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Relations in Categories

Stefan Milius

A thesis submitted to the Faculty of Graduate Studies in partial

fulfilment of the requirements for the degree of

Master of Arts

Graduate Program in Mathematics and Statistics

York University

Toronto, Ontario

June 15, 2000

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Relations in Categories

by Stefan MiIius

a thesis submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of

Permission has been granted to the LIBRARY OF YORK UNIVERSITY to lend or seIl copies of this thesis, to the NATIONAL LIBRARY OF CANADA to microfilm this thesis and to lend or seIl copies of the film, and to UNIVERSITY MICROALMS to publish an abstract of this thesis. The author reserves other publication rights, and neither the thesis nor extensive extracts from it rnay be printed or otherwise reproduced without the author's written permission.

Abstract

This thesis investigates relations over a category C! relative to an (&, M)-factori-

zation system of e. In order to establish the 2-category Rel(e) of relations over C

in the first part we discuss suficient conditions for the associativity of horizontal

composition of relations, and we investigate specid classes of morphisms in

Rel(e). Attention is particularly devoted to the notion of mapping as defhed

by Lawvere. We give a sigdicantly simpi5ed proof for the main result of

PavloviC, namely that C 2i Map(Rel(C)) if and O* if E C RegEpi(C) . This

part also contains a proof that the category Map(Rel(e)) is hitely complete,

and we present the resdts obtained by Kelly, some of them generalized, Le.,

without the restrictive assurnption that M Mono(C) .

The oext part deah with factorization systems in Rel(e). The fact that

each set-relation has a canonical image fâctorization is generalized and shown

to yield an (E, %)-factorkation systern in Rel(e) in case M E Mono(C). The

setting without this condition is studied, as weU. We propose a weaker notion of

factorization system for a 2-category, where the commutativity in the universal

property of an (&, M)-factorization systern is replaced by coherent 2-ceIls.

In the last part certain limits and colimits in Rel(e) are investigated. Co-

products d t in ReI(C!) and are given as in (2 provided that C! is extensive. How-

ever, finite (co)completeness fails. FinalZy we show that colimits of w-chains do

not &st in Rel(e) in general. However, it turns out that a canonical construc-

tion with a 2-categorial universal property &sts if e has well-behaved colimits

of w-chahs. For the case € _C Epi((2) we give a necessary and suEcknt condition

that forces our construction to yield colimits of w-chains in Map(Rel(e)).

Acknowledgements

1 would like to express my deep gratitude to Walter Tholen, who was my su-

p e ~ s o r during my stay a t York University. Without his support and the many

useful comments on earlier versions of this text this thesis would not have been

possible.

1 dso would like to thank J= A d h e k , who made me acquainted with Cate-

gory Theory in the first place. He aiso suggested to investigate initial F-dgebras

in categories of relations; this was the motivation for start ing this whole project

on relations in categories.

Findy, I would like to thank the other members of my Exarnining Com-

mittee, Nantel Bergeron, fianck van Breugel, and Joan Wick Pelletier, for the

effort they put into reading this work, and for the useful suggestions they made

to improve the h a 1 version.

This thesis has been typeset using the typesetting system Tjj?C of Donaid

E. Knuth, and using B w of Leslie Lamport. The commutative diagrams have

been set using Kristoffer H. Rose's w-pic.

Toronto, June 2000.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A bit of history 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 About this thesis 2

2 (1. M)-structured categories 6

3 Spans and Relations 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spans 12

. . . . . . . . . . . . 3.2 Relations relative to a factorization system 14

. . . . . . . . . . . . . . . . . . . . . . . 3.3 Composition of relations 15

. . . . . . . . . . . . . . . . . 3-4 Associativity and identity relations 17

. . . . . . . . . . . 3.4.1 Images and inverse images of relations 19

. . . . . . . . . . . . . . . . . . . . 3.4.2 Legs and leg-stability 20

. . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Structure of Rel(C!) 25

. . . . . . . . . 3.6 Rel(C) as synunetric monoidal closed 2-category 28

. . . . . . . . . . . . . . . 3.7 A caiculus of relations using elements 30

4 Maps 36

. . . . . . . . . . . . . . . . . . . . 4.1 Maps in arbitrary bicategories 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Induced relations 39

. . . . . . . . . . . . . . . . . 4.3 Maps in the bicategory of relations 42

. . . . . . . . . . . . . . . . . 4.4 Convergence and the graph functor 47

. . . . . . . . . . . . . . . . . . 4.5 Total and single-valued relations 50

. . . . . . . . . . . . . . . . . . . . . . . 4.6 b c t i o n comprehension 57

. . . . . . . . . . . . . . . . 4.7 Finite completeness of Map(Rel(e)) 60

5 Functional relations 68

. . . . . . . . . . . . . . . . . . . . . 5.1 Maps as functional relations 68

. . . . . . . . . . . . . . . . . . . . . . . 5.2 Total relations revisited 70

. . . . . . . . . . . . . . . . . 5.3 A characterization of stabiiity of E 71

. . . . . . . . . . . . . . . . . . . 5.4 Special relations for E C Epi(C) 76

. . . . . . . 5.4.1 Total, single-valued and isomorphic relations 76

. . . . . . . . . . . . . . . . 5.4.2 Sections and Monomorphisrns 78

6 Relations induced by maps 83

. . . . . . . . . . . . . . . . . . . . . . . . 6.1 The monic arrows of & 84

. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Maps as fiactions 85

. . . . . . . . . . . . . . . . . . . . . 6.3 Regularity of Map(Rel(e)) 86

. . . . . . . . . . . . . . 6.4 The category of bicategories of relations 88

. . . . . . . . . . 6.5 Cornparison of Rel(e) and Rel(Map(Rel(e))) 89

7 Factorization systems in Rel(C) 92

. . . . . . . . . . . . . . . . . . 7.1 Factorization in the general case 92

. . . . . . . . . . . . . . . . . . . . . . . . 7.2 A functorial approach 100

. . . . . . . . . . . . . . 7.3 Factorization systems for M C Mono(C) 112

8 Some limits in Rel(C) 117

. . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Extensive categories 117

. . . . . . . . . . . . . . . . . . . . . . . . . 8.2 2-Limits and bilimits 119

. . . . . . . . . . . . . . . 8.3 Initial and terminai objects in Rel(C) 121

. . . . . . . . . . . . . . . . 8.4 Products and Coproducts in ReI(C) 123

. . . . . . . . . . . . . . . . . . . . . . . . . 8.5 2-products in Rel(C) 128

. . . . . . . . . . . . . . . . . . . 8.6 Rel(e) is not finitely complete 130

. . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Application to maps 130

9 Coffmits of w-chains in Rel(e) 134

. . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 A counterexample 135

. . . . . . . . . . . . . . . . . . . . 9.2 Lax adjoint b i t s of w-chains 138

. . . . . . . . . . . . 9.2.1 Construction of the canonical cocone 140

. . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Commutativity 143

. . . . . . . . . . . . . . . . . . . . 9.2.3 The universal property 143

. . . . . . . . . . . . . . . . . . . . . 9.2.4 The weak uniqueness 148

. . . . . . . . . . . . . . . . . 9.3 Consequences and Open Problems 152

A Allegories 157

. . . . . . . . . . . . . . . . . . . A.1 Prelirninaries and Tenninology 158

. . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Basic defkitions 159

. . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Special Morphisms 160

. . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Tabular Allegories 164

. . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Unitary Ailegories 168

. . . . . . . . . . . . . . . . . . . . . A.6 h c t o r s between allegories 171

List of symbols

The following tables contain some of the symbols that are used most fkequently

in this thesis. Of course, it is far fiom being complete. However, it shodd

contain all the symbols that are used more than only locally, that means more

than in one section. For each symbol there is a short explanation and a reference

to its definition or 6rst place of occurrence.

Classes of morphisms

M r Epi (e)

the class of isornorphisms of the category e Def. 2-1

class of monomorphisrns of e Prop- 2.7

class of epimorphisrn of e Prop. 2.4

class of sections of e (morphisms with a left Prop- 2.4

inverse)

class of extremal monos of e (A monic arrow m Prop. 2 -4

is called extremal if m = f e with e epic implies

that e is an iso.

class of extremal epis of e (dual notion of ex- Cor. 2-5

tremal mono)

class of strong epis of e (An epimorphism is Cor. 2.5

c d e d strong if it has the unique diagonaliza-

tion property W. r. t. dl monos.)

class of regular epis of C (An arrow is a regu1a.r Prop. 2.7

epi if it is the coequalizer of a paralfel pair of

arrows.)

class of equivalences of the 2-category 23 p. 107

classes of morphisms that fonn an (&, M)-struc- Def. 2.1

ture

classes of relations induced by the canonical p. 93

factorization of relations

abbreviation for E n Mono(e) Sec. 6.1

Note that we have the following chain of inclusions:

Iso(C!) C Sect(e) C RegMono(e) C StrongMono(C!) C ExtrMono(e)-

By duality, the same is true for the respective classes of epimorphisms.

Categories and 2-categories

Set

category of spans between objects A and B Sec. 3.1

category of relations berneen objects A and B Sec. 3.2

2-category of spans over the category e Sec- 3.1

2-category of relation over the category (3 Sec. 3.3

subcategory of maps (1-cells with a right ad- Def. 4.2,

joint) of Rel(C) Cor. 4.9

category of sets and functions Ex. 3.1

CAT (2-) category of (srnail) categories, functors

(and natural transformations)

category of topologicai spaces and continuous

functions

category Ti-spaces and continuous fùnctions

2-categories of fkitely complete categories wit h

a (proper) stable (&, M)-structure

Reg full sub-Zcategory of regular categories in X

m, m~ 2-categories of 2-categories of relations over

categories in X: and X p respectively

O t her s y m b 01s

symbols for natural, rational, and real numbers

notation for the unique arrow C : A x B induced

b y a : C + A a n d b : C + B

notation for the unique arrow A + B + C induced b y c : A + C a n d d : B + C

kernel pair of an arrow f

notation for arrows of M (of an (8, Ml-structure)

in diagrams

notation for arrows of E (of an (8, M)-structure)

Ex. 3.15

Ex. 3.2

Ex- 4.24

Sec. 6.4

Sec. 6.4

Sec. 3-1

Sec. 8.4

Sec. 4.2

in diagrams

f'7 f'

b o a

&A

S O T

L A

constant (2-)functor

functor im : Span(A, B) + Rel(A, B) given by

(8, M)-factorizing

image and inverse image of relations

composite of the spans a and b

identie span given by (lA, lA) : A + A x A

composite of the relations r and s

identity relation aven by image im(bA) of an iden-

tity span

opposite of the relation r

local product of the relations r and s

notation for pointwise calculus of relations

graph im(1, f ) of an arrow f , graph functor

simpIif5ed notation for 2-ceils in Rel(e) if M c

Mono(C!)

Prop. 2.8,

Sec. 8-2

Sec. 3.3

Def. 3.3

Sec. 3.1

Sec- 3.1

Sec. 3.3

Sec. 3-4

Sec. 3.5

Sec. 3.5

Prop. 3.17

Def- 4.4,

Sec. 4.4

Sec. 5.1

1 Introduction

1.1 A bit of history

Relations between sets as w d as the equident concept of multi-dued h c -

tions have been an important tool in mathematics for a long time. The calculus

of binary relations played an important role in the interaction between algebra

and logic since the rniddle of the nineteenth century. The first adequate de-

velopment of such calculi was given by de Morgan and Peirce. Their work has

been taken up and systematically extended by Schroder in [24J. More than 40

years later, Tarski started with [26] the exhaustive study of relation algebras,

and more generaliy, of Boolean algebras with operators.

Categorial generalizations of calculi of relations have been playing a role in

many works for quite a whiie, too. TraditionaUy the relation R Ç A x B defined

set theoretically is substituted by a monomorphism r : R + A x B in a category

e, where, moreover, r often Lies in a special class Sul: of monornorphisms belong-

ing to a pullback stable (E, M)-factorization system of the category e. The first

categorial treatment of relations is due to MacLane (cf. [18]). He axïomatizes

additive relations between (Ieft) modules over a fixed ring. His results appear

a t about the same time as the axiomatization of relations in Abelian categories

of Puppe (cf. [23] and [4] for a more extensive treatment). The notion of rela-

tions relative to a factorization system f h t appears in [15], stili with slightly

distorted terrninology. It is full]- developed with the introduction of the bicate-

gorg Rel(C) of relations over a category (2 for the e s t tirne in [2O]- However,

both of these papers impose conditions on the ( E , M)-factorizations system of

e, namely E E Epi(e) and additionally M C Mono(e) in the first paper. I t

seems that most of the Iater investigations on relations, Iike for example, in [14]

and [IO], always use one or both of these assumptions. But as the interest of

theoretical cornputer science in relations grew, these conditions became a great

obstacle for considering certain important examples.

In bis important work [21], Pavlovie shows how to obtain a reasonable theory

avoiding all assumptions on E and M other than necessary. Admittedly, this had

to be done at the cost of making the proofs quite involved, and finally i t resulted

in an even more general treatrnent of relations relative to regular fibrations in

the sequel [22] of [21] by the sanie author.

Meanwhile, the work of F'reyd and Scedrov (cf. [8]) led to an axiomatization

of relations over regular categories, the so-called allegories. Some authors have

used this setting to investigate relations further. A very recent example of this

is [27]. Here we shall mainly stick to relations relative to an (&, M)-factorization

system.

1.2 About this thesis

This thesis starts by recalling a few basic facts about (E, M)-factorization struc-

tures in categories. In the third section the results of Pavlovit ([21]) and

Jayewardene and Wyler ([IO]) will be used to define the Zcategory Rel(C)

of relations over an (E, M)-structured category e. We shall investigate two suf-

ficient conditions for associativity of the horizontal compositions of relations.

One of these is that E is stable under pullback, The other one is a weaker con-

dition. These conditions are known to be equivalent in case & s Epi(e). As a new result we add that they are necessary for the associativity in this case.

The fourth section is devoted to the notion of a mapping as defhed by

Lawvere, i.e. the class of l-cells with a right adjoint- Of great interest is the

question under what circumstances the category e can be recovered via the

isomorphism

The answer to this is that these categories are isomorphic precisely when E C

RegEpi(e). In principle this section presents the main result of [21]. However,

it was possible to significantly sirnpli& the proofs and to remove an error in

the argument for the main result and its technical lemma. Parts of the credit

for this has to go to PavloviC himself since his more general main result of [22],

which characterizes maps, has a very easy proof in our setting, as we shall show

here. The last part of the section adopts a proof from the theory of allegories

(cf. [8]) and shows that the category Map(ReI(C)) is fbitely complete; hence,

we generalize a similar result of Jayewardene and WyIer (cf. [IO]), and of Kelly

(cf. [14]) respectiveiy.

In the fifth section we investigate other important classes of special relations

in the setting of [IO], i. e., where M Mono(e), and, in the second part of the

section, with the additional condition that & Epi(e). That section presents

the result of [IO] and some of [25].

The sixth section further investigates the category Map(Rel(C)) of map-

pings in Rel(C). Kelly proved in [14] that in the setting where e has a so-caiied

proper stable (&, M)-factorization system, so that hZ: 2 Mono(C) and E C Epi(e)

is stable under pdback, there is an isornorphism

Kelly's proof of this is presented here, and we analyze wbere the conditions on

M and & are used. It tums out, that some of the results in [14] do not need

these conditions or only E S Epi(C)-

In the seventh section we turn our attention to factorization systems in

Rel(e), It is well-known that any relation between sets factorizes through its

image when considered as multivalued function. We shall show that this can be

generalized. In fact, the factorization gives rise to classes and fi so that we

obtain an ( E , %)-factorkation system in Rel(C) provided that M C Mono(e).

Moreover, even without this condition a weaker 2-categod universal property

still holds for the canonical factorization. However, the question whether there

is an (E,fi)-factorization system in 'B := Map(Rel((2)) such that Rel(e) =

Rel(23) without any condition on M remains open.

In the last two sections we investigate limits and colimits in the ordinary

as weU as in the 2-category Rel(C'). W e shail show that the existence of well-

behaved coproducts in e implies the existence of coproducts in Rel(e). More

precisely, if e îs an eztensiue category (cf. [6]), then the coproducts in Rel(C) are

given as in e. Moreover, Map(Rel((2)) is closed under coproducts in Rel(C).-

Unfortunately, Rel(e) is not fbitely (CO) complete in general. As open problems

we l a v e the questions whether the coproducts in Map(Rel(e)) are extensive

and whether Map(Rel(C)) is cocomplete if e is so.

Last, but not least, we shed some ligfit on colimits of w-chains. These are

of particular interest especially in theoretical cornputer science, because they

allow the iterative construction of initial algebras of w-cocontinuous functors.

Initial algebras can be used as a model for recursively specified data types. (cf.

[19]). Being able to construct initial algebras in a category of relations yields

a powerful tool for the specification of non-deterministic problems, for example

optimization problems (cf. [2]).

Unfortunately, the desired colimits do not exist in general in Rel(C!). How-

ever, if we impose certain conditions on C, then there is a canonical construction

with a weaker (2-categorial) universal property. As in the case of coproducts,

these suEcient conditions simply Say that colirnits of w-chains in C! must &st

and be well-behaved. To be precise, colimits of w-chahs in C! have to be univer-

sa1 and they need to comrnute with pullbacks. It is somewhat unfortunate that

the construction seerns to force us to deal only with rnonic relations, so that ?Y€

has to consist of monomorphism. For maps, however, this is not a problem at

ali because it is automatically true. Moreover, in case & E Epi(C) our canonical

construction yields colimits of w-chah in Map(Rel(C)). It rernains an open

problem though, whether the condition & C Epi(e) is necessary for this result.

But now let us begin our treatment of relations in categories by quickly

recalling the basics about (&, M)-factorization systems.

2 ( E , M)-struct ured cat egories

The (E, M)-stnictured categories discussed in this section give the appropriate

environment to derive a calculus of relations in categories. Therefore the most

important results about (E, M)-structured categories used in this thesis are listed

here. The definitions and theorems are al1 standard. The proofs are almost al1

omitted. They can be found for example in [l].

De5ït ion 2.1. Let & and M be classes of rnorphisms in a category (2. (&, M)

is culled a factorization structure for morphisrns in e and C! i s called (E, M)-

structured prouided t ha t

1. each of & and ?YI i s closed under composition with isomorphisrns, i.e.

i f e E &, h E Iso(e), and i f he exists, then he E E,

i f rn E M, h E Iso(e), and if m h exists, then mh E M,

2. C hm (&, M) -factorizations of morphisrns; i. e., each morphisrn f in e has

a factorization f = me, with e E & and rn E M, and

3. C has the unique ( E , M)-diagonalization property; i. e., for each commuta-

tive square

with e E E and m E M, there ezist a unique diagonal d : B 4 C such that

Note that if e is (E, M)-structured, then eOp is (M, &)-structured.

Proposition 2.2. If e i s ( E , M)-structured, then the following hold:

2. each of E and M is closed under composition,

3. E and M detemine each other via the diagonalziation property; i e., a

rnorphisrn rn belongs to M i f m d only i f for each commutative square of

the form (1) with e E & there is a diagonaP. O

Proposition 2.3. If fg and f are bath in M, then g is in M.

Proposition 2.4. In an (&,M)-structured category C! unth products of pairs of

objects the following are equivalent:

3. Sect(C) Ç M ,

4- for each C-object A the diagonal r5A = (lA, lA) : A + A x A belongs to M,

5. f g E SYt implies that g f M,

6. fe E M and e € & imply that e E Iso(e). O

Corollary 2.5. If C is (8, Mono(C)) -structured, and hlrs binary products, then

E = StrongEpi(e) = ExtrEpi(e). O

l Note that uniqueness of the diagonal is not necessary hem.

The following results 2.6 and 2.7, which will be needed in Section 6 are taken

from [14]. Parts of them can &O be found in [12].

Let (2 be a category with pdbacks, so that the strong epimorphisms coincide

with the extrema1 ones. Recall that the pullback of a pair x, y along a morphism

g is the M t

of the diagram given by x, y and g. Since it is forrned by taking three pullbacks

and setting h = pire = qorr, u = poro, and v = qlrl , it foilows that h is

epimorphic if every pullback of g is epimorphic.

Lemxna 2.6. If m g has the sarne kemel-pair crs g , and every pullbock of g is

epimorphic, then rn is monomorphic.

Proof. It is easy to see that the pullback u, v of the kernel-pair x, y of rn dong

g as in (2) is the kernel-pair of mg. By hypothesis, this is the kernel of g. Thus

xh = gu = gv = yh, and therefore x = y since h is epic, whence m is monic. O

Now recall that in a category with puUbacks an epimorphism is regular,

precisely when it is the coequalizer of its kernel-pair.

Proposition 2.7. Suppose that C! is a category in which pullbacks of extrernal

epimorphisrns are epimorphic. Then

if C either

(a) admits wequaliiers, o r

(b) i s finitely complete and (ExtrEpi, Mono) -structured.

Proof. (a) If e is an extrernal epi and g the coequalizer of e's kernel-pair, then

e = mg for some morphism m. By Lemma 2.6, m is monic, and therefore, by

extremaLity of e, an iso, whence e is a coequalizer of its kemel-pair.

(b) Let e be an extremal epi again, let ko, kl be its kernel-pair, and let f be

a morphism with f = f ki. Factorize (e, f ) = (m, n ) g . Clearly the kernel-pair

of mg = e coincides with that of (e, f), and since (m,n) is monic with that

of g. Hence, by Lemma 2.6, m is rnonic and, by extremality of e, an iso. So

f = ng = nm-'el which shows that e is regdar.

Note that (b) implies that a 6nitely complete (ExtrEpi, Mono)-structured

category C is regular as soon as extremal epimorphisms are stable under pull-

back- From (a) we get that in every (&, M)-structured category with M Ç

Mono(e) and 1 C Epi(C) stable under pdlback (also cdled a category with

a proper and stable factorization system) the extremal epirnorphisms coincide

with the regular ones, because ExtrEpi(e) C E, by the dual of Proposition 2.4.

Proposition 2.8. In any ( E , M)-stnrctured category the c l w M (as a full sub-

category of C2) is closed under al1 limits.

Proof. Let C? be an (E, 3t)-structured and T ) be a category. Further Iet F :

D + C and G : 9 I, e be diagrams and let X : AA -+ F and v : AB + G

respectively be their Limits. Finally, let p : F -t G be a natural transformation

We m u t show that the unique arrow f : A + B that makes the diagram

Af AA-AB

commutative lies in M. In order to see this (€,?YI)-factorize f = me. By the

diagonaiization property there is a unique arrow di : E + Fi for all i E D such

that the following diagram cornmutes:

Note that, by uniqueness, the 4 form a cone d : A E + F,

Tberefore there exists a unique arrow h : E + A such that A - Ah = d. Then

since X = d Ae and since X is a mono source, he = lA. To see that eh = LE,

r e c d that f = me is ao ( E , M)-factorkation and consider the diagram

where the lower right triangle cornmutes since

by diagram (3) and diagram (4).

This proves that m is an isomorphism, whence f E M-

Corollary 2.9. The class E i s closed under al1 colimits in any ( E , M) -strirctured

Proposition 2.10. If C is ( E , M) -structured then M is stable under the forma-

tion of pullbacks, i- e. giuen a pullback square

9 P - B

3 Spans and Relations

The introduction into relations given in this section and the discussion on maps

in Section 4 in principle follow the work of Jayewardene and Wyler (cf. [IO])

and Pavlovit (cf, (211) respectively. However, the proofs presented here may

difFer fkom the original papers and the materid is presented in a different or-

der. Moreover, some of the results have been refined here and others are new.

DifFerent proofk and r e h e d or new results wlll be indicated where they appear,

Now let us begin by introducing spans.

3.1 Spans

Let e be a finitely complete category. For objects A and B of C! we form the

category Span(A, B) = C'/A x B consisting of equivalence classes of isomorphic objects of C'/A x B, Hence, an object of Span(A, B) can be represented by a

pair of arrows r = (ro, r l ) : R + A x B, which will be denoted by r : A t , B.

We will also take the fieedom and refer to such an arrow as a span, instead of

considering the equivalence classes.

Arrows in Span(A, B) are denoted by a : r + s, for spans r : R + A x B

and s : S + A x B; hence, a is represented by an arrow a0 : R + S such that

7- = sao.

To compose spans r : A u B and s : B t, C horizontally, one fonns a

pullback as shown in the next diagram

and puts s O r := (rom, $1 pl). This composition is clearly functoriai and associative, up to isornorphism at the level of niorphisms in e, hence strictly

so at the level of equidence classes. The identities are represented by the

It is easy to check that we obtain a 2-category Span(e). Its O-cells are the

objects of e, its 1-cells are the spans and its 2-cells a re the arrows of the hom-

categories Span(A, B) .

Example 3.1. Consider the bicategory Span(Set). A function f : X + Y in

Set can be viewed as a functor F : Y + Set assigning to each object of the

discrete category Y its preimage under f . Hence, a span r = (ra , rl ) can be

represented by a set-vdued matrix

M = (r-' (a, b ) ) ;;:.

Composition of M : A 5, B and N : B t, C then resembles matrix multiplica-

tion:

The identity on A is of course represented by the identity matrk whose entries

are singleton sets containing the elements of A dong the diagonal, and whose

other entries are empty.

Let us close this section with the remark that a calculus of sparts, and more

generdy a calculus of relations as presented below, does not depend on the ex-

istence of binary products in C?. Assuming that they e d t forces e to be fkitely

complete, since the existence of pullbacks is required to deûne horizontal com-

position, However, dropping the existence requirement for binary products in e

greatly increases the technical &ort necessary to discuss relations. Therefore we

will aiways assume that (3 has binary products, and that it is therefore finitely

complete.

3.2 Relations relative to a factorization system

Let (2 be any ( E , M)-structured category. For an object X we denote by M/ X the

full subcategory of e / X fonned by the arrows of M. By considering equivalence

classes of isomorphic elements of M / X we get the equivalent categov sub(X) of

M-subobjects of X. Rather than using these equivalence classes we shall refer to

elements of M/X as M-subob jects in Lieu of the equidence classes represented

by them, and we mi te rn = n for m,n E M/X if there exists an isornorphism i of e with mi = n. Note that if M Mono((2), then M/X becomes a preordered

class with respect to the order defined by

Furtherrnore, in this case sub(X) is a partiaily ordered class.

Now let C! be a h i te ly complete (&, M)-structured category. For objects A

and B of e let us form the category of relations fkom A to B as Rel(A, B) =

sub(A x B). Note that (ho, AU) where Ail denotes the class of ail morphisms

and Iso denotes the cl- of isomorphisms of e, is a factorization structure for

every category e. Thus Span(A, B) is a special case of a category of relations

fiom A to B. We therefore adopt all the notation introduced in 3.1 for relations.

As for spans we shall refer to objects of M/A x B as relations, again meaning

the equivalence class represented by them.

FinaUy observe that although our definition of relations and their composi-

tion follows [21] notationally, it in fact is slightly different since Pavlovie does

not work with equivalence classes but rather with objects of %€/A x B.

3.3 Composition of relations

For ob jects A and B of (2 the (E, M)-factorization system gives rise to a functor

im : Span(A, B) + Rel(A, B),

which migns to an object of Span(A, B) represented by s : S + A x B the

object of Rel(A, B) obtained by ( E , M)-factorizing S. This is clearly well-defhed

and functorial. We denote by in the inclusion b c t o r . h order to compose

relations one uses the adjunction im

in the diagram

-i in : Rel(A, B) + Span(A, B) as shown

Rel(A, B) =- M/Ax B.

The functors i~ and ino are obtained by choosing representatives of the

equidence classes in Span(A, B) and Rel(A, B) respect i~ei~

The composite of two relations r : Ai+ B and s : B t, C is thus dehned as

follows:

This means that first a pullback is formed as in diagram (5), and then the

image of ( r o m , s i p i ) is taken. Note that composition clearly defines a functor

from Rel(B, C) x Rel(A, B) to Rel(A,C). Moreover, any 2-cell a : r + r'

in Rel(A, 8) induces an arrow a : (ropo, s i p l ) -+ (ropb, s i p i ) in Span(A, C ) ,

where (&, d) is obtained fiom r' and s in a diagram like (5). The 2-ce11 s O a :

s 0 r -t s o r' is now the arrow im(a) : im(ropo, s l , p l ) + im(rbpb, sip',) in

Rel(A, B).

Given ,6 : s + s', the 2-cell P O r : s O r + s' O r is obtained similady. The

constructions s O (-) and (-) O r are easily seen to be functorial. Furthermore

ac and p are natural in the sense that

(/3 O #)(s O a) = (SI O a)@ o r )

holds. This is taken to be the 2-cell B O a.

Fkom now on, we will no longer distinguish between im and ho. For ex-

ample, the notation r = im( f) , where f E e / A x B represents a span and

r E M/A x B represents a relation, shail be used frequently.

Taking as its objects the objects of e, as its hom-categories the categories

Rel(A, B) and the horizontal composition as defined above, the only ingredi-

ents rnissing to make Rel(C) a bicategory would be associativity and the identity

laws. Unfortunately it turns out that, without m y restrictions on (Cl M), a s s e

ciativity fails, so that Rel(e) is in general not a bicategory. An example will

be given in Section 3.4. However, once associativity and the identity laws are

established, Rel(C!) will even be a 2-category. Nevertheless, we shall refer to it

as a bicategory of relations.

3.4 Associativity and identity relations

Composition of relations is in general not associative. We will first give an

example for this (which is due to Klein 11151) and then derive sdücient conditions

for relational composition to be associative, with identities given by

In the following example the opposite relation of a relation r = (ro, r i ) shall

be denoted by r0 = ( r i , ro).

Example 3.2. Consider the category Top of topological spaces and continuous

functions. Every morphism f : X + Y of Top factors

where e has dense range and m is a closed embedding. Taking M to be the class

of closed embeddings and E the cIass of continuous maps with dense range, it is

not difficult to check that Top is ( E l M)-stmctured.

Denote by î f the relation M(1, f) . Let i : $ + IR be the usual embedding

of the rationals into the real line. Clearly i E E. Fbrther let j : $ + IR be

dehed by j(x) = ax for some fked h t i o n a l number a. Note that we do not

need to take images for any of the relations ri, ïj and bR since IR is Hausdorff,

which implies that graphs are closed. Now clearly ri o (ri)" = 6 ~ , and then

( r i O (ri)") o i'j = rj. On the other hand we clearly get (r i)o o ï j = ( i q , iQ) : O -t $ x $, where again taking images is not necessary since the empty set is

closed in $ x $. But then ri O ((ri)O O r j ) ( iQ , iR) : O + $ x IR which is in ?Y€-

Recall that rj = (1, j) is clearly not the empty relation from $ to R. Therefore

( r i O (ri)O) O r j $ ri O ((ri)O O l? j).

For the more restrictive setting of a pmper (E, M)-factorization system, i. e.,

with 3M: E Mono(e) and & E Epi(e), it has been shown by Klein in [15] that

associativity of relational composition is equivalent to E being stable under di

pullbacks. Note that in this setting the identities are git-en as in Span(e) by

the diagonals : A + A x A since & Ç Epi(e) is equivalent to al1 diagonals

being in M (see Proposition 2.4).

PavloviC in [21] claims that in the generd case with no restrictions on & and

M pdback stability of E is s t i l l necessary and &cient for relational composi-

tion to be associative and the identities on A to be im(JA). He only proves a

very small part of this statement and does not prove necessiSr of the condition

at d.

Jayewardene and Wyler analyse the situation much more thoroughly in [IO].

They prove stabiliw of E is still sdcient but that a weaker condition also

suffices. The results of the next two subsections except 3.9 and 3.10 are due to

P O 1 -

3.4.1 Images and inverse images of relations

Definition 3.3. Given an arrow f : A + B of e and d a t i o n s r = (ro,rl) :

A t-, C and s = (so, si) : B t-, C, we define the image functor f-' :

Rel(A, C) + Rel(B, C) 6 y

Further we define the inverse image functor f + : Rel(B, C) + Rel(A, C)

by

f "3 = (4, ~lf'),

where 98 and f' are obtained by fonning a pullback as in

Note that (8) is a pdback if and only if

is a pdback. That means that we need not take images in the last definition

since M is stable under puilback.

Let us now note some properties of the functors just defined.

Proposition 3.4. (i) f-) -i fC for any C!-amw f.

(ii) I f g f is defined in (2, then (gf)-' = g-' of-' and (gf)+ ft O g+.

(iii) For relations r = (ro, rl) : A -t+ B und s : B W C we have s o r 21 rO)rts.

Proof, Items (ii) and (iii) are immediate consequences of the definitions. For (i)

where f : A + B let r = (ro,rL) : A t , C and s = (sol s l ) : B t+ C- We must

show that

is a natural isomorphism. To a given k : r + f +s assign k H f 'k : (f ro, r i ) -+ s

of Span(B, C), where f' is obtained as in (8). This yields a natural isomorphism

Then use the adjunction

to get a natural isomorphism

hom((fro,r1),s) = Y.

3.4.2 Legs a n d leg-stability

D a t i o n 3.5. For a relation r = (ro, r l ) : A -t-, B we cul1 r o and rl the legs

of r. We say that e has stable legs, or that (2 is leg-stable, if E is stable under

pullback dong legs of relations.

Observe that (8) shows legs to be pdback stable. Further note that since

C is assumed to have products, legs are composites nm of a projection n of a

binary product and an arrow m E M. Therefore the following result holds:

Proposition 3.6. e i s leg-stable if and only i f

(i) For eo, el E E , the product eo x el is in C.

(ii) € is stable under pullback along M-arrows.

The next result gives us sufEicient conditions for composition of relations to

be associative, with identities k ( d A ) .

Theorem 3.7. For a finitely complete, (&, M)-structured categow C the follow-

ing are equivalent:

(i) r O im(a) z= i m ( r O a) for al1 spans a and relations r.

(ii) im(b) o r im(b o r ) for al1 spans 6 and relations r .

(iii) T 0 im(ao, a l ) = a z a t r for all spans a = (m, a l ) and relations r .

(iu) e+etr = r for al1 relations r and all e E &.

If these conditions are satisfied, then composition of relations i s associative

with identities LA = im(dA) : A t+ A.

Proof. Considering opposite reIations it is clear that (i) and (ii) are equivalent.

Using these we shall prove the statements at the end about relational composi-

tion. In (i) and (i) choose a and b to be identity spans. Then

for any reIation r : A t , B. To see associativity of reIational composition

consider

t o ( s o r ) t o i m ( s o r ) = i m ( t o ( s o r ) ) = i r n ( ( t o s ) o r ) cz i m ( t o s ) o r zz ( t u s ) o r .

2 1

Now in (iii) let af-r = (uo , ul) . Then s O (ao, al) = (aouo, ul) and

+ t im(r o (a~, al)) = im(aouo, ul) = a, a, r-

Therefore if (i) is true, then the left-hand side is equal to r o im(m, al). Con-

versely, if (iii) is true, then the right-hand side is equal to r 0 im(m, al)- This

shows that (i) and (üi) are equident.

FinaUy, consider e E E with codomain A. Then im(e, e) = i1n(6~). So

if (iii) holds then (9) implies (iv). Conversely, let (m, al) = (so, sl)e be an

(8, M)-factorization. Then

i. e., using (iv), (iii) is shown. O

Now let us investigate how leg-stability of C! is connected to the last result.

Theorem 3.8. If C is leg-stable, then

fo r all spans a = (a~, al) and b = (bo, 61) such that the composition is defined

and ai o r bo is a Zeg. Furthermore 3.7 (il-(iv) are valid.

Conuersely if M C Mono(e) and 3.7 (i)-(iu) are true, then C! h m stable legs.

Proof. Conditions (i) and (i) of 3.7 are clearly special cases of (10).

To prove (10) let (mi al) = (ro, rI)e, and (bo, bi) = (sol si)eb be (E, M)-

factorizations. J?urther let alt squares in the diagram

be pullbacks. Now observe that a o b = (roe,qouo, slebq3u1). Recali that legs of

relations are pdback stable. Hence, po and pi are legs, and then ql and q2 are

in & by leg-stability of (2.

If ai = rl e, is a leg, then pl ql is a Ieg and therefore uo is in E. But then

An analogous argument applies if bo is a leg.

Conversely, let r = (ro,rl) : A* B be a relation, and let e E E have

codornain A. Now form a pdback squaze in e:

Thus we have

By 3.7 (iv), this is isornorphic to r . Hence, there is an ( E , M)-factoriza-

tion (ropl, ripl) = (ro, ri)et . Since we assume that M Mono(e) this implies

pl = e E E. Therefore e has stable legs. O

Corollary 3.9. For a finitely complete (8, M)-structured category e with E sta-

ble under pullback the image finctor satisfies

for spans a and b such that i%e composition is defined. In particdar, the com-

position 5" of relations is crssociative, urith identities given by LA : AuA-

Proof. Considering diagram (11) we have, by pullback stabiliiy of E, that uo,

u l , ql and q2 are in E. As before this irnplies the result. O

Corollary 3.10. If E Epi(e), then composition of relations is associative if

and only $1 is stable under puflback.

Proof. One direction is 3.9. For the converse note that & C Epi(e) implies that

a l l arrows of the form (1, f ) lie in M. Suppose

is a p d b a c k square with e E E. I t is easy to see that

On the other hand (e, 1) o (1, e) = 6, which implies that r E (f, 1) by associativ- ity. Hence, ( fp i ,p1)e1 = (f, 1) for some e' E E. Since (f, 1) is monic, pl e' = 1,

whence pl lies in E. O

The last result shows that if E C Epi(e), then Ieg-stabiIity of (3 is equivalent

to pullback stability of &. Moreover, observe that the results obtained essentially

generalize those of Klein (cf, [15]).

If & does not consist of epimorphisms, then it is possible that E is not stable

under pullback but (2 is stiU leg-stable (see [IO], 1-8.1 for an example). However,

there remain two unsolved problems:

1. Can relational composition be associative, with identities LA, but 3.7 (i)-

(iv) not valid?

2. Can 3.7 (i)-(iv) be valid if (2 does not have stable legs?

3.5 Structure of Rel(e)

The hom-categories Rel(A, B) of the bicategory of relations Rel(C) are finitely

complete. Pullbac1.s are lifked from e. The terminal object is l A x B : A x B -t

A x B which clearly is in M. The product of r : A t , B and s : A t + B in

Rel(A, B) is denoted by r A s and is given by forming the pdlback

By the pdback stability of ?Y€ (see Proposition 2.10) the arrow r A s is in M.

Composition of relations can be presented using the local product just de-

scribed and the global product of e:

where ?r : A x B x C -+ A x C denotes a projection. To see this note that

( l A x S) and (r x lc) are in M, recall diagram (5) and consider the following

diagram:

where the 4 inner squares are pdbacks.

Every bicategory of relations is self-dual. Consider the global assignrnent

that leaves the objects and 2-cells unchanged, while taking an M-relation r =

(ro, TI) : R + A x B into the opposite relation r0 = (ri, ro) : R + B x A, which

indeed is in Sri: since A x B E B x A. It defines a functor for each hom-category.

Furthemore composition of 1-cells is preserved. So ( - ) O is a bifunctor.

In any bicategory Rel(C!), where C! is (E, M)-structured with & stable under

pullback, Fkeyd's Modular Law (cf. [8]) holds true, in the sense that for any

relations r : A t, B, s : B -i+ C and t : A t, C there exist a 2-ce11

The relation on the left-hand side is obtained by factorizing the arrow Pt +

A x C obtained by fonning the pdback

where (ropO, s i p l ) is obtained as in diagram (5). Now let us consider s0 0 t,

which is obtained using

To get r A sO O t we form the pullback

Now observe that t l p f = s l p l x . So there is a unique arrow h : P' -t Q with

(40, q l ) h = (p', p r ~ ) . Now (toqo, sopi) h = (top', s O p l x ) = rmx. Hence, there is

a unique arrow h' : Pr + Q' with yh' = h and q'hf = pox. FinaUy, consider the

which induces the relation s O (r A sO O t). Then rl q'h' = soql h = s o p ~ x , whence

there exists an arrow

which gives us the desired 2-cell by the universal property of the (El M)-facto-

rization system.

Finally, note that a bicategory of relations need not yield an allegory in the

sense of Fteyd and Scedrov (cf. 181) if we want the local product to coincide with

the "intersectionn of relations (see Example A.3 in Appendix A).

3.6 Rel(C') as symmetric monoidal closed 2-category

It is weU-known that Rel(Set) c m be viewed as a symmetric monoidal closed

2-category with the tensor given by the cartesian product. This fact can be

generalized. For every (E, M)-stnictured category e with E stable under pull-

back, Rel(e) is symmetric monoidal closed. For the deîmition of a syrnmetric

monoidal closed category consult [13].

In ReI(f2) the tensor

is given by the binary product on e, that means that A@ B := A x B on objects

a n d r @ s := ( T ~ x s ~ , ~ ~ X S ~ ) : A@B + A'@B1 forrelationsr = ( r o , r l ) : A t , A 1

and s : (so, si) : B u Br. Clearly @ is 2-functorial since products commute with

pullbacks in e and E and J'Y€ are closed under products by stability. Note that

the unit object is the terminal object 1 of (2. Associativi~ and symmetry of

@ as well as the necessary coherence conditions follow now easily from those of

x : e x e + e. As for closedness note the following result.

Proposition 3-11. The finctor - @ A : Rel(C) + Rel(e) is self-adjoint for

euery object A of Rel(C) .

Proof. The components of the counit eB : (B x A) x A t , B are given as es :=

im(ls x 6 A , ~ o ) , where TO : B x A -+ B is a product projection in e. For any

relation r : ((ro, rr), 4 : C x A -H B we defhe X(r) := (ro, (rz, ri)) : C tt B x A-

In order to see that e O (X(r) @ L ~ ) c~ r consider the following diagram

R x A B x A

C x A (B x A) x A

where the square is evidently a pullback. As for the uniqueness of X(r) suppose

that s = (so, (s2, sl)) : C -+t B x A is a relation with e o ( s @ 1 ~ ) = r. Note that the left-hand side of this is formed as in (13) with all ri replaced by si . Hence,

((sol sL), s2) = T, which means that s - X(r) by associativity of x in e. O Clearly we cannot be satisfied with the universal property as shown in the

last result in sr 2-category. But unfortunately we have to impose a condition

either on M or on E to obtain the universal property for 2-cells.

Corollary 3-12, The 2-functar - @ A is self-adjoint for every object A, i f

M E Mono(e) o r E C Epi(e).

Proof, Suppose t = ((to, ti), ta) : C x A t, B is a relation and a! : t + r with

r as above is a 2-ceU. Then a : X(t) + X(r) is a 2-ceU with eg O (a @ c A ) = a:

because the 2-cd on the ieR-hand side of this equation can be formed by pulling

back a x lA dong ( I R , rl ). But clearly the square

is a pullback.

Uniqueness is clear for M C Mono(e). If & Epi(e) and 0 : X ( t ) + X(r) is a

2-cell with e B o ( P @ 6 ~ ) = a, then, since r@6A is &en by ( ro x lA, ( r 2 , rl) x lA) :

C x A+ (B x A) x A without taking an image, the composite eB 0 (p @ JA)

must be given by a pullback like (14) with a x lA replaced by P x LA- But then

a = p is obvious.

Note that the conditions on M and E respectively are only used to show the

uniqueness of the 2-cell X ( t ) + X(r).

3.7 A calculus of relations using elements

Under certain circumstances it is possible to obtain a convenient ca.Iculus of

relations using generalized elements. The convenience of this lies in the fact

that relations rnay be treated almost as if they were relations between sets by

referring to their elements.

This has for example been observed by Kelly in [14] for the case of a category

C! with a stable proper (&, ?Y€)-factorizations system. Kelly's results however may

be further generalized, by dropping the condition E Ç Epi(e).

But first let us clarify what is meant by an element or generalized element.

Definition 3.13. Let C be an (&, M) -structured category and B an object of C.

(i) An arrow b : X -t B is called an X-element of the object B.

(ii) If m : M + B ts an M-subobject of B we Say an X-element b belongs to

- mx. m if there m k t an a m w x : X + M such that b -

The next proposition is the key observation that dows us to develop a

calculus of relations using elements.

Proposition 3.14. Let e be an ( E , ?Y€)-stnrctured category with M c Mono(e) - Then any M-subobject is determined by the knowledge for al1 objects X of the

X-elements that belong to it.

Proof. First observe that the elements of an M-subobject do not depend on the

representation of that subobject. Let rn : M + B and m' : M' + B represent

the same subobject, i. e-, there is an isomorphism i : M + Mt with rn'i = m.

Now if b : X + B belongs to m, tbat means that TRX = b for some arrow

x : X + M, then m'ix = 6, whence b belongs to m'.

Note that every M-subobject m belongs to itseif. To complete the proof

suppose that m and m' have the same X-elements for every object X. In

particular m' belongs to m and vice versa. That means there &st arrows

i : M + M' and j : M' + M such that m'i = m m d m j = m'. Using

M C Mono(C), it is easy to see that m = m', whence m and m' represent the

same M-subob ject. O

The second part of the proof of the previous result does not in general remain

true if the condition M E Mono(e) is dropped, as the following example shows.

Example 3.15. Consider the category CAT of (smd) categories and functors

between them equipped with a factorizations system as follows. Let 1 be the

class of functors that are bijective on objects and let M be the class of full and

faithful functors. Then clearly every functor F : A. -+ 'B factors through the cat-

egory e which has the same objects as A and with AI, A*) := 'B(FA1, FA2).

Obviously F gives nse to functors E : A. + (2 in E and M : e + 23 is M with

F = ME. It is furthermore easy to check the universal property. Finally note

that CAT is finitely complete and that E is stable under pdback. So CAT

admits a calculus of relations. Observe however, that neither E Epi(e) nor

M C Mono(e) hold true.

To see that the second part of the previous proof fails, consider the functors

M : 2 + 1 and Idl : 1 -+ 1 where 1 denotes a discrete category with one object

and 2 is the category given by

Clearly both functors are in M, hence represent difFerent M-subobjects since

obviously 2 is not isomorphic to 1. Rowever, Idr and M have the same X-

elements. ClearLy all X-elements of Idl belong to M since 1 is a subcategory

of 2. Moreover if B : X + 1 belongs to M, then B belongs to Idl, too, since

Proposition 3.16. Let C! be an (El M)-stnrctured category with E stable under

pullbacL Then an X -element b of B belongs to the image of an arrow f : A -P B

i f and only i f there i s an arrow p : Y -+ X in & and an Y-element a of A such

that the square

Proof. Let f = me be an (E, M)-factorization of f. If b belongs to m, tha t

means that mi = b for some arrow i, then pulling back e along i yields the

desired p and a.

Conversely, if we have a square f a = bp with p E &, then the universal

property of the factorization systems gives an arrows i with mi = b so that b

belongs to m. O

The previous result holds true, whenever E is stable under pullback. If

M C Mono(e), then the statement is even equivalent to stability of E under

pullback. In fact, suppose that

a puliback square with e in &, then f belongs to the image of e. So if we

have p in E and an element a with eu = fp we get a unique h with gh = a

and e'h = p, whence e' E & by the dual of Proposition 2.4. For the rest of this

section we assume that & is stable under pdback.

Now let us introduce some more notation. An element (a, b) : X + A x B

belongs to a relation r : R -t A x B if and only if there is an x : X + R with

(a, b) = rz. In this case we shall mite b(r)a.

Proposition 3.17. Let r : A t + B and s : B t+ C be relations and let (a, c) :

X + A x C be an X-element. men c(s O r )a if and only i f the= &t an arrow

p : Y + X in E and an Y-element b with cp(s)b and b(r)ap.

Proofi (+) The composite s O r is obtained by forming a pullback as shown in

the diagram

Let me = (rom, sipl) be an (E, M)-factorization. Now if c(s o r)a, then there

is an arrow x with m x = (a, c). Form the pullback of e dong x to obtain an

arrow p : Y + X in E and an arrow y : Y + P as shown in the diagram

P Y - X V I + y P y - A x C . m

Define b := rlpo y = sopi y. Then observe that (ap, b) = (rom y, r lpo y) = rpo y

(+) Conversely, suppose we have cp(s)b and b(r)ap, FPhich means that there

are arrows z : Y + R and y : Y + S with rz = (ap,b) and sy = (b,cp). So

r l x = b = soy, whence there is a unique h : Y -+ P with poh = x and plh = y.

Thus we obtain a commutative square

where p E E. Thus, by Proposition 3.16, (a, c) belongs to rn, i. e., c(s o r)a. Ci

Note that the second part of the result still holds if we omit p. To be precise,

c(s)b and b(r)a imply c(s O r )a but in general not conversely. Also observe that

b(r)a impiies bx(r)az for al1 arrows x such that the composition is defked and

conversely bp(r)ap impIies b(r)a if p E & using the universal property of the

factorkation system.

Baving established an "elementwisen caiculus of relations one can now easily

prove associativity of relation composition sirnilar as in Rel(Set). This is a

straightforward computation which may be done in 4 lines by the Reader-

To illustrate the strength of this calculus let us reprove Freyd's Moddar

Law, which needed quite a bit of diagram chasing, when we proved it in the

general case (see 3.5)- First note that for any relations x and y, b ( z A y)a if

andonlyif b(x)a and &)a, Nowlet T : A - B , s : B t ) C a n d t : A++C be

relations. In order to show

it is d c i e n t to show that b(s o rAt )a irnplies 6(s O ( rhsO O t ) )a . But this c m be

checked easily enough. If b(s o r A t)a, then b(s O r )a and b(t)a. By Proposition

3.17, there is a p E & and an element c such that bp(s)c and c(r)ap. ClearIy

bp(t)ap, whence c(sO o t)ap, which implies c(r A sO o t)ap. Finally, by Proposition

3.17, b(s O (r A so o t)a.

4 Maps

In Subsection 4.5 below the categorical notion of totality and single-valuedness

of a relation will be introduced. One of the standard results about regdar

categories, i. e., categories that are (RegEpi, Mono) structured, is that every

total and single-valued relation (also called a nzap) of Rel(C) corresponds to a

unique arrow of C- That means that every regular category is isomorphic to its

category of maps:

In [21] PavloviC gives a condition for an arbitrary ( E l %€)-factorkation system

to allow an isomorphisrn like this. The condition is simply that E must consist

of regular epis- Although easily stated, the proof of this result is quite involved.

Here we present Pavlovi?~ results in a slightly different order, which enables us

to s i m p w some of his proofs, Moreover, it was possible to remove an error in

the proof of the main result of [21] and its technical lemma. We give a proof

of the main result of [22] for our setting. This result turns out to be useful for

many subsequent results. Finally, note that the results of Section 4-7 do not

appear elsewhere in this form.

For this section it is assurned that, for a given fhïtely complete (El?@-

structured category C!, the class E is stable under all pullbacks. CoroUary 3.9

then dows us to form the bicategory Rel(C). Before we start let us just note

a rather simple but very usehi result. Its proof is immediate and therefore

omitted.

Proposition 4.1. Let C! be an (E , M)-structured category with & stable under

pullback. Further let

be a wmrnutatiue square where the inner squares are fomed by taking ( E , M ) -

factorizations o f f and g and then using the universal property of the factoriza-

t ion to obtain d.

Then the outer square is a pullback i f and only i f the two inner ones are. 0

Now let us follow Pavlovit and investigate maps in such a bicategory of

relations.

4.1 Maps in arbitrary bicategories

Definition 4.2. (Lawvere) A 1-ce11 in a bicategory is called a rnap i f it has a

right adjoint.

That means for a map r : A t , B that there exists a 1-cell r* : B t, A and

2-celh 7 : LA + r* O r and E : r O r* + LB such that the adjunction equations

hold.

Note that in every bicategory the maps form a subbicategory-they are

closed under composition and contain the identities. Now let us prove a very

useful lemma showing that two maps become isomorphic as soon as they are

connected by a 2-ce& comxnuting with the adjunction structure.

Lemma 4.3- Let 23 be any bicategory. Let r, s E B(A, B) be maps with right

adjoints r' and s*, units 77p and qs, and counits E~ and E,. Further let cr : r -F s

and a* : T' + s* be 2-cells such that

are satisfied Then a is an isornorphism so that r and s are isomorphic to each

other. (Dually a* is an isomorphisrn.)

Proof. First of ail, let us show that a is split monic. Consider the diagram

The right-hand square of this cornmutes by condition (18), the left-hand

square simply by the bicategorial naturality Iaw ((7) on page 16). The adjunc-

tion r -I r' says that ( E ~ O r ) ( ~ O vr) = Ir. Therefore

is a left inverse of a. Since s i s* , a diagram dual to (19) yields a left inverse

ai* of a' defhed by

Then a third version of (19) obtained by switching r and s and replacing a by

ii and a' by ai' yields a left inverse of 6. In this diagram the commutativity

of the Mt-hand square will again just be naturality. In order to prove that the

right-hand square is commutative, one has to show that

To see this start with E,(G O Gr) and plug in the definitions of ai and 6'. Then

use natwality, condition (17) and the adjunction equations (15) and (16), the

former in the form 7, = (r' O E, O r) (77, O rjt).

Fhally, since oi has both a left and right inverse, it must be an iso. Thus a

is an iso, too.

4.2 Induced relations

Before we further investigate maps in a bicategory of relations, let us divert

for a whiIe and introduce the notion of a graph of an arrow of e and List some

properties that wiU simplify our investigation.

Firstly, some prerequisites. R e d 1 that a kernel of an arrow f : A + B is a

pair of arrows ker(f) = (ko, kl) : K + A x A obtained by pulling back f along

itself. That means that ker(f) is a mono-subobject of A x A. Then, of course,

kernels of arrows with the same domain are partially ordered. Fùrthermore it

is easy to see that

ker(f) 5 ker(g) if andonlyif (fx = f y + g z =gy)

for all x, y.

Deftnition 4.4. For any arrow f E C(A, B), the relation îf := im(1, f) is

called the graph of f or the relation induced by f .

First note the following result, which wil l be needed in the proof of Lemma

4.6- It is a generalized version of 4. l (b) in [21],

Lemma 4.5. If f i s monomorphic in e and f = me is an (E, M)-factorkation,

where E ii stable under puliback, then m is monomorphic, too.

Proof. We shall show that the kernel pair of r n consists of two equd isomor-

phisms. In order to see this let f : A + B be monic and consider the diagram

where al1 the squares are pullbacks. By stability of the factorization, ?i~ lies in

M. Since khe" = 1, kk E E by the dual of 2.3. But then, since koel = ekh, ko

lies in E for the same reason. Hence, ko is an isomorphism. Being a kernel pair,

ko and ki have a common right inverse. Thus, ko and kl mmt be equai. O

The next lemma lists some important properties of graphs. Note that item

(iv) is due to [IO], that no proof of item (i) appears in [21] and that for (ii) we

use the more general Lemma 4.5.

Lemma 4.6. (i) For any C-morphism f, î f is a map.

(22) As an arrow of C, a graph is always monic. In particular, every identity

relation c = im(1,l) is monic.

(iii) Any relation r = (ro, r i ) : A t, B can be written as the composite r =

ïri O (rr0)O.

(iv) For any Chzorphisrn f and for relations r and s the following hold:

and dually

rf o r = (f4r0)" and (rf)O O s = (f t 9 ) O O -

Proof. (i) We s h d show that (rf)O is a right adjoint of rf in Span(e), i. e.

(1, f ) 4 (f, 1). Then the resdt follows from the fact that the image functor im

is a homomorphism of bicategories.

Note that (f,l) O (1, f ) = (ko,ki) = ker(f) : K + A x A. The common

right inverse of and kl will be the unit 77 : ~5~ -+ (ï f)O O î f of the adjunction.

The counit E : rf o( ï f )O+ 6B is s i m p l y ~ = f : (f, f ) +de viewedas 2-cell.

Let us check the adjunction equation

It is easy tosee that I'f 0(1rf)~oI'f = ( b , fkl). Then weget that ïf 0 7 = q :

(1, f ) + (ko,fk~) and ~ o r f = ko : ( b , f h ) + (W. But koq = 1 ~ .

The second adjunction equation

holds by self-duality of Span(C) simply because the first holds.

(ii) For any arrow f of e, (1, f ) is monic. fience, I'f = im(1, f ) is rnonic by

Lemma 4.5.

(iü) This is immediate knowing that im(b O a) = im(b) o im(a) for all spans

a, b.

(iv) This is again obvious fkom the definitions of f+, ft, and relation

composition. O

Note that (ii) is not restricted to graphs. By Lemma 4.5, every span that is

rnonomorphic in e induces a relation that is s a

4.3 Maps in the bicategory of relations

Observe that in light of Lernma 4.6(ii) every graph of an arrow can be the

codomain of a t most one 2-cell. This already greatly simplifies Lemxna 4.3 in a

bicategory of relations, narnely conditions (17) and (18) are redundant.

Note that this result is not given in [21]. However, it simplifies the proofs of

the important results 4.8 and 4.9 significantly. The tedious checking of condi-

tions (17) and (18), which were omitted in [21], become obsolete, and the work

to prove 4.9 will be nil. This iç the payoff for having fVst proved Lemma 4.6

about graphs, especially item ci).

Corollary 4.7. Let r, s : A t , B be maps in a bicategory of relations with right

adjoints r' and s', units vp and und counits ep and E ~ . Further let a : r + s

and a' : r* -+ s* be 2-cells. Then a is an isornorphisrn.

Proof. Proceed exactly as in the proof of 4.3 to produce left inverses of 6 and

oi= of a and a' respectively. Note that condition (18) holds now automaticaiiy

since E, (a: O a') and e, have monic codomain t ~ .

For the last step of the proof one no longer needs a Iengthy diagram chase to

prove the right-hand side of the third version of (19) commutative- This tirne

the required equation

€,.(ai 0 ai*) = E8

holds automatically. O

The next result shows that in a bicategory of relations a rnap r h a s a canon-

ical right adjoint, namely its opposite relation rO.

Proposition 4.8. If r : A t , B às a map in the biccrtegory of relations Rel(C),

its right adjoint is ro : B + A .

Proof. For a simpler notation let us denote a given right adjoint of r by s : B U

A. Dualizing r i s, we have s0 t r0 : B t+ A. The plan is to construct Zcells

a : s0 + r and a* : r0 + S. Applying Corollary 4.7 we c m then conclude that

r0 = s, which is the result.

The 2-cell a : s0 + r will be obtained by dualizing

where E is the counit of the adjunction r i S. The 2-cell K- : LA + ro o r will be

induced by an arrow k : 6A + r0 O r in Span(A, A). So Iet us constxuct k.

The following diagram in (2 summarizes the construction:

Now let us analyse how this cornes about- The arrow d in (1) is induced

using the universal property of the product r A r in Rel(A, B) , which is given

by the pullback

Note that d is simply the diagonal (Ir, 1,). The arrow q that makes (II) com-

mutative is then obtained using the universal property of the pullback in C that

forms the product (lA x rO) A (r x lA) in (?/A x B x C. Apply this property

to (Po, rorpo) : P + R x A and (rorpo, po) : P + A x R where po is the left

projection of r A r and 1r0 the left projection of A x B.

Square (III) is easily seen to be a pdback square. Squares (IV) and (V) are

now obtained forrning an ( E , M)-factorization of Tor = ro and

Observe that by (12) on page 25 the latter indeed yields r0 0 r. Furthemore

(E, M)-fxtorize m(r A r ) to form (VII) and (VIII), In order to obtain d' which

actually separates them, use the diagonalization property of the factorization

system. Findy use that property again to obtain 6-

Next we shall show that im(ro) has a section j. Having done this it is possible

to define

k := q'd'j : bA -+ r0 O r. (22)

In order to construct j we shaü first construct an arrow p' : s 0 r -t im(ro x

f A). The following diagram shows this construction:

Note that z denotes ( l A x S ) A (r x lA). The arrow p in (1) is one of the

projections of this product in e / A x B x A, Square ( I I ) c m easily seen to be a

pullback if extended to the right by the 'leftl' product projections. Of course,

( I I I ) is still a pullback square, and (TV) and (V) are again (8, M)-factorizations.

F'urthennore, ( E , M)-factorize (m x I A ) (r x lA) = (ro x 1 ~ ) to get im(r0 x 1 ~ ) .

The arrows w and p' are £inaLly obtained by using the universal property of the

factorization again. Observe that since (LI) and (III) are pdbacks both sqiiares

glued together fonn a pullback, too. Since (VI) is obtained by (E, M)-factorizing

two opposite sides of this puUback it is a pdback by 4.1.

The universal property of this puUback now produces the desired section j

of im(ro). Just let bA = LAe be an (€,?kt)-factorkation of the diagonal and

consider the diagram

Recalling definition (22) we get k : bA + r0 o r, whence tc : CA -+ r0 O r of

The other 2-cell a" : ro + s can be constructed similady as

where x : LA + s o s0 is derived from 77O : &A + r0 O sO just as K. : LA -+ ro 0 f

was derived from 7 : LA + s o r. O

Corollary 4.9. Let r, s be mups in a bicategory Rel(f2) of relations. If there

is a 2-ce11 r + s, then r S.

Proof. We have r 4 ro and s -i sO by 4.8- Now every 2-cell a : r + s induces a

2-celi a" : r" -+ sO. Thus by 4.7, r = S. Cl

Note that Proposition 4.8 significantly simplifies checking mhether a relation

r is a map. Firstly r0 is the oniy candidate for a right adjoint and secondy, one

has to check only one of the adjunction equations since (15) and (16) become

dual to each other:

The last result shows that maps in a bicategory of relations are abçolutely

rigid. The only 2-ceh between them are isomorphisms. Therefore the maps of

Rel(C) form even a category Map(Rel(e)). Its objects are the same as those

of Rel(e), and its arrows are equivalence classes of isomorphic maps in Rel(e).

Proposition 4.8 &O dows a characterization of equivalences (or equivalently,

isomorphisms) of Rel(e) similar to 3.8 of [IO], but generalized.

Proposition 4.10. A relation r is an isomorphism (necessarily with inverse

rO), i f and only if r and ro are maps.

Proof. (+) If s in an inverse of r : A t , B, then L A E sr and T S = LB show r to be a niap with right adjoint S. Hence, by 4.8, s E rO. Moreover, r0 is a map.

(¢=) Conversely, if r and r0 are maps, then r0 o r and r o ro are maps, too.

But there are 2-cells 77 : LA + r0 o r and E : r O r0 + LB . Hence, by 4.9, LA r0 or

and r O r0 LB, whence r is an isomorphism with inverse rO.

4.4 Convergence and the graph functor

CoroUary 4.9 and the remark foilowing it give us a finctor

which maps objects identicaiiy and which takes an arrow f to the equidence

class rf = im(1, f).

The following notions are again relative to an (E, M)-factorization system.

Definition 4.11. A map r : A t+ B is said to converge to an arrow f : A + B

~ f e g r = rf. An object B of C is separated if a map to i t can converge to ut most one

a w w ; t, e-, î f rg implies f = g for al1 f,g E e(A, B).

The next result wil l d o w a characterization of the faithfulness of i? in terms

of a condition on the (&,M)-factorkation system. It is a combination of the

results 4.6. and 4.9. of [21]. Item (ii) which is inherent in Pavlovik's proofs is

stated separately here because it will be helpful for many subsequent results.

Proposition 4.12. The following are equivalent:

(i) B is sepanzted

(ii) For the identity morphism LB = , L ~ ) , LO = LI holds.

(iii) The diagonal 6B is in M.

(itl) For any pair f , g E e(A, B ) the ezistence of a 2-ce11 im(f, g ) + LB irnplies

Proof. (i) + (ii): Fonn the kernel pair of LB as shown in the diagram

where both squares are pullbacks. Being a component of ker(tB), ko has a right

inverse h : Ig 3 K. Hence, in e/rB x B, one has the morphisrnph : (1, L ~ ) + I'b

and im(ph) : l 7 L l -, rco by (E,M)-factorizing. Applying Corollary 4.9 we have

i ? ~ ~ = ï L 1 . Now, since B is separated, b = L I .

(ii) + (Ki): Because 6B = t ~ e is an (C, M)-factorkation with some e E &,

lB = b e follows. Then consider the square

which clearly cornniutes. By the universal property of the factorization we rnust

have e b o = lr,. So e is an iso, whence LB bg, which means that bs E M.

(iii) + (i): Note that for every arrow f : A + B, ( l ~ , f ) can be obtained

by pulling back bB dong f x LB. SO if 6B iS in ?d, then ( l ~ , f ) is in M, too.

Therefore rf = (1, f ) . But now (1, f ) = (1, g) easily implies f = g. Hence B is

separated.

(Yi) (iv): If dg f M, then LB cz dB. I f further a : im(f,g) + bB is a

2-cell, then im( f, g) = (a, a) . But then f = ae = g where (f, g) = im( f , g)e is

an (&, M)-factorization-

(iv) 3 (Il): Take (f, g) = (b, L ~ ) to conclude CO = L I . O

The next result is in principle a corollary of Proposition 4.12. However, we

s h d give an alternative proof due to an earlier preprint of [IO], which does

not appear in the actual paper. Note that this proof does not use stability of

E under pullback, as Proposition 4.12 does, but only the weaker condition of

Theorem 3.7(iv).

Corollary 4.13. For a finitely complete (&,M)-structured category (2, the fol-

ZounBg are equivalent:

1- the gmph finctor I' : C + Map(Rel((2)) is faithful,

2. & Epi(C).

Pro05 (i) + (ii) Suppose f e = ge for e E &. Then

by 4.6(iv). Applying e-) to both sides we get ïf zz ïg by 3.7(iv), whence f = g

by faithfuhess of ï. Thus e is epimorphic-

(ii) + (i) Let l? f and r g be graphs. By 2.4, I'f zz (1, f ) for all arrows f of

e. Hence, I'f zz rg implies f = g. O

4.5 Total and singlevalued relations

The notion of a map as a relation with a right adjoint tries to capture the idea

of total and singlevalued relations being maps. Since the 2-ce& in a bicategory

of relations generalize inclusion of relations, the following definitions emerge.

Definition 4.14- A relation r : A t, B is called

(i) total if there is a 2-ce11 LA + r0 o r,

(ii) single-valued if there is a 2-ce11 r o ro + &B,

(iii) injective if ro is single-valued,

(fi) surjective if ro is total.

Clearly every map is total and single-vzlued. The units and counits provide

the needed 2-celIs,

The following easy fact not given in 1211 wi l i again simpliff the proof of the

forthcornhg result of [21], and it will &O be useful later.

L e m m a 4.15. If r = (rl,ro) : R -t A x B is a nonomorphism, the follouring

are equiualent:

(i) r o is monic,

(ii) ker(ro) 5 ker(rl)

Proof. I f r o is monic, the second statement clearly holds since ker(ro) =

(IR, IR)-

Conversely, suppose that ker(ro) < ker(r1). But this is equivalent to saying

for ail z, y. So since (ro,ri) is xnonic, rox = roy implies 3: = y.

Proposition 4.16. A relation r = (ro, r i ) : R -+ A x B is

(5) total if and only if im(ro) is a split epi,

(ii) single-valued if ker(r0) 5 ker(rl); the converse holds tme $ and only i f B

is separated.

Proof.. (i) A 2-ce11 77 : LA -+ r0 O r induces a section of ixn(ro) as shown in the

proof of Proposition 4.8. Just exactly repeat the construction shown in diagrams

(23) and (24) with s replaced by rO.

Conversely, let j be a section of im(ro). Note that the composite r0 o r is

given by im(rob, rlkl), where (ko, kl) = ker(rl). That means that there is a

common right inverse h of ko and kl. Then we get a commutative square

where e and eo are the coimages of (ro ko, ri kl) and ro respectively. The universal

property of the (E, M)-factorization systern induces d so that the next diagram

is commutative, and 7 : LA + r0 o r is again induced by the universal property

of the factorization system:

(ii) The statement ker(ro) 5 ker(rl) actually is equident to the identis

for (ko, ki) = ker(ro). In other words, there is an arrow rl ko : (rl k ~ , rr ki) + bB.

Applying the image functor yields

Conversely, if there is a 2-ceU im(rl k ~ , rl kl) -+ LB , then since B is separated

we have ri ko = rl ki by Proposition 4.12.

Finaily suppose that single-valuedness of a relation r = (ro, r i ) implies that

ker(ro) 5 ker(r1). In particdar the identity relation LB = (L* , L ~ ) is single-

d u e d . So ker(@) 5 ker(Ll) and since LB is monic, is monic by Lemma 4.15.

But LO is also a split epi because LB = im(JB), which shows that 4 e = 1 = LI e

for some e E E. Therefore Q is an iso; hence, LO = L I . Thus B is separated by

4-12.

The 1 s t result is a slightly strengthened version of Lemma 5.2 of 1211- Note

that separatedness of B is not only sufficient but also necessary for the converse

of item (ii) to hold.

Using ideas of the previous proof we can add another property of graphs to

our List in Lemma 4.6.

Proposition 4.17. For un a m w f : A + B of e, the graph Ir f is injective $

f is monic. The converse holds true if A is sepanzted.

Proof. Our task is to find a 2-celi (I'f)O O rf + LA. Recall that (l? f)O O r f

can be obtained as im((f, lA) O (lA, f)) = irn(ker(f)). Bu t if f is monic, then

ker(f) = aA, whence im(ker(f)) = LA. Conversely suppose that A is separated. Then 6-4 E ?Y€- So if there is a 2-ce11

77 : ( r f )*orf = irn(ker(f)) + 8A, we have ker(f) = im(ker(f))e = dAqe. SO the

kernel pair o f f consists of two morphisms that are equal. Thus f is monic. O

The characterizations of Proposition 4.16 show that not all total and single-

valued relations are maps. To see this consider the following result about spans

taken £kom [5], We omit the proof because it wi l l be a corollary of Theorem

4.20.

Proposition 4.18. A span r : ( ro , r i ) : A u B is a mup in Span(e) if and

only if fro i s an iso. O

Proposition 4-16 shows that a span r = (ro , T I ) is total and single-valued as

soon as to is a split epi and ker(ro) ker(rr). This is clearly not enough to

make ro art iso. Take for example any non-monic split epi f. Then (f, f ) is total

and single-valuecl but not a rnap in Spa@).

This means that in general maps cannot be reduced to total and single-valued

relations. Totality and single-valuedness have to be suitably connected by the

adjunction equations. However, a total and single-valued relation is a rnap as

soon as it is monic as an arrow of C. Moreover, it turns out that al1 maps are

monic arrows of e. This is essentially Theorem 5.3 of [22]. In this sequel of [21],

Pavlovit works in an even more general setting than that of (E, M)-structured

categories, namely in the context of regular fibrations. Here we give an easy

proof of his result for relations relative to a factorization system.

Theorem 4.19. A relation r in Rel(e) is a rnap if and only if i t is total,

single-ualued, and monic as an a m w of e.

Proof. A total and single-valued relation that is monic is obviously a rnap since

(E O r ) (T 0 q) : r + r must be the identity.

Conversely, every rnap r : A tt B is total and single-valued. We shall show

that r Ar is a map, too. The relation r A r is given as the arrow p in the pullback

square

T I P - R

R - A x B-

Furthemore, let d : r + r A r be a diagonal in Rel(A, B). Then clearly,

q d = 1, for i = 0,l. D e h e 2-cells by rl, := (do O d ) ~ , : LA + p0 o p and

q, := ~ ( q o T:) : p O p0 + LB- NOW consider the diagram

Note that since LB is monic in (2, and therefore ~ ( m o nz) = €(?rl O nf), both

squares are commutative by definition for i = 0,l- But the lower side of the

whole rectangle is an identity since r is a map, Thus the upper side must be

an identity, too, since m, RI are product projections in Rel(A, B). Hence, p is

a map. But now, no : p + T is a 2-cell between maps, whence an isomorphism

by 4.9, which shows that ?ro = ?ri, which irnplies that r is monic in e. O

In a posetal bicategory maps correspond exactly to the total and single-

valued relations because in a poset ali arrows are monic. The hom-categories

ReI(A, B) are posets precisely if M c Mono(e) (cf. [20], 3.7). This setting will be studied in Section 5.

The last theorem allows another characterization of maps. Note that this

result çtrengthms the technical L-a 5.4 of [21] but has a much easier proof.

Theorem 4.20. Let r = (ro,rl) : A U B be a relation. Ifro E E and ker(ro) _<

ker(rl), then r i s a map; the conuerse holà& true if and only if B i s separated.

Proof- If ro E E and ker(ro) 5 ker(rr), then r is total and single-valued by

Proposition 4-16. We shall show that r is monic as an arrow of e. It is weii-

b o w n that the kernel pair b, ki of r is given by pulling back ker(ro) dong

ker(rl) , i. e., by fonning the intersection of both kernels. But by hypothesis,

(ko, kl) = ker(r) = ker(ro). Being the puilback of r dong itseif, ko and kl lie

in M. But both arrows lie in E, too, since they also can be obtained by pulling

back ro d o n g itself. Hence k~ and ki are isomorphisms. Being a kernel pair,

ko and ki have a cornmon right inverse, which implies that they must be equal.

Thus r o and therefore r are monic.

Conversely, suppose that B is separated and r is a map. By Theorem 4.19,

r is total, single-valued, and monic in e. Applying Proposition 4.16 we see

that if ro = me is an (&,?Y€)-factorization, then m is split epic. Moreover,

ker(ro) _< ker(rl) implies that ro must be monic since r is so. Hence, m is

rnonic by Lemma 4.5. Thus m is an iso, which shows that TO lies in E.

That separatedness of B is necessary for the converse to be true can be seen

as in the proof for single-valuedness in Proposition 4.16. O

The last result together with Propositions 4.10 and 4.16 allows the following

important CoroUary, which does not appear in [21].

Coroliary 4-21. The following are equivalent:

(ii) a relation is single-valued if and only i f ker(ro) 5 ker(ri),

(iii) a relation is injective if and only i f ker(ri) 5 ker(ro),

(iv) a relation r = (ro, rr) is a rnap if and only i f r o E E fi Mono(C),

(u) a relation is an àsornorphism i f and only i f ro , rl E & n Mono(C!). 0

4.6 Function comprehension

Here we discuss two more notions relative to a factorization system.

Definition 4.22. A n object B of C! is said to be Cauchy-complete i f a map to

it converges tu a unique a m w .

The category C! is function comprehensive i f al1 of its objects are Cauchy-

complete.

First of all, observe that every Cauchy-complete object is separated. Fur-

thermore, the graph functor I: is faithful if and only if al1 objects of (2 are

separated.

Now let us characterize function comprehension in terms of a condition on

the factorization system, The result will then also produce a condition for the

desired isomorphism C! + Map(Rel(e)) of categories.

Theorem 4.23. For a finitely complete (€, M)-structured categort~ C with E

stable vnder pullback, the following are equiualent:

(i) the gntph functor î is an isornorphism of categories,

(ii) the category C is function comprehensive,

Proof. (i) (ii): The e s t and second statements are equivalent, since, by

definition, l? maps ob jects ident idy, and Cauchy-cornpleteness of al1 ob jects

of e is equivalent to l? being fuil and f a i tm .

(u) =+ (iii): Suppose that e is function comprehensive, i-e., ail objects of e

are Cauchy-complete, whence separated. But th i s implies that & C Epi(C), or

equivalently that all diagonal 6,4 are in M.

We must show that E C RegEpi(e). So let e : X + A be in E , and let

g : X + B be an arrow of e that equates the kernel (ho, hi) of e in e, Le.,

gho = ghl - We have to construct a unique arrow f such that g = f e. Then e is a coequaiizer of its kernel pair, whence a regular epi-

Now let r = {ro, ri) = im{e, g). Clearly r o is in E since roet = e for some

et in E. We wiil now show that ker(ro) 5 ker(rl). This together with r o E E

ailows to apply Theorem 4.20 to conclude that r is a map. First we relate

(ho, hi) = ker(e) and ( k ~ , kl) = ker(ro). To do this consider the diagram

Clearly both squares in the upper half are pullbacks; hence, by stabiiity of E

under pullback, ë is in €- Therefore

But since 9h-0 = ghl, there is a 2-cd im(gho, ghl) + LB = 6B, whence riko = rlkl. But this is equivalent to ker(ro) 5 ker(ri). Thus r is a map. Using

function comprehension, we conclude that there is a unique arrow f of e with

(1, f ) r. R e d 1 that (1, f ) is in M since & Ç Epi(e). That means, we can

choose r = (1, f) , which implies that (e,g) = (1, f )e is an (E, M)-hctorization

that gives us r. Therefore we have a unique f with g = f e.

(iii) + (ii): Suppose & C RegEpi(C). For every map r : A t , B we must

find a unique arrow f