Topological functors and factorizations

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<ul><li><p>Vol. XXVI, 1975 1 </p><p>Topological Functors and Factorizations </p><p>By </p><p>I=~UDOLF-E. HOFFMANN </p><p>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]): </p><p>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. </p><p>The first attempt 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. </p><p>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]). </p><p>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 ... at most one topological functor from ~to ~. </p><p>The terminology is taken from [6], [7], and [11]. In Section 0 we compile some facts on topological funetors and factorizations. </p><p>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]. </p><p>0. 0.1. For a functor V: ~ ---&gt; ~ a V-datum (T; D, be) consists of a diagram T: Z -+~ and a cone (D, be: Dr--&gt; VT) in ~; S is called the type of the V-datum (and of be or T). A cone (C, 2: Cz --&gt; 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: Cz -+ 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 . </p><p>Archiv der Mathematik XXVI 1 </p></li><li><p>2 R.-E. HOFPMANN ARCH. MATH. </p><p>0.2. A functor V: ~ -&gt; ~ 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 </p><p>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). </p><p>Examples of topological functors can be found in [5, 6, 8, 9, 10, 12a, 12b, 13a, 13b] (Top --~ Ens, TopGroups ---&gt; Groups, etc.). </p><p>Idt. functors were introduced first in [6]: the obvious functor Mod (-~ all modu- les) --~ Rings, and the functor "Object": cat --&gt; 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 -&gt; 1 is topological, iff ~ is equivalent to a complete lattice). (For the verification some non-trivial criteria of [6] are needed.) </p><p>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. </p><p>(a) A(b') ~ (b). </p><p>0.3 depends on the fact, that topological functors are automatically faithful ([6] 3.2.2). </p><p>0.4. I f a funetor V: @ --&gt; ~ lifts isomorphisms, (a) is equivalent to (a') The fibres of V have small skeleta (~ is assumed to have small horn-sets [6] </p><p>3.1.2). 0.5. For topological functors a fundamental duality theorem holds (due to [la, </p><p>lb] and [10]): V: ~r -+ ~ is topological, iff Vop is. (Idt. functors generally are not co-idt., but </p><p>they are inclined to be -- cf. [6] 5.2.7.) </p><p>0.6. V: ~-~ has V-idt. lifts for every V-datum, whose "functor part" is T:X- -&gt;~, iff the induced functor V/T: C~/T--&gt;~/VT has a fully faithful left adjoint (where cCf/T denotes the category of co-cones (2: T ---&gt; Cz, C) in c~ with the obvious morphisms (2, C) --&gt; (2', C') induced by some f: C --&gt; C' with A = A']z). </p><p>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). </p><p>Let V: ~ -&gt; ~ 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). </p><p>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 </p><p>(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 .-&gt; At}l) there is an E-morphism e: A --&gt; B and an </p><p>M-cone (B, {m,}1)with ]t = role. A~--~C </p><p>(3) For every commutative diagram ~ ~g, (i e I) where (B, {rot}x) e M and B-~7, D, </p></li><li><p>Vol. XXVI, 1975 Topological Functors and Factorizatlons 3 </p><p>e eE there is (exactly) one morphism h: C -&gt; B in ~ with he ---- ] (hence m~h ~- ---- g~). ( Dla~onal condition".) </p><p>(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 --&gt; </p><p>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]. </p><p>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. </p><p>b) I f e' e e E and e is epic, then e' e E. </p><p>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. </p><p>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 --&gt; B, where B is an M-object, there is an isomorphism k, such that ke ---- ~A, cf. [7]. </p><p>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: </p><p>a) Every cone {et: A--&gt;Ai)x with e~eE (i e I ; I =~0) has a multiple pushout (~- co-intersection) (/~: Ai --&gt; B}r, and/~ e E (i e I). </p><p>b) A ~-~ C in ~ with e e E has a pushout (/': B --7 D, e': C --&gt; D), and e' e E, cf. [7] </p><p>B </p><p>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--&gt; ~ denotes the left adjoint of the embedding of the (E, M)-germ ~ into ~, then there is an equivalence J : ~[E-1]--&gt; ~ with V = JP (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]. </p><p>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 part of the concept by the existence of limits (of d i sc re te type), then generally one cannot help presupposing some kind of smallness ([6] 4.1.8, [7] 0.8). </p><p>1" </p></li><li><p>4 R.-E. HOFFMANN ARCH. MATH. </p><p>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. </p><p>(c) The/unctor J: ~[E-1] --&gt; 9, such that V ---- J P (where P denotes the projection c~ _~ C~[E-1 ] _ c[. [11] w 19), is an equivalence. </p><p>Proof . (a) Since V is faithful, E consists of bimorphisms. (b) Since Vop: r#op _&gt; ~op is topological too (duality theorem), E induces also </p><p>a factorization in pop. (c) P has a fully faithful right adjoint (because E admits a terminal calculus of </p><p>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.) </p><p>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. </p><p>1.3. Theorem. Let E be a class o/bimorphisms in ~ inducing a factorization in c~ as well as in pop. </p><p>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: </p><p>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. </p><p>V is "uniquely" determined: I / W: ~--&gt; ~ 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 ~: JV ~ W. </p><p>Proof . W. 1. o. g. the left adjoint V: ~- -~ of the embedding ~-&gt;~ 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 </p><p>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) </p><p>Ct -~L A~ </p><p>B -~V VAi </p><p>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--&gt; B and morphisms hi: C--~Ai with hc = eihi. Since B ~ Ob ~, by 0.9 there is an isomorphism j: VC --&gt; 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. </p></li><li><p>u XXVI, 1975 Topological Funetors and Faetorizations 5 </p><p>.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. </p><p>By 1.1 there are equivalences K: P [E -1] - -~ , L: ~[E -1] - -~ with KP = V, LP ..~ W. Since K is an equivalence, there is an equivalence G: ~-~ qr [E-l] and a natural isomorphism ?: GK--~id, hence K?P: LGV .-+ W is an isomorphism, and J :----LG is an equivalence. </p><p>1.4. Remark. From the proof of 1.3 (and from 1.1) the following statement is immediate: </p><p>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. </p><p>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/ </p><p>bimorphisms and E induces a/actorization o/cones in P as well as in pop. </p><p>Proof . 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: ~--&gt; p[E- I ] with GV ~--- P. </p><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: </p><p>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. </p><p>(A non-constructive characterization of topological functors in terms of factoriza- tions is due to [5].) </p><p>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: </p><p>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. </p><p>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). </p><p>Herefrom it is clear, that 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). </p><p>The forgetful functor T1--&gt; Ens is in fact a "pseudo-topological" funetor [6] w167 2, 3, 5. This suggests the following question: </p><p>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- </p></li><li><p>6 R.-E. HOFFMANN ARCH. MATH. </p><p>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) ? </p><p>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/al l bimorphisms in </p><p>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- </p><p>valence J : ~ --&gt; ~ ' with JV ~ W. In particular, i /~ is balanced, V is an equivalence. </p><p>Proof . 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 --&gt; C be a bimor- phism in ~, then b is mono in W, because ~ is E-reflective in W (0.9). I f / , g: C -&gt; X in %0 with ]b = gb, then ~x/b = ~xgb in ~ for the unit ~x, hence ~x/---- ~x...</p></li></ul>