Vol. XXVI, 1975 1

Topological Functors and Factorizations

By

I=~UDOLF-E. HOFFMANN

As is known (cf. [6, 12a, 12b]; meanwhile this result is in the folklore) topological functors, which lift isomorphisms uniquely, can be generated by the topological theories of O. Wyler [13a, 13b] (cf. also [8]) being special pseudofunetors as defined by A. Grothendieck [4] (a survey can be found in [3]):

topological theories are (contravariant) functors: ~op _+ C-ord ( = complete lat- tices and inf-preserving maps); hereby the domain category ~ and the forgetful functor from ~ to the base category ~ is constructed.

The first a t t empt to reconstruct the base category ~ of a topological functor V from its domain category ~ was made -- to my knowledge - - by P. Faehling [2], who axiomatized the class of those ~-morphisms, which are taken by V into identities.

Here we consider the class of those ~-morphisms, which are taken by V into isomorphisms. For our characterization the concept of factorization of cones ([5, 7, 9]) and its characterization by means of certain colimits ([7]) turns out to be central, on the other hand we make extensive use of the so-called calculus of fractions ([11] w 19), what is very much s]~ortening our proofs (compared with [2]).

As an application we obtain an improvement of our uniqueness theorem in [6]: I f ~ is topological over a balanced category ~ , then ~ is uniquely determined up to an equivalence by ~, and there is up to ... a t most one topological functor from ~ t o ~ .

The terminology is taken from [6], [7], and [11]. In Section 0 we compile some facts on topological funetors and factorizations.

Unfortunately we need some pages, but all these things are really needed in Sec- tions 1, 2. We are going to prepare an abstract of [6] for publication including a de- tailed bibliography on the subject, but we hope that the survey in Section 0 is of interest in itself. 0.1, 0.2, 0.3, 0.4, 0.6 are taken from [6].

0. 0.1. For a functor V: ~ ---> ~ a V-datum (T; D, be) consists of a diagram T: Z - + ~ and a cone (D, be: Dr--> VT) in ~ ; S is called the type of the V-datum (and of be or T). A cone (C, 2: Cz --> T) in ~ is called V-co-identifying ( = V-eo-idt.), iff for every cone (B, ~?) with codomain T in ~ and every morphism u: V (B) --~ V (C) with (V * 2)u = V * ~ there is exactly one morphism g: B - ~ C with V(g)= u and 2gz = 7. (C, 2: C z - + T; h) is called a V-co-idt. lift of (T;D, be), iff (C, 2) is a V-co-idt. cone in ~ and h: V(C) -+D is an isomorphism with bthz = F * 2 .

Archiv der Mathematik XXVI 1

2 R.-E. HOFPMANN ARCH. MATH.

0.2. A functor V: ~ -> ~ is called topological, iffit satisfies the fo]lowing two axioms: (a) For every D e Ob ~ every set of non-isomorphic objects C e Ob cd with V (C) ~_ D

is U-small (where U denotes the fixed universe). (b) For every V-datum of U-small type there is a V-co-idt. lift.

I f V satisfies only (b), V is called a co-idt, flmctor (Vop is called an idt. functor). Examples of topological functors can be found in [5, 6, 8, 9, 10, 12a, 12b, 13a, 13b]

(Top --~ Ens, TopGroups ---> Groups, etc.). Idt . functors were introduced first in [6]: the obvious functor Mod (-~ all modu-

les) --~ Rings, and the functor "Object": cat --> En8 are both idt. and co-idt, func- tors, but not topological. For a category W the unique functor ~ - + 1 is an idt. funetor, iff ~ is co-complete (W -> 1 is topological, iff ~ is equivalent to a complete lattice). (For the verification some non-trivial criteria of [6] are needed.)

0.3. For topological functors it is sufficient to have (a) and

(b') For every V-datum of small discrete type there is a V-co-idt. lift.

(a) A(b') ~ (b).

0.3 depends on the fact, that topological functors are automatically faithful ([6] 3.2.2).

0.4. I f a funetor V: @ --> ~ lifts isomorphisms, (a) is equivalent to (a') The fibres of V have small skeleta (~ is assumed to have small horn-sets [6]

3.1.2). 0.5. For topological functors a fundamental duality theorem holds (due to [ la ,

lb] and [10]): V: ~r -+ ~ is topological, iff Vop is. (Idt. functors generally are not co-idt., but

they are inclined to be -- cf. [6] 5.2.7.)

0.6. V: ~ - ~ has V-idt. lifts for every V-datum, whose "functor par t" is T : X - - > ~ , iff the induced functor V/T: C~/T-->~/VT has a fully faithful left adjoint (where cCf/T denotes the category of co-cones (2: T ---> Cz, C) in c~ with the obvious morphisms (2, C) --> (2', C') induced by some f: C --> C' with A = A']z).

This yields a characterization of topological functors (remember 0.2a), in par- ticular (X = 0) they have both a fully faithful left and right adjoint (cf. 0.5).

Le t V: ~ -> ~ be topological, then W: ~ -~ :Y is topological, iff W V is (this is immediate from the analogous statement on functors having a fully faithful left adjoint).

0.7. Let E be a class of epimorphisms of ~ with Iso c~ __C E, and let M be a class of cones of discrete type in ~, then (E, M) is called a factorization of cones in [5, 7, 9] ( ~ " ~ is an (E, M)-category"), iff the following conditions hold: (1) ~ is E-co-well-powered; E is closed under composition with isomorphisms; if

(A, {m,}x) e M and k: A ' -~ A is an isomorphism, then (A', {mCk}D e M. (2) For every cone (C, {/4 : A .-> At}l) there is an E-morphism e: A --> B and an

M-cone (B, {m,}1)with ]t = role. A~--~C

(3) For every commutative diagram ~ ~g, (i e I) where (B, {rot}x) e M and

B-~7, D,

Vol. XXVI, 1 9 7 5 Topological Functors and Factorizatlons 3

e e E there is (exactly) one morphism h: C -> B in ~ with he ---- ] (hence m~h ~- ---- g~). ( Dla~onal condition".)

(In the sequel we consider only cones indexed by discrete graphs ----- sets.) In every category ~ there is a trivial factorization with E ---- Iso ~. I f V: d -->

is a topological functor, and ~ is an (E, M)-category, then V lifts this factorization in a natural way: E ' consists of those d-morphisms, which are mapped by V into E, M' consists of those V-co-idt. cones in d , which are taken by V into M-cones [5].

0.8. a) M consists exactly of those cones which do not factor over a non-isomorphic E-morphism. M does not necessarily consist of monic cones.

b) I f e' e e E and e is epic, then e' e E.

c) The definition of (E, M) can be replaced by: 1) E C Epi ~ is compositive, Iso ~ C E, 2) M is compositive, Iso ~ _C M, 3) existence of the factorization, 4) "uniqueness" of the faetorization, 5) ~ is E-co-well-powered.

0.9. In an (E, M)-category there is a smallest E-reflective subcategory, called "(E, M)-germ" of ~, consisting of the "M-objects" (i.e. 0-indexed cones e M). By 0.Sa) an object C of an (E, M)-category is an M-object, iff every E-morphism with domain C is an isomorphism. The unit ~A of this adjunction is obtained by an (E, M)- factorization of A e 0b ~ (considered as an 0-indexed cone). Since this factorization is "unique" (up to ...), for any E-morphism e: A --> B, where B is an M-object, there is an isomorphism k, such that ke ---- ~A, cf. [7].

0.10. A compositive class E of epimorphisms in an E-co-wellpowered category with Iso ~f C E induces a factorization (E, M) (M according to 0.Sa), ff and only if the following conditions are satisfied:

a) Every cone {et: A-->Ai)x with e~eE (i e I ; I =~0) has a multiple pushout (~- co-intersection) (/~: Ai --> B}r, and/~ e E (i e I).

b) A ~-~ C in ~ with e e E has a pushout (/': B --7 D, e': C --> D), and e' e E, cf. [7]

B

0.11. I f W is an (E, M)-category, then by 0.8--10 E admits a terminal calculus of fractions [11] w 19. I f V: W--> ~ denotes the left adjoint of the embedding of the (E, M)-germ ~ into ~, then there is an equivalence J : ~[E-1]--> ~ with V = J P (where P denotes the projection W-+ ~[E-1]). The saturation X of E consists exactly of those morphisms / of ~, for which there is an e e E with e] e E, cf. [7].

0.12. In the above definitions (of topological functors and of factorizations) we have restricted ourselves to U-small indexing sets assuming some kind of smallness (for V, ~). One can avoid this considering U-large sets I. But if one tries to replace par t of the concept by the existence of limits (of d i s c r e t e type), then generally one cannot help presupposing some kind of smallness ([6] 4.1.8, [7] 0.8).

1"

4 R.-E. HOFFMANN ARCH. MATH.

1. 1.1. Lemma. Let V: ~ --+ ~ be a topological ]unctor, and let E denote the class o/ all morphisms in ~ which are talcen by V into isomorphisms, then the /ollowing statements hold:

(a) E consists o/bimorphisms.

(b) E induces a/actorization o/cones in ~ as well as in pop.

(c) The/unctor J: ~[E-1] --> 9 , such that V ---- J P (where P denotes the projection c~ _~ C~[E-1 ] _ c[. [11] w 19), is an equivalence.

P r o o f . (a) Since V is faithful, E consists of bimorphisms. (b) Since Vop: r#op _> ~op is topological too (duality theorem), E induces also

a factorization in pop. (c) P has a fully faithful right adjoint (because E admits a terminal calculus of

fractions 0.11) and also V has, hence J has too. By [11] 19.3.1 (a) J reflects iso- morphisms, hence it is an equivalence. (A functor having a fully faithful right ad- joint and reflecting isomorphisms is an equivalence.)

1.2. Remark. I f in 1.1 (E, M) denotes the induced factorization in ~, then M is exactly the class of V-co-idt. cones.

1.3. Theorem. Let E be a class o/bimorphisms in ~ inducing a factorization in c~ as well as in pop.

Then there is a category ~ and a topological [unctor V : ~ -+ ~ , such that E consists exactly of those morphisms which are taken by V into isomorphisms:

Let (E, M) denote the [actorization in ~ induced by E, and let ~ be the (E, M)-germ o / ~ , then the le/t ad~oint o/ the embedding ~--+ ~ is topological and induces E. O/ course, there is an analogue/or the (E, M')-germ in ~op.

V is "uniquely" determined: I / W: ~--> ~ is topological, such that E consists ex- actly o/those morphisms which are mapped by W into isomorphisms , then there is an equivalence J: ~ --~ ~ and a natural isomorphism ~: J V ~ W.

P r o o f . W. 1. o. g. the left adjoint V: ~ - - ~ of the embedding ~ - > ~ can be chosen, such that for a unit ~/of the adjunction holds ~B ---- 1B for every B e Ob ~ . Consequently V lifts isomorphisms ; the fibres of V have small skeleta, if ~ is E-well- powered (i.e. ~op is E-co-well-powered). Now let {A/}i~x be a family of objects of

with units ei : = r]A,: A~ ~ VAt in E, and let (B, {/~: B --~ VA~}I) be a cone in ~ . Since E induces a factorization of cones in ~op, we have pullbacks (because of 0.10)

Ct -~L A~

B -~V VAi

with ci ~ E, and we have a multiple pullback (intersection) for the family {ci}i (w.1. o.g. I :~ 0) defining an E-morphism c: C--> B and morphisms hi: C--~Ai with hc = eihi. Since B ~ Ob ~ , by 0.9 there is an isomorphism j: VC --> B with ~ c = c, hence/t~ = Vh~. That (C, {h~}~; ~) is a V-co-idt. lift of the given V-datum, can be seen by straightforward computation.

u XXVI, 1 9 7 5 Topological Funetors and Faetorizations 5

.By 0.11 V maps exactly the class 2: of those morphisms / into isomorphisms, for which there is an e ~ E, such that ef e E. Since E induces a factorization in pop, by 0.8b / e E, hence E ~-Z.

By 1.1 there are equivalences K: P [ E -1] - - ~ , L: ~ [ E -1] - - ~ with K P = V, L P ..~ W. Since K is an equivalence, there is an equivalence G: ~ - ~ qr [E-l] and a natural isomorphism ?: GK--~id, hence K ? P : L G V .-+ W is an isomorphism, and J :----LG is an equivalence.

1.4. Remark. From the proof of 1.3 (and from 1.1) the following statement is immediate:

Let E be a class of bimorphisms in W inducing a faetorization of cones in r if E induces a factorization of 0-indexed cones (=- objects) in P, then E induces a factorization of all cones in %0.

1.5. Coronary. Let I ~ be a class o~ morphisms in P: The projection P: (d--~ P [ F -1] is topological, ill the saturation E of I" consists o/

bimorphisms and E induces a/actorization o/cones in P as well as in pop.

P r o o f . I f P is topological, 1.1 applies. I f E induces factorizations in P as well as in ~op, then we have a topological functor V: ~--~ ~ inducing E by 1.3, and by 1.1 there is an equivalence J : qg[/~-l] _--~[E-1] __~ ~ with J P - ~ V, hence there is an equivalence G: ~- -> p [ E - I ] with GV ~--- P.

In [7] 1.1 (3) we have defined for each compositive class E C Mor ~ with Iso P C E a certain functor, such that E induces a factorization in q~, iff this functor is topo- logical:

The above corollary 1.5 and 1.1 state a reverse containment giving a constructive characterization of topological functors by (factorizations and) a category of fractions.

(A non-constructive characterization of topological functors in terms of factoriza- tions is due to [5].)

1.6. Example. In the category of Tl-spaces (and also in To, T2) one has a dual factorization induced by the class E of continuous bijections (the corresponding class of dual cones consists of those co-cones in T1 inducing the final topology with respect to Top), but this class does not induce a factorization in TI:

I f M is defined as in 0.Sa, then M is not compositive -- cf. [6] 2.1. Since in T1 epimorphisms are known to be surjective (other than in To, T2), bimorphisms are bijeetive.

Since the dualized forgetful functor Vop: T~p --~ EnsOp has a fully faithful right adjoint, V~ essentially coincides with the projection P: T~ ~ --+ T~P[E -1] (using the same arg-uments as in the proof of 1.1).

Herefrom it is clear, tha t in 1.3, 1.4, 1.5 it does not suffice to assume E to induce a factorization in pop (because the forgetful functor T1 --+ Ens is not topological).

The forgetful functor T1--> Ens is in fact a "pseudo-topological" funetor [6] w167 2, 3, 5. This suggests the following question:

1.7. Problem. Develop a theory of factorizations (E, M) of cones, such that E is compositive, but M is not necessarily compositive, cf. 0.8 c. Pseudo-topological fune-

6 R.-E. HOFFMANN ARCH. MATH.

tors lift the trivial factorization of the base category (E ~- Iso) into a "weak fac- torization" of this kind. Can these functors be characterized by an analogue of 1.3 ? What about a generalization of [7] 1.1 (see 0.10 here) ?

2. 2.1. Theorem ("Uniqueness Theorem"). Let c# be a category: There is a balanced category ~ and a topological/unetor V: c~ ._+ ~ , ill the class E o /a l l bimorphisms in

induces a/actorization in c~ as well as in pop. I / W: cg ..+ ~ , is any topological/unetor and ~ ' is balanced, then there is an equi-

valence J : ~ --> ~ ' with J V ~ W. In particular, i / ~ is balanced, V is an equivalence.

P r o o f . I f ~ is balanced, then a topological functor V maps W-bimorphisms into ~-bimorphisms = ~-isomorphisms, because V preserves epis and monos (0.5,6). Consequently by 1.3 we need only show that the (E, M)-germ ~ of W is balanced (where E denotes the class of bimorphisms in ~). Now let b: B --> C be a bimor- phism in ~ , then b is mono in W, because ~ is E-reflective in W (0.9). I f / , g: C -> X in %0 with ]b = gb, then ~x /b = ~xgb in ~ for the unit ~x, hence ~x/---- ~xg. Since ~;r is a bimorphism in ~, / -- 9, and b turns out to be epic in ~. Since B is an M-object and b: B -> C is an E-morphism, by 0.9 b is an isomorphism, hence is balanced.

Since W is faithful (0.3), W reflects bimorphisms; by (0.5,6) W preserves epis and monos, hence bimorphisms; i.e. a C#-morphism is taken by W into an isomorphism, iff it is a bimorphism. ~qow 1.3 applies.

2.2. Corollary. Let V, W: c# .._> Ens be two topological/unctors, then V ~ W.

The p r o o f is immediate from 2.1 and from the following

2.3. Lemma. I] J : E n s - > Ens is an equivalence, then there is an isomorphism

: lena --~- J.

P r o o f . J(1)---~ 1, because J preserves limits. Now for M - - I _ I 1 we find an M

isomorphism M ~ J ( M ) induced by 1--~ J(1), because J preserves eoproduets. 2.2 was originally obtained in [6] 4.2.3 (where the proof is based on a completely

different idea).

2.4. Example. The funetors AO ("domain") and A1 ("range"): [2, s _+s are both identifying and co-identifying functors, if :~ is complete and co-complete (by [6] 4.1.8).

For ~ = Ens A0 is not isomorphic to A 1.

2.5. Proposition ("Minimality"). I] V: c~ _~ ~ is a topological/unctor, and i/ is balanced, then /or an arbitrary topological /unctor R: ~--> 9f there is a /unetor S: ~ --> ~ with S ~ ~ V.

P r o o L Let E denote the class of ~-morphisms which are taken by R into iso- morphisms: since E consists of bimorphisms, V takes every E-morphism into an isomorphism. 5~ow by 1.1 c S with S R ~-- V can be found (universal property of P). By 0.6 S is topological.

2.6. Remark. I f V: ~ --~ ~ is topological and ~ is balanced, then V is minimal in the sense of 2,5. For an arbitrary category ~ consider the family {Ei}i of those

Vol. XXVI, 1 9 7 5 Topological Functors and Factorizations 7

classes Ei of bimorphisms inducing a factorizat ion in ~ as well as in Wop, E : = ~ J Ei. I

For ~ there is a "minimal" topological functor, iff the project ion P : ( d - + ~ [ E -1] is topological.

2.7. Example. We wang to give an example of a nombaIanced category c#, such tha t every topological functor V with domain W is an equivalence:

;b a

Let E denote the class of all those morphisms which are taken by V into isomorphisms. Since/~ induces a faetorization in ~op, b does no t belong to E, because for the above diagram there is no pul lback (cf. 0.10b) - - analogously a ~ E, hence E = Iso ~'. Since V: ~ ~ N is essentially the project ion P : ~ --~ ~'[E-1], V is an equivalence.

2.8. Remark. One can avoid to use the a rguments based on the calculus of frac- tions (even in 2.5).

References

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[2] P. FA~,n-Lr~G, Kategorien mit ausgezeichneten Morphismenktassen. Thesis FU Berlin 1973. [3] J .W. GRAY, Fibred and cofibred categories. In: Proc. Conf. La Jolla 1965 on Categorical

Algebra, pp. 21--83. Berlin-Heidelberg-New York 1966. [4] A. GROTHENDI:ECK, Cat6gories fibr6es et descente. S6minaire de g6om6trie alg6brique de

l'I. H. E. S. Paris 1961. [5] H. HERRLICH, Topological Functors. General Topology Appl., to appear. [6] R.-E. H O F F ~ r Die kategorielle Auffassung der Initial- und Finaltopologie. Thesis Bo-

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Math. Soc. 118, 303--315 (1965). [9] TH. MA~y, Rechts-Bikategoriestrukturen in topologischen Kategorien, Thesis FU Berlin

1973. [10] J .E . ROBOTS, A characterization of initial ftmctors. J. Algebra 8, 181--193 (1968). [11] H. SCn-~BV.RT, Categories. Berlin-Heidelberg-New York 1972. [12a] M. Wiscn~EwsKY, Initialkategorien. Thesis Miinchen 1972. [12b] M. WiscmcEwsxr, Partielle Algebren in Initialkategorien. Math. Z. 127, 83--91 (1972). [13a] O. W~v.R, On the categories of general topology and topological algebra. Arch. Math. 12,

7--17 (1971). [13b] O. W~v,~, Top categories and categorical topology. General Topology Appl. 1, 17--28

(1971).

Eingegangen am 15. 3. 1974

Anschrift des Autors:

Rudolf-E. Hoffmann Mathematisehes Institut der Universit~t D-4 Diisseldorf/West Germany Moorenstr. 5