A MODEL STRUCTURE ON THE CATEGORY OF PRO-SIMPLICIAL SETS DANIEL C. ISAKSEN Abstract.We study the category pro-SSof pro-simplicial sets, which arises in 'etale homotopy theory, shape theory, and pro-finite completion. We e* *stab- lish a model structure on pro-SSso that it is possible to do homotopy th* *eory in this category. This model structure is closely related to the strict * *structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give s* *ev- eral alternative descriptions of weak equivalences, including a cohomolo* *gical characterization. We outline dual constructions for ind-spaces. 1.Introduction If C is a category, then the pro-category pro-C [1, Expos'e 1, Section 8] is * *the category whose objects are small cofiltered systems in C (of arbitrary shape) a* *nd whose morphisms are given by the formula Hom (X, Y ) = limscolimtHomC(Xt, Ys). While investigating the 'etale homotopy functor, Artin and Mazur [2] studied the category pro-Ho(SS), where Ho(SS) is the homotopy category of simplicial sets. They also introduced a notion of weak equivalence of pointed connected pro-spac* *es that involved isomorphisms of pro-homotopy groups. However, an Artin-Mazur weak equivalence is not the same as an isomorphism in pro-Ho(SS). This suggests that pro-Ho(SS) is not quite the correct category * *for studying 'etale homotopy. Around the same time, Quillen [14] developed the fundamental notions of homo- topical algebra by realizing that model structures allow one to do homotopy the* *ory in many different categorical contexts. A model structure on a category is a ch* *oice of three classes of maps (weak equivalences, cofibrations, and fibrations) sati* *sfying certain axioms. The weak equivalences are inverted to obtain the associated hom* *o- topy category, while the cofibrations and fibrations serve an auxiliary role. Q* *uillen [14, II.0.2] observed that the homotopy theory of pro-spaces would be an intere* *sting application of model structures. At least two model structures for pro-spaces are already known to exist. Ed- wards and Hastings [6] established a "strict" model structure on pro-spaces for* * the purposes of shape theory and proper homotopy theory, but their weak equivalences did not generalize those of Artin and Mazur. ____________ Date: June 18, 2001. 1991 Mathematics Subject Classification. Primary 18E35, 55Pxx, 55U35; Second* *ary 14F35, 55P60. Key words and phrases. Closed model structures, pro-spaces, 'etale homotopy. The author was supported in part by an NSF Graduate Fellowship. 1 2 DANIEL C. ISAKSEN Also, Grossman [9] described a different model structure for pro-spaces that * *are countable towers. The weak equivalences of this theory are appropriately related to the Artin-Mazur equivalences. The category of towers is suitable in many app* *li- cations from proper homotopy theory because it is reasonable to assume that the neighborhoods at infinity have a countable basis. However, applications of pro-spaces to the algebro-geometric concept of 'etale cohomology require more general pro-spaces. General cofiltered systems of spaces are necessary for essentially the same reason that the sheaf theory of Grothend* *ieck topologies is necessary to define 'etale cohomology. This paper gives a generalization of Grossman's model structure to the entire category of pro-spaces. Weak equivalences between pointed connected pro-spaces are precisely Artin-Mazur weak equivalences. The Edwards-Hastings strict struc- ture is an intermediate stage to building our structure. Our homotopy theory is* * the P -localization of the strict homotopy theory, where P is the functor that repl* *aces a space with its Postnikov tower. Our weak equivalences have several alternative characterizations, one of which uses twisted cohomology. This is important because it is often difficult to ch* *eck that a map of pro-spaces induces an isomorphism on homotopy groups. Usually it is much easier to verify a cohomology isomorphism and then conclude that the map is a weak equivalence. Another characterization of weak equivalences is in terms of "eventually n- connected" maps (see Theorem 7.3 (d)), which is a very convenient property in practice. The equivalence of this property with the definition of weak equivale* *nce is not obvious. One must use the full power of the model structure to prove the equivalence. We mention two applications of this homotopy theory of pro-spaces. First, the model structure gives a more convenient category for studying 'etale homotopy b* *e- cause it allows the reinterpretation of the central ideas of the theory in term* *s of the established notions of model structures. It is also a start towards the def* *inition of generalized cohomology of pro-spaces. For example, define K0 to be represent* *ed by the constant pro-space BU. The realization of K-theory as a generalized co- homology theory requires more understanding of the category of pro-spectra. The 'etale K-theory of a scheme [5] is probably most clearly expressed as a general* *ized cohomology theory applied to the 'etale homotopy type. Second, pro-spaces arise in the study of pro-finite completion [15] [4] [13].* * Again, the new model structure provides a better category in which to study such compl* *e- tions. For example, Mandell [11] has used the model structure to compare his new algebraic construction of p-adic homotopy theory to Sullivan's p-pro-finite com* *ple- tion. We describe briefly the model structure; the formal definitions appear in Sec* *tion 6. Given a pointed pro-space X (i.e., a map from the one-point constant pro-spa* *ce * to X or equivalently a pro-object in the category of pointed spaces), one can* * apply the functor ßn(-, *) to each space in the pro-system to get a pro-group ßn(X, ** *). The most obvious notion of equivalence of pro-spaces X ! Y is the requirement that the map induce an isomorphism of pro-homotopy groups ßn(X, *) ! ßn(Y, *) for every n 0 and every point * of X. However, points of a pro-space are rather awkward. In fact, some non-trivial * *pro- spaces have no points whatsoever. Local systems permit the discussion of homoto* *py A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 3 groups without choosing basepoints. Grossman's work on towers [9] inspired this trick. Cofibrations of pro-spaces are maps that are isomorphic to levelwise cofibrat* *ions of systems of spaces of the same shape. The model category axioms then force the definition of fibrations. These fibrations are similar to those of Edwards * *and Hastings [6, 3.3], but they satisfy an extra condition that compares the homoto* *py groups of the total pro-space to the homotopy groups of the base pro-space. The model structure has a few interesting aspects. For example, it is not cof* *i- brantly generated. This means that the proof of the model axioms is quite diffe* *rent from the standard arguments. One result of this fact is that the factorizations* * are not functorial. Almost all naturally arising model structures are cofibrantly g* *ener- ated. Hence pro-spaces are an interesting example of a non-cofibrantly generated model structure. Also, we only know how to work with pro-simplicial sets, not pro-topological spaces. We use in a very significant way the fact that finite dimensional skele* *tons are functorial for simplicial sets. Since relative cell complexes do not have f* *unctorial skeletons, the same proofs do not apply. Finally, note that the category of ind-spaces has a similar model structure. * *We do not provide details in this paper because we have no application in mind. Se* *veral comments throughout the paper explain where the significant differences occur. The paper is divided into three main parts. Sections 2-5 give some background material and introduce language and tools for later use. Sections 6-10 describe the model structure, state the main theorems, and make comparisons to other homotopy theories. Sections 11-19 provide proofs of the main theorems. We assume familiarity with model structures. See [10] or [14] for background material. Many of the important ideas in this paper come from Grossman [9]. I thank Peter May, Bill Dwyer, Brooke Shipley, Greg Arone, Michael Mandell, and Charles Rezk for many helpful conversations throughout the progress of this work. 2. Preliminaries on Pro-Categories First we establish some terminology for pro-categories. Definition 2.1.For a category C, the category pro-C has objects all small cofil- tering systems in C, and Hom pro-C(X, Y ) = limscolimtHomC(Xt, Ys). Objects of pro-C can be thought of as functors from arbitrary small cofilteri* *ng categories to C. See [1] or [2] for more background on the definition of pro-ca* *tegories. We use both set theoretic and categorical language to discuss indexing categori* *es; hence "t sä nd "t ! s" mean the same thing. The word "systemä lways refers to an object of a pro-category, while the word "diagram" refers to a diagram of pro-objects. A subsystem of an object X : I ! C in pro-C is a restriction of X to a cofilt* *ering subcategory J of I. A subsystem is cofinal if for every s in I, there exists so* *me t in J and an arrow t ! s in I. A system is isomorphic in pro-C to any of its cof* *inal subsystems. A directed set I is cofinite if for every t, the set of elements of I less th* *an t is finite. Except in Section 11, all systems are indexed by cofinite directed * *sets 4 DANIEL C. ISAKSEN rather than arbitrary cofiltering categories. This is no loss of generality [6,* * 2.1.6] (or [1, Expos'e 1, 8.1.6]). The cofiniteness is critical because many construct* *ions and proofs proceed inductively. Whenever possible we avoid mentioning the structure maps of a pro-object X explicitly. When necessary the notation (X, OE) indicates a system of objects {* *Xs} with structure maps OEts: Xt! Xs. Definition 2.2.Let I be a cofinite directed set. For each s in I, the height h(* *s) of s is the value of n in the longest chain s > s1 > s2 > . .>.sn starting at s* * in S. In particular, h(s) = 0 if and only if there are no elements of I less than s. We always assume that systems have no initial object or equivalently that any system has objects of arbitrarily large height. This is no loss of generality s* *ince it is possible to add isomorphisms to a system so that it no longer has an initial ob* *ject, and this new system is isomorphic to the old one. We frequently consider maps between two pro-objects with the same index cat- egories. In this setting, a level map X ! Y between pro-objects indexed by I is given by maps Xs ! Ys for all s in I. Up to isomorphism, every map is a level m* *ap [2, Appendix 3.2]. A map satisfies a certain property levelwise if it is a level map X ! Y such * *that each Xs ! Ys satisfies that property. Lemma 2.3. A level map A ! B in pro-C is an isomorphism if and only if for all s, there exists t s and a commutative diagram At _____//_Bt | ---- | | -- | fflffl|""fflffl|-- As _____//Bs. Proof.The maps Bt! As induce an inverse. |___| 3. Preliminaries on Simplicial Sets Now we review some definitions and results about simplicial sets. Let SS be t* *he category of simplicial sets. We use the expressions "spaceä nd "simplicial se* *t" interchangeably. For simplification, we often use the same notation for a basepoint and its im* *age under various maps (e.g., ßn(X, *) ! ßn(Y, *)). Definition 3.1.A map f : X ! Y of simplicial sets is an n-equivalence if for all basepoints * in X, f induces an isomorphism ßi(X, *) ! ßi(Y, *) for 0 i < n and a surjection ßn(X, *) ! ßn(Y, *). The map f is a co-n-equivalence if for a* *ll basepoints * in X, f induces an isomorphism ßi(X, *) ! ßi(Y, *) for i > n and an injection ßn(X, *) ! ßn(Y, *). Definition 3.2.Set Jn = { mk! m |m 0} [ {@ m ! m |m > n}. A map of simplicial sets is a co-n-fibration if it has the right lifting proper* *ty with respect to all maps in Jn. A map of simplicial sets is an n-cofibration if it h* *as the left lifting property with respect to all co-n-fibrations. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 5 In other words, a co-n-fibration is a Jn-injective, and an n-cofibration is a* * Jn- cofibration [10, 12.4.1]. Note that co-n-fibrations and n-cofibrations are char* *acter- ized by lifting properties with respect to each other. When n = -1 or n = 1, the definitions reduce to the usual definitions of trivial cofibrations and fibrati* *ons or to the definitions of cofibrations and trivial fibrations. We will see below that n-cofibrations are just maps that are both cofibrations and n-equivalences. Also, co-n-fibrations are just maps that are both fibration* *s and co-n-equivalences. These facts motivate the terminology. Proposition 3.3.For any n, each map f of simplicial sets factors as f = pi, where i is an n-cofibration and p is a co-n-fibration. Proof.Apply the small object argument [10, 12.4]. |___| Lemma 3.4. A map f : E ! B is a co-n-fibration if and only if f is a fibration and co-n-equivalence. Proof.First suppose that f is a co-n-fibration. The generating trivial cofibrat* *ions are contained in Jn, so f is also a fibration. Now we must show that f is also a co-n-equivalence. Consider test diagrams of the form @ m _____//E< n. Let g : m ! B be a constant map with image *, and let F be the fiber of f over *. Since lifts exist in the test diagram, ßm F = 0 for m n. U* *sing the long exact sequence of homotopy groups of a fibration, it follows that f is* * a co-n-equivalence. Now suppose that f is a fibration and co-n-equivalence. It follows from the l* *ong exact sequence of homotopy groups that ßm F = 0 for m n, where F is any fiber of f. There are lifts in the test diagrams shown above for m > n because * *the __ obstructions to such lifts belong to ßm-1 F . Hence f is a co-n-fibration. * * |__| Definition 3.5.If A ! X is a cofibration of simplicial sets, then the relative n-skeleton X(n)is the union of A and the n-skeleton of X. Lemma 3.6. A map f : A ! X is an n-cofibration if and only if f is a cofibrati* *on and n-equivalence. Proof.Since trivial fibrations are co-n-fibrations, n-cofibrations are also cof* *ibra- tions. Suppose that f : A ! X is a relative Jn-cell complex [10, 12.4.6]. Then A ! X(n)is a weak equivalence since X(n)is obtained from A by gluing along maps of the form mk! m . Note that X(n)! X is an n-equivalence, so A ! X is also an n-equivalence. Arbitrary n-cofibrations are retracts of relative Jn-cell com* *plexes [10, 13.2.9], so all n-cofibrations are n-equivalences. Conversely, suppose that f is a cofibration and n-equivalence. We show that f* * is an n-cofibration by demonstrating that it has the left lifting property with re* *spect to all maps that are both fibrations and co-n-equivalences. By Lemma 3.4, this means that f has the left lifting property with respect to all co-n-fibrations. 6 DANIEL C. ISAKSEN Factor f as A __j_//_Yq__//X, where j is a trivial cofibration and q is a fibration. Note that q is also an * *n- equivalence. Let p : E ! B be a map that is both a fibration and co-n-equivalen* *ce, and consider a diagram A _____________//E77nn j|| n n n |p| fflffl|nnn fflffl| Y __q__//X____//_B. The dashed arrow exists because j is a trivial cofibration and p is a fibration. This gives the diagram A _____//Y_____//_E f || |p| fflffl| fflffl| X _____//X_____//B. There is no obstruction to lifting over X(0)since ß0A ! ß0X is surjective. Obstructions to finding a lift over the higher relative skeletons of X belong* * to ßm F , where F is some fiber of p. These obstructions lie in the image of ßm G, where G is a fiber of q. For every m, either ßm G or ßm F is zero. Hence there_* *are no obstructions to lifting. |__| Remark 3.7.For ind-spaces, consider the set In = { mk! m |m 0} [ {@ m ! m |m < n} to define n-fibrations and co-n-cofibrations. All n-fibrations are both fibrati* *ons and n-equivalences, but the converse is not true. If f : A ! X is a co-n-cofibratio* *n, then f is a cofibration and the induced map X(n-1)! X is a weak equivalence. 4. Local Systems The language of local systems is necessary in order to state the idea of homo* *topy groups for pro-spaces. Recall that a local system on a space X is a functor X ! Ab, where X is the fundamental groupoid of X and Ab is the category of abelian groups. Denote by LS(X) the category of local systems on X or equivalently the category of locally constant sheaves on X. For example, nX is a local system on X for n 2. It is defined by ( nX)x = ßn(X, x) with isomorphisms ( nX)x ! ( nX)y given by the usual maps on ho- motopy groups induced by paths. Occasionally we refer to local systems with values in non-abelian groups. For example, 1X is such a local system. We emphasize the notational distinction between X (a groupoid) and 1X (a local system). If f : X ! Y is a map of spaces, then f induces a map of local systems nX ! f* nY on X, where f* is the pullback functor LS(Y ) ! LS(X). Recall that the functor f* is exact in the sense that it preserves finite limits and colimits, * *and it is also exact in the sense that it preserves exact sequences. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 7 Lemma 4.1. Let f : X ! Y be a map of spaces. Then f is a weak equivalence if and only if ß0f is an isomorphism and nX ! f* nY is an isomorphism of local systems on X for all n 1. Proof.This is a restatement without reference to basepoints of the definition_of weak equivalence of simplicial sets. |__| We now extend the definitions to pro-spaces. Definition 4.2.A local system on a pro-space X is an object of colimsLS(Xs). If L is a local system on (X, OE) represented by a functor Ls : Xs ! Ab and M is another local system on X represented by a functor Mt : Xt ! Ab, then a map from L to M is an element of colimuHom LS(Xu)(OE*usLs, OE*utMt). Denote by LS(X) the category of local systems on X. A local system on X is represented by a local system on Xs for some s. For example, for n 1, nXs is a local system on X for each s. A map between two local systems on X is a map of representing local systems pulled back to some Xu. For example, for n 1 and t s, nXt ! nXs is a map of local systems on (X, OE) because nXt! OE*ts nXs is a map of local syste* *ms on Xt. Let f : X ! Y be a map of pro-spaces, and let L : Ys ! Ab be a local system on Ys. Choose any map fts: Xt ! Ys representing f and consider the functor L O fts: Xt ! Ys ! Ab . This gives a well-defined functor colimsLS(Ys) ! colimtLS(Xt). Definition 4.3.Let f : X ! Y be a map of pro-spaces. The pullback f* : LS(Y ) ! LS(X) is the functor colimsLS(Ys) ! colimtLS(Xt). Lemma 4.4. If f : X ! Y is a map of pro-spaces, then f* : LS(Y ) ! LS(X) is an exact functor in the sense that it preserves finite limits and colimits. Proof.Without loss of generality, we may assume that f is a level map. Given a finite diagram of local systems L on Y , there is an s such that each Liis repr* *esented by a local system Lison Ys. Then for each i, f*Li is represented by f*sLis. Now colimLi in LS(Y ) is represented by colimLisin LS(Ys), so f*(colimLi) is repre- sented by f*s(colimLis). Also, colimf*Li in LS(X) is represented by colimf*sLis* *in LS(Xs). But f*scommutes with finite colimits, so colimf*Li and f*(colimLi) are isomorphic since they are represented by the same local system on Xs. __ An identical argument shows that f* commutes with finite limits. |__| It follows from the lemma that f* is exact in the sense that it preserves exa* *ct sequences. 5. Homotopy Groups With the notions of local systems in place, we can define homotopy groups of pro-spaces as pro-objects in a category of local systems. The local systems are necessary to avoid mentioning basepoints. Definition 5.1.If X is a pro-space and n 2, then nX is the pro-local system on X given by { nXs}. Also, ß0X is the pro-set given by {ß0Xs}, and 1X is the pro-local system on X with values in non-abelian groups given by { 1Xs}. 8 DANIEL C. ISAKSEN Note that a map of pro-spaces f : X ! Y induces a map nX ! f* nY in pro-LS(X). Lemma 5.2. If f : X ! Y is a map of pro-spaces and ß0f is an epimorphism in the category of pro-sets, then a map g of pro-local systems is an isomorphism in pro-LS(Y ) if and only if f*(g) is an isomorphism in pro-LS(X). Proof.Without loss of generality, assume that f is a level map. Note that pro-L* *S(Y ) is an abelian category [2, Appendix 4.5]. Since f* is exact, it suffices to con* *sider a local system L on Y such that f*L = 0 and conclude that L = 0. A pro-local system M is zero if and only if for every i, there exists j i s* *uch that the map Mj ! Mi is trivial. For any i, choose j i so that f*Lj ! f*Li is trivial. Let Lisand Ljsbe local systems on Ys representing Li and Lj respective* *ly. Choose s large enough so that f*sLjs! f*sLisis a trivial map of local systems on Xs. Since ß0f is an epimorphism, there exists some t and a map ß0Yt! ß0Xs such that the map ß0Yt ! ß0Ys factors through ß0Xs. Since f*sLjs! f*sLisis trivial, the map Ljs! Lisis trivial when restricted to the components of Ys in the image of Xs. Now the image of ß0Xs in ß0Ys contains the image of ß0Yt, so Ljs! Lis becomes trivial when pulled back to Yt. Hence the map Lj ! Li is trivial. This_ means that the pro-local system L is zero. |__| Remark 5.3.A similar statement is true for pro-local systems with values in non- abelian groups. Lemma 5.4. A map of pro-spaces f : X ! Y induces an isomorphism of pro-sets ß0f : ß0X ! ß0Y if and only if f is isomorphic to a level map f0 : X0 ! Y 0such that f0 induces a level isomorphism ß0f0 : ß0X0! ß0Y 0. Proof.One direction is clear because level isomorphisms are pro-isomorphisms. Assume that ß0f is an isomorphism. We may also assume that f is a level map. Define X0= X and Y 0= Y xß0Yß0X. Here we identify pro-sets with pro-spaces of dimension zero. Then Y 0is isomorphic to Y since ß0X ! ß0Y is an isomorphism. Let f0 : X0! Y 0be the map induced by f : X ! Y and the projection X ! ß0X. Pullbacks can be constructed levelwise in pro-categories, so for all s, Ys0= Ys xß0Ysß0Xs and* * f0s is induced by fs : Xs ! Ys and Xs ! ß0Xs. Note that f0sinduces an isomorphism_ ß0X0s! ß0Ys0. Hence f0 is the desired map. |__| If f : X ! Y is a map of spaces such that ß0f is an isomorphism and ß1f is an isomorphism for every basepoint, then f* induces an equivalence of categories LS(Y ) ! LS (X). The following lemma makes an analogous statement for pro- spaces. Lemma 5.5. If f : X ! Y is a map of pro-spaces such that ß0f and 1X ! f* 1Y are isomorphisms, then the functor f* : LS(Y ) ! LS(X) is an equivalence of categories. Proof.We only prove that f* is essentially surjective in the sense that every o* *bject L of LS(X) is isomorphic to f*M for some M in LS(Y ). We leave the rest of the proof to the interested reader. We will use only the surjectivity in this work. With no loss of generality, we may assume that f is a level map. By Lemma 5.4, we may also assume that ß0f is a level isomorphism. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 9 Let L be a local system on (X, OE) represented by a local system Ls on Xs. Th* *ere exists t s and a commutative diagram 1Xt _______//_f*t 1Yt | qqqqq | | qqqq | fflffl|xxqq fflffl| OE*ts 1Xs___//OE*tsf*s 1Ys of local systems on Xt. Choose one point xi in each component of Xt. Let yi be the image of xi in Yt; this is a choice of one point in each component of Yt. Let x0ibe the image of x* *i in Xs. By evaluating the above diagram at xi, the map ß1(Xt, xi) ! ß1(Xs, x0i) facto* *rs as ß1(Xt, xi)___//_ß1(Yt,_yi)gi//_ß1(Xs, x0i). The maps gi : ß1(Yt, yi) ! ß1(Xs, x0i) and the local system Ls determine a local system Mt on Yt by setting (Mt)yi= Lx0iwith the ß1(Yt, yi)-action induced by gi. Let M be the local system on Y represented by Mt. Note that f*tMtis isomorphi* *c_ to OE*tsLs. Hence f*M is isomorphic to L. |__| Remark 5.6.A similar statement applies to local systems with values in non-abel* *ian groups. 6.Model Structure Now we explicitly describe the model structure on the category of pro-spaces. Definition 6.1.A map of pro-spaces f : X ! Y is a weak equivalence if ß0f is an isomorphism of pro-sets and nX ! f* nY is an isomorphism in pro-LS(X) for all n 1. In Corollary 7.5 we will see that for pointed connected pro-spaces, a level m* *ap X ! Y is a weak equivalence if and only if ßnX ! ßnY is a pro-isomorphism for all n. This works because there is no need for arbitrary basepoints. Hence Artin-Mazur weak equivalences [2, Section 4] are also weak equivalences. Definition 6.2.A map of pro-spaces is a cofibration if it is isomorphic to a le* *vel- wise cofibration. Definition 6.3.A map of pro-spaces is a fibration if it has the right lifting p* *roperty with respect to all trivial cofibrations. Theorem 6.4. The above definitions give a proper simplicial model structure on pro-SS(without functorial factorizations). This model structure is not cofibran* *tly generated. Proof.The axioms for a proper simplicial model structure are verified in Sectio* *ns 11 through 17. Limits and colimits exist by Proposition 11.1. The two-out-of-three axiom is Proposition 13.1. Retracts preserve weak equivalences because weak equivalences are defined in terms of isomorphisms. Retracts preserve fibrations because retr* *acts preserve lifting properties. Corollary 12.2 is the retract axiom for cofibrati* *ons. Propositions 15.1 and 15.2 are the factoring axioms, while Proposition 15.4 is * *the 10 DANIEL C. ISAKSEN non-trivial lifting axiom. The axioms for a simplicial model structure are demo* *n- strated in Proposition 16.3. Proposition 17.1 shows that the model structure i* *s __ proper, and Corollary 19.3 states that it is not cofibrantly generated. * * |__| The model structure can be considered in two stages. The "strict" structure of Edwards and Hastings [6, 3.3], in which the weak equivalences are defined level* *wise, is an intermediate step; see Section 10 for details. The situation is not unlik* *e the Bousfield-Friedlander strict and stable model structures for spectra [3]. We assume Theorem 6.4 for the rest of this section and for the next three sec* *tions. Sections 11-19 contain the details of the proof of the theorem. In practice we need a more concrete description of fibrations. The next defin* *ition and proposition provide such a description. Definition 6.5.A map is a strong fibration if it is isomorphic to a level map of pro-spaces X ! Y indexed by a cofinite directed set such that for all t, Xt! Ytxlims s and a commutative diagram Yt _____//_Zt | ____| | __ | fflffl|~~__fflffl| Ys ____//_Zs. By restricting to cofinal subsystems, we may assume that such a diagram exists * *for every t > s. Let J be the directed set of indecomposable arrows of I. The domain and range functors J ! I are both cofinal since I is cofinite. For each OE : t ! s in J, * *factor the map Zt! Ys as iOE pOE Zt ____//_AOE___//Ys, where iOEis a cofibration and pOEis a fibration. Let BOEbe the pullback XsxYsAOE, and let COEbe the pushout WtqZtAOE. These objects fit into a commutative diagram 18 DANIEL C. ISAKSEN Xt _____//Yt____________//_Zt___//_Wt | | """"| | | | "" | | fflffl| || ~~"" || fflffl| BOE____________//_AOE___________//COE | " | | | "" | | | | "" | | fflffl| fflffl|~~"" fflffl| fflffl| Xs _____//Ys____________//Zs___//_Ws. Note that BOE! AOEis a weak equivalence because it is a pullback of a weak equivalence along a fibration. Also, AOE! COEis a weak equivalence because it is a pushout of a weak equivalence along a cofibration. Hence the composition B ! A ! C is a levelwise weak equivalence. This composition is isomorphic_to ghf since B ~=X, Y ~=A ~=Z, and W ~=C. |__| The above proof works for any pro-category pro-C provided that C is a proper model category. In fact, a minor variation of the proof works when C is either * *left proper or right proper. It is possible to prove the other parts of the two-out-* *of-three axiom for strict weak equivalences with similar techniques. Proposition 10.5.[6] A map of pro-spaces is a strict fibration if and only if i* *t is a retract of a strong strict fibration. The relationship between the strict model structure and our model structure is expressed in the following results. Let Hostrict(pro-SS) be the homotopy category associated to the strict struct* *ure. Proposition 10.6.The category Ho(pro-SS) is a localization of Hostrict(pro-SS). Proof.Every levelwise weak equivalence is a weak equivalence in the sense_of De* *fi- nition 6.1. |__| Corollary 10.7.If f is a fibration, then f is also a strict fibration. Proof.The class of trivial cofibrations contains the class of strictly trivial_* *cofibra- tions by Proposition 10.6. |__| Corollary 10.8.A map of pro-spaces is a trivial fibration if and only if it is a strictly trivial fibration. Proof.Cofibrations are the same as strict cofibrations. |_* *__| Proposition 10.9.Let X be a pro-space, and let Y be a pro-space such that each Ys has only finitely many non-zero homotopy groups. Then [X, Y ]pro~=Hom Hostrict(pro-SS)(X, Y ). Proof.The condition on Y ensures that its strictly fibrant replacement Y 0is al* *so a fibrant replacement. To calculate morphisms from X to Y in either homotopy cat- egory, consider morphisms from X to Y 0modulo the simplicial homotopy relation._ Hence the morphisms are the same. |__| For example, the proposition applies when Y is a system of Eilenberg-Maclane spaces. Edwards and Hastings described a relationship between homological algeb* *ra and the strict homotopy theory of such pro-spaces [6, Section 4]. It follows t* *hat the relationship works just as well for our homotopy theory of pro-spaces. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 19 11. Limits and Colimits The rest of the paper concentrates on the technical details of the main theor* *ems stated in Section 6. We provide specific constructions of limits and colimits in pro-categories. T* *he existence of all colimits seems to be a little-known fact. In this section only* *, consider pro-objects indexed by arbitrary cofiltering categories, not just ones indexed * *by cofinite directed sets. Proposition 11.1.If C is complete, then pro-C is also complete. If C is cocom- plete, then pro-C is also cocomplete. Proof.Artin and Mazur [2, Appendix 4.2] showed that pro-C has all equalizers (resp. coequalizers) provided that C does. It suffices to construct arbitrary p* *roducts (resp. coproducts) when these exist in C. Let A be a set and let Xffbe a pro-objectQfor each ff in A. Let Iffbe the cof* *iltering index categoryQfor Xff. Define X = ff2AXffto be the cofiltering system with objects ff2BXffsffand index category I consistingQof pairs (B, (sff)) where B* * is a finite subset of A and (sff) is an element of ff2BIff. A morphism of I from (B, (sff)) to (C, (tff)) corresponds to an inclusion C B and morphisms sff! t* *ff in Ifffor all ff in C. Use of finite subsets B of A is essential because we use* * the fact that finite products commute with filtered colimits. Direct calculation shows that for any Y in pro-C, Y Hom pro-C(Y, X) ~= Hom pro-C(Y, Xff). ff2A Thus arbitrary products exist. ` To construct`the coproduct, define X = ff2AXffto beQthe cofiltering systemQ with objects ff2AXffsffand index category {(sff) 2 ff2AIff}. Note that If* *fis cofiltering since each Iffis. In order to show that X has the correct universal mapping property, it suffic* *es to see that Y Hom pro-C(X, Y ) ~= Hom pro-C(Xff, Y ). ff2A for Y any object of C (i.e., Y is a constant system in pro-C). This can be chec* *ked directly, using the fact that colimits indexed on product categories commute wi* *th_ the relevant products. Thus arbitrary coproducts exist. |_* *_| 12.Retract Axioms The class of fibrations is obviously closed under retracts. The class of weak* * equiv- alences is closed under retracts because weak equivalences are defined in terms* * of isomorphisms of pro-local systems. We must show that the class of cofibrations * *is also closed under retracts. We prove a general result and then apply it to cofi* *bra- tions. Proposition 12.1.Let C be a category, and let C be any class of maps in C. Defi* *ne the class D as those maps in pro-C that are isomorphic to a level map that belo* *ngs to C levelwise. Then D is closed under retracts. 20 DANIEL C. ISAKSEN Proof.Suppose that f : W ! Z is a retract of g : X ! Y , where g is a level map that belongs to C levelwise. Hence there is a commutative diagram W _____//X_____//W f || g || |f| fflffl| fflffl|fflffl| Z _____//_Y____//_Z where the horizontal compositions are identity maps. We must show that f also belongs to D. Choose a level representative for f. By adding isomorphisms to the systems for f or g, make the cardinalities of t* *he index sets equal. Choose an arbitrary isomorphism ff from the index set of f to the index set of g. Define a function t(s) inductively on height satisfying several conditions. F* *irst, choose t(s) large enough so that Xt(s)! Ws and Yt(s)! Zs represent respectively the maps X ! W and Y ! Z. Also, choose t(s) large enough so that t(s) ff(s). Finally, choose t(s) large enough so that for all u < s, t(u) < t(s) with a com* *muting diagram Xt(s)__________//Ws vvv ____|| --vvv || ""__ | Xt(u)____________//Wu | | | || || | fflffl| | fflffl| | Yt(s)____ |____//Zs | v | _ | vvv | ___ fflffl|--vv fflffl|""__ Yt(u)___________//Zu. __ __ __ Now_the function t defines cofinal subsystems X and Y of X and Y where X and Y have the same index sets_as_W_and Z. Repeat this_process_on _g: X ! Y to obtain another function u inducing cofinal subsystems W and Z of W and Z. The result is a level diagram _______//______// W X W _f| _ | | | g | |f _fflffl|_fflffl|fflffl|_ Z _____//_Y____//_Z representing (up to isomorphism) f as a retract of g. Since W ! X ! W and Z ! Y_!_Z are_identity maps, u_can_be_chosen so that the composites Wu(s)= W s ! X s! Ws and Zu(s)= Zs ! Y s! Zs are structure maps of W and Z for all s. Since g belongs to C levelwise, the same is true for _g. Define a pro-space ^* *W by starting with the system_W and replacing the single map Wu(s)! Ws with the pair of maps Wu(s)! X s! Ws. Define ^Zsimilarly. Note that W and Z are cofinal subsystems of W^ and ^Zrespectively, so W is isomorphic to ^W and Z is isomorphic to ^Z. Thus it suffices to show that the l* *evel map ^f: ^W! ^Zbelongs to D. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 21 __ The subsystem of ^W on objects {X s}_is_also a cofinal subsystem, and the same is_true for the subsystem on_objects {Y s} in ^Z. Beware that the subsystem of * *^Won {X s} is not isomorphic_to X because_the structure maps are different. The same warning applies to {Y s} and Y.__ __ Restrict ^fto the subsystem {X s} ! {Y s}. This last map belongs to C lev- elwise, so it belongs to D. Hence f also belongs to D because D is closed_under isomorphisms. |__| Corollary 12.2.The class of cofibrations of pro-spaces is closed under retracts. Proof.Apply Proposition 12.1 to the class of all cofibrations in SS. * * |___| 13. Weak Equivalences We begin with the two-out-of-three axiom. Proposition 13.1.Let f : X ! Y and g : Y ! Z be maps of pro-spaces. If any two of the maps f, g, and gf are weak equivalences, then so is the third. Proof.For n 1, the map nX ! f*g* nZ factors as nX ! f* nY ! f*g* nZ. Also, the map ß0X ! ß0Z factors through ß0Y . This immediately proves two of the three cases. For the third case, suppose that f and gf are weak equivalences. Then f* nY ! f*g* nZ is an isomorphism for all n 1. By Lemma 5.2, nY ! g* nZ is also_ an isomorphism for all n 1. |__| The following lemma is a surprising generalization to pro-groups of an obvious fact about groups. Bousfield and Kan [4, III.2.2] stated without proof a speci* *al case. Lemma 13.2. Let U be the forgetful functor from pro-groups to pro-sets. Then a map f of pro-groups is an isomorphism if and only if U(f) is an isomorphism of pro-sets. Proof.For simplicity write the group operations additively, even though the gro* *ups are not necessarily abelian. We may assume that f : X ! Y is a level map. If f is an isomorphism, then Uf is also an isomorphism by Lemma 2.3. Now suppose that Uf is an isomorphism. By Lemma 2.3 applied twice, for every s, there exist u t s and a commutative diagram Xu _____//Yu | g ____| | __ j| |fflffl~~fflffl|__ft Xt _____//Yt __ OE||h____ || |fflffl~~fflffl|__ Xs _____//Ys where the diagonal maps are not necessarily group homomorphisms. 22 DANIEL C. ISAKSEN However, the composite map Yu ! Xs is in fact a group homomorphism. For every x and y in Yu, hj(x + y) = h(jx + jy) = h(ftgx + ftgy) = hft(gx + gy) because of commutativity in the top square. Also, hft(gx + gy) = OE(gx + gy) = OEgx + OEgy because of commutativity in the bottom square. Finally, OEg = hj. Therefore, there is a commutative diagram of groups Xu _____//Yu | ----| | -- | fflffl|""fflffl|-- Xs _____//Ys. By Lemma 2.3, f is an isomorphism. |___| The formal nature of Definition 6.1 is often too abstract for comfort in tech* *nical situations. The following proposition shows that conditions (a) and (b) of Theo* *rem 7.3 are equivalent, thus giving a less natural but more concrete equivalent def* *inition of weak equivalence. Proposition 13.3.A level map of pro-spaces f : (X, OE) ! (Y, j) is a weak equiv- alence if and only if for all n 0 and for all s, there exists some t s such* * that for all basepoints * in Xt, there is a commutative diagram ßn(Xt, *)____//_ßn(Yt, *) | qqqqq | | qqq | fflffl|xxqq fflffl| ßn(Xs, *)____//ßn(Ys, *). Remark 13.4.By the argument in the proof of Lemma 13.2, it is not important whether we assume that the diagonal map is a group homomorphism or just a map of sets. For convenience, we assume that it is a group homomorphism. Note that the diagonal map is not geometrically induced; it just exists abstractly. The c* *hoice of t may depend on n and s, but it must work for every basepoint. Proof.First suppose that f is a weak equivalence. Since ß0f is an isomorphism, Lemma 2.3 gives the conclusion for n = 0. For n 1, Lemma 2.3 implies that, for every s, there exists a t s and a commutative diagram nXt _______//_f*t nYt | | | | fflffl| fflffl| OE*ts nXs____//OE*tsf*s nYs of local systems on Xt. In particular, for every basepoint * in Xt, there is a commutative diagram ßn(Xt, *)____//_ßn(Yt, *) | qqqqq | | qqq | fflffl|xxqq fflffl| ßn(Xs, *)____//ßn(Ys, *). A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 23 This proves the ö nly if" part of the claim. Now suppose that the diagrams in the statement of the proposition exist. By Lemma 2.3, ß0f is an isomorphism. For n 1, we use the fact that a local syste* *m L on a space Z is determined up to isomorphism by its value Lx as a ß1(Z, x)-modu* *le for one point x in each component of Z. For every s, there exist u t s such that for every basepoint * in Xu there are commutative diagrams ßn(Xu, *)____//_ßn(Yu, *) ß1(Xu, *)_____//ß1(Yu, *) | qqqqq | | qqqqq | | qqq | | qqq | fflffl|xxqq fflffl| fflffl|xxqq fflffl| ßn(Xt, *)_____//ßn(Yt, *) ß1(Xt, *)____//_ß1(Yt, *) | qqqqq | | qqqqq | | qqq | | qqq | fflffl|xxqq fflffl| fflffl|xxqq fflffl| ßn(Xs, *)_____//ßn(Ys, *) ß1(Xs, *)____//ß1(Ys, *). A diagram chase like that in the proof of Lemma 13.2 shows that the map ßn(Yu, *) ! ßn(Xs, *) is actually a map of ß1(Xu, *)-modules, even though the diagonal maps in the left diagram above are not maps of ß1(Xu, *)-modules. This defines a commutative diagram nXu _______//_f*u nYu | ppppp | | pppp | fflffl|wwpp fflffl| OE*us nXs____//OE*usf*s nYs of local systems on Xu. __ Hence nX ! f* nY is an isomorphism by Lemma 2.3. |__| Corollary 13.5.Suppose that f : X ! Y is a level map of pro-spaces indexed by a cofinite directed set I for which there is a strictly increasing function n :* * I ! N such that fs : Xs ! Ys is an n(s)-equivalence. Then f is a weak equivalence. Proof.For any s in I and any n 0, choose t s such that n(t) > n. For every point * in Xt, there is a commutative diagram of solid arrows ~= ßn(Xt, *)____//_ßn(Yt, *) | q q | | q q | fflffl|xxq fflffl| ßn(Xs, *)____//ßn(Ys, *). Since the top horizontal map is an isomorphism, this diagram can be extended to include the dashed arrow. Thus f satisfies the condition of Proposition 13.3,_s* *o it is a weak equivalence. |__| 14.Fibrations The following lemma states some useful properties of strong fibrations that f* *ollow directly from the definition. Lemma 14.1. If f : X ! Y is a strong fibration, then for all t the maps ft: Xt! Yt and gt: lims a(s) for all s < t. Now select t2N-1, t2N-2, . .,.t1* *, t0 so that ti> ti+1and there exist commutative diagrams ßn(Ati, *)____//_ßn(Ati+1,7*)7 pp | ppp | | pppp | fflffl|pp fflffl| ßn(Xti, *)____//ßn(Xti+1, *). for all 0 n N and all basepoints * in Ati. This is possible since j is a we* *ak equivalence and there are only finitely many conditions on the choice of each t* *i. Finally, choose a(t) so that a(t) > t0 and there exists a commutative diagram ß0Aa(t)_____//_ß0At0:: | tttt | | ttt | fflffl|tt fflffl| ß0Xa(t)____//_ß0Xt0. Functorially factor each map Ati! Xtias Ati__ai_//Ytibi//_Xti, where ai is a trivial cofibration and bi is a fibration. Choose a basepoint * in Yti. Since ai is a weak equivalence, there exists a basepoint ] in Atisuch that there is a path in Ytifrom * to the image of ]. For 0 n N, there is a diagram of solid arrows 28 DANIEL C. ISAKSEN ßn(Ati, ])___________//_ßn(Ati+1,2])2eee sss | eeeeeeeeeeeeppppp | yysssse|eeeeeeee xxpp | ßn(Xti, ])_________//_ßn(Xti+1, ]) | | | | | | | | | fflffl| | fflffl| | ßn(Yti, *)_____|_____//ßn(Yti+1, *) | s |e e e2p2 | ssss e e e e | pppp fflffl|yysseeee fflffl|xxpp ßn(Xti, *)_________//_ßn(Xti+1, *) where the vertical arrows are induced by the choice of path. Since the vertical arrows are all isomorphisms, the dashed arrow also exists. For similar reasons, there is also a commutative diagram ß0Aa(t)_____//_ß0Yt0:: | tttt | | ttt | fflffl|tt fflffl| ß0Xa(t)____//_ß0Xt0. Now we have Aa(t)____//At0____//At1____//._._._//At2N____//W | -=|= | a0| a1| a2N| - | | | | | - | | fflffl| fflffl| fflffl|- | | Yt0_____//_Yt1___//._._._//Yt2N |q | | | | | | | | b0| b1| b2N| | fflffl| fflffl| fflffl| fflffl| fflffl| Xa(t)_____//Xt0____//Xt1____//._._._//Xt2N____//Z where the dashed arrow exists because a2N is a trivial cofibration and q is a f* *ibration. So we need only find a lift for the diagram Aa(t)_____//Yt0___//_Yt2___//._._._//Yt2N-2____//_Yt2N___//W | || | | fflffl| fflffl| Xa(t)____//_Xt0___//_Xt2___//._._._//Xt2N-2____//Xt2N____//Z. Lemma 14.4 tells us that this diagram satisfies the hypotheses of Lemma 14.3._ Hence the desired lift exists. |__| 15. Lifting and Factorization Axioms We now prove the lifting and factorizations axioms. Proposition 15.1.If f : X ! Y is a map of pro-spaces, then f factors (not functorially) as p X __i_//_Z___//_Y, where i is a trivial cofibration and p is a strong fibration. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 29 Proof.We may assume that f is a level map. We construct the factorization in- ductively. Assume for sake of induction that the factorization is already constructed on* * all indices less than t. Recall the height function h(t) from Definition 2.2. Factor Xt! YtxlimsE>__=__//E">> "" "" """ i||" |p| "" fflffl|fflffl|" A _____//_Y____//B>>">>~ ~~ j|| "" q||~=~ fflffl|"fflffl|~~ X _____//B, the lift in the lower left square exists because of the strict model structure,* * and the lift in the upper right square exists by the definition of fibrations. The_comp* *osition X ! Y ! E is the desired lift. |__| 16.Simplicial Model Structure Recall that a model structure on a category C is simplicial if for every X in* * C and every simplicial set K, there are functorial constructions X K ("tensor") * *and XK (öc tensor") in C satisfying certain associativity and unit conditions. Also* *, for every X and Y in C, there is a simplicial function complex Map (X, Y ). These t* *hree constructions are related by the adjunctions Map (X K, Y ) ~=Map (K, Map(X, Y )) ~=Map (X, Y K). Finally, Map (-, -) must interact appropriately with the model structure as fol* *lows. If i : A ! X is a cofibration in C and p : E ! B is a fibration in C, then Map (X, E) ! Map (A, E) xMap(A,B)Map (X, B) is a fibration that is a weak equivalence if either i or p is a weak equivalenc* *e. We begin with a general proposition showing that tensors and cotensors defined for finite simplicial sets automatically extend to all simplicial sets. Proposition 16.1.Suppose that C is a model category with a simplicial function complex Map (-, -). Also suppose that the tensor X K and the cotensor XK are defined for all objects X of C and all finite simplicial sets K so that the axi* *oms for a simplicial model structure are satisfied when they make sense. Then the definit* *ions of tensor and cotensor can be extended to provide a simplicial model structure * *for C. Proof.For any simplicial set K, let Kfinbe the filtering system of finite subsp* *aces of K. For an object X of C, define X K to be colims(X Kfins) and XK to be limsXKfins. Using the fact that the system Kfinx Lfinis cofinal in the system (K x L)fin, the required isomorphisms X (K x L) ~=(X K) L and Map (X K, Y ) ~=Map (K, Map(X, Y )) ~=Map (X, Y K) can be verified directly. |___| A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 31 Definition 16.2.If X and Y are pro-spaces and K is a simplicial set, define Map (X, Y ) = Hom pro-SS(X x o, Y ) = limscolimtMap(Xt, Ys), X K = colims(X x Kfins), and fin Y K = lims(Y Ks ). For an arbitrary pro-space X and a simplicialfsetiK,nX x K can be constructed as theflevelwiseiproductnwith K. Also, limsY Ks can be constructed as the system {YtKs }, indexed by all pairs (s, t) in the product of the index categories. Note that X K is not in general isomorphic to X x K because finite limits do not always commute with filtered colimits in the category of pro-spaces. Howeve* *r, if K is finite, then X K is isomorphic to X x K since K itself is the terminal object of Kfin. Also, when K is finite, Y K is the system {YsK} with the same i* *ndex category as that of Y . Proposition 16.3.The above definitions make pro-SSinto a simplicial model cat- egory. Proof.By Proposition 16.1, it suffices to check the axioms only for finite simp* *licial sets. Most of the axioms are obvious; we verify only the non-trivial ones here. Let X and Y be arbitrary pro-spaces, and let K be a finite simplicial set. We* * use the fact that Hom SS(K, colimsZs) is equal to colimsHom SS(K, Zs) for any filte* *red system Z of simplicial sets because K is finite. It follows by direct calculati* *on that Map (X K, Y ) ~=Map (X, Y K) ~=Map (K, Map(X, Y )). We now show that the map f : Map (B, X) ! Map (A, X) xMap(A,Y )Map(B, Y ) is a fibration whenever i : A ! B is a cofibration and p : X ! Y is a fibration and that this map is a trivial fibration if either i or p is trivial. We proce* *ed by showing that f has the relevant right lifting property. Let j : K ! L be a generating cofibration or a generating trivial cofibration. Note that K and L a* *re finite simplicial sets. By adjointness, it suffices to show that the map g : A L qA K B K ! B L is a cofibration that is trivial if either i or j is trivial. We may assume that i is a levelwise cofibration. For every s, As ! Bs is a cofibration. Therefore, the map As L qAs K Bs K ! Bs L is also a cofibration. This is a standard fact about simplicial sets. Thus g * *is a levelwise cofibration. In order to show that g is trivial whenever i or j is, it suffices to show th* *at the map A K ! B K is trivial if i is trivial and that the map A K ! A L is trivial if j is trivial. This reduction follows from the two-out-of-three a* *xiom, the fact that trivial cofibrations are preserved by pushouts, and the commutati* *ve diagram 32 DANIEL C. ISAKSEN A K __________//_A JLJ | | JJJJ | | JJ |fflffl // fflffl| JJJJJ _____YA L qA K B K B K YYYYYYYYY UUU JJJ YYYYYYYYY UUUUUU JJJ YYYYYYYYUUUUUJJJ YYYYYYYJJ%%,,Y**UUU B L. First suppose that j is trivial. The map A K ! A L is a levelwise weak equivalence, so it is a weak equivalence of pro-spaces. Now suppose that i is trivial. Since A K ! B K is constructed by levelwis* *e__ product with K, condition (b) of Theorem 7.3 is easily verified. * *|__| 17. Properness We now show that the model structure of Theorem 6.4 is proper. Recall that a model structure is left proper if weak equivalences are preserved under pusho* *ut along cofibrations. Dually, a model structure is right proper if weak equivalen* *ces are preserved under pullback along fibrations. Proposition 17.1.The simplicial model structure of Theorem 6.4 is left and right proper. Proof.Left properness follows immediately from the fact that all pro-spaces are cofibrant. We must show that the model structure is right proper. Let p : E ! B be a fibration and let f : X ! B be a weak equivalence. Use Theorem 7.3 to suppose that p and f are level maps with the same cofinite direc* *ted index set I for which there is a strictly increasing function n : I ! N such th* *at fs is a n(s)-equivalence. Let P be the pullback X xB E, which is constructed levelwis* *e. We must show that the projection P ! E is a weak equivalence. We start with a special case. First suppose that p is a levelwise fibration. * *Let * be a basepoint in Ps. This yields a diagram F _____//Ps___//_Xs =|| || |fs| fflffl| fflffl| fflffl| F ____//_Esps__//Bs in which the rows are fiber sequences. From the 5-lemma applied to the long exa* *ct sequences of homotopy groups of the fibrations, Ps ! Es is also an n(s)-equival* *ence. By Theorem 7.3, P ! E is a weak equivalence. Now let p be an arbitrary fibration. By Proposition 6.6, there exists a stro* *ng fibration q : E0! B such that p is a retract of q. Note that q is a levelwise f* *ibration by Lemma 14.1. Consider the commutative diagram A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 33 P __________//P_0_______//_P ~~ ___ """ | ~~~~~||__//0""__||___//_~~"" || E E E | | | | | | | || fflffl|||__/fflffl|||/__/fflffl|/_ | ~X | _X | "X | ~~ | __ | """ fflffl|""~~ fflffl|~~__ fflffl|~~" B _________//_B________//_B, where P 0= X xB E0. This diagram is a retract of squares in the sense that all of the horizontal compositions are identity maps. The map P 0! E0 is a weak equivalence by the special case. Since weak equivalences are closed under retra* *cts,_ the map P ! E is also a weak equivalence. |__| 18.Alternative Characterizations of Weak Equivalences We finish here the proof of Theorem 7.3 describing weak equivalences in other terms. For expository clarity, we split the theorem into several parts. The equ* *iva- lence of (a) and (b) was shown in Proposition 13.3. Proposition 18.1.A map of pro-spaces is a weak equivalence if and only if it is isomorphic to a level map g : Z ! W indexed by a cofinite directed set I for wh* *ich there is a strictly increasing function n : I ! N such that gs : Zs ! Ws is an n(s)-equivalence. Proof.Corollary 13.5 showed that a map g satisfying the conditions of the propo- sition is a weak equivalence. Now suppose that f is a weak equivalence. We may assume that f is a level map. Use Proposition 15.1 to factor f as p X __i_//_Z___//_Y, where i is a trivial cofibration and p is a trivial fibration. By Corollary 10.* *8, p is also a strictly trivial fibration. In particular, p is isomorphic to a levelwis* *e weak equivalence. The proof of Proposition 15.1 indicates that i satisfies the condi* *tions of the proposition. By an argument similar to the proof of Proposition 10.4, f_* *also_ satisfies the conditions of the proposition. |_* *_| Recall the Moore-Postnikov functor P from Definition 7.2. Lemma 18.2. The canonical map X ! P X is a weak equivalence for any pro- space X. Proof.Condition (b) of Theorem 7.3 is easily verified. |_* *__| Proposition 18.3.A map of pro-spaces f : X ! Y is a weak equivalence if and only if P f is a strict weak equivalence. Proof.Suppose that P f is a strict weak equivalence. Then it is also a weak equ* *iv- alence. The maps X ! P X and Y ! P Y are weak equivalences by Lemma 18.2, so f is also. Now suppose that f is a weak equivalence. By Proposition 18.1, we may assume that f is a level map indexed by a cofinite directed set I for which there is a* * strictly increasing function n : I ! N such that fs : Xs ! Ys is an n(s)-equivalence. 34 DANIEL C. ISAKSEN Consider the subsystem X0 = {Pn(s)Xs|s 2 I} of X and the subsystem Y 0= {Pn(s)Ys|s 2 I} of Y . Note that X0 and Y 0are cofinal in X and Y . Let f0 be t* *he level map X0 ! Y 0induced by f, so f0 is isomorphic to f. Since Xs ! Ys is an n(s)-equivalence, the map Pn(s)Xs ! Pn(s)Ys is a weak equivalence. Hence f0 is * *a__ levelwise weak equivalence, so f is a strict weak equivalence. * *|__| Proposition 18.4.A map of pro-spaces f : X ! Y is a weak equivalence if and only if ß0f is an isomorphism of pro-sets, 1X ! f* 1Y is an isomorphism of pro-local systems on X, and for all m and all local systems L on Y , the map Hm (Y ; L) ! Hm (X; f*L) is an isomorphism. Proof.Let f be a weak equivalence. By Proposition 18.1, we may assume that f is a level map indexed by a cofinite directed set I for which there is an incre* *asing function n : I ! N such that fs is an n(s)-equivalence. By the Whitehead theorem, fs induces a cohomology isomorphism in dimen- sions less than n(s) for any local system on Ys. Hence f induces an isomorphism Hm (Y ; L) ! Hm (X; f*L) in the colimit for every m. Now suppose that f satisfies the conditions of the proposition. Factor f as p X __i__//Y_0___//Y, where i is a cofibration and p is a strictly trivial fibration. Since p induce* *s co- homology isomorphisms by the first part of the proof, the map i still satisfies* * the hypotheses of the proposition. Therefore, we may assume that f is a level map t* *hat is a level cofibration. Note that M = (X x 1) qX Y is weakly equivalent to Y since M is constructed levelwise and M is levelwise weakly equivalent to Y . We prove the proposition by showing that for every strongly fibrant pro-space Z, the map Map (M, Z) ! Map (X, Z) is a weak equivalence. A retract argument then shows that the map Map (M, Z) ! Map (X, Z) is a weak equivalence for all fibrant pro-spaces Z. Assume that Z is an arbitrary strongly fibrant pro-space. Note that each Zs is a fibrant simplicial set with only finitely many non-zero homotopy groups. Recall that Map (X, Z) = limsMap(X, Zs). Also recall that for every t, the map Zt! lims tZs is a fibration since Z is fibrant. Therefore, the map Map (X, Zt) ! limsMtap(X, Zs) = Map (X, limsZts) is a fibration. It follows that Map (X, Z) is weakly equivalent to the homotopy* * limit holimsMap(X, Zs). Similarly, Map (M, Z) is weakly equivalent to the homotopy limit holimsMap (M, Zs). Since homotopy limits are invariant under levelwise weak equivalence, we only need show that Map (M, Zs) ! Map (X, Zs) is a weak equivalence of simplicial se* *ts for each s. Therefore, we may assume that Z is a fibrant simplicial set with on* *ly finitely many non-zero homotopy groups. By adjointness, to show that Map (M, Z) ! Map (X, Z) is a weak equivalence, it suffices to find lifts in the diagrams of pro-spaces A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 35 X _____//_Z N. Hence, the desired lifting exists when k 1. Now consider k = 0. The argument given for k 1 does not work. The trouble is that we cannot lift over p1 with obstruction theory because the first homoto* *py groups of the fiber are not necessarily abelian. When k = 0, the map p is just the map Z ! *, so we need to find an s and a factorization of Xs ! Z through Ms. Note that such factorizations are the same as factorizations up to homotopy of Xs ! Z through Ys. Artin and Mazur [2, Section 4] constructed such factorizations when Z is con- nected. Their argument works even when Z is not connected provided that ß0X ~= ß0Y . Here we use the fact that [X, Z]pro= Hom pro-Ho(SS)(X, Z) by Lemma 8.1._ This proves the result. |__| 19. Non-Cofibrantly Generated Model Structures We prove in this section that the model structure of Section 6 is not cofibra* *ntly generated. The same argument shows that the strict model structure [6] is also * *not cofibrantly generated. See Section 10 for a description of the strict structure* *. We start with a general lemma about cofibrantly generated model structures. A MODEL STRUCTURE FOR PRO-SIMPLICIAL SETS 37 Lemma 19.1. Suppose that a model structure on a category C is cofibrantly gen- erated with a set of generating cofibrations I. Let T be the set of targets of * *maps in I, and let X be any cofibrant object of C not isomorphic to the initial object.* * Then there exists some Y in T not isomorphic to the initial object with a map Y ! X in C. Proof.Let X be a cofibrant object of C. Then X is a retract of another object X* *0, where X0 is a transfinite composition of pushouts of maps in I [10, 14.2.12]. S* *ince there is a map from X0 to X, it suffices to find a map from some object of T to* * X0. Since X is not the initial object, X0 is also not the initial object. Hence X0 * *is a non-trivial transfinite composition of pushouts of maps in I. Let Z ! Y be a ma* *p__ in I occurring in the construction of X0. Then there is a map from Y to X0. * *|__| The next proposition gives a construction of specific pro-sets with special p* *rop- erties. Proposition 19.2.Let F be a small family of pro-sets (i.e., a set of pro-sets) * *not containing the empty pro-set. Then there exists a pro-set X such that for every* * Y in F , there are no maps Y ! X of pro-sets. Proof.Choose an infinite cardinal ~ larger than the size of any of the sets occ* *urring in any of the objects of F . Let S be a set of size ~. Define a pro-set X as follows. Consider the collection of all subsets U of S * *whose complements Uc are strictly smaller than S. Note that this implies that the size of U is ~, but the converse is not true. These subsets form a pro-set, where t* *he structure maps are inclusions. This system is cofiltered because (U \V )c = Uc[* *V c is strictly smaller than S when Uc and V care. Let Y be an object of F . Suppose that there is a map f : Y ! X of pro-sets. Then there exists a t and a map ft,S: Yt! S representing f. Let A be the image * *of ft,S, so A is strictly smaller than S since Ytis strictly smaller than S. Consi* *der the set S - A, which occurs as an object in the system X. Since f is a map of pro-s* *ets, there exists a u t such that the composition Yu ! Yt! S factors through S - A. Since Ytand S -A have disjoint images in S, this is only possible if Yu is the * *empty set. However, Yu cannot be the empty set because Y is not the empty pro-set._By contradiction, the map f cannot exist. |__| Corollary 19.3.There are no cofibrantly generated model structures on pro-spaces for which every object is cofibrant. Proof.We argue by contradiction. Suppose that there exists a cofibrantly genera* *ted model structure for which every object is cofibrant. Let I be the set of genera* *ting cofibrations, and let T be the set of targets of maps in I. Apply ß0 to T to ob* *tain a small family of pro-sets F . Let X be the pro-set constructed in Proposition 19.2. We can think of X as a pro-space by identifying a set with a simplical set of dimension zero. 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May, Simplicial objects in algebraic topology, Van Nostrand Mathemati* *cal Studies, vol. 11, Van Nostrand, 1967. [13]F. Morel, Ensembles profinis simpliciaux et interpr'etation g'eom'etrique d* *u foncteur T, Bull. Soc. Math. France 124 (1996), 347-373. [14]D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, * *Springer Verlag, 1967. [15]D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of * *Math. 100 (1974), 1-79. [16]G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics* *, vol. 61, Springer Verlag, 1978. Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany E-mail address: isaksen@mathematik.uni-bielefeld.de