topological functors and factorizations

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  • 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 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.

    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 ... at most one topological functor from ~to ~.

    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: 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 .

    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 part" 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).

    Let 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, 1975 Topological Functors and Factorizatlons 3

    e eE 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 = 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].

    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).

    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.

    Proof . (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 th

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