalization of the construction of a "ca-

nonical" extension of the action of a subgroup to an action of the whole

group. It is also related to the construction of "induced representations".

Both techniques are very important in the theory of actions of compact

groups. See [9] and [40]. The methods, used in [39] also indicate how exam-

664

ples can be constructed which show that K does not preserve all coproducts.

3. SUBCATEGORIES OF TTG

3.;. Let A and B denote subcategories of TOPGRP and TOP, respectively, and

set X: = K§ x B]. Then X is a subcategory of TTG. The restrictions and

corestrictions of the functors K,G and S will be denoted by the same sym-

bols; so we have K: X § A x B, G: X § A and S: X § B.

3.2. If one wants to shc~ that K: X § A • B is monadic using the same meth-

ods as in 2.3, one has to require, among others, that G x X is an object in

B for every object (G,X) in A x B. This condition appears to be rather

harmless at first sight. However, a large portion of Topological Dynamics

deals with actions of discrete groups on compact Hausdorff spaces (cf.[]6]).

So one might try to apply category theory in this field by taking B: = COMP

(the category of all compact Hausdorff spaces) and A a category having dis-

crete groups as objects. Then the above condition is only fulfilled if the

objects of A are all finite. For Topological Dynamics, the restriction to

finite groups is unacceptable. For other parts of the theory of ttg's, ac-

tions of finite groups on compact spaces is very important: it is one of

the corner stones of the general theory of actions of compact Lie groups

(cf.[29~, p.222).

Although monadicity of the functor K: X ~ A x B may be unattractive

in view of practical purposes, K and X do have nice properties under rather

mild conditions. The proofs of the following propositions can be found in

[39], section 4.

3.3. PROPOSITION. Suppose that the inclusion functor of B into TOP prese~ses

all limits. Then the functor K: • + A x B creates all limits. Hence, if A

and B are complete, then so is X, and all limits and monomorphisms are pre-

served and reflected by K.

3.4. PROPOSITION. Suppose B is a full subcategory of TOP. If either A S B

or B is productive and closed hereditary, then K preserves and reflects

monomorphisms. D

665

3.5. If one wants to show that X is cocomplete or that K: X § A x B pre-

serves epimorphisms, then the proofs given for [[G cannot be adapted to

the present situation, unless the restricted functor 51: X + TOP actually

sends X into B. So this brings us directly to the first problem, mentioned

in section 1. In addition, something must be known about the epimorphisms

in a.

Although the question of which nice properties of the phase space of

a ttg are inherited by the orbit space (and under what circumstances:) is

very interesting, the following proposition avoids this problem (for details,

cf.[39], section 4).

3.6. PROPOSITION. Suppose that the following conditions are fulfilled:

(i) Epimorphisms in A have dense ranges;

(ii) B is a full subcategory of HAUS, having a terminal object;

(iii) If Y is an object in B and A is a closed subset of Y, then YUAY is

inB.

Then the functor K: • § A x B preserves and reflects epimorphisms. B

3.7. The conditions (ii) and (iii) are rather mild. Yet the above proposition

is of restricted applicability, because of condition (i). Indeed, it is

still unknown to me whether the category HAUSGRP satisfies condition (i).

It is known, however, that COMPGRP does~ Of course, the preceding proposi-

tion can also be applied to the very important case (which we neglected

untill now) that the category a has only one object, a fixed topological

group G, and only one morphism, namely I G. In that case, the category X

will be denoted B G (the category of all G-spaces with phase space in B).

3.8. I want to make now a few remarks concerning reflective subcategories

of IYG. I shall restrict myself to the question of the "preservation of

reflections" by the funetor K for only one particular case. In view of the

following lemma, it is the functor S which causes difficulties.

3.9. LEMMA. Suppose that X: = K+[A x B] is a reflective subcategory of TTG

and that B has a final object. Then A is a reflective subcategory of TOPGRP.

In addition, if <G,X,~> is an object in TYG and

666

<~,f> : <G,X,~> + <H,Y,O>

is its reflection into X, then 4: G + H is a reflection of G into A.

PROOF. This is a consequence of the fact that under rather mild conditions

a functor having a right adjoint (c.q. the functor G) preserves reflections.

For a different proof, see [39], p. 133.

3.10. It is not difficult to show that K§ • COMP] is a reflective

subcategory of TYG. So by the preceding lemma, the reflection of an object

<G,X,~> of TTG into this subcategory has the form

<~G,f> : <G,X,~> + <Gc,y,o>,

where aG: G § G c is the Bohr compactification of G. In general, f: X + Y

is not the reflection of X into COMP. In fact, there are examples which

show that Y can be a one-point space even if X is a non-trivial compact

Hausdorff space. The problem whether Y is trivial or not has been important

in Topological Dynamics, and is related to many interesting questions. This

follows from the following observation (cf.[39], p.141):

Using the above notation with X a compact Hausdorff space, the morphism

<IG,f> : <G,X,~> § <G,y,oa>

where sa(t,y): = a(a(t),y), (t,y) c G x y, is just the maximal equicontin-

uous factor of <G,X,~>.

For definitions and results concerning maximal equicontinuous factors,

cf. []7], [18] and [35].

3.]]. Using the notation explained in 3.7, it is not difficult to show that

COMP G is a reflective subcategory of T0P G, for any topological group G.

Let the reflection of the G-space <G,X,~> into COMP G be denoted by

<IG,k> : <G,X,~> § <G,Z,~>.

667

If G is discrete, k: X § Z is just the reflection of X into COMP. There

are examples which show that for non-discrete groups (e.g. G=~ ) k: X § Z

may be not the reflection of X into COMP. See [11]; it can also be shown

that <G,G,%> gives such an example, provided RUC*(G) @ C*(G). For the in-

equality RUC*(G) @ C*(G), cf. [13]. The reflection of <G,G,%> into COMP G

plays an important role in Topological Dynamics; there it is called the

greatest (or maximal) G-~nbit. See [10] and the references given there.

I do not know whether the mapping k: X § Z is a topological embedding

if X is a Tychonoff space. I have some partial results, including the cases

that X is locally compact Hausdorff or that <G,X,~> has the form

<G,(G/H)• with ~t(sH,y): = (tsH,y), t, s �9 G, y e Y, where H is a closed

subgroup of G. See [37].

4. THE CATEGORY TTG. AND GENERALIZATIONS

4.1. The objects of TTG, are the same as the objects of TTG, viz. the ttg's.

The categories differ from each other with respect to their morphisms. We

shall first give a brief motivation for the definition of the morphisms in

TTG,. The idea stems from the following problem: Given a ttg <G,X,~>, does

there exist a ttg <H,Y,o> such that

(i) Y is a topological vector space ;

(ii) Each o t, t �9 H, is an invertible continuous linear operator on Y*);

(iii) X can be embedded in Y as an invariant subset in such a way that

I t �9 H} ~[G] = {Otix 6

If <G,X,~> is effective (i.e. t # s if t @ s), then it follows from (iii)

that we obtain a homomorphism ~: H § G such that for every t e H the fol-

lowing diagram commutes

~(t) X ~X G

o Y' >Y H

*) Such a ttg <H,Y,o> will be called a liz~gar ttg.

668

Here f: X + Y is the embedding mapping of X into Y.

For more details about the above mentioned linearization problem we

refer to section 5. At this point we are only interested in the diagram

which expresses the relationship between ~ and f. We shall use it in the

following definition:

4.2. The object class of TTG, is the class of all ttg's. A morphism in TTG,

from <G,X,~> to <H,Y,o> is a morphism (c~ (G,X) + (H,Y) in the catego-

ry TOPGRP ~ x TOP such that for every t c H the diagram in 4.1 commutes

(now f is not necessarily an embedding). Notation: <c~ <G,X,~> *

<H,Y,o>. The composition of the morphisms in TTG, is defined coordinate-

wise.

4.3. The obvious forgetful functor from TTG, to TOPGRP ~ x TOP will be de-

noted by K,. It can be shown that this functor preserves all colimits.

Using this, it is fairly easy to construct an example which shows that the

category TTG, is not complete (the example is related to the one in 2.13).

It can also be shown that K, preserVes all monomorphisms. In particular

it follows that TTG, is well-powered. We shall see in 4.7 below that TTG,

has a coseparator. According to Theorem 23.14 in [24], a complete, well-

powered category having a coseparator is cocomplete. It follows, that TTG

is not complete.

4.4. We want to say something more about the existence of coseparators in

TTG, , also because such objects are related to the general embedding prob-

lem, mentioned in 4.1. The notation will be similar to the notation in sec-

tion 3, except for some obvious modifications. Thus, A and B are subcate-

gories of TOPGRP and TOP, respectively, and X.: = K.§ ~ x B]. Moreover, for any object (G,X) in TOPGRP ~ • TOP, the evaluation mapping f F-~ f(e):

G Cc(G,X ) + X will be denoted by ~X"

4.5. LEMMA. For every object (G,X) in TOPGRP ~ x TOP with G a locally com- op G

pact Hausdorffgroup, the pair (<G,Cc(G,X),~> , (I G ,~X )) is a co-universal

arrow for (G,X) with respect to the functor K,.

669

PROOF. Consider the following diagram:

. op .X. (I G ,6 G)

<G,Cc(G,X),~> (G,Cc(G,X)) > (G,X)

1 I 1 i

<~op,foa(_)o~> Ii / / % (~op fo2 (_)o~)I op) [ i f ) i I I I

<H,Y,o> (H,Y)

_ = : H + Y, so that foa(-)o@: y ~-+ foo o@: Here for every y e Y, a(y): Oy _ y

G § X. Observe, that local compactness of G is needed to ensure that

~: G • Cc(G,X) § C (G,X) is continuous. D C

4.6. COROLLARY. If A has a generator G which is locally compact Hausdorff

and if B has a coseparator x such that Cc(G,X) is an object in B, then

<G,Cc(G,X),~> is a coseparator in X..

PROOF. Apply the dual of Prop. 31.11 of [24].

4.7. EXAMPLES.

(i) Let E be the indiscrete 2-point space. Then the ttg <Z,~,~> is a

coseparator in TTG,.

(ii) <Z,[0,]~,~> is a coseparator for the full subcategories of TTG,,

determined by all ttg's with a Tychonoff, resp. with a compact T2,

phase space.

(iii) If G is a fixed locally compact Hausdorff group, then <G,Cc(G,[0,1],~>

is a coseparator for the full subcategory of T0P G defined by all

Tychonoff G-spaces (not for COMP G, unless G is discrete).

4.8. In the remainder of this section, A shall denote the full subcategory

of TOPGRP, defined by all locally compact Hausdorff groups, and X,: :

K +[A ~ x TOP] (so we take B: = TOP). In this case, it follows i~ediately

from 4.5, that the functor K : X + A ~ x TOP has a right adjoint. This

follows also from our next theorem:

670

4.9. THEOREM. The functor K,: X, § A ~ x T0P is comonadic. Consequently,

• is a finitely cocomplete (for so is A ~ • IOP), and all existing co-

limits and all epimorphisms are preserved and reflected by K,.

PROOF. Details will be published elsewhere. We only mention that in the

proof essential use is made of the canonical homeomorphisms

Cc(G• ~ Cc(G,Cc(G,X)) ,

C (GxX,X) = Cc(X,Cc(G,X)) C

G any locally compact Hausdorff space and X an arbitrary topological space. U

4.10. If G is a fixed locally compact Hausdorff topological group, then

the category T0P G can be considered as a subcategory of TTG,. Similar to

4.9, one shows that T0P G may be considered as a category of coalgebras over

a comonad in TOP. On the other hand, similar to 2.3, T0P G may be considered

as a category of algebras over a monad in TOP. The monad and the comonad

considered here are nicely related: it can be shown that they are adjoint

to each other according to the definition given in ~15]. Although this

seems to be known, I could find no references to this fact in the literature.

4.]]. One might conclude from the above remarks that ttg's with locally

compact phase groups are the nice objects which deserve further study. Al-

though this conclusion is true as far as it concerns the applications, from

a categorical point of view there is a much nicer class of objects. Indeed,

the homeomorphisms used in the proof of 4.9 are an indication of the fact

that we should work in the cartesian closed category KR df all k-spaces.

The proper objects are the systems [G,X,~] where G is a k-group (i.e. a

group G with a k-topology making the mapping (s,t)~-~st-l: G | G § G con-

tinuous), X is a k-space and ~: G | G § X is a continuous mapping satisfy-

ing the usual equations (here | denotes the product in the category KR:

the k-refinement of the cartesian product). With the class of these k-ttg's

as object class, one can form the categories k-TTG and k-TTG (morphisms

similar to TTG and TTG,, respectively). The study of these categories is

initiated in [39].

671

5. LINEARIZATION OF ACTIONS

5.1. The general problem which we described in 4.1 has a trivial solution

if we work in the category TTG,. Indeed, lemma 5.2 below shows that the

only condition which must be imposed on a ttg <G,X,~> in order that the

action can be linearized, is that X can be embedded in a topological vector

space. If we restrict ourselves to Hausdorff topological vector spaces, this

means exactly that X is a Tychonoff space. In the following len~na, G d is

the group G with the discrete topology, and ~: G d § G is the identity map-

ping.

5.2. LEMMA. ff <G,X,~> is a ttg with X a Tychonoff space then there exists

a morphism <1~ <G,X,~> § <Gd,V,~> in TTG. such that V is a topological

vector space, ~ is a linear action, and f: x § v is a topological embedding.

PROOF. There exists a topological embedding g: X + A m for some cardinal

number K. Moreover, the mapping E: x~-+ ~x: X § Cc(G,X) is a topological

embedding. Hence f: = Cc(G,g)o E is a topological embedding of X into the

topological vector space Cc(G,~K ) =: V. Plainly, <Gd,Cc(G,~,~> is a ttg,

and <1~ is a morphism in TTG,. 0

5.3. In order to make the problem more interesting, the following extra

conditions will be imposed. First, a G-space <G,X,~> should be linearized

in a G-space rather than in a Gd-space. Second, if X is a metric space,

then <G,X,~> should be linearized in a Hilbert G-space. And finally, a large

class of G-spaces should be linearized simultaneously in one and the same

linear G-space.

The proof of the following theorem is a modification of the proof of

lemma 5.2 above. Observe, that the apparent relationship with lemma 4.5

has a categorical background (cf. [24], Prop. 19.6).

5.4. THEOREM. Let G be a locally compact Hausdorff group and let K be a

cardinal number. Then every G-space <G,X,~> with X a Tychonoff space of

weight <- K can equivariantly be embedded in the linear G-space <G,Cc(G,I~)K

~K> . 0

672

Using similar ideas, in combination with results from [5] and [32], the

following theorem can be proved. Here H(m) is the Hilbert sum of K copies

of the Hilbert space L2(G), and the action a(K) induces on each copy of

L2(G) a "weighted" right translation. For a proof, see [36].

5.5. THEOREM. Let G be a sigma-compact locally compact Hausdorffgroup and

let K be a cardinal number. Then there exists a linear G-space <G,H(K),s(~)>

in which every G-space <G,X,~> with X a metric space of weight ~ K can be

equivariantly embedded. D

5.6. Let G be as in 5.5. and assume, for convenience, that G is infinite.

The weight of the Hilbert space H(K) equals m. ~(G), where ~(G) is the

weight of G (for compact groups this is well-known; the proof for the non-

compact case can be found in [36]). If we take K = W(G), then H(m) is iso-

morphic to L2(G). So there is an action ~ of G on L2(G) such that <G,L2(G),o>

is a linear ttg in which every metric G-space <G,X,~> with W(X) ~ W(G) can

be equivariantly embedded. No explicit description of ~ can be given in

this case. However, there is an action T of G on L2(GxG) which can easily

be described explicitly, such that <G,L2(GxG),T> is a linear ttg in which

every metric G-space <G,X,~> can equivariantly be embedded, provided

w(X) ~ /(G), the LindelSf degree of G). The proof is highly non-categorical;

see [38].

5.7. One of the most notable early results on linearizations of actions

is BEBUTOV's theorem. See [26] and also [23]. These results have applica-

tions in the theory of differential equations. Also related to differen-

tial equations are the results in [12]. In these three papers, only actions

of ~ are linearized. Actions of Lie groups are considered, among others,

in [33]. See also [34] and the references given there. Linearizations

in Hilbert spaces of actions of more general locally compact groups appear

in [3] and [5], where earlier work of COPELAND and DE GROOT (E|4],[20])

was generalized. More information about the history of these results can

be found in [3] and in [6].

673

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Math. Annalen ]44 (1961), 80-92.

674

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Illinois J. Math. 9(1965), 381-398.

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ON MON01DAL CLOSED TOPOLOGICAL CATEGORIES I

Manfred B. Wischnewsky_

The aim of this paper initiated by L.D. Nel's talk at the Con-

ference on Categorical Topology at Mannheim (1975) is to give two dif-

ferent characterizations of monoidal closed topological categories.

Both of these characterizations include as special instance the

Herrlich-Nel results on cartesian closed (relative) topological cate-

gories*over the base category Sets ([11], ~3~ ). The main tools in this

paper are a generalization of the Special Adjoint Functor Theorem -

the Relative Special AdJoint Functor Theorem - and the Dubuc-Tholen

theory of Adjoint Triangles. As corollaries we obtain a characteri-

zation of monoidal closed (relative) Top-categories over wellbounded

(= locally bounded) categories and over cocomplete, wellpowered and

cowellpowered monoidal closed categories.

It is assumed that the reader is familiar with the notation and

the content of ~12]. All other notions which are used in this paper

are briefly recalled.

w I Factorizations and Generators, Review.

Let us first recall some of the basic notions and propositions on

factorizations and generators (see e.g. ~9], ~6], ~8a]). Let A be a cate-

gory. For two morphisms e : A-~B and m : C-~D we write e~m if

every commutative diagram ge = mf can be made commutative by a unique

morphism w : B---+C. If PcMor(A) then let P~:= [e: e~m for all mEP]and

P+:=[m: eCru for all e~P).Let E,McMorA.Then (E,M) is called a prefactori-

nation in A if E ~= M and M@= E. A prefactorization (E,M) is called a fac-

torization if every morphism f in A is of the form f = me with e ~E

and m ~M. A factorization (E,M) is called proper if every e ~ E is an

*) Relative topological category ~ E- reflective subcategory of an

(absolute) topological category.

677

eplmorphism and every m 6M is a monomorphism. We say that a category

has a M-factorization for a class M of A__-morphisms if A has a (Me, M) -

factorization.

(1.1) LEMMA. Let (E,M) b_ee a prefactorization i_nn A and consider the

following assertions:

(a) Every E is an epimorphism.

(b) pq EM implies qQM.

(c) Every equalizer is in M.

(d) Every section is in M.

(b)

Then (a) ~ ~ (d)

while (d) )(a) if A admits either finite products or weak cokernel

pairs.

(1.2) LEMMA. Let (E,M) be a prefactorization. Then

(a) M contains all isomorphisms and is closed under composition.

(b) Every pullback of a M is a M.

(c) The fibred product of_ m i : A i ~ B, i6I, is a M if each m i is a M.

(d) If mn is a M so is n provided m is either a M or a monomorphlsm.

In the sequel let A be a category with a proper (E,M)-factorization.

By a subobject of AEA I mean a morphism i : U >A in M. A is M-well-

powered if for each A~ A the class of subobjects of A is small up to

Isomorphisms.(Dually E-cowellpowered). If the u i : U i ~A, i E I,

are M-subobJects of A and if the monomorphlsms u i admit an intersection

then the intersection is again an M-subobject of A. AnE-generator of A

is a small full subcategory G of A such that for each A6 A_ the family

of all morphisms G >A with domain G6G is in E. We say that a family

678

qi : Bi )A, i~ I, is in E if whenever we have m : C )D in M and

morphisms u i : B i ~C and f : A ~D such that for each i~ I

fqi = mui " then there is anA-morphism w : A >C with f = mw (Freyd-

Kelly [9]). Dually one defines M-cogenerator.

(1.3) LEMMA Let A be a category wlth a proper (E,M)-factorization.

If A has smE-generator and admits finite intersections of sub-

obJeots, it is M-wellpowered([ )

(I .4) PROPOSITION ([9]). Let A admit coproducts or be finitely complete.

Consider the following statements:

(a) G is a small dense subcategory of A.

(b) G is a generator with respect to a proper (E,M)-factorization.

(c) Whenever f ~ g : A ~B there is a GQG and an h : G >A such that

Cgh.

Then (a) ) (b)----~(e). Moreover (c)---~(b) if the factorizatlon (E,M)

is the (epi, ext mono)-factorization.

2 The Relative Special Adjoint Functor Theorem (RSAFT).

The classical Special Adjoint Functor Theorem (SAFr) - a powerful

categorical tool - replaces the solution set condition of the Adjolnt

Functor Theorem by the condition that the domain category has a co-

generator and is wellpowered. Typical applications of the SAFT are

Watt's Theorem (see e.g. 4 or the existence of the Cech - Stone -

Compactiflcation. If one wants to apply the dual of the SAF2 then one

often has the situation that the domain category has a generator but

is unfortunately not cowellpowered. Hence the SAlT can not be applied.

In this paragraph I'll prove a useful generalization of the SAlT - the

Relative Special Adjoint Functor Theorem (RSAFT) by assuming that all

data are relative to a proper (E,M)-factorization. The proofs are

679

straightforward and hence only sketched or omitted.

(2.1) THEOREM (Relative Special Initial Object Theorem).

Let A be a complete category with a proper (E,M)-factorization. Assume

that A has a M-cogenerator Q and that every class of M-subobjects

of ~ 0bject A in A has an intersection. Then A has an initial object.

Proof: Let Qo:=~Q be the product of all objects in Q and let I be Q~Q

the intersection of all M-subobjects of Qo" Then I is an initial object

in A. The proof is the same as in the classical case (see~V w Theo-

rem I) if one takes into consideration LEMMA 1.1 and LEMMA 1.2.

Let X ~ X be an object in an arbitrary category X, G : A ~X be a

functor and denote by Q : (XCG) ~ A the projection functor from the

comma category (X%G) to A.

(2.2) I~MMA. Let A be a category with a proper (E,M)-factorization.

Then (XSG) has a proper factorization (E G, ~) which is preserved by

Q (i.e. Q creates Q-IM - factorizations from M - factorizatlons).

where EG:= Q-IE and ~:= Q-IM. Furthermore if A is M-wellpowered

(E- cowellpowered) then (X~G) i_ssM G - wellpowered (E G - cowellpowered).

Proof: Straightforward, if one takes into consideration that equality

of morphisms in (X~G) means equality as morphisms in A__.

(2.3) THEOREM (Relative Special Adjoint Functor Theorem).

Let A be a complete category with a proper (E,M)-factorization, a

M- cogenerator and with the property that every class of M-subobjects

has an intersection. Then a functor G : A )X has a left adjoint if

and only if G preserves all limits and all intersections of classes

680

of M-morphisms.

Proof: We have to show that each category (X~G) has an initial object

i.e. a G- universal morphism. Since G preserves limits and A is complete

and has a proper (E,M)-factorization, each category (X~G) , X QX, is

complete and has a proper factorlzation (E G, MG).It is easy to see that

the subcategory of (XSG) consisting of all objects k : X >GQ , Q~ Q,

is a M G- cogenerator. Then continue in the same vein as in ~ V w

Theorem 2 .

(2.4) COROLLARY. Let A be a complete category with a proper (E,M)-

factorization. If A is M-wellpowered and has a M- cogenerator then

a functor G : A > X has a left adjoint if and only if G preserves

limits. In particular every continuous functor G : A--~Sets is re-

presentable.

(2.5) COROLLARY. Let A be a category with a proper (E,M)-factorizatlon.

If A is complete and M-wellpowered and has a M- cogenerator then A is

also cocomplete.

w 3 Wellbounded Categories

Wellbounded categories - a generalization of locally presentable

categories - play an important role in the theory of Categorical Uni-

versal Algebra as P. Freyd and M. Kelly showed in the fundamental paper

on "Categories of continuous functors I " ([9]).

Let K be a complete and cocomplete category with a proper (E,M)-

factorization. A M-subobject m : U ~K of a K-object K is the M-union

of M-subobjects u i : Uol ~K ,i~ I, if u i~- m for all iQ I and if for

every i u i = mf i implies that the family of K-morphisms fi ' i ~ I , is

681

in E. The union of a family u i : Ui---,K , i 61 , is denoted by ~iUi @

Let r be a regular cardinal. An ordered set I is r- directed if every

subset of I of cardinality r has an upper bound in I. An r- directed

family of M-subobjects u i : U i b K , iCI , is a family of M-subobJects

u i where I is r- directed and where u i ~ uj whenever i5 j . Then the

M- union~JU, of the M- subobjects u. is called an r-directed union. i 1

An object K~ K is said to be bounded for a regular cardinal r if any

morphism from K into an r- directed unionUU i factors through some U i.

K ~ K is bounded if K is ~ounded for a regular cardinal. The category K

is bounded if each object K in K is bounded.

(3.1) DEFINITION. A bicomplete category K with a proper (E,M)-factori-

zation is wellbounded if it is bounded, E- cowellpowered and possesses

an E- generator.

(3.2) ~xA~s.

(I) Every locally presentable category in the sense of Gabriel-Ulmer

is wellbounded (Freyd-KellyL9 ~ ), as for example the categories

of sets, groups, rings, Lie-algebras, sheaves over Sets, Grothen-

dieck- categories with generators, or the dual of the category of

compact spaces.

(2) Let K be a Top-category over a wellbounded category. Then K is

again wellbounded (Wischnewsky [19] ). So for instance the catego-

ries of topological, measurable, uniform, compactly generated

or limit spaces, groups, rings,.., are wellbounded.

(3) Let U be an E K - reflective subcategory of a wellbounded category

K. Then U is wellbounded (Wischnewsky[19~). So for instance all

epireflective subcategories of a Top-category over Sets are well-

bounded as the categories of TO-, TI-, T2-, T 3- spaces, of zero

dimensional or completely regular spaces.

682

(4) The categories of coalgebras, bialgebras, Hopf-algebras or formal

groups are wellbounded (R~hrl- Wischnewsky [I 7] ).

4 Monoidal Closed Topological Categories Over Wellbounded Categories

Recall that a monoidal category V = <V,Q ,E,~ k p> consists of a

category_V, a bifunctor u : Vw V V, an object E 6 V and three natural

i s omo rph i sins c~ : AU(BnC)~ ( A n B ) n C

k : E~A~A

p : A m E ~ A

satisfying the usual coherence axioms ( Mac Lane ~2~).

A monoidal category V is called syrmmetrical if there is a functorial

and coherent isomorphism

y : ANB =~BmA .

A closed category or monoidal closed category is a symmetrical monoi-

-oV dal category such that for any V6 V the functor V ~V has a

right adJoint V (_)V ~V_. I refer for examples to [I], [2~,[3~,[5]

Let T : K bL be a fibresmall functor. T is called topological or

an initialstructure functor if for all (small)categories D and all

functors I : D > K the canonically induced functor between the con~ua-

categories (Ar and (A~TI)

T I : (~r ~(A+TI)

<K,~ : A~ ~I> , +<TK, T~ : AT~ >TI>

has a right adJoint with the identity as counit. The cones in the

image category of the right adJoint right-inverse are called T- initial

cones.

(4.1) REMARKS: The above definition allows at once some important

generalizations by simply restricting the classes of admissible index-

683

categories D or of admissible functors I or of admissible cones.

(I) If one restricts oneself to small categories D with card0b(D) s

where a is a fixed regular cardinal then one obtains the ~-restrlc-

ted Top- categories. For instance the category of pseudo-metric

spaces is an ~- Top- category (~ech[4]).

(2) Take any class M of cones in thecategories of type (A~TI).

A M-Top functor T : K--bL is a functor which generates only

INS- cones of M- cones. The most important examples are the rela-

tive topological functors in the sense of H.Herrlich (~0~) which

correspond to E- reflective subcategories of Top- categories.

More general every M-Top-category U over a category Lcan be em-

bedded "initialstructure compatibly" into a Top- category over

the category L (Wischnewsky~to be published elsewhere).

(4.2) THEOREM. Let K be a cocomplete category with a proper (E,M)-fac-

torization. If K has an E- generator and is E- cowellpowered then for

any (symmetrical) monoldal category <K, 0, E ,a,k,p~ over K there

ar___e equivalent:

(i) K is monoidal closed.

(ii) For any K~ K the functor -~K :

(iii) (a) For any K ~ K the functor -

(b) The tensor product

regular epimorphism.

K ---~K preserves colimits.

K preservesc~roducts.

of any two regular epimorphisms is a

Proof: One has only to prove (ii)~(i). But this is obvious from

the Relative Special Adjoint Functor Theorem (w THEOREM 2.3).

We obtain now immediately as a corollary the following theorem

which shows that

(I) the base category Sets in Herrlich-Nel's results can be replaced

684

(2) by an arbitrary wellbounded category, and that

the characterization remains the same if we replace the cartesian

product by an arbitrary tensor product

(4.3) THEOREM. Let K be a relative topological category over a well-

bounded category L i.e. an E- reflective subcategory of a Top- category

over L. Then for any (symmetrical) monoidal category ~K, 0 ,E,a,k,p~

over K there are equivalent:

(i) K is monoidal closed.

(ii) For any KE K the functor -m K preserves colimits.

(iii) (a) For any K~K_the functor -~K preserves coproducts.

(b) The tensor product of any two regular epimorphisms is a

regular epimorphism.

If one of ths is fulfilled then we obtain for all K,K' ~.K

~(K, K') ~ ~(E, (K')K).

In particular if T : K )Sets is relative topological over Sets and

E is the"free object"over a one point set then

K(K, K') ~ T((K')K).

Proof: Any relative topological category over a wellbounded category L

is again wellbounded.

w 5 Monoidal Closed Topological Categories And Adjoint Triangles

In this paragragh I consider a slightly different situation. I assume

that the base category L is already monoidal closed (as for example

the category of sets or the category of R- modules) and that the Top-

functor from a monoidal Top-category K over L is strict i.e. preserves

the monoidal structure (see e.g. ~I~ ). More exactly let K = <K~ ,E,~,kp>

and L = <L, N ,E,~ k p> be monoidal categories over K resp. over L.

685

Let T : K >L be a Top-functor which is moreover strict monoidal .

In this case K is called a strict monoidal Top-category over L. In the

sequel I will apply W.Tholen's generalizations ([18b~) of Dubuc's re-

suits on adjoint triangles to the following Dubuc triangle:

-DK

Tholen's Adjoint Functor Theorem for Dubuc-triangles applied to the

above situation delivers at once the following theorem.

(5.1) THEOREM (Monoidal Closedness Theorem for Topological Categories).

Let K be a strict monoidal Top-category over a monoidal closed category

L. If K has coproducts and a proper (E,M)-factorization and if for any

K~ K and e~ E the K-morphism e ~ K is T-final and if finally K is M-well-

powered then K is monoidal closed if and only if for any K~ K the functor

- OK preserves coproducts.

This THEOREM has now some important corollaries.

(5.2) COROLLARY. Let K be a strict monoidal Top-category over a monoidal

closed category L. Let L have coproducts,let L be wellpowered and let

every morphism in L factorize through a regular epimorphism and a mono-

morphism.Then there are equivalent:

(i) K is monoidal closed.

(ii) For any K 6 K the functor - ~K preserves coproducts and regular

epimorphisms.

(5.3) COROLLARY. Let K be a strict monoidal Top-category over a monoidal

closed category L. Let L be cocomplete,wellpowered and cowellpowered. Then

K is monoidal closed if and only if the functors - ~K preserve colimits.

686

(5.4) COROLLARY.Let K be a Top-category over a wellpowered, cowellpowered

and cocompletec~rtesian closed category L. Then K is cartesian closed

if and only if all functors -~<K, KC K, preserve colimits.

This COROLLARY contains again Herrlich's result as a special

instance.

REFERENCES

I BASTIANI,A.,EHRESMANN, C.: Categories of sketched structures. Cahier Topo. Geo. diff. XIII, 2, 105 -214 (1972)

2 BENTLEY, HERRLICH, ROBERTSON : Convenient categories for topologists Preprint (1975).

3 BINZ, E.KELLER : Funktionenr~ume in der Kategorie der Limesr~ume. Ann. Acad. Sci. Fenn. Sec. AI 383 ,I - 21 (1966).

4 ~ech,E. : Topological spaces. Prague 1966.

5 DAY, B.: K reflection theorem for closed categories. J. of Pure and Applied Algebra 2 (I 972) I - 11.

6 DUBUC, E., PORTA, H. : Convenient categories of topological algebras their duality theory. J. Pure Appl. Algebra 1, 281 - 316 (1970).

7 EILENBERG, S.,KELLY, G.M.: Closed Categories. Proc. Conf. on Categorical Algebra. La Jolla, Springer, Berlin, Heidelberg, New York 1966.

8 FRANKE, D. : Funktionenalgebren in kartesisch abgeschlossenen Katego- rien. thesis.Free Univ. Berlin 1975.

9 FREYD, P.,KELLY, G.M.: Categories of continuous functors I. J. of Pure a. Appl. Alg. 2, 169 - 191 (1972).

I 0 HERRLICH, H. : Topological functors.General Topology Appl. _4,125-145 (I 974 ).

11 " : Cartesian closed topological categories.Math. Colloqu. Univ. Cape Town I~, I - 16 (1974).

12 MAC LANE, S.: Categories for the working mathematician. Springer, Berlin, Heidelberg, New York 1971 .

13 NEL, L.D.: Recent results on cartesian closed topological categories. (this volume).

14 PAREIGIS,B. :Categories and functors.Academic Press New York 1970.

15 PUMPL[~, D., THOLEN, W. : Covollst~ndigkeit vollst~ndiger Kategorien Manuscripta math. 111, 127 - 140 (1974).

16 RINGEL, C.M.: Diagonalisierungspaare I (II) .Math.Z. 117. 248 - 266 (1970) (Math. Z. 122, 10- 32 (1971).)

17 ROHRL, H.,WISCHNEWSKY, M.B. : Universal Algebra over Hopf-algebras. Algebra-Bericht Nr.26 ,I - 35, (1974).

687

17a SCHUBERT,H.: Categories,Springer, Berlin, Heidelberg, New York 1974.

18a THOLEN, W.: Relative Bildzerlegungen und algebraiscbe Kategorien. thesis.M~nster 1974.

18b " : Adjungierte Dreiecke, Colimites und Kan-Erweiterungen. to appear in Math. Ann.

19 WISCHNEWSKY, M.B.: On the boundedness of topological categories. Manuscripta math. 12, 205 - 215 (1974).

20 WYLER, 0.: Convenient categories for topo!ogy.Gen. Topo. Appl. , 225 - 241 (1973).

M.B.Wischnewsky

Mathematisches Institut

der Universit~t

8 M~nchen 2, Theresienstr. 39

W-Germany

ON TOPOLOGICAL ALGEBRAS RELATIVE TO

FULL AND FAITHFUL DENSE FUNCTORS

Manfred B. Wischnewsky

In this paper I consider algebras in the sense of Diers ([5]) in

topological categories ([~, [~, [I~, [I~ , [I~, [I~, ~, [18], [I~, [20],

~, ~5] ). This notion of an algebra generalizes at the same time the

notion of an algebra for a monad and the notion of a sheaf on a

Grothendieck-topology. The underlying algebraic theories are defined in

a natural way with respect to full and faithful dense functors. The

results given here combine and generalize in an easy way some of the

basic theorems of the theory of Topological Universal Algebra whose funda-

mental results and techniques can be found for instance in the author's

survey article ([23]). The proofs are only sketched or even omitted

since only standard techniques of Categorical Topology are used.

w I Algebraic Theories And Algebraic Categories

In this paragraph I'll recall Diers' notion of an algebraic theory

resp. category ([5]) which generalizes at the same time the notion of

an algebra for a monad and the notion of an algebra on an esquisse in

the sense of Ehresmann (see e.g. [I]) resp. of a sheaf on a Grothen-

dieck-topology. 0me uses the notion of relative adjoint functors ([I~)

Let J : Ao ~ A be a functor, A~A an object, and JA : (J~A) ~ A -- -- - - O

the forgetful functor from the comma category (J~ A) to Ao and

~A (J) : JJA ~ AA the canonical transformation (X,f)~ ~ f. J is dense

at A ([7],[17],[27]) if ~A(J) is a colimit cone. If J is dense at all

A ~ A then J is called dense.

Let now J : Ao- ) A be a full and faithful dense functor.

A J-theory (Linton[13]) is a pair (T , T) consisting of a category T,

689

having the same objects as ~o'

the identity on the objects.

and a functor T : A ~ > T which induces --0

(I .I) DEFINITION (Diers). A J-theory ~,T) is said to be algebraic if

the functor T ~ : _A o- > T ~ has a J-right adjoint in the sense of

Ulmer ([17j).

The category Th(J) of all algebraic J-theories is the full subcategory

of (A~P~ Cat) having as objects the J-algebraic functors. The category

AT of all T-algebras is defined by the following pullback:

A T ~ [ _T, Sets]

A ~ [_A~ Sets]

A(J(-), �9 )

If (T,T) is an algebraic J-theory then the functor U T : A T > a

has a J-left adJoint. Furthermore U T creates limits and J-absolute co-

limits. A J-absolute colimit in A is a cocone @ : D---~AA over A such

that for any A o in $o the cocone A(JAo,~) : A(JAo, D~ >AA(JAo,A)

is a colimit in Sets.

(1.2) REMARK. The above notion of an algebraic J-theory can be genera-

lized in the following way:

Let Abe a category with a factorization (E,M) and with a(not

necessarily small)E-generator G. By Definition G is a subcategory

J : G > A. Then a J-theory (T,T)(in the sense of Linton)is G-alge-

braic if the functor T ~ : G >T ~ has a J-right adjoint.

(I .3) DEFINITION. A T-algebra with values in a category K is a functor

P : T ~K such that the functor P~176 has a J-right adJoint.

The category AIg(T,K) of all T-algebras in K is the full subcategory

of[ ,K] having as objects the T-algebras in

690

(I .4) EXAMPLES of algebraic J-theories resp. algebraic J-categories (~])

I) Algebraic theories in the sense of Lawvere (see e.g.[28]).

Let Cardfin be the full subcategory in Sets of all finite cardinals

and let J : Cardfin--~Sets be the inclusion. A functor F : Cardfin

> K has a J-right adjoint if and only if F preserves finite sums.

Hence each algebraic J-theory is a theory in the sense of Lawvere

and conversely. The category Alg(T,K) is the category of all T-alge-

bras in K in the sense of Lawvere.

2) Algebraic theories in the sense of Benabou ([2~).

Let I be a set and denote by Cardfin (I) the full subcategory of Sets I

having as objects all l-families (X i, i s I) for which Xi~Cardfin and

[i 6 I, X i # ~} is finite and denote by J : Cardfin (I) > Sets I the

inclusion. A functor F : Cardfin (I) > K has a J-right adjoint if

and only if it commutes with finite sums. An algebraic theorywith

respect J is an algebraic theory in the sense of Benabou and con-

versely ([2]). The category AIg(T,K) is the category of all T-alge-

bras in K in the sense of Benabou.

3) Eilenberg-Moore Categories.

Let (T,q,~) be a monad on A. Denote %y_A T the corresponding Kleisli-

. op op category and by F T : A >~ the free functor. Then (~T ,F~ ) is an

algebraic theory with respect to the identity id A on A. The category

AT is isomprphic to the Eilenberg-Moore category induced by the

monad (T,q,~).

4) Esquisse in the sense of Ehresmann ([I]) resp. algebraic theories

in the sense of Gabriel-Ulmer (~7]).

Let (C,Z) be a small category C together with a class Z of colimits.

Then there exists in a canonical way ([5] 6.5 ) a full and faithful

dense functor J : Ao--)[c#P,sets] and an algebraic J-theory (c~

such that a functor S : C ~ )K is a C-algebra if and only if S

i s a ( C , Z ) - s h e a f .

691

Let F : A > X be a J-left adjoint of U : X > A. Let F I : A - > C --O . . . . O

and F 2 : C >X be respectively the full coimage and the full image of

F. Then F I is J-left adjoint to UF 2. (c~ is an algebraic J-theory

It is said to be generated b~ F. Furthermore there exists a canonical

comparison functor C : X ~A T such that U T C = U. The pair (X,U) is

called J-algebraic if the functor C is an equivalence. With these nota-

tions I can give now Diers' characterization of algebraic categories.

(1 .5) THEOREM (Diers). A pair (X,U) is J-algebraic if and only if U

has the following properties:

(i) U has a J-left adjoint.

(il) U reflects isomorphisms and

(iii) U reflects J-absolute colimits.

w 2 Topological Algebras Relative To Full And Faithful Dense Functors

Let J : ~6- >A be a full and faithful dense functor. Furthermore

we assume that A is complete in order not to complicate the following

presentation. Let (T,T) be an algebraic J-theory.

(2 .1) LEMMA. The inclusion functor E : AIg(T,K) >IT, K]creates limits.

Proof: Let D : p--~AIg(T,K) be a diagram and let A(IimED)----~ ED be a

limit cone of ED in[T,K]. Let S d be a J-right adjoint of D(d)~ ~

where d6D. Then D induces a functor D* : _D ~[_K~ the assignment

d ~--> S d �9 Let S be a limit of D* inIK~ Then we obtain:

A(JAo,SK) = A(JAo,limSd(K)) ~ limA_(JAo, Sd(K)) ~ llm~(K,O(d)TA o) = _K(K, IimD(d)TA o)

Hence S is a J-right adjoint of (IimED)~ ~ i.e. limED is a T-algebra

in K.

692

(2.2) LEMMA. Let F : K ) L be a right adjoint functor. Then the induced

functor [T,F] : [_T,_K] ~[T,L~ : A~-~ FA

factors through Alg (T, K) and Alg (T, L).

Proof: Let A be a T-algebra in K and let S : _K ~ >A be a J-right ad-

joint of A~ ~ Let D be a left adjoint of F. Then

L(L, FAT(Ao) ) ~K(DL, AT(Ao)) ~A(JAo,SD(L)) for all L(L and A~A o

Hence SD ~ is a J-right adjoint of F~176 ~

In the sequel let us furthermore assume that the base categories of

topological categories are complete.

(2.3) THEOREM. Let (T,T) be an algebraic J-theory and F :

Top-functor. Then the functor

Alg(T,~) : A!g(T,K) ~AIg(T,L) : A1 ~FA

is again a Top-functor provided AIg(T,F) is fibresmal!.

K >L be a

Proof: Let ! be the right adjoint right inverse functor of F. Let A be

a T-algebra in L with J-right adjoint S. Then !A is a T-algebra

in K with J-right adjoint SF ~ One can show easily that the assignment

A I >IA defines a right adjoint right inverse of AIg(T,F). Since AIg(T,K)

and AIg(T,L) are complete (LEMMA 2.1) and AIg(T,F) preserves obviously

limits we obtain the above theorem by applying Hoffmann's characteri-

zation of Top-functors ([I ~ ).

(2.4) REMARK. The condition that AIg(T,F) is fibresmall is fulfilled in

all examples given in w I , in particular if ~o is small or AIg(T,L)

is an Eilenberg-Moore category. For the rest of this paper we will

always assume that AIg(T,F) is fibresmall.

693

Let F : K---*Land F' : K--~L be Top-functors and M : K----*K' and

N : L---*L' be arbitrary funetors. Recall that (M,N) : (K,F) ~ (K',F')

is initial continuous or an initial morphism if (II) F'M = NF and

(12) for every INS-cone AK > T , T E [D,K], the cone AMK~>MT is again

an iNS-cone. If L and L' are complete then (M,N) is initial continuous

if and only if M preserves limits and codiscrete objects ([2~THEOREM I .~

(2.5) THEOREM. Let (T,T) be a small algebraic J-theory and let F : K-*L

be a Top-functor. Then the pair of inclusion functors

z K : A Z g ( T , K ) and

E L : A-Ig(T,~) b[T,~l iS initial continuous.

Proof: Clear from the above characterization.

(2.6) COROLLARY. Notation as above.

E K is adJoint if and only if E L is adjoint.

Proof: One has only to prove that E K is adjoint provided E L is adjoint.

But this is trivial since E K preserves limits and since Alg(T,F)-IRLT, FA

is obviously a solution set for A6[T, Kqwhere R L denotes a coadJoint of

E L �9

(2.7) ~SMm~KS. I ) The above COROLLARY follows also immediately from Wyler's taut lif-

ting theorem ([24]) by using THEOREM 2.5.

2) If Alg(T,L) is for example wellpowered and cowellpowered then the

above COROLLARY can also be derived from the following Dubuc-trlangle

(see e.g.[261Thm 28.12 or[16]):

Alg(T, K_) EK ~ T, K_

694

For instance all categories of continuous functors with values in a

locally presentable category L or with values in the dual of a locally

presentable category fulfill this assumption.

A standard example for the above THEOREM rasp. COROLLARY is given

by the following

(2.8) COROLLARY (Wischnewsky[20],[23]). Let (C,Z)be an esquisse in

the sense of Ehresmann resp. in the sense of Gabriel-Ulmer and let L

be a locally presentable category. Let K be an arbitrary Top-category

over L. Then we obtain the following assertions:

(I) The inclusion functor Alga, K) >[C,K]is reflective.

(2) The inclusion functor Coalg(C, K)---->[C, _K ~ op is coreflective.

(3) The categories Alg(C,K) and Coalg(C,K) are complete,cocomplete,

wellpowered and cowellpowered and have a generator.

(2.9) THEOREM. Let (T,T) be an algebraic J-theory. Then the pair of

evaluation functors

U~ : AIg(T,K) ----§ K : A I--~AC an___d

U~ : AIg(T,L) ~L : A ~-bAC where C6T is

initial continuous.

(2.10) COROLLARY. Notation as above.

@iS adJoi__nt if and only if is adjoint

(2.11) COROLLARY. Let F : K >L be a Top-functor and (T,q,~) be a

monad over L. Then the underlying functor

U : A_lg(T,K) > K is monadlc.

Denote by Top(L) the category of all topological functors over L

695

and all initial continuous morphisms between Top-categorles over L as

morphisms. Then we obtain

(2.12) THEOREM. Let (T,T) be an algebraic J-theory. Then there exists

a functor AIg(T,-) : Top(L)

F : K---~L

H K > K'

L

) Top (Alg (_T, L) )

AIg(T,F) : Alg(T,K) ' >AIg(T,L)

AIg(T,K) -- AIg(T,H)~ AIg(T,K')

AIg(T, L)

where AIg(T,H) is defined by A ~-~HA (Take into consideration that if

S is a J-right adJoint of A~ ~ then SR ~ is a J-right adjoint of

H~176 ~ where R is a coadjoint of the initial continuous functor H).

Proof: Easy calculation.

(2.13) COROLLARY. Let H : K--+K' be initial continuous over L. Then

the functor AIg(T,H) has a left adjoint.

A standard example is given by the initial continuous functor

uniform spaces ~ topological spaces.

(2.14) DEFINITION. Let Ao be a category together with a class Z of co-

limits. A full and faithful dense functor J : ~o ~ & is said to be

Z-dense if for every category K and for every functor F : Ao ~ K holds;

F has a J-right adjoint if and only if F preserves the colimlts in Z.

Examples of Z-dense functors can be found in (I .4) I, 2 .

(2.15) THEOREM. Let J : _A o > A be a full and faithful Z-dense functor

and (T,T) an algebraic J-theory. Then (T,T) induces a functor

Top(CAT) > Top(CAT)

696

F K - "~ L Alg(T,K) -~ ilg(T,S)

M ~ i n i t . cont.~ N , ) Alg(T,M)~ AZg(T,N ) F'

_K' b L__' Alg (% K' ) --> Alg (T, L' )

In particular AIg(T,M) is adjoint if and only if AIg(T,N) is adjoint.

Let H : (T',T') --~ (T,T) be a morphism between algebraic J-theories.

Then H induces a functor AIg(H,K) : Alg(T,K)---~AZg(T',K) : A ~-~bHA .

Alg(HjK) is called a J-algebraic functor. In the same vein as

THEOREM 5.2 inL2 ~ one can prove the following

(2.16) THEOREM. Let F : K~ L be s Top-functor and let H : (T',T')-~

(T,T) be amorphism of J-theories. Then we obtain the following state-

ments:

(I) The pair of functors (Alg(H,K),Alg(H,L)) is initial continuous.

(2) Alg(H,K) is adJoint if and only if Alg(H,L) is adjoint.

(2.17) Final Observation.

In the same vein as for instance in[21~ and[23]we can now study T-alge-

bras in reflective or coreflective subcategories of Top-categories.

One obtains similar results. Hence one can state the following

METATHEOREM. Replace theory in (i6], ~I~, [I~ ,~,~2~ ,~2~, [2~ ) by

algebraic J-theory then you will get the same results for algebras

over Top-categories.

697

REFERENCES

I BASTIANI, A.,EHRESMANN, C.: Categories of sketched structures. Cahier Topo. Geo. diff. XIII,2, 105 - 214 (1972).

/ . / .

2 BENABOU, I. : Structures algebrlques dans les categories. Cahier Topo. Geo. diff. X,1, I - 126 (1968).

3 BRU~MER3G.C.L, : A categorical study of initiality. Thesis. Cape Town (1971).

4 DIERS,Y." Type de densit~ d'une sous-cat~gorie pleine. Preprint Universlt~ de Lille 1975.

5 " : Foncteur pleinement fiddle dense classant les alg~ebres. Preprint. Universit~ de Lille. 1975.

6 ERTEL, H.G. :Algebrenkategorlen mit Stetigkeit in gewissen Variablen familien, thesis, Univ. DGsseldorf, 1972.

7 GABRIEL, P.,ULMER, F. : Lokal pr~sentierbare Kategorien. LN 221, Springer, Berlin, Heidelberg, New York (1971).

8 HERRLICH, H. : Topological functors. General Topology and Appl., 4 ( 1 9 7 4 ) .

9 " : Cartesian Closed Topological Categories. Math. Coll. Univ. Capetown,9 (1974).

10 HOFFMANN, R.E. : Die kategorielle Auffassung der Initial- und Final- topologie.thesis, Univ. Bochum 1972.

11 HUSEK, M.: S-categorles. Comm. Math. Univ. Carol. 5 (1964).

12 KENNISON, J.F.: Reflective functors in general topology and else- where. Trans. Amer. Math. Soc. 118)303 - 315 (1965).

13 LINTON, F.W.: An outline of functorlal semantic, LN 80, Sprlnger 1968

14 ROBERTS,J.E. : A characterization of initial functors. J.Algebra 8 , 181 - 193,(1968).

15 TAYLOR, J.C. : Weak families of maps. Canad. Math. Bull. 8,77-95, (1968)

16 THOLEN, W. : Relative Bildzerlegungen und algebraische Kategorien, thesis, Univ. MGnster, 1974.

17 ULMER, F. : Properties of dense and relative adjoint functors. J.Algebra 8 , 77 - 95 (1968).

18 WISCHNEWSKY, M.B. : Algebren und C~algebren in Initial- und Gabriel- kategorien. Diplomarbeit, Univ. ~'~iuchen 1971.

19 " : Partielle Algebren in Initialkategorien, Math. Z. 12___7, 83 - 91 ( 1 9 7 2 ) .

20 " : Generalized Universal Algebra in Initialstructure categories. Algebra-Berichte Nr. 10 (1973) I - 35.

21 " : On regular topological algebras over arbitrary base categories. Algebra- Berichte Nr. 16 (1973) I -36.

22 " : On the boundedness of topological categories,

Manuscripta math. 12, 205- 215 (1974).

23 " : Aspects of Universal Algebra in Initialstructure categories. Cahier Topo. Geo. diffo X ~, I - 27 (1974).

698

24 WYLER, 0.: On the categories of general topology and topological algebra. Arch. d. Math., 2_2/I, 7 - 17 (I 971).

25 " : Top categories and categorical topology. General topology and its applications !, 17 - 28 (1971).

Books on Category Theory

26 HERRLICH, H.,STRECKER, G.E.: Category theory, Allyn and Bacon, Boston, 1973.

27 MAC IANE, S.: Categories for the working mathematician. Springer, Berlin, heidelberg, New York 1971.

28 PAREIGIS,B.: Categories and ~unctors. Academic Press,New York (1970) /

29 EHRESMANN, C.: Categorles et structures. Dunod, Paris, (1965). 30 SCHUBERT, H.: Categories. Springer, Berlin, Heldelberg, New York (1973)

M.B. Wischnewsky

Mathematisches Institut

der Universit~t

8 MGnchen 2

Theresienstr. 39

W - Germany

ARE THERE TOPOI IN TOPOLOGY ?

Oswald Wyler Department of Mathematics Carnogls-Mellon University

Pittsburgh, PA 15213

ABSTRACT. The straight answer is no. Topoi are too set-like to occur as

categories of sets with topological structure. However, if A is a category of

sets with structure, and if A has enough substructures, then A has a full and

dense embedding into a complete quasitopos of sets with structure. There is a

minimal embedding of this type; it embeds e.g. topological spaces into the

quasitopos of Choquet spaces. Quasitopoy are still very set-like. They are car-

tesian closed, and all co~imits in a quasitopos are preserved by pullbacks. Thus

quasitopoi are in a sense ultra-convenient categories for topologists. Quasi-

topoi inherit many properties from topoi, For example, the theory of geometric

morphisms of topoi remains valid, almost without changes, for quasitopoi.

ARE THERE TOP01 IN TOPOLOGY ?

Oswald Wyler

Introduction

Topoi were introduced in SGA 4 [21] as categories of set-valued sheaves.

Crothendieck stated in the introduction of SCA 4 that topologists should be con-

cerned with the topos of sheaves instead of the underlying topological space, but

this advice was not followed. Lawvere and Tierney recognized the set-like and

logical properties of topoi, and they introduced elementary topoi as categories

with these properties. Tierney [20], Kock and Wraith [13], and Freyd [9] gave

introductions to elementary topoi. The latest and simplest version of the axioms

for an elementary topos will be found in w i of this paper.

For topological purposes, topoi are too set-like. They can serve as base

categories for non-standard topology, as in L. Stout's thesis [19], but this

seems to be their only use. On the other hand, there has been an intensive

search for topological categories more set-like -- or more convenient as Steen-

rod [18] called them -- than topological spaces; see Herrlich [I0] for a survey

of this search. ~onvenient categories should be at ~east cartesian closed;

B. Day [6] suggested that all categories T/A , for objects A of a convenient

categor~ ~ , should be cartesian closed. A category ~ with finite limits and

this property is called span-closed.

701

We go one step further. We show that the span-closed categories which occur

in topology are in fact quasitopoi and thus very set-like indeed. Quasitopoi

were introduced by J. Penon [i~] as a generalization of elementary topoi. The

generalization is broad enough to allow topological examples~ but not too broad

so that quasitopoi retain many useful properties of topoi. Thus q~sitopoi are

useful and convenient for topologists, and we obtain a quasi-affirmative answer

to the title question of this paper by studying quasitopoi in topology.

We begin in w i by defining topoi and quasitopoi and stating some of their

basic properties. w 2 describes categories of P-sieves for a set-valued func-

tot P . These categories were invented by P. Antoine [i], [2], as cartesian

closed completions of concrete categories. Day [6] showed that categories of

P-sieves are span-closed; we show that P-sieves form a qmasitopos if P allows

enough subobject inclusions. Thus every topological category with enough sub-

spaces can be densely embedded into a quasitopos.

Quasitopoi of P-sieves are quite large ; thus we devote w167 3 and 4 mainly

to the construction of smaller quasitopoi from a given quasitopos. In w 3,

we describe the general theory of geometric morphisms of quasitopoi$ this is

essentially a generalization of the corresponding theory for topoi. In w 4,

we apply the results ef w ~ to categories of P-sieves. This generalizes the

results of B. Day [6] on closed-span categories of limit spaces. Our main result

is that every concrete category with enough subobjects has a minimal quasitopos

extension, resulting from a canonical Grothendieck topology. For topological

spaces, this minimal extension has been k~nown in ~o forms: it is the category

of pseudotopological or Choquet spaces [4], and also M. Schr~der's category of

solid convergence spaces [l~]. The observation that these are the same category

seems to be new.

702

In presenting our theory of quasitopoi in topology, we suppress most of the

proofs. Some of the proofs are quite involved, but only a few new ideas seem to

be required. Thus the interested reader may be able to supply the proofs, using

the existing literature on topoi as a guide, i plan to describe the theory with

more details, and with full proofs, in a set of lecture notes.

I. Topoi__aand_g_uasitopoi

I.i. An elementary topos can be defined as a category E with finite

limits and with powerset objects. E has finite limits if E has pullbacks and

a terminal object; powerset objects represent relations in E as follows.

1.2. We define a relation (u,v) : A ..... IB in E as a "span" or pair

of morphisms A u > X v > B with ~ommon domain, ~nd with ~u,v) : X >

A )<B monomorphic. We call ~o relations (u,v) and (u',v') equivalent and

write (u,v)_~(u',v') if u' = u x and v' = v x for an isomorphism x , but

we do not identify equivalent relations. We say that (u,v) is a partial mor-

phism if u is a monomorphism.

For f : A---~B in E and a relation (u,v) : B - IC in E , we define

a composition (u,v)O f as a relation

(u,v)of = (u',vf') : A .C ,

where

f!

f .>

is a pullback square in E .

Now a powerset object for an object A of $ is given by an object P A

and a relation ~A : P A , A ,

(u,v) : X ~ A with codomain A

703

with the universal property that every relation

has exactly one factorization (u,v)~Aof

with f : X ~PA in E . The mornhism f

acteristic morphism of the relation (u,v) .

PA

thus obtained is called the char-

such that

_~.4.

1.4.1.

coequalizer.

1.4.2.

I_~.3. The categories of sets and of finite sets are elementary topoi, with

the set of all subsets of A , and with ~A given by all pairs (X,x)

x~X and X CA . Categories of set-valued sheaves are also topoi.

We note some basic properties of a topos E .

Every monomorphism of E is an equalizer, and every epimorphism a

E has finite colimits.

1.4.~. E is cartesian closed.

1.4.4. Partial morphisms in E can be represented (see 1.6).

1.5. In a topological situation, or in a lattice regarded as a category,

not every monomorphism is an equalizer, and not every epimorphism a coequalizer,

but the remainder of 1.4 and other properties of topoi may still be valid. This

led J. Penon [IZ] to define quasitopoi.

We recall first that a monomorphism m

if for every commutative square m u = v e

is called strong [12] or strict [i~]

with e epimorphic~ there is a mot-

phism t such that u = t e and v = m t . Strong epimorphisms are defined

dually. Strong monomorphisms are closed under composition and pullbacks, and

every equalizer is a strong monomorphism. We say that a partial morphism (mtf)

is strong if m is a stron~ monomorphism.

704

1.6.

QT i.

QT2.

We define a quasitopos as a category

E has finite limits and colimits.

E is cartesian closed.

E with the following properties.

QT 3. Strong partial morphisms of ~ are represented.

The last statement means that for every object A of

h ~ partial morphism : A z A such that every strong partial morphism

X---r A factors (m,f).--~AO ~ for a unique morphism @ : X ) A in

follows that id wi h i <id id

there is a strong

(m,f) :

E .

1.7. Topoi clearly are quasitopoi, and a quasitopos E is a topos if every

monomorphism of ~ is strong. The limit spaces of Kowalsk7 []4] and Fischer [8]

form a quasitopos. Heyting algebras, also called relatively pseudocomplemented

lattices [3] are quasitopoi. We shall obtain other examples.

1.8. The "fundamental theorem for topoi" is valid for quasitopoi. This

means that every pullback functor f* : E/B ~/A , for f : A > B in E ,

has not only the usual left adjoint ~-f : ~/A > E_~B , given by~--f u = f u

for an object u : X >A of E=/A , but also a right adjointT~f .

As Day [6] has shown, right adjoints for all pullback functors f* exist

if and only if every category E_/A is cartesian closed.

1.9. We list some additional basic properties of a quasitopos ~ .

I.~.I. Every strong monomorphism of ~ is an equalizer, and every strong

epimorphism a coequalizer.

1.9.2. Every morphism f of E has a factorizatien f = m u e with e

a strong epimorphism, m a strong monomorphism, and u epimorphic and monomor-

phic. Pullbacks in E preserve this factorization.

705

1.9.~. Strong relations in E , i.e. relations (u,v) with <u,v> a

strong monomorphism, are representable.

1.9.~. Strong equivalence relations in ~ are kernel pairs of their char-

acteristic morphisms.

1.9.5. Pullbacks preserve col~nit cones in E .

2. Quasitopoi of sieves

We consider in this section a concrete category A , i.e. A is equipped

with a faithful functor P : { > En__~s to the category of sets. We assume for

convenience that P has skeletal fibres, i.e. if P u is an identity mapping

for an isomorphism u of A , then u is an identity morphism. By the usual

abus de langage, we often use the same symbol for a morphism f of ~ and its

underlying mapping P f .

2.1. We define a P-sieve on a set E as a pair (~, E) consisting of E

and a subfunctor ~ of the contravariant functor E ns(P - , E) : ~op ) E~_ .

In this situation, ~ is given by assigning to every object X of ~ a set

~X of mappings u : P X ~E , with the property that v~ P f always is in

~X for f : X )Y in ~ and v in ~Y . With built-in abus de langage,

we define a morphism h : (~, E) ) (~, F) of P-sieves as a mapping h :

E---~F such that always h u ~YX for u in ~X .

This defines a category of P-sieves which we denote by ~cr (for "c~ible"),

and putting pcr ($, E) = E defines a forgetful functor per : ~cr ~Ens .

Composition of morphisms in ~cr is composition of the underlying mappings,

706

2__~.2. The functor pCr : Acr= > Ens admits all possible initial and final

structures. Thus ~cr is a top category over sets, in the sense of [22] and

[25], except that fibres of pcr may be large. The large fibres do not malter,

however, since pcr admits initial and final structures for all admissible fami-

lies of data, large or small.

For a family of P-sieves (~i' Ei) and of mappings fi : E ----> E i , the

initial structure (~, E) for per has ~X consisting of all mappings u :

P X ~--~E such that every fi u is in the corresponding ~i X . Final struc-

tures are obtained dually, with "every" replaced by "some".

2. 3 .. For an object A of A= , we define a P-sieve ~A on P A by

l tti X be the set of all Ff for f in tting

~g = P g :~A--~B for g : A >B in A = then defines a functor ~ :

A ) A cr = = , with p = pcr . By abus de langage, we may use P to identify

subsieves of ~A with subfunctors of A(~, A) , i.e. with sieves on A in the

usual sense of the word. Antoine [i], [2], who introduced P-sieves, proved that

~ preserves all initial structures which P admits, and he obtained the fellow-

ing Yoneda lemma.

2.4. PROPOSITION. For a P-sieve (r E) , ama_~ u : P A > E

A -~ A er is a full embedding.

i_Es

2.5. For P-sieves (~, E) and (~b, F) , we construct a P-sieve

[(~,~] on the set F E by letting [r X consist of all mappings Q:

P X >F E such that the corresponding mapping u : P X >~ E " >F is a morphism

u :~X~ (d~, E) >(~, F) of P-sieves. By 2.4, this is the only way to

707

construct a cartesian closed structure for P-sieves, and it works. In fact,

~cr is not only cartesian closed but span-closed.

We often can say more. We say that A has P-inclusions [24] if P admits

an initial structure for every subset inclusion E C P A .

2.6. THEORF2~. Let P : A > Ens be a faithful functor. If A

P'inclusions, then the category ~cr of P-sieves is a quasitopos.

has

Proof. A cr has limits and colimits, and we have seen that A cr is carte-

sian closed. Strong monomorphisms m : (~, E) > (~, F) of ~cr are injec-

rive mappings with ~ the initial structure for m and ~2, i.e. u :

P X >E is in ~X iff m u~2X .

For sets, E is E with one point added, and is the inclusion. For

P-sieves, we claim that ~E = : (~' E) >(~, E) with constructed

as follows. For u : P X > E , we construct a pullback square

E' u'> E

PX u > ~

of sets, with a set inclusion at left. If X' is the initial structure for P

and this inclusion, then we put u ~ X iff u'~X' .

2.7. We say that ~ has constant morphisms if ~ satisfies the following

two conditions. (i) ~ has a terminal object A 1 with P A 1 a singleton, and

every constant mapping f : PA 1 >P X lifts to a morphism f : A 1 ~X

of =A . (ii) A= has an object A ~ with PA ~ empty, and an object A ~ with

this property is an initial object of ~ . The categories occurring in topology

usually satisfy these two conditions.

708

We obtain a category of P-sieves with constant morphisms in two steps.

We denote by Aci the full subcategory of ~cr with those P-sieves (~ E)

as objects for which the unique u : P A---~ E is in ~A if PA is empty,

and by ~c the full subcategory of ~ci with the P-sieves (~, E) as objects

for which every @X contains all constant mappings P X > E . Ac has con-

stant morphisms, and we have a Yoneda embedding ~ : A= >A c= , with pC~= p

for the forgetful functor pC : A c ) Ens , if A has constant morphisms.

2.8. PROPOSITION. If A cr is a quasitopos, then A c is a quasitopos.

Proof. The reflector ~cr > ~ci adds the unique u : P A ) E to ~A

if necessary, for P A empty. This functor preserves limits; thus ~ci is a

quasitopos by 3.7 below.

If (@, E) is an object of ~ci , let E' be the set of all xE E such

that the constant mapping u : P X >E with range {X} is in ~X , f~r

every object X of { with P X not empty, and let (+', E') be the initial

P-siev~ for ~ and the inclusion E' >E . This is an object of A c= , and if

h : (~, F) (~, E) with (h~, F) an object of ~c , then h factors

through E' and hence through (~' E') Thus A c is eoreflective in A ci

and isomorphic to the categor2 of coalgebras for an idempotent comonad. As this

comonad is exact, A e is a quasitopos, by 3.1 below, if A ci is one.

2. 9 . REMARKS. (i) We note that pCr preserves the full quasitopos struc-

ture. pC does not. (ii) If A has constant morphisms, and if P A and E

are both not empty, then a P-sieve (~, E) is an object of ~c iff the map-

pings u : P A ~E in CA are collectively surjective. (iii) If ~ has

constant morphisms, then the functor P : A---~Ens preserves limits.

709

3. Geometric morphisms of quasito~oi

Geometric morphisms of quasitopoi are defined in the same way as for topoi:

they are adjunctions f* If. : E~>~ such that f* is left exact, i.e.

preserves finite limits. It follows that f* preserves monomorphisms and strong

monomorphisms. We consider in this section only the geometric morphisms used to

construct quasitopoi of coalgebras and of sheaves.

3.1. A comonad (G, g,~tl) o n a category

functor G preserves finite limits.

T}~OREM.

qategory

a left exact co=o=d (a, ~ ~, y) For

of coalgebras is a quasitopos.

is called left exact if the

on a quasitopos E , the

(A,~O ~ for a G-coalgebra.

in E , with Cff~'~.G~ = G~,~. G G+A.~ A

The forgetful functor U G : E creates finite limits since G is left

exact, Thus we have an equalizer fork

(A,~) ~ e

Proof. We indicate only the construction of

have a st~on~ ~rtial morphism I G ~, a A) G ~ --+~ We

CA >A

e A > a

and hence a pullback

710

of coalgebras, with G~Ao~(= e %~ for ~A~ " (A~) > (A,~O "~.

3.2. We define a topology of a quasitopos E as a natural closure operator

for monomorphisms of ~ . Thus we require the following.

(i) m and ~m have the same codomain, and ~m~m' if m~m'

(ii) m~ ~,m , and ~m__~gm .

(iii) f* (~m)~__~(f * m) if m and f have the same codomain.

We recall that f* m is the pullback of m by f . Axioms (i) - (iii) suffice

for topoi; for quasitopoi we need an additional axiom.

(iv) ~m is strong if m is strong.

3.3. A monomorphism m is called closed for a topology ~ if ~ m_~m ,

and m is called dense if ~m is an isomorphism. We note some elementary pro-

perties of closed and dense monomorphisms.

f* m

3.3.1. If m = m I m' , then m'

3.3.2. If f* m is 8efined, then

is closed if m is closed.

3.3.3.

is dense iff ml_~<~m .

f* m is dense if m is dense, and

The composition of dense monomorphisms is dense, and the composition

of closed monomorphisms is closed.

3.~.4. ~(m I {~ m 2) ~ gm I ~ Y~2 if m I and

3.3.5, If m I u = v m' with m I closed and

and m I t = v for a unique morphism t of ~ .

m 2 have the same codomain.

m' dense, then u = t m'

~.4. An object S of ~ is called separated for a topology ~ of E if

~(d,S) is injective for every dense monomorphism d , and an object F is

called a sheaf for ~ if ~(d,F) is bijective for every d~se monomorphism d .

711

If F is a sheaf for ~ , then we define a closure operator for strong

partial morphisms (m,f) : A / F . If m = ~m. d , then f = ~ d for a

unique ~ , and we put ~(m,f) = (~m, ~) : A ~ F . This closure operator

is idempotent and natural in A . Thus

y ((m,f)Oh) ~ ~(m,f) ~ h

if (m,f)o h is defined. It follows that

(m,f) ~ (t~,id)~ ---> ~ (re,f) r~ (~F,id) o JFT

for JF : F ~F given by ~ ,id) ~ ,id) e JF "

3.5. TIfiEOREM. l__ff ~ is a topology of a quasitopos E , then separated

objects and sheaves for ~ define full reflective subcategories Sepg~ and

Sh~ E= __~ E= . The category S~ E= of sheaves for ~ is a quasitopos, and the

reflector E ~Sh~ E preserves finite limits.

Proof. In order to obtain ~T 3 for sheaves, we construct an equalizer

if F is a sheaf.

struction of JF '

i d ) : F. - - - T F t 3

e ~" J F. > F F . " J i d ~ F

T h i s f u r n i s h e s a s h e a f F . S~nce JF = by t h e c o n - 3

w e

represents strong partial m~rphisms in Sh~ E .

3.6. Strictly full reflective subcategories of ~ can be characterized by

idempotent monads (T,~, id T) on E , with T T = T . An objec~ F of E

is in the reflective subcategory determined by T iff ~F is an isomorphism.

If =E is a quasitopos and T obtained from a category Sh~ E= of sheaves for a

topology ~ , then T is left exact, i.e. T preserves finite limits.

712

Conversely, if T is left exact, then putting ~T m___~m for a pullback

m >i

; L, \ T m) T A

in E = defines a topology ~T of E = . An object F of E = is a sheaf for ~T

if and only if ~F is isomorphic in ~ , and we have the f$11owing result.

3.7. THEOREM. If E is a quasitopos and F a reflective full subcategory

of E with left exact reflector, then F is a quasitopos.

~.8. It remains to compare ~ with ~T if s is a topology of a quasi-

topos =E and (T,~, id) the left exact idempotent monad on E= obtained from

the category Sh~ ~ of sheaves for ~.

We note first that ~ and ~T produce the same separated objects and the

same sheaves. In 3.6, T m and hence m are closed for ~ . Thus ~m~g~T m ,

and ~T is coarser than ~ . If m is strong, then ~T m__~m by ~he usual

argument for topoi, and it is easy to see that ~T m ~___~m if the eodomain of m

is separated for ~ . We do not know whether always ~ m ~ m , so that ~T

and ~ are equivalent, or whether it is possible to obtain the same quasitopos

of sheaves from two topologies which are not equivalent.

~._~. In a quasitopos ~ , every monomorphism has a factorization m = m I u

with m I a strong monomorphism and u epimorphic and monomorphic. Pullbacks

preserve this factorization; thus putting ~ m__~m I defines a topology of ~ .

Closed monomorphisms for ~ are strong; thus Sh~ is a topos. This is the

topos of coarse objects of ~ , obtained by Penon [16] in a different way.

713

4. Quasitopoi in topolog~

All categories in this section are assumed to be concrete, with constant

morphisms (2.7).

If a commutative triangle of faithful funetors

A C ~B

Ens

is given, then we call G a dense embeddin~ of ~ into ~ if G is full, and

every object B of B has the final structure for the functor Q and all mor-

phisms u : G X ~B in B .

In this situation, tbe morphisms u : G X ~B form a colimit cone in B .

Thus a dense embedding is a dense functor as defined e.g. in [15].

4~ PROPOSITION. A dense embedding

tures and limits. If A is co~olete, then

which preserves underlying sets.

G : A ~B preserves initial struc-

G has a left adjoint left inverse

4~.. PROPOSITION. ~ : A >A c is a dense embedding. If G : A )B

a dense embedd ing , t h e n ~ = G c G f o r a dense_.embeddinA, G c : =B ~ A c=

is

Proofs. The proof of the first part of 4.2 is straightforward; the second

part follows from [22; 6.3].

is a dense embedding by the definitions and 2.4, used for ~ : ~ > ~e

If C : ~ ~ is given as in 4.1, then we let G c B be the P~sieve over Q B

with u : P X--->Q B in (G c B) X iff u : C X---->B in B . This defines a

714

functor G c : ~ __~Ac , and one sees easily that G c

that G e C = ~.

We need a s p e c i a l c a s e o f a theorem of Day [ 5 ] .

is a dense embedding such

4.4. PROPOSITION. l_~f G : A ~ B is a dense embedding with left adjoint

H : B > A , and if B is cartesian closed, then A is cartesian closed if

and only if H preserves f.inite products.

4.5. A dense embedding G : ~ >,~ induces a dense embedding G A : ~/A

> ~/GA for every object A of ~ . A left adjoint H of G induces a left

adjoint H A for every functor G A , and 4.4 is valid for these adjunctions.

If H preserves finite limits, then every H A preserves finite products.

Combining this information with 4.3, we see that we obtain dense embeddings

of ~ into complete span-closed categories by looking for full reflective sub-

categories of A c w~th left exact reflectors which preserve underlying sets.

By 3.7, these subcategories are quasitopoi if A c is a quasitopos, and we are

looking for topologies ~ of ~c with a reflector ~c ) Shu c which pre-

serves underlying sets, and such that every object ~A is a sheaf for ~ .

The associated sheaf functor ~c > Sh~ c preserves underlying sets only

if every dense monomorphism for ~ is bijective at the set level, Conversely,

if this is the case, then every object of ~c is separated for ~ , hence

densely embedded into a sheaf, and thus an associated sheaf functor which pre-

serves underlying sets exists.

c 4.6. Let P : A >C be the forgetful functor. If P 14. is bijective

for every g-dense monomorphism ~t of A c then the topology ~/ of A r is

equivalent to a unique topology ~ of A= c such that pC (~)= pe~ for every

715

v monomorphism ~ of ~c . We say that ~ with this property is a P-topology

of A c . Since every object of A c Js seoarated for a P-topology, we have by

3.8 a bijection between P-topologies of A c and the corresponding categories of

sheaves. These are the strictly full reflective subcategories of ~c with a

reflector which preserves finite limits and underlying sets.

Embeddings (~, P A)--9~ A in AC correspond bijectively to subfunctors

of the functor A( - , A) , i.e. to sieves on A in the usual sense, if we

allow only those sieves which contain all constant merphisms with codomain A .

With the same restriction on sieves, we obtain a bijection between P-topologies

of A c and Grothendieck topologies of A as follows.

If a P-topology ~ of ~c is given, then we denote by J~A the class of

all sieves on an object A of ~ for which the corresponding (~, P A) >~A

is dense for ~ . This defines a Crothendieck topology J~ of ~ . If J is a

Crothendieck topology of ~ , and if~ = m : (~, E)--) (~, F) is a monomor-

phiem of ~c , then we put ~j~= m : (~, E) ,,~(~, F) , with u :

P A >E in ~A iff m u ~ A and u*~J A , where (u*~) X consists

of all f : X--~A in A such that u ,P f is in CX . This defines a

P-topology yj on ~c �9 One verifies easily that the correspondences ~--->J~

and Jl >~j are inverse bijections.

4.7. If R is a sieve on an object A of A and ~ a P-sieve on a

set E , both with constants, then we put RA-~ if a mapping u : P A ~E

is in ~A whenever u -P f is in ~X for every f : X >A in a set R X .

If g is a P-topology of ~c and J the corresponding Grothendieck topo-

logy of ~ , then the sheaves for ~ are the P-sieves erthogonal to J for

the relation -i- just defined, and J is the orthogonal complement of the class

716

of all ~-sheaves;for _L . The fact that R_]_~ for R in J A and every

sheaf (~, E) for V can be expressed by saying that ~ A has the final

structure for the morphisms f : X >A in R and the category of ~-sheaves.

if

4.8. We say that a sieve R on an object A of ~ is a ~uotient sieve

A has the final structure for P and all morphisms f : X ~A in a

set R X . With J and ~ as in 4.7, we have the following corollary of 4.7.

PROPOSITION. Every object ~A of = A c is a sheaf for ~ if and only if

e v e r y s i e v e i n J i s ,a q u o t i e n t s i e v e .

With set inclusion as order relation, Grothendieck topologies of A form

a complete lattice. There is thus a largest Grothendieck topology of A which

consists of quotien~ sieze~ this is the canonical topology of A . The cor-

responding category of sheaves in A c is the smallest complete quasitopos into

which ~ can be densely embedded -- provided that ~c is a quasitopos. By 2.8

and 2.6, this is the case if A has P-inclusions.

~A~" Let Top denote the category of topological spaces and Lim the

category of limit spaces. The embedding T_~o~ >Lim is dense, and Lim is a

quasitopos [16]. Thus we can identify Lim with the category of sheaves for a

P-topology of Top c , by a functor G c . If we do this, then the category of

sheaves for the canonical topology of T__qop becomes a category of limit spaces.

Theorem 1 of [7] can easily be generalized from quotient maps to quotient

sieves; ~hus we can describe th~ canonical topology of T o~ as follows.

THEOREN. For a quotient sieve R on a topolo~.ical space Y , the fol-

lowing three conditions are logically equivalent.

717

(i) R is in J Y for the canonical Grothendieck topology of To~ -

(ii) If an ultrafilter G o_~n Y converges to y i__nn Y , then there ar 9

always f : X >Y i__n_n R , an ultrafilter F o_~n X and x in X such th~$

F converges to x i_~n X , and f(F) = G and f(x) = y .

(iii) If we assign to ever 2 f : X > Y i__nn R an open cover (Uf, i) of

f-l(y) i_.nn X , for a point y of Y , then there is always a finite union o<

sets f(Uf,i) which is a neighborhood.of y i_qn Y .

4.10. With the notation of 4.7, condition (ii) in the Theorem above says

that R J_ (E,q) for every Choquet soace (E,q) , and condition (iii) says that

RD_(E,q) if (E,q) is a solid convergence space [17], i.e. a limit space which

satisfies the following axiom L 3'.

L 3'. If (Fi)i eI is ~ family of filters on E converging to x in E

and if G is a filter on E such that for every family of sets A i~ F i , one

for every i ~I , a finite union of sets A belongs to G , then G q x . 1

Solid limit spaces and Choquet spaces both define strictly full reflective

subcategories of Lim , with reflectors which preserve underlying sets and

finite limits. Thus we have the following result from 4.7 and 4.9.

THEOREM. Solid limit spaces and Choquet spaces define the same str&ctly

full subcategory of L'im . This category is a quasitopos, isomorphic to the

category of sheaves in ~op c for the canonical topology of T~_ .

718

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Vol. 529: J. Grandell, Doubly Stochastic Poisson Processes. X, 234 pages. 1976. Vol. 531: Y.-C. Wong, The Topology of Uniform Convergence on Order-Bounded Sets. VI, 183 pages. 1976.

Vol. 532: Theorie Ergodique. Proceedings 197311974. Edite par J.-P. Conze and M.S. Keane. VIII, 227 pages. 1976.

Vol. 534: C. Preston, Random Fields. V, 200 pages. 1976.

Vol. 535: Singularites d'Applications Differentiables. Plans-sur-Sex. 1975. Edite par 0. Burlet et F. Ronga. V, 253 pages. 1976.

Vol. 538: W. M. Schmidt. Equations over Finite Fields. An Elementary Approach. IX, 267 pages. 1976.

Vol. 537: Set Theory and Hierarchy Theory. Bierutowice, Poland 1975. A Memorial to Andrzej Mostowski. Edited by W. Marek, M. Srebrny and A Zarach. XIII, 345 pages. 1978.

Vol. 538: G. Fiacher, Complex Analytic Geometry. VII, 201 Seiten. 1976.

Vol. 539: A Badrikian, J. F. C. Kingman et J. Kuelba, Ecole d'Ete de Probabilites de Saint Flour V-1975. Edite par P.-L. Hennequin. IX, 314 pages. 1978.

Vol. 540: Categorical Topology, Proceedings1975. Edited by E. Binz and H. Herrlich. XV, 719 pages. 1976.