COLIMITS FOR THE PRO CATEGORY OF TOWERS OF SIMPLICIAL SETS DAVID BLANC Revised: January 18, 1995 Abstract.We describe a certain category of Ind-towers into which the Pro* * category of towers of simplicial sets embeds, and in which all colimits (rather t* *han just the finite ones) may be constructed explicitly. 1.Introduction The Pro category of towers of spaces (and of other categories) has been stud* *ied in several contexts, and used for a variety of applications in homotopy theory,* * shape theory, geometric topology, and algebraic geometry - see for example [AM , B* *K , DF , EH , F, G , GV , H , HP , MS ]. Our interest in it first arose, in [BT ], in * *the study of vn-periodicity in unstable homotopy theory (cf. [Bo , D, Md , MT ]). One problem in the usual version of the Pro category of towers is that, whil* *e fi- nite limits and colimits exist, and may be constructed in a straightforward (le* *velwise) manner, the same does not hold for infinite colimits; and these were needed for* * the application we had in mind in [BT ]. It is the purpose of the present note to * *improve on the rather ad hoc solution to this difficulty presented in [BT , x3] (in ter* *ms of of what were there called "virtual towers"), by enlarging the Pro category of towe* *rs in such a way as to allow a straightforward construction of arbitrary colimits. On* *e object of this is to enable us to then provide a suitable framework for studying perio* *dicity in unstable homotopy theory in terms of a Quillen model category structure for * *our version of the Pro category of towers (see [Bc ]). The construction we provide embeds (a suitable subcategory of) the Pro categ* *ory Tow of towers of simplicial sets in a certain category Net of strict Ind-towe* *rs, in which we have explicit constructions for all colimits, as well as finite limits* *. This category Net can thus be thought of as a cocompletion of the Pro category of * *towers of spaces. We shall show in [Bc ] how this construction can also provide a "hom* *otopy theory of finite simplicial sets" (compare [Q , II, 4.10, remark 1]); it may be* * of use in other contexts, too. There are other cocomplete categories in which Tow may be embedded - for example, the category P ro-S? of all pro-simplicial sets (cf. [AM , A.4.3 & * *A.4.4]), or the full category Ind-Tow of all inductive systems of towers (cf. [J, VI, Thm* *. 1.6] - this is actually "the" cocompletion of Tow, in an appropriate sense: see [J, * *VI, x1] or [TT ]. One advantage of the approach described here is that one obtains a sm* *aller, _____________ 1991 Mathematics Subject Classification. Primary 55U99; Secondary 18A30. Key words and phrases. Pro-space, Pro-simplicial set, tower, net, Ind-tower, * *colimit, cocompletion. 1 2 DAVID BLANC and more mangeable, cocompletion, in this special case, and the construction of* * the colimits may be made quite explicitly. A side effect of our approach is the elimination of certain "phantom phenome* *na" from the Pro category of towers (see x2.10(b) and x4.13 below). 1.1. conventions and notation. Let T? denote the category of pointed topological spaces, S the category of simplical sets, and S? that of pointed simplicial s* *ets (see [My ]). We shall refer to the objects of S? simply as spaces. A finite simplic* *ial set Xo is one with only finitely many non-degenerate simplices (in all dimensions toge* *ther). (For technical convenience we prefer to work with simplicial sets, rather th* *an topo- logical spaces. This makes no difference for our purposes, since T? and S? * * have equivalent homotopy theories, in the sense of Quillen - see [Q , I, x4].) The category (ordered set) of natural numbers will be denoted by N, the ca* *tegory of abelian groups by AbGp, and the category of R-modules (for a commutative r* *ing R) by R-Mod. For any category C we shall denote by Ind-C the category of Ind-objects ov* *er C - that is, diagrams F : J ! C, where J is a small filtered category (cf. [GV , * *Defs. 2.7 & 8.2.1]) - with the appropriate morphisms (see [GV , Def. 8.2.4]). Similarly,* * P ro-C denotes the category of Pro-objects over C (i.e., diagrams F : J ! C where * *Jop is filtered - cf. [GV , Def. 8.10.1]). For any functor F : I ! C we denote the (inverse) limit of F simply by li* *mF or lim IF , (rather than lim ), and the colimit (=direct limit) by colimF . The * *(co)limit is finite if the category I is such (finitely many objects and morphisms betwee* *n them). All limits and colimits in this paper are assumed to be small - i.e., Ob(I) * * is a set. A category C is called pointed if it has a zero object (i.e., one which is b* *oth initial and terminal). This object will be denoted by *C (or simply *). 1.2. organization. In section 2 we give some background on towers of simplicial* * sets, their Pro category Tow, and the finite (co)limits in Tow. In section 3 we d* *efine "good" subcategories" - a concept which merely codifies those properties of * *Tow which are needed to construct its cocompletion. In section 4 we show that the * *category Net, consisting of certain strict Ind-objects over such a good subcategory of * *C, serves as a cocompletion for C, in the sense of having all colimits (and all finite li* *mits). 1.3. acknowledgements. I am grateful to the referee for his comments, and in pa* *rtic- ular for providing references to previous work, and suggesting Remark 4.12. I * *would also like to thank Emmanuel Dror-Farjoun, Bill Dwyer, Haynes Miller, and Brooke Shipley for several useful conversations. 2. The category of towers In order to fix notation, we recall the definition of the usual Pro category* * of towers of spaces: 2.1. towers of spaces. The objects we shall be studying are towers in S? - i.* *e., sequences of pointed spaces and maps pn pn-1 p0 X = { : :X:[n + 1] -! X [n] -! X [n - 1] ! : :-:! X [0] }; COLIMITS FOR TOWERS OF SIMPLICIAL SETS 3 where the space X [n] is called the n-th level of X (n 0), and the map pn* * is called the n-th level map (or bonding map) of X. We denote such towers by Goth* *ic letters: X; Y; : :.: For any n > m, the iterated level map pnm: X [n] ! X [m] is defined to b* *e the composite of pn-1 pm X [n] -! X [n - 1] ! : :!:X [m + 1] -! X [m] (so pn+1n= pn). We set pnn= idX [n]. Definition 2.2. Such towers are simply objects in the functor category SN? of * *dia- grams in S? indexed by the ordered set (N; ) of natural numbers. Thus a morphism f : X ! Y between two towers pn qn X = {: :X:[n] -! X [n - 1] : :}:and Y = {: :Y:[n] -! Y [n - 1] ! : :}: is just a sequence (f[k] : X [k] ! Y [k])1k=0of maps such that qkO f[k + 1] = f* *[k] O pk for k 0. The category SN? has all limits and colimits, of course. However, we are in* *terested rather in the Pro category of towers of spaces: Definition 2.3. Let N N! denote the set of sequences (ns)1s=0of natural number* *s, such that ns max {ns-1; s} for all s > 0. We shall denote the elements of * *N by lower case Greek letters, with the convention that = (ms)1s=0, = (ns)1s=0, * * and so on. The set N is partially ordered by the relation , ms ns for all s 0 - in fact, is a lattice. N has a least element ! = (s)1s=0(though o* *f course no maximal elements). Moreover, N is a monoid under composition (where = O is defined by ns = `ms), with ! as the unit and ; O . Definition 2.4. Given a tower X and a sequence = (nk)1k=02 N , we define the -spaced tower over X, denoted X<>, to be: pnk+1nk pn1n0 : :X:[nk+1] -! X [nk] ! : :!:X [n1] -! X [n0]: In particular X = X. Note that (-)<> is a functor on SN?. If in N , there is an SN?-map p : X<> ! X<> defined by pnkmk: X [n* *k] ! X [mk ] for all k 0. Such a p will be called a self-tower map (with respect* * to X). If = ! we write simply p for p , and call p a basic self-tower map (* *for X). For = we have p = id, and the composite of two self-tower maps (with res* *pect to the same X), when defined, is a self-tower map. Definition 2.5. We define now define the category of towers, denoted Tow, in wh* *ich the self-tower maps have been inverted: the objects of Tow are towers of spa* *ces (as in x2.1) and its morphisms, called tower maps, are defined for any X, Z by: Def (2.6) HomTow(X; Y) = colim HomSN?(X<>; Y) 2 N (compare [GV , x8.2.4]). Equivalently, one may define Pro maps of towers (cf. * *[EH , 2.1]) by: HomTow(X; Y) ~= lim colim HomS?(X [n]; Y[s]): s 2 N n 2 N 4 DAVID BLANC (It is not hard to see this is equivalent to the above, since any Pro map of* * towers from X to Y is represented by a sequence of maps f s : X [ns] ! Y [s] (s 0) * *in S?, which are compatible in the sense that for each s there is an ms ns; ns+1 suc* *h that qsO fs+1O pmsns+1= fsO pmsns: X [ms ] ! Y [s]. This defines an SN?-map f : X* *<> ! Y for 2 N defined by k0 = n0 and ks+1 = max {ms; ks}. One readily verifies * *this correspondence between the two definitions of tower maps is bijective.) Proposition 2.7. The category Tow has all finite limits and colimits. Proof. This is well known; for completeness we recapitulate the proof: To show that Tow has all finite limits, it suffices to show that Tow has a t* *erminal object and pullbacks (cf. [Bx1 , Prop. 2.8.2]). The tower * (with *[n] con* *sisting of a single point for each n) is clearly a terminal object in Tow. In order to de* *fine the pullback in Tow of two tower maps: ^f ^g (2.8) X -! Z - Y ; f g choose any two SN?-maps X -! Z - Y representing (2.8); their pullback P* * (in f[n] g* *[n] SN?) is defined levelwise (i.e., P [n] is the pullback of X [n] -! Z [n] -* * Y [n] in S?), and similarly for the structure maps i : P ! X and j : P ! Y such that f O i = g O j. The level maps of P are induced from those of X and Y by the na* *turality of the pullback in S?. Now given a basic self-tower map p : X<> ! X, denote by "P the pullback * *(in fOp g SN?) of X<> -! Z - Y. Again by the naturality of the pullback we can fit suitable spacings of P an* *d "P together as follows: _________________ | nk | | pk | | |? : :":P[nk]_-P [nk]__-"P[k]__-P [k] | n |6 | "pkk | |________________| implying that P and "Pare isomorphic; this shows that the pullback ^P of (2.8* *) in Tow is well-defined by taking the pullback P in SN? of any representatives of* * ^f, ^g. Next, given tower maps ^h: W ! X and ^k: W ! Y with ^gO ^k= ^fO ^h(in T* *ow), there are SN?-representatives h : W ! X and k : W ! Y with g O k = f O h * * in SN? (for suitable spacings of X, Y and W). Thus by the universal property in S? * * the SN?-maps h and k factor through the unique "universal" SN?-map l : W ! P. Conversely, given a tower map ^l2 HomTow(W; P) such that (2.9) ^iO^l= ^h and ^jO^l= ^k in Tow ; one can find SN?-representatives for the maps in question such that i O l =* * h and j O l = k in SN?, so that the SN?-map l : W ! P is the "universal map" as above. Thus it suffices to check that any two universal SN?-maps l : W ! P * *and l0 : W ! P<> represent the same tower map; but this follows readily from the uniqueness of the universal maps in SN?. COLIMITS FOR TOWERS OF SIMPLICIAL SETS 5 To show that Tow has all finite colimits, we show analogously that it has a* *n initial object and pushouts, again defined levelwise. ___|_|_ Remark 2.10. The category Tow may be embedded in a category with all colimits (and all filtered limits) - namely, the category P ro-S? of all pro-simplici* *al sets (see [AM , A.4.3 & A.4.4] or [GV , Props. 8.9.1 & 8.9.5]). The problem is that the* * limit or colimit of an infinite diagram of towers will not itself be a tower, and is rat* *her difficult to construct explicitly. Note that the naive (levelwise) construction of colimits in Tow can fail i* *n two different ways: (a) If `{Xff}ff2A is some (infinite) collection of towers, and we define a* * tower Y by Y [n] = ff2AXff[n], then Y is "too small" - in general, there will be * *maps ^fff: Xff! Z in Tow such that for any choice of representatives fff: Xff ! Z in SN?, the set of numbers {nff0}ff2A is unbounded. Thus there will be no way t* *o define a SN?-representative of the putative corresponding map ^f: Y ! Z which restri* *cts to ^fffon Xff. (b) On the other hand, let (Ai)1i=0be some sequence of non-trivial spaces* *, and define towers (Xi)1i=0by letting X i[n] = Ai, `and (pi)n = IdAi, if n i, * * and X i[n] = * otherwise. If again we set Y [n] = 1i=0Xi[n], we see Y is now "to* *o big": For a given tower Z, any collection of maps (fi : Ai ! Z [i])1i=0yields a* * unique SN?-map f : Y ! Z in the obvious way, and two such choices (fi)1i=0and (gi)* *1i=0 yield equivalent tower maps (^f= ^gin Tow) if and only if there is an N such* * that fi= gi for i N (at least for suitable Z - e.g., if Z is constant) . Thus * *there are many such tower maps ^f: Y ! Z; but the corresponding maps ^fi: Xi! Z are all trivial in Tow. (In some sense the maps ^fso defined may be thought of as "ph* *antom tower maps" - compare [GM ]). 3. Good subcategories We now describe those properties of the category Tow which are needed to con* *struct the extension. Since this construction is also needed for [Bc ], we describe it* * in greater generality than required for our immediate purposes. Definition 3.1. Let C be a pointed category, and F a small full subcategory. Fo* *r each A 2 C, let FA denote the subcategory of the over category C=A (cf. [Bx1 , * *x1.2.7]), whose objects are monomorphisms i : F ,! A with F 2 F and whose morphisms are (necessarily monic) maps j : F ! F 0such that i0O j = i. Similarly, let* * FA denote the subcategory of the under category A=C, whose objects are epimorphi* *sms q : A!!F with F 2 F, and whose morphisms are (epic) maps p : F ! F 0with p O q = q0. We say that F is a good subcategory of C if: (a) F is closed under taking subobjects and quotient objects. (b) F is finite-complete and -cocomplete, and the inclusion I : F ! C is @0-(c* *o)con- tinuous (i.e., any finite diagram (Fff)ff2A (|A| < 1) over F has a limit L a* *nd a colimit C in C, with L; C 2 F). 6 DAVID BLANC (c) For any F 2 F the category FF is co-artinian - that is, given a seque* *nce of quotient maps q0 q1 q2 qn qn+1 F-! ! G0-!! G1-!! : :G:n-1-!! Gn-!! : :;: there is an N such that qn is an isomorphism for n N (compare [GV , x8.12.* *6]). (d) Any morphism f : F ! C, with F 2 F and C 2 C, has an epimorphic image Im(f) (see [Bx1 , Def. 4.4.4]) - which is necessarily in FC. The inclusions iG : G ! C thus induce a natural bijection: (3.2) F;C: colim HomF(F; G) -=! HomC(F; C): G 2 FC Definition 3.3. Let Towst denote the category of (essentially) strict towers o* *f sim- plicial sets (cf. [GV , x8.12.1]) - that is, the full subcategory of Tow wh* *ose objects are towers X for which there is an N such that all level maps pn : X [n + 1] !* * X [n] are epimorphisms for n N. (We think of these as being "good" towers, because they avoid the pathologies mentioned in 2.10(b)). Note that Towst has all fi* *nite colimits and products, but not all pullbacks. Let F = FTow denote the full subcategory of Towst whose objects are tower* *s X such that each X [n] is a finite simplicial set (x1.1), and there is an N such * *that pn is an isomorphism for n N. We denote by Sp(X) the finite simplicial set limk* *X [k] (which is naturally isomorphic to X [n] for n N). Proposition 3.4. F = FTow is a good subcategory of Towst. Proof. (1) Given Y 2 F, let ^f: X ! Y be a monomorphism in Towst, with a representative f : X<> ! Y. For simplicity of notation let X = X<>. Now let Z = {: :Z:[n] -sn!Z[n - 1] ! : :}: be the (levelwise) pullback (in SN?) of f f X -! Y - X ; with h1; h2 : Z ! X the two projections. Since f O h1 = f O h2 and ^fis a monomorphism in Tow, there is a 2 N * *such that h1 O s = h2 O s : Z<> ! X in SN?. Now let (x0; x1) 2 Z [k]t X [k]tx X [k]t be a pair of k-th level t-simpli* *ces of X. Since the level maps pn : X [n + 1] ! X [n] of X are epimorphisms, ther* *e are t-simplices x0; x12 X [nk]t such that pnkk(xi) = xi (i = 0; 1). Thus qnkk(f(x0)) = f(pnkk(x0)) = f(x0) = f(x1) = f(pnkk(x1)) = qnkk(f(x1)),* * and since Y 2 F, each level map qn of Y is monic, and so (f(x0)) = (f(x1)), -* * i.e., (x0; x1) 2 Z [nk]t, with snkk(x0; x1) 2 Z [nk]t = (x0; x1). But then x0 = x* *1, since v1 O s = v2 O s . Thus f is levelwise monic. But this implies that each qn O f[n + 1] = f[n] O pn is monic, so pn is,* * too, and since Y 2 F we see X 2 F, too. (2) If ^f: X ! Y is any epimorphism in Towst, we shall show more generall* *y that ^fmay be represented by a levelwise epimorphism: without loss of generality, ^f* *has an SN?-representative f : X ! Y; by factoring f via its (levelwise) image, Im(f), * *we may COLIMITS FOR TOWERS OF SIMPLICIAL SETS 7 assume that f is levelwise monic. Now set Z = Y=X to be the (levelwise) push* *out (in SN ? ) of f * - X -! Y ; with two SN?-maps: g : Y!!X the quotient map, and * = h : Y!!X the trivial map. Clearly g O f = * = h O h (in SN?), so there is a 2 N such* * that g O q = h O q : Y<> ! Z, since ^fis an epimorphism in Tow. But then g O q* * = *, so q factors as f O q (for q : Y<> ! X), with q O f<> = p , so that f;* * q are inverese to each other in Tow, and thus f is an isomorphism. As before we con* *clude that if X 2 F then also Y 2 F. (3) Given a finite diagram over F, its limit and colimit in Tow may be de* *fined levelwise by Proposition 2.7, so are in F. (4) Since epimorphisms in Towst are actually levelwise surjections, the cat* *egory FY is equivalent to a finite category for any Y 2 F - so in particular it is c* *o-artinian. (5) Given ^f: X ! Y in Towst, with X 2 F, one can define Im(f) = Z to be the levelwise image tower for some SN?-representative f : X<> ! Y of ^* *f. This is independent of the representative f chosen, since given another represe* *ntative f0: X<> ! Y, there is a ; such that f0O s = f O s , and thus both maps have the same (levelwise) image, because Z is in F by (2), and thus the level m* *aps sn of Z are epimorphic. We write i : Z ! Y for the inclusion, with f : X!!Z such * *that f = i O f. Now if ^j: W ! Y is another monomorphism in Towst, equipped with a tower map ^g: X ! W such that ^jO^g= ^fin Tow, we may assume without loss of generality t* *hat ^gis represented by g : X<> :! W with j O g = f = i O f, and moreover by fa* *ctoring g itself through its image we may assume g is levelwise epimorphic, so W 2 F * * by (2), being a quotient of X. Finally, factoring our chosen SN?-representative j : W ! Y through Im(j) * * (which is in F, by (2)), we find that the SN?-map W!!Im(j) is an isomorphism, as in* * (a); but since Im(j) ,! Y is a levelwise monomorphism, by the universal property of * *Im in S? (and thus in SN?) there is a (levelwise) monomorphism k : Z ! W thro* *ugh which g and i factor, showing that f = i O f is indeed initial in Towst among* * the factorizations ^f= ^jO ^gof ^fwith ^jmonic. ___|_|_ 3.5. good generating subcategories. We shall in fact be interested in good sub- categories F C which generate C (cf. [Bx1 , Def. 4.5.1] or [Me , V, x7]) - * * that is, such that for any object C 2 C, {f : F ! C}f:F!C;F2F is an epimorphic family ([GV , x10.3]). Note that because of 3.1(d) and (3.2), this is equivalent to requiring that,* * for all C; D 2 C, there be a canonical natural inclusion of sets: (3.6) JC;D : HomC(C; D) ,! lim colim HomF(F; G); F 2 FC G 2 FD induced by the restrictions f|F for any f : C ! D and the correspondences * *-1F;D of (3.2). Example 3.7. (i) The category of pointed sets is generated by the good subcate* *gory of finite pointed sets. 8 DAVID BLANC (ii) The category S? of pointed simplicial sets is generated by the good su* *bcategory Sf ? of finite pointed simplicial sets (x1.1). (iii) The category of torsion groups is generated by the good subcategory o* *f finite groups. (iv) The category AbGp is generated by the good subcategory of finitely gen* *erated abelian groups, and more generally R-Mod is generated by the good subcategory* * of f.g. R-modules for any noetherian ring R. In these cases the natural inclusion JC;D of (3.6) is actually bijective. Proposition 3.8. The category Towst is generated by the subcategory FTow. Proof. Let ^f6= ^g: X ! Y be two different tower maps, with X 2 Towst; with* *out loss of generality we may assume they have SN?-representatives f; g : X<> ! Y respectively. By definition 2.5, there is a k 0 such that for all n k we* * have f[n] O pnk6= g[n] O pnk, so in particular for each n k there is a t-simplex x* *n 2 X [n]t (t independent of n) such that f[n](xn) 6= g[n](xn). Since X 2 Towst, the leve* *l maps pn of X are surjective, and we may evidently assume pn(xn+1) = xn for all n* * N. To each xn 2 X [n]t there corresponds a map 'xn : [t] ! X [n]t, and let Z * *[n] 2 S? denote the simplicial set Im('xn). Then Z = {: :Z:[n] -sn!Z[n - 1] ! : :}* *: is in fact a sub-tower of X, with sn = pn|Z [n+1], and because each Z [n] is a quo* *tient of both [t] and Z [n + 1], for sufficiently large n the maps sn must be isomorph* *isms (since [t] has only finitely many non-isomorphic quotients), so that Z 2 FX. Clearly f|Z 6= g|Z, and both have images in F by Definition 3.1(d) and Pro* *position 3.4 - which proves JX;Y is indeed one-to-one. ___|_|_ It may be useful to think of the finite subtowers Z ,! X (Z 2 F) as the a* *nalogue of the stable cells of a CW -spectrum - compare [A , III, x3]. Remark 3.9. Note that in general our category C will not be locally generated b* *y the subcategory F, in the sense of [GU , xx7,9], because C need not be cocomplete * *- and we are interested precisely in such cases, because only then will the cocomplet* *ion of C be of interest. C need not even be @0-accessible in the sense of [Bx2 , Def.* * 5.3.1], because we do not assume that FC has all colimits for arbitrary C 2 C. 4. Nets and cocompletion When C is generated by a good subcategory F, it embeds in the category Inds* *t-F of strict Ind-objects over F; by constructing all colimits for Indst-F, (or r* *ather, for an equivalent subcategory Net), we show that this can serve as a cocompletion * *for C. In analogy with the completion of a metric space, the objects of Net are the* *mselves directed systems of suitable towers; one should think of these as representing * *their colimit (which may not exist in Tow). Definition 4.1. A strict Ind object over a category G is a diagram X : I ! G indexed by a small filtered partially ordered category I, such that all bonding* * maps X(f) : Xff! Xfi (for f : ff ! fi in A) are monomorphisms (cf. [GV , Def. 8.1* *2.1]). The full subcategory of Ind-G whose objects are strict will be denoted by In* *dst-G. In order to simplify our constuctions, it is convenient to consider the subc* *ategory Net Indst-G defined as follows (this is actually equivalent to Indst-G, under * *suitable assumptions - see Fact 4.4 below): COLIMITS FOR TOWERS OF SIMPLICIAL SETS 9 Definition 4.2. If G is a pointed category, a net over G to be a strict Ind-obj* *ect (X ff)ff2A indexed by a lattice with least element 0, such X0 = * **, and for each ff; fi 2 A, the square: iff^fi;ff Xff^fi_____-Xff | | | iff^fi|;fi |iff;ff_fi | | |? ifi;ff_f|?i Xfi_______Xff_fi_- Figure 1 is both cartesian and cocartesian. (Since we required the bonding maps of the * *net to be monic, this simply means that Xff^fiis the intersection Xff\ Xfiof Xffand Xfi and Xff_fiis their union Xff[ Xfi (cf. [Bx1 , Def. 4.2.1 & Prop. 4.2.3]) Definition 4.3. If (Xff)ff2A and (Yfi)fi2B are two nets over G, a proper net * *map between them is a pair , where OE : A ! B is a order-preserv* *ing map with OE(0) = 0, and for each ff 2 A, fff: Xff! YOE(ff)is a morphism in G. * *We require that for all ff fi in A, the diagram: fff Xff______-YOE(ff) | | | iff|;fi |iOE(ff);OE(fi) | | |? ffi |? Xfi_____YOE(fi)_- commutes. If YOE(ff)= Im(fff) for all ff 2 A - in other words, each fff i* *s epic - we say is a minimal proper net map. Two proper net maps ; < ; (gfi)fi2B> : (Xff)ff2A! (Yfi)fi2Bar* *e equivalent - written ' < ; (gfi)fi2B> - if for each ff 2 A there is* * an ae(ff) such that OE(ff) _ (ff) ae(ff) (in the lattice B) and the diagram fff Xff___________-YOE(ff) | | | gf|f |iOE(ff);ae(ff) | | |? i (ff);ae(|?ff) Y (ff)________Yae(ff)_- commutes. Note that if G is a category with images, then each equivalence class* * of proper net maps will have a unique minimal representative. The category of nets over G, with equivalence classes of proper net maps as * *mor- phisms, will be denoted NetG. We shall sometimes use the notation f : (Xff)f* *f2A! (Yfi)fi2B to denote a morphism of nets (i.e., an equivalence class of proper ma* *ps) - cf. [GV , x8.2.4-5]. Fact 4.4. If G has finite unions and intersections, every object in Indst-G i* *s isomor- phic to one in NetG. Proof. By the dual of [MS , I, x1, Thm. 4] every (strict) Ind-object over G is* * Ind- isomorphic to a (strict) Ind-object (X )2 indexed by a directed ordered set <* *; > 10 DAVID BLANC which is closure finite - i.e., the set of predecessors of every 2 is fin* *ite. Now let A be the free lattice generated by , and set X S ff^fi= flff;fiXfl, with * * Xff_fi defined by Figure 1. Since is cofinal in A, we actually have an Ind-isomorphi* *sm (X )2 ,! (Xff)ff2A. ___|_|_ This shows that we could assume, if we wish, that our nets are always indexe* *d by closure finite lattices (and this will in fact be the case for FTow, of cours* *e, because in this case FF will be a finite category for each F 2 F), but this is not * *needed for our constructions. Proposition 4.5. If F C is good (so in particular has all finite colimits), * *then N etF has all colimits. Proof. It suffices to show N etF has coproducts and pushouts (cf. [P , x2.6, P* *rop. 1 & 2]): I. Given any`collection {(Xiff)ff2Ai}i2I of nets over F (indexed by an arbi* *trary set I), let B = i2IAi denote the coproduct lattice - so that the elements of B * *are of the form fi = ffi1_ : :_:ffin for ffij2 Aij (and ij 6= ik for j 6= k). The coproduct net is then defined to be n_ ( Xijffij)ffi1_:::_ffin2B; j=1 and the universal property for the coproduct evidently holds. II. Given two net maps with minimal proper representatives: (Xff)ff2A____________-(Yfi)fi2B | | < ; (gff)ff|2A> | |? (Zfl)fl2C Figure 2 For each fi 2 B and fl 2 C, let A(fi;fl)= {(ff 2 A | OE(ff) fi & (ff) f* *l} (a sublattice of A), and for each ff 2 A(fi;fl), let W ff= Wfffi_fldenote the p* *ushout in: fff iOE(ff);fi Xff_______-YOE(ff)________-Yfi g | pp ff| pp |? pp Z (ff) pp i | pp (ff)|;fl pp |? |? Zflpppppppppppppppppppppppppp_-Wfffi_fl Now for each fi0 _ fl0 in the coproduct lattice B q C let [ [ Lfi0_fl0= {(ff; fi; fl) 2 A x B x C | ff 2 A(fi;fl)}: fi0fifl0fl COLIMITS FOR TOWERS OF SIMPLICIAL SETS 11 For any (ff; fi; fl) 2 Lfi0_fl0, the bonding maps ifi0;fiand ifi0;fiinduce * *a map qfi0_fl0(ff;fi;fl): Yfi0q Zfl0! Wfffi_fl: We let U(ff;fi;fl)= Ufi0_fl0(ff;fi;fl)denote Im(qfi0_fl0(ff;fi;fl)) Wfffi_fl. Note that, for fixed fi0 _ fl0 2 B q C, the objects U(ff;fi;fl)form a dia* *gram in F indexed by the (possibly infinite) filtered set Lfi0_fl0, and set Def fi0_fl0 (4.6) Wfi0_fl0= colim U(ff;fi;fl): (ff; fi; fl) 2 Lfi0_fl0 This limit exists in F - in fact, in F0 = FYfi0qZfl0, by [Bx1 , Prop. 2.16.* *3] - since F0 is co-artinian by Def. 3.1(c), and thus has all filtered colimits. The natur* *al map ifi0_fl0;fi1_fl1: Wfi0_fl0! Wfi1_fl1 (induced by the fact that each qfi0_fl0(ff;fi;fl)factors through qfi1_fl1(ff;fi* *;fl)) is always a monomor- phism. Thus we have defined as net (Wfi_fl)fi_fl2BqC) over F. (Had we not req* *uired that our nets be strict Ind-objects, we could have defined Wfi0_fl0more simply* * as the colimit of the objects Wfffi0_fl0for ff 2 A(fi0;fl0)). We claim that this net is the pushout for the diagram in Figure 2: given a * *commu- tative diagram in NetF (Xff)ff2A____________-(Yfi)fi2B | | | | < ; (gff)ff|2A> | | | (Z|?fl)______________-l2C|?(V")"2E (where we may assume the proper representatives indicated make it commute on the nose), we define a net map : (Wfi_fl)fi_fl2BqC-! (V")"2E* * as follows: set Def _ 0 0 o(fi _ fl) = ae(fi ) _ oe(fl ): (ff0;fi0;fl0)2L^fi_fl We then have `fi_fl: Yfiq Zfl-! Vo(fi_fl)induced by the appropriate bonding ma* *ps, and if OE(ff) fi and (ff) fl, the diagram fff iOE(ff);fi Xff_______-YOE(ff)______-Yfi g | | || ff| | | |? |? | Z (ff)_____-Wfffi_flpppff|i O hfi pppp`fi_fl| i (ff)|;fl pppp | | pppp| |?__________________s_-i|O?kfl Zfl Vo(fi_fl) commutes, so `fi_flinduces a map `fffi_fl: Wfffi_fl-! Vo(fi_fl), and thus `* *fi_fl: Wfi_fl! Vo(fi_fl). One may also verify that has the appropria* *te universal property. ___|_|_ 12 DAVID BLANC Proposition 4.7. If F C is good (so in particular has all finite limits), the* *n N etF has all finite limits, too. Proof. It suffices to show that N etF has pullbacks (it clearly has a terminal* * object - namely, the zero net indexed by the zero lattice). Thus, given two net maps: (Yfi)fi2B | | | | < ; (gfl)fl2C|?> (Zfl)fl2C____________-(Xff)ff2A where we assume the indicated representatives are minimal, for each (fi; fl) 2* * B x C, set W(fi; fl) to be the pullback of gfl ffi Zfl-! X (fl)_OE(fi)-Yfi: It is readily verified that this defines a pullback net with the required unive* *rsal prop- erty. ___|_|_ Proposition 4.8. If C is generated by a good subcategory F C, then there is * *an embedding of categories I : C ! NetF, defined I(C) = FC = (F )F2FC. (Note that FC is both the lattice indexing the net I(C) 2 NetF, and the net* * itself) Proof. For any f : C ! D in C and F 2 FC a subobject of C which is in F, the image Im(f|F) is in FD by Def. 3.1(e). Thus we may define a proper net map : FC ! FD by OEf(F ) = Im(f|F) 2 FD and fF = f|F : F ! Im(f|F) D, for any F 2 FC. This defines I on morphisms. Definition 3.1(e) a* *lso implies that I : HomC(C; D) ! HomNetF(FC; FD ) is monic, since if f; g : C !* * D satisfy f|F = g|F for all F 2 FC, then f = g. ___|_|_ We may summarize our results for the Pro category of towers of spaces in the* * fol- lowing Theorem 4.9. The functor I : Tow ! NetF, defined by I(X) = FX restricts to an embedding of Towst in the cocomplete and finite complete category of nets o* *ver FTow. I preserves all finite limits, and the functor I|Towst preserves all co* *limits. Proof . If W is the pullback in Tow of ^f ^g (4.10) Z -! X - Z ; (which may not be in Towst, even if (4.10) is), then (as in the proof of Propos* *ition 2.7) W may be constructed as the levelwise pullback of any SN?-representatives of (* *4.10), so W [n] is a subobject of Y [n] x Z[n] (by the usual construction in S?). Th* *us any finite subtower of W is just a finite subobject of Y x Z, satisfying the ap* *propriate (levelwise) compatibility condition - so that FW is isomorphic to the pullb* *ack net for F^f F^g FZ -! FX - FZ constructed in the proof of Proposition 4.7. COLIMITS FOR TOWERS OF SIMPLICIAL SETS 13 Similarly, if W is the pushout in Towst of ^f ^g (4.11) Z - X - ! Z ; then W may be may be constructed as the levelwise pushout of any representative* *s of (4.11) and W [n] is thus a quotient of Y [n] q Z[n]. Note that the structur* *e maps of the pushout induce an epimorphism ^h: Y q Z!!W. Now if U is a finite subobject of W, then it is in fact a quotient of some f* *inite subobject V0q V00,! Y q Z, with V0 2 FY and V002 FZ, as in the proof of Proposition 3.8. But since W [n] ~=(Y [n] q Z[n])= ~, where the equivalence r* *elation ~ is generated by f[n](x) ~ g[n](x), we see that any finite subspace U[n] * *W [n] (and thus U W) is obtained form a finite subspace V0q V 00 Y [n] q Z[n] by* * a finite colimit as in the proof of Proposition 4.5. This shows that FW is iso* *morphic to the pushout net for F^f F^g FZ - FX -! FZ: Remark 4.12. The fact that NetF serves as a cocompletion of C, when F is a g* *ood subcategory generating C, follows directly from more general results: By [J, VI, Thms. 1.6 & 1.8] we know that Ind-C is the cocompletion of C (ass* *uming C itself is finite-cocomplete), and it is easy to see that C embeds in Ind-F * *(as in the proof of Proposition 4.8), so that Ind-C embeds cocontinuously in Ind-(Ind-F* *), which is equivalent to Ind-F (see [GV , Cor. 8.9.8]). Because F is co-artinia* *n (Def. 3.1(c)), Ind-F is equivalent to Indst-F (see [GV , x8.12.6]), which is equ* *ivalent in turn to NetF by Fact 4.4 and Def. 3.1(b). However, we believe that the explicit description fo the colimits in NetF gi* *ven above may be more useful than that obtained form unwinding the above chain of equival* *ences. The results relating specifically to towers of simplicial sets - Proposition* *s 3.4 & 3.8 - may also be extended to other Pro categories of towers over categories C gen* *erated by a good subcategory F, such as towers of sets (cf. x3.7). Remark 4.13. The example in 2.10(b) shows that the functor I : Tow ! NetF of t* *he Theorem fails to be an embedding, since the tower Y defined there has no non-tr* *ivial finite subobjects, and thus I(Y) = FY = *, even though there are non-trivial * *maps Y ! Z in Tow. This is not a serious flaw, since one often chooses to work with the "good" * *towers of Towst in applications. In fact, there is a certain advantage to this fact,* * from our point of view, since it yields a version of the Pro category of towers from whi* *ch we have eliminated the phantom phenomena (as in the case of Towst), but still have fi* *nite limits (and have actually added infinite colimits). Question 4.14.Although only colimits were needed for our application in [BT ], * *one can obviously ask the same question regarding the completion of Tow - that is, embe* *dding the Pro category of towers in one where arbitrary limits (ideally: both limits* * and colimits) may be constructed. While the categorical part of our construction c* *ould presumably be dualized, it is not clear that the category Tow, or any other ver* *sion of the Pro category of towers, will indeed satisfy the required assumptions, since* * specific 14 DAVID BLANC properties of Tow and S? were used in the proof of Proposition 3.8 and Theo* *rem 4.9. Note however that for any small finite-complete category C, the category Ind* *-C has all limits (as well as all colimits), and the inclusion C ,! Ind-C preserves * *all limits which exist in C, by [J, VI, Prop. 1.7]. References [A] J.F. Adams, Stable Homotopy and Generalised Homology, Chicago Lec. Math* *., U. of Chicago Press, Chicago-London, 1974. [AM] M. 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Lib. 26, North-Holland, Amsterdam, 1982. COLIMITS FOR TOWERS OF SIMPLICIAL SETS 15 [My] J.P. May, Simplicial Objects in Algebraic Topology, U. of Chicago Press* *, Chicago-London, 1967. [P] B. Pareigis, Categories and Functors, Academic Press Pure & Appl. Math.* * 39, New York-- London, 1970. [Q] D.G. Quillen, Homotopical Algebra, Springer-Verlag Lec. Notes Math. 20,* * Berlin-New York 1963. [TT] W. Tholen & A. Tozzi, "Completions of categories and initial completion* *s", Cah. Top. Geom. Diff. Cat. 30 (1989) No. 2, pp. 127-156. University of Haifa, 31905 Haifa, Israel E-mail address: blanc@mathcs.haifa.ac.il