FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES CHARLES REZK Abstract.We show that homotopy pullbacks of sheaves of simplicial sets o* *ver a Grothendieck topology distribute over homotopy colimits; this generali* *zes a result of Puppe about topological spaces. In addition, we show that inve* *rse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the notion of a quasi-fibration of spaces, and see* *ms to be of independent interest. 1.Introduction Dold and Thom [3] introduced a class of maps called quasi-fibrations. A map f :X ! Y of topological spaces is called a quasi-fibration if for each point y * *2 Y the fiber f-1 (y) is naturally weakly equivalent to the homotopy fiber of f ove* *r y. Thus, quasi-fibrations behave for some purposes of homotopy theory very much li* *ke other types of fibrations; for example, there is a long exact sequence relating* * the homotopy groups of X, Y , and f-1 (y). A notable feature of quasi-fibrations is* * that (as shown by Dold and Thom) quasi-fibrations defined over the elements of an op* *en cover of a space Y can sometimes be "patched" together to give a quasi-fibration mapping to all of Y . In this paper we study a class of maps called sharp maps. In our context, a m* *ap f :X ! Y will be called sharp if for each base-change of f along any map into t* *he base Y the resulting pullback square is homotopy cartesian. We are particularly interested in sharp maps of sheaves of simplicial sets. We shall show that sharp maps of sheaves of simplicial sets have properties analog* *ous to those of quasi-fibrations of topological spaces. In particular, they can be "pa* *tched together", in a sense analogous to the way that quasi-fibrations can be patched together. We give several applications. 1.1. Applications. Let E denote a Grothendieck topos; that is, a category equiv- alent to a category of sheaves on a small Grothendieck site. The category sE of simplicial objects in E admits a Quillen closed model category structure, as was shown by Joyal (unpublished), and by Jardine in [5] and [6]. Let X :I! sE be a diagram of simplicial sheaves indexed on a small category I. We say that such a diagram is a homotopy colimit diagram if the natural map hocolimIX ! colimIX is a weak equivalence of objects in sE, where hocolimde- notes the homotopy colimit functor for simplicial sheaves, generalizing that de* *fined by Bousfield and Kan [1] for simplicial sets. ____________ Date: November 3, 1998. 1991 Mathematics Subject Classification. Primary 18G30; Secondary 18B25, 55R* *99. Key words and phrases. simplicial sheaves, fibrations, homotopy colimits. 1 2 CHARLES REZK Given a map f :X ! Y of I-diagrams of simplicial sheaves, for each object i of Ithere exists a commutative square Xi _____//colimIX (1.2) fi|| || fflffl| fflffl| Y i_____//colimIY and for each morphism ff: i ! j of Ithere exists a commutative square Xi _Xff_//Xj (1.3) fi|| fj|| fflffl|Y fflffl|ff Y i_____//Y j The following theorem essentially says that in a category of simplicial sheav* *es, homotopy pullbacks "distribute" over homotopy colimits. Theorem 1.4. Let f :X ! Y be a map of I-diagrams of simplicial objects in a topos E, and suppose that Y is a homotopy colimit diagram. Then the following two properties hold. (1) If each square of the form (1.2) is homotopy cartesian, then X is a homoto* *py colimit diagram. (2) If X is a homotopy colimit diagram, and each diagram of the form (1.3) is homotopy cartesian, then each diagram of the form (1.2) is also homotopy cartesian. The proof of (1.4) is given in Section 7. This result is well-known when sE i* *s the category of simplicial sets: Puppe [9] formulates and proves a version of the a* *bove result for the category of topological spaces, which can be used to derive (1.4* *) for simplicial sets; see [4, Appendix HL] for more discussion of Puppe's result. Al* *so, Chacholski [2] has proved a result of this type in the category of simplicial s* *ets using purely simplicial methods. As another application we give the following. Let p: E ! E0 be a geometric morphism of Grothendieck topoi, and let p*: E0 ! E denote the corresponding inverse image functor. This functor prolongs to a simplicial functor p*: sE0! s* *E. Theorem 1.5. The inverse image functor p*: sE0! sE preserves homotopy carte- sian squares. The proof of (1.5) is given in Section 5. An example of an inverse image func* *tor is the sheafification functor L2: PshC ! ShC associated to a Grothendieck topology on C. Thus, (1.5) shows in particular that sheafification functors preserve hom* *otopy cartesian squares. 1.6. Organization of the paper. In Section 2 we define sharp maps and state some of their general properties. In Section 3 we recall facts about sheaf theo* *ry and the model category structure on simplicial sheaves. Section 4 gives several use* *ful characterizations of sharp maps of simplicial sheaves, which are used to prove a number of properties in Section 5, as well as the proof of (1.5). In Section 6 we prove a result about how sharp maps are preserved by taking t* *he diagonal of a simplicial object. This result is used in Section 7 to prove a si* *milar fact about how sharp maps are preserved by homotopy colimits; this result is us* *ed FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 3 in turn to give a proof of (1.4). Section 8 does the hard work of showing that * *sharp maps which agree "up to homotopy" can be glued together, thus providing lemmas which were needed for Section 6. Section 9 proves a result about sharp maps in a boolean localization which was needed in Section 4. In Section 10 we prove that in a boolean localization the local fibrations ar* *e the same as the global fibrations, a fact which is used at several places in this p* *aper. 1.7. Acknowledgements. I would like to thank Phil Hirschhorn, Mike Hopkins, Mark Johnson, Brooke Shipley, and Carlos Simpson for their comments on various parts of this work. 2.Sharp maps In this section, we define the notion of a sharp map in a general closed model category, and prove some of its general properties. I learned about the notion * *of a "sharp" map from Mike Hopkins, who was originally led, for different reasons, to formulate the dual notion of a "flat" map. Let M be a closed model category [10], [11]. We say that a map f :X ! Y in M is sharp if for each diagram in M of the form A __i__//A0___//_X | | | | | f| |fflffljfflffl| fflffl| B _____//B0___//_Y in which j is a weak equivalence and each square is a pullback square, the map * *i is also a weak equivalence. It follows immediately from the definition that the cl* *ass of sharp maps is closed under base-change. 2.1. Proper model categories. A model category M is said to be right proper if for each pullback diagram in M of the form __i__// X0 X | | | |f fflffl|jfflffl| Y 0_____//Y such that f is a fibration and j is a weak equivalence, then i is also a weak e* *quiv- alence. The categories of topological spaces and simplicial sets are two well-k* *nown examples of right-proper model categories. There is an dual notion, in which a model category for which pushouts of weak equivalences along cofibrations are weak equivalences is called left proper. A model category is proper if it is both left and right proper. Since the class of fibrations in a model category is closed under base-change* *, we have the following. Proposition 2.2.A model category M is right proper if and only if each fibrati* *on is sharp. 4 CHARLES REZK 2.3. Homotopy cartesian squares. Let P _____//Y | | (2.4) | |g fflffl|fflffl|f X _____//B be a commutative square in M . Say such a square is homotopy cartesian if for some choice of factorizations X ! X0 ! B and Y ! Y 0! B of f and g into weak equivalences followed by fibrations, the natural map P ! X0xB Y 0is a weak equivalence. It is straightforward to show that the choice of factorizations do* *es not matter. Clearly, any pullback square of the form (2.4) in which f and g are already fibrations is homotopy cartesian. Any square weakly equivalent to a homotopy cartesian square is itself homotopy cartesian. Lemma 2.5. In a right proper model category, a pullback square as in (2.4) in which g is a fibration is a homotopy cartesian square. Proof.Choose a factorization pi of f into a weak equivalence i: X ! X0 followed by a fibration p: X0! B. Then we obtain pullback squares j P _____//P_0___//Y (2.6) | | |g | |h | |fflfflifflffl|pfflffl| X _____//X0____//B in which j is a weak equivalence by (2.2), since it is obtained by pulling back* * the weak equivalence i along the fibration h. Thus the square (2.4) is weakly equiv* *alent_ to the right-hand square of (2.6) which is homotopy cartesian. |* *__| The following proposition gives the characterization of sharp maps which was alluded to in the introduction; it holds only in a right proper model category. Proposition 2.7.In a right proper model category, a map g :Y ! B is a sharp map if and only if each pullback square (2.4) is a homotopy cartesian square. Proof.First suppose that g is sharp. As in the proof of (2.5) choose a factoriz* *ation pi: X ! X0! B of f into a weak equivalence i followed by a fibration p, obtaini* *ng a diagram (2.6). Then the right hand square of this diagram is homotopy cartesi* *an by (2.5), and i and j are weak equivalences, since j is the pullback of the weak equivalence i along the sharp map g. Conversely, suppose g is a map such that each pullback along g is a homotopy cartesian square. Given a diagram of pullback squares as in (2.6) in which i is a weak equivalence, it follows that j is also a weak equivalence, since both the right-hand square and the outer rectangle are homotopy cartesian squares which * * __ are weakly equivalent at the three non-pullback corners. Thus g is sharp. * *|__| Example 2.8.The category of topological spaces is a right proper model category. The class of sharp maps of topological spaces includes all Serre fibrations, as* * well as all fiber bundles. Every sharp map is clearly a quasi-fibration in the sens* *e of Dold and Thom [3]. It is not the case that all quasi-fibrations are sharp; inde* *ed, the class of quasi-fibrations is not closed under base change, see [3, Bemerkun* *g 2.3]. I do not know of a simple characterization of sharp maps of topological spaces. FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 5 3. Facts about topoi In this section we recall facts about sheaves and simplicial sheaves. Our main reference for sheaf theory is Mac Lane-Moerdijk [8]. 3.1. Grothendieck topoi. A Grothendieck topos E is a category equivalent to some category ShC of sheaves of sets on a small Grothendieck site C . Among the many properties of a Grothendieck topos E, we note that E has all small limits * *and colimits, and that E is cartesian closed. The internal hom object in E is denot* *ed by Y X. Example 3.2. 1. The category Setis a Grothendieck topos, since it is sheaves on a one-point space. 2. The presheaf category PshC , defined to be the category of functors C op! Set, is the category of sheaves of sets in the trivial topology on C , and* * thus is a Grothendieck topos. 3. The category Sh(T ) of sheaves of sets on a topological space T is a Groth* *en- dieck topos. A geometric morphism f :E ! E0 is a pair of adjoint functors f* :E0AE E :f* such that the left adjoint f* preserves finite limits. The left adjoint f* is c* *alled the inverse image functor, and f* the direct image functor. 3.3. Boolean localizations. Let B be a complete Boolean algebra. Then B, viewed as a category via the partial order on B, has a natural Grothendieck top* *ol- ogy, and hence gives us a Grothendieck topos ShB. (This topos is discussed in m* *ore detail in Section 10.) A Boolean localization of a topos E is a geometric morphism p: ShB ! E such that the inverse image functor p*: E ! ShB is faithful. Example 3.4. 1. The category of sets is its own Boolean localization, since it is equivale* *nt to sheaves on the trivial Boolean algebra. 2. For a category C, let C0 C denote the subcategory consisting of all objec* *ts and all identity maps. Then p: PshC0 ! PshC is a boolean localization, where p*: PshC ! PshC0 is the obvious restriction functor; this is because PshC0 is equivalent to the category of sheaves on the boolean algebra P(ob* *C ), the power set of obC . 3. For a topological space T , let T ffidenote the underlying set T with the * *discrete topology. Then Sh(T ffi) Sh(PT ffi) is a boolean localization of Sh(T );* * the inverse image functor p*: Sh(T ) ! Sh(T ffi) sends a sheaf X to the collec* *tion of all stalks of X over every point of T . Remark 3.5.In each of the examples above, the Boolean localization turned out to be equivalent to a product of copies of Set. However, there exist topoi E which* * do not admit a Boolean localization of this type. Boolean localizations have the following properties. 1. Every Grothendieck topos has a Boolean localization. 6 CHARLES REZK 2. The inverse image functor p* associated to a Boolean localization functor p: ShB ! E reflects isomorphisms, monomorphisms, epimorphisms, colimits, and finite limits. 3. The topos ShB has a "choice" axiom: every epimorphism in ShB admits a section. 4. The topos ShB is boolean; that is, each subobject A X in ShB admits a "complement", namely a subobject B X such that A [ B = X and A \ B = ?. Property (1) is shown in [8, IX.9]. See Jardine [6] for proofs of the other pro* *perties. 3.6. A distributive law. For our purposes it is important to note the following relationship between colimits and pullbacks in a topos E. Proposition 3.7.Let Y :I ! E be a functor from a small category I to a topos, and let A ! B = colimI(i 7! Y i) be a map. Then the natural map colimI(i 7! A x Y i) ! A B is an isomorphism. This proposition says that if an object is pulled back along a colimit diagra* *m, then that object can be recovered as the colimit of the pulled-back diagram. It makes sense to think of this as a "distributive law". In fact, in the special c* *ase in which B = X1 q X2, and A ! B is the projection (X1 q X2) x Y ! X1 q X2 the proposition reduces to the usual distributive law of products over coproduc* *ts: (X1 x Y ) q (X2 x Y ) (X1 q X2) x Y . To prove (3.7), note that it is true if E = Set, and thus is true if E = PshC* * . The general result now follows from the properties of the sheafification functor L2: PshC ! ShC . Remark 3.8.Consider the diagram X defined by i 7! A xB Y i. It is equipped with a natural transformation f :X ! Y with the property that for each ff: i ! j in I the map Xff is the pullback of Y ff along fj. One can formulate the following "converse" to (3.7) which is false. Namely, g* *iven a natural transformation f :X0! Y of I-diagrams such that for each ff: i ! j in* * I the map X0ff is the base-change of Y ff along fj, one may ask whether the natur* *al maps X0i ! AxB Y i are isomorphisms, where A = colimIX0. A counterexample in E = Setis to take Ito be a group G and X0! Y to be any map of non-isomorphic G-orbits. (1.4, part 1) may be viewed as a homotopy theoretic analogue of (3.7). (1.4, * *part 2) may be viewed as a homotopy theoretic analogue of the "converse" to (3.7). 3.9. Simplicial sheaves. We let sE denote the category of simplicial objects in a Grothendieck topos E. Note that sE is itself a Grothendieck topos. The full subcategory of discrete simplicial objects in sE is equivalent to E; thus, we r* *egard E as a subcategory of sE. For any topos E there is a natural functor Set ! E sending a set X to the corresponding constant sheaf. This prolongs to a functor sSet! sE, and we will thus regard any simplicial set as a constant simplicial sheaf. FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 7 3.10. Model category for simplicial sheaves. We will make use of the elegant model category structure on simplicial sheaves provided by Jardine in [6]. We summarize here the main properties of this structure which we need. Let sE deno* *te the category of simplicial objects in a topos E. Let p: ShB ! E denote a fixed boolean localization of E. A map f :X ! Y in sE is said to be 1. a local weak equivalence (or simply, a weak equivalence) if (L2Ex 1 p*f)(b): (L2Ex 1 p*X)(b) ! (L2Ex 1 p*Y )(b) is a weak equivalence for each b 2 B. Here Ex1 :sShB ! sPshB denotes the functor obtained by applying Kan's Ex1 functor [7] at each b 2 B, and L2 denotes the simplicial prolongation sPshB ! sShB of the sheafification functor. 2. a local fibration if p*f(b): p*X(b) ! p*Y (b) is a Kan fibration for each b 2 B. It should be pointed out that local fibrations are not in general t* *he fibrations in the model category structure on sE; but note (3.14). 3. a cofibration if it is a monomorphism. 4. a global fibration (or simply, a fibration) if it has the right lifting pr* *operty with respect to all maps which are both cofibrations and weak equivalences. Note that the definition of local weak equivalence simplifies when E = ShB, s* *ince ShB is its own boolean localization. Furthermore, a map f in sE is a local weak equivalence if and only if p*f in sShB is a local weak equivalence. Theorem 3.11. (Jardine [6], [5]) The category sE with the above classes of cof* *i- brations, global fibrations, and local weak equivalences is a proper simplicial* * closed model category. Furthermore, the characterizations of local weak equivalences, * *local fibrations, and global fibrations do not depend on the choice of boolean locali* *zation. Example 3.12. 1. When sE = sSet this model category structure coincides with the usual one, and local fibrations coincide with global fibrations. 2. For sE = sPshC , a map f :X ! Y is a local weak equivalence, cofibration, * *or local fibration if for each C 2 obC, the map fC :X(C) ! Y (C) is respectiv* *ely a weak equivalence, monomorphism, or Kan fibration of simplicial sets. 3. For sE = sSh(T ), a map f :X ! Y is a local weak equivalence, cofibration, or local fibration if for each point p 2 T the map fp: Xp ! Yp of stalks is respectively a weak equivalence, monomorphism, or Kan fibration of simplic* *ial sets. We also need the following property. Proposition 3.13.[6, Lemma 13(3)] Let E be sheaves on a Grothendieck topos. If fi:Xi ! Yi is a family`of local`weak equivalences in sE indexed by a set I, then the induced map f : i2IXi! i2IYi is a local weak equivalence. We need one additional fact about fibrations in a boolean localization. Proposition 3.14.In the category sShB of simplicial sheaves on a complete bool- ean algebra B, the local fibrations are precisely the global fibrations. The proof of (3.14) is given in Section 10. Finally, we note that if f :E ! E0is a geometric morphism, then the induced inverse image functor f* :sE0 ! sE preserves cofibrations and weak equivalences; 8 CHARLES REZK this is because the composite * p* sE0 f-!sE -! sShB must preserve such if p: ShB ! E is a boolean localization of E. 3.15. Model category for simplicial presheaves. Although we will not make much use of it here, we note that if E = ShC for some Grothendieck site C , then Jardine [6, Thm. 17] constructs a "presheaf" closed model category structure on sPshC related to that on sE (and not to be confused with the "sheaf" model category structure obtained by applying the remarks of the previous section to E = PshC ). In this structure on sPshC , the cofibrations are the monomorphisms, and the weak equivalences are the maps in sPshC which sheafify to local weak equivalences in sE. Furthermore, a map in sPshC is called a local fibration i* *f it sheafifies to a local fibration in sE. The natural adjoint pair sPshC o sE ind* *uces an equivalence of closed model categories in the sense of Quillen; in particula* *r, the homotopy category of sPshC (induced by the presheaf model category structure) * *is equivalent to the homotopy category of sE. Thus, many results stated for sE such as (1.4) carry over to the presheaf model category of sPshC without change. 4. Local character of sharp maps of simplicial sheaves The following theorem provides several equivalent characterizations of sharp maps in sE. There are two types of such statements: (5) and (6) say that sharpn* *ess is a "local condition", i.e., sharpness is detected on boolean localizations, w* *hile (2), (3), and (4) say that sharpness is detected on "fibers", i.e., by pulling back * *to the product of a discrete object and a simplex. Theorem 4.1. Let f :X ! Y be a map of simplicial objects in a Grothendieck topos E. The following are equivalent. (1) f is sharp. (2) For each n 0 and each map S ! Yn in E, the induced pullback square P _______//_X | | | f| fflffl| fflffl| S x [n] _____//Y is homotopy cartesian. (3) For each n 0 there exists an epimorphism Sn ! Yn in E such that the induced pullback square P ________//_X | | | |f fflffl| fflffl| Sn x [n] _____//Y is homotopy cartesian. (4) For each n 0 there exists an epimorphism Sn ! Yn in E such that for each map ffi :[m] ! [n] of standard simplices, the induced diagram of pullback FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 9 squares P _____h_____//_P_0_____//X | | | | | f| fflffl|1xffi fflffl| fflffl| Sn x [m] _____//Sn x [n]____//_Y is such that h is a weak equivalence of simplicial sheaves. (5) For any boolean localization p: ShB ! E, the inverse image p*f :p*X ! p*Y of f is sharp in sShB. (6) There exists a boolean localization p: ShB ! E such that the inverse image p*f :p*X ! p*Y of f is sharp in sShB. Proof. (1) implies (2): This follows from (2.7), and the fact that sE is right prope* *r. (2) implies (3) and (4): Let Sn = Yn. either (3) or (4) implies (5): This will follow from (9.1), since p*: E ! ShB preserves pullbacks and epimorphisms. (5) implies (6): This is trivial, since every E has a boolean localization (3* *.10). (6) implies (1): If p: ShB ! E is a boolean localization, and p*f is sharp, t* *hen since p*: sE ! sShB preserves pullbacks and reflects weak equivalences (3.10),_* *it follows that f is sharp. |__| Remark 4.2.In the case when sE = S, and f :X ! Y a map of simplicial sets, the above theorem implies that the following three statements are equivalent. (1) f is sharp. (2) For each map g :[n] ! Y the pullback square of f along g is homotopy cartesian. (3) For each diagram of pullback squares of the form P ___h___//P_0____//X | | | | | |f fflffl|ffifflffl| fflffl| [m] _____//[n]_____//Y the map h is a weak equivalence. Note that characterization (2) is reminiscent of the definition of quasi-fibrat* *ion of topological spaces. A sharp map to a simplicial set Y induces a "good diagram" indexed by the simplices of Y , in the sense of Chacholski [2]. Remark 4.3.Recall from (3.4) that if E = PshC is a category of presheaves on C , then a suitable boolean localization for E is PshC0 . This implies using part * *(6) of (4.1) that a map f :X ! Y of presheaves on C is sharp if and only if for each object C 2 C the map f(C): X(C) ! Y (C) is a sharp map of simplicial sets. Remark 4.4.Recall from (3.4) that if E = Sh(T ) where T is a topological space, then a boolean localization for E is ShT ffi. This implies using part (6) of (4* *.1) that a map f :X ! Y of sheaves over T is sharp if and only if for each point p 2 T t* *he induced map fp: Xp ! Yp on stalks is a sharp map of simplical sets. 10 CHARLES REZK Remark 4.5.The statement of (4.1) remains true if we replace sE with sPshC equipped with the presheaf model category structure of (3.15), and replace bool* *ean localizations ShB ! E with composite maps ShB ! ShC ! PshC . That this is the case follows easily from the observation that f :X ! Y 2 sPshC is sharp if and only if L2f :L2X ! L2Y 2 sShC is sharp, the proof of which is straightforward. 5. Basic properties of sharp maps of simplicial sheaves In this section we give some basic properties of sharp maps in a simplicial t* *opos. Theorem 5.1. The following hold for simplicial objects in a topos E. P1 Local fibrations are sharp. P2 For any object X 2 sE the projection map X ! 1 is sharp. P3 Sharp maps are closed under base-change. P4 If f is a map such that the base-change of f along some epimorphism is sha* *rp, then f is sharp. P5 If maps fffare sharp for each ff 2 A for some set A, then the coproduct qf* *ff is sharp. P6 If p: E ! E0 is a geometric morphism of topoi, the inverse image functor p*: sE0! sE preserves sharp maps. Proof.Property P1 follows from part (6) of (4.1), the fact that global fibratio* *ns are sharp (2.2), and the fact that local fibrations are global fibrations in a * *Boolean localization (3.14). Property P2 follows immediately from the fact weak equivalences in sE are pre- cisely those maps f such that (L2Ex 1 p*f)(b) is a weak equivalence for each b * *2 B (where p: ShB ! E is a boolean localization), together with the fact that the functor L2Ex 1 p* preserves products. Property P3 has already been noted in Section 2. To prove property P4, consider the pull-back squares Q ______q_____//Q0_______//P____//X | | | | | | |g f| fflffl|1xffi fflffl| fflffl|pfflffl| Cn x [m] ____//_Cn x [n]____//C____//Y where g is sharp and p is an epimorphism. Then q is a weak equivalence since 1 * *x ffi is, whence f is sharp by part (4) of (4.1), since the map Cn ! Yn is an epimorp* *hism in E. To prove property P5, let fff:Xff! Yffbe a collection of sharp maps, and let f = qff2Ifff. Then P5 follows from the fact that a coproduct of weak equivalenc* *es is a weak equivalence (3.13) and using part (4) of (4.1). Property P6 follows easily from part (4) of (4.1), and the fact that inverse_* *image_ functors preserve pullbacks, epimorphisms, and weak equivalences. |_* *_| We can now easily prove (1.5). Proof of (1.5).Recall (3.10) that any homotopy cartesian square is weakly equiv- alent to a pullback square in which all the maps are fibrations. Since fibrati* *ons are sharp by (2.2), the square obtained by applying the inverse image functor p*: E0 ! E is a pullback square in which the maps are sharp by P6, and hence FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 11 is a homotopy cartesian square by (2.7). Since p* preserves weak equivalences_t* *he conclusion follows. |__| Remark 5.2.Parts P1-P5 of (5.1) remain true if we replace sE with sPshC equipped with the presheaf model category of (3.15), for the reasons discussed in (4.5). 6.Diagonal of a simplicial object Let X :op ! sE be a simplicial object in sE; we write [n] 7! X(n) where X(n) 2 sE. The diagonal |X| of X is an object in sE defined by [n] 7! X(n)n. Theorem 6.1. Let p: X ! Y be a map of simplicial objects in sE such that each p(n): X(n) ! Y (n) is sharp, and each square X(n) _____//X(m) | | | | fflffl| fflffl| Y (n)_____//Y (m) is homotopy cartesian. Then |p|: |X| ! |Y | is sharp. We prove this theorem using the following well-known inductive construction of the diagonal of a simplicial object. Namely, |X| colimnFn|X|, where F0|X| = X(0) and each Fn|X| is obtained from Fn-1|X| by a pushout diagram of the form [ X(n) x @[n] Ln-1X x [n] _____//X(n) x [n] Ln-1Xx@[n] | (6.2) | | | | fflffl| fflffl| Fn-1|X|____________________//Fn|X| where Ln-1X denotes the subobject of X(n) which is the union of the images of all degeneracy maps si:Xn-1 ! Xn for 0 i n. 6.3. Colimits on posets of proper subsets. Before going to the proof of (6.1) we collect some facts about colimits of diagrams indexed by the subsets of a fi* *nite set. These facts will also be needed in Sections 8 and 9. If S is a finite set, let PS denote the poset of subsets of S, and let PS den* *ote the poset of proper subsets of S; we regard PS and PS as categories with T ! T 0 if T T 0 S. Given a functor X :PS ! sE and a subset S0 S, we define X|S0:PS0 ! sE to be the restriction of X to PS0 via the formula X|S0(T ) = X(T ) for T S0. We a* *lso speak of the restriction X|S0:P S0! sE to PS0. We say that a functor X :PS ! sE (resp. a functor X :PS ! sE) is cofibrant if for each subset (resp. proper subs* *et) T S the induced map colimPTX|T ! X(T ) is a monomorphism. (The cofibrant functors are in fact the cofibrant objects in a model category structure on the categories of functors PS ! sE and PS ! sE.) Say that S = {1; : :;:n}, and let S0 = {1; : :;:n - 1}. Define X0:P S0! sE by the formula X0(T ) = X(T [ {n}) for T S0. There is a natural map X|S0 ! X0 of diagrams indexed by PS0. 12 CHARLES REZK Proposition 6.4.Suppose X :PS ! sE is a functor. There is a natural pushout square colimPS0X|S0_____//colimPS0X0 | | | | fflffl| fflffl| X(S0) ________//_colimPSX and if f is a cofibrant functor, then both vertical maps in the above square are monomorphisms. Proof.This is a straightforward induction argument on the size of S, using the_* *fact that in a topos, pushouts of monomorphisms are again monomorphisms. |__| Corollary 6.5.Given a cofibrant functor X :PS ! sE such that for all T T 0 the map X(T ) ! X(T 0) is a weak equivalence, the induced map colimPSX ! X(S) is a weak equivalence. Proof.This is proved by a straightforward induction argument on the size_of_S, using (6.4). |__| Corollary 6.6.Given cofibrant functors X; Y :P S ! sE and a map f :X ! Y such that each map X(T ) ! Y (T ) is a weak equivalence, then the induced map colimPSX ! colimPTY is a weak equivalence. Proof.This is proved by induction on the size of S, using (6.4) and the fact_th* *at sE is a left proper model category. |__| 6.7. The proof of the theorem. The object Ln-1X has an alternate description using the above notation. Let S = {1; : :;:n}. Define a functor F :PS ! sE sending T S to X(#T ), and sending i: T ! T 0to the map induced by the simplicial operator oe :[#T 0] ! [#T ] defined by oe(0) = 0 and oe(k) = max(` |* * i(k) `) for 0 < k #T 0. Then Ln-1X = colimPSF . Furthermore, F is a cofibrant functor Proof of (6.1).Each map @[n] ! [n] is mono, as are the maps Ln-1X ! X(n), and the top horizontal arrow in (6.2). The proof is a straightforward induction following the inductive construction of diagonal given above and using (8.1) to* *gether with (6.4). That is, suppose by induction that Fn-1|X| ! Fn-1|Y | is sharp. Using (8.1, 3) one shows that Ln-1X ! Ln-1Y is sharp. Then using (8.1, 2) one shows that the induced map from the upper left-hand corner of (6.2) to the upper left-hand cor* *ner of the corresponding square for Y is sharp. Applying (8.1, 2) to the whole squa* *re (6.2) gives that Fn|X| ! Fn|Y | is sharp. Finally, (8.1, 1) shows that |X|_! |Y* * | is sharp, as desired. |__| Remark 6.8.If f :X ! Y is a map of simplicial objects in sE such that in each degree n the map f(n): X(n) ! Y (n) is a weak equivalence, then one may show by using the above inductive scheme together with (6.6) that |f|: |X| ! |Y | is a * *weak equivalence, since sE is a proper model category and the cofibrations are preci* *sely the monomorphisms. FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 13 7.Homotopy colimits Let X :I ! sE be a diagram of simplicial sheaves. As in [1] the homotopy colimit of X, denoted hocolimIX, is defined to be the diagonal of the simplicial object in sE given in each degree n 0 by a [n] 7! Xi0; i0!...!in where the coproduct is taken over all composable strings of arrows in Iof lengt* *h n. From (3.13) and (6.8) it follows that hocolimIX is a weak homotopy equivalence invariant of X. Let (I # i)denote the category of objects over a fixed object i in I. Given an I-diagram X, one can define an I-diagram eXby eXi = hocolim(I#i)X. Thus, eXi is the diagonal of the simplicial object in sE given by a [n] 7! Xi0: i0!...!in!i There is a natural map Xe! X of I-diagrams, and an isomorphism of simplicial sheaves hocolimIX colimIeX. (This is the construction of [1].) Theorem 7.1. Let f :X ! Y be a map of I-diagrams of simplicial sheaves such that (1) each map fi: Xi ! Y i is sharp for i 2 obI, and Xi ____//_Xj (2) each square || || for ff: i ! j 2 Iis homotopy cartesian. fflffl| fflffl| Y i____//Y j Then (a) the induced map hocolimIf :hocolimIX ! hocolimIY is sharp, and eXi____//_hocolimX (b) for each object i in Ithe square || || is a pull-back square. fflffl| fflffl| eYi____//hocolimY Proof.First, we note that (b) follows without need of the hypotheses (1) and (2* *). This is because for each n 0, the square a a Xi0 _____// Xi0 i0!...!in!i i0!...!in | | | | a fflffl| a fflffl| Y i0____// Y i0 i0!...!in!i i0!...!in is a pullback square by the distributive law (3.7), and taking diagonals of bi- simplicial objects commutes with limits. 14 CHARLES REZK To show (a), we consider the square a a Xi0 _____// Xj0 i0!...!in j0!...!jm | | | | a fflffl| a fflffl| Y i0_____// Y j0 i0!...!in j0!...!jm where the horizontal arrows are induced by a map ffi :[m] ! [n] 2 . The vertical arrows are sharp by (5.1, P5), and the square is homotopy cartesian using (7.2)* * . (In fact, the square is a pullback square except when ffi is a simplicial opera* *tor for which ffi(0) 6= 0, in which case the square is only homotopy cartesian.) The_re* *sult then follows from (6.1). |__| Lemma 7.2. In sE, an arbitrary coproduct of homotopy cartesian squares is ho- motopy cartesian. Proof.A coproduct of weak equivalences is a weak equivalence by (3.13), and a coproduct of pullback squares is a pullback square by the distributive law (3.7* *). Thus it suffices to factor the sides of each square into a weak equivalence fol* *lowed by a fibration and demonstrate the result for the resulting pullback squares; s* *ince fibrations are sharp (2.2), the coproduct of sharp maps is sharp (5.1, P5), and* * __ pullbacks along sharp maps are homotopy cartesian, the result follows. * *|__| Proof of (1.4).To prove (1), choose a factorization colimIX -j!W 0p-!colimIY such that j is a weak equivalence and p is sharp (e.g., a fibration). Define an I-diagram X0 by X0i = W 0 x Y i; by the distributive law (3.7) we see that colimY colimIX0 W 0. Note also that the induced map Xi ! X0i is a weak equivalence, since p is sharp and by the hypothesis that each square (1.2) is homotopy carte* *sian. In the diagram __i__// ____//_ colimeX ~ colimfX0 colimeY |k| || `|~| fflffl|j fflffl|p fflffl| colimX __~__//colimX0___//_colimY the map p is sharp and the indicated maps are weak equivalences; that i and ` a* *re weak equivalences follows from the homotopy invariance of homotopy colimits and the hypothesis that Y is a homotopy colimit diagram. Thus to show that k is a weak equivalence, and hence that X is a homotopy colimit diagram, it suffices to show that the right-hand square is a pull-back square. Since each X0i is defined to be the pullback of colimX0! colimY along a map Y i ! colimY , we see that eXi is the pullback of colimX0 along the composite m* *ap eYi ! Y i ! colimY . The assertion that the right-hand square is a pullback squ* *are now follows using the distributive law (3.7). To prove (2), choose a factorization of f :X ! Y into X -j!X0 p-!Y , in which j is an object-wise weak equivalence and p is an object-wise fibration, and hen* *ce FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 15 object-wise sharp. Then the square fX0i____//_colimIfX0 | | | | fflffl| fflffl| eYi_____//colimIeY is homotopy cartesian by (7.1), and since X is by hypothesis a homotopy colimit diagram it follows that this square is weakly equivalent to (1.2), and we_get t* *he desired result. |__| 8. Lemmas on sharp maps of special diagrams In this section we show that for special kinds of maps of diagrams, the induc* *ed map of colimits is sharp. These results were the key element of the proof of (6* *.1). Proposition 8.1.Let I denote a small category, and p: X ! Y a map of I- diagrams in sE. Suppose that pi: Xi ! Y i is sharp for each i 2 obI, and that Xi _Xff_//Xj pi|| pj|| fflffl|Y fflffl|ff Y i_____//Y j is a homotopy cartesian square for each ff: i ! j in I. Then under each of the following cases (1)-(3), the induced map colimIX ! colimIY is sharp, and for each i 2 obI, the square Xi ____//_colimIX | | | | fflffl| fflffl| Y i_____//colimIY is homotopy cartesian. (1) Iis the category obtained from the poset N of non-negative integers, and e* *ach map X(n) ! X(n + 1) and Y (n) ! Y (n + 1) is a monomorphism. (2) I= (i1 -ffi0 fi-!i2) and Xfi and Y fi are monomorphisms. (3) Iis the category obtained from the poset PS of proper subsets of a finite * *set S, and X and Y are cofibrant diagrams in the sense of (6.3). Lemma 8.2. Consider a countable sequence of maps over B Y (0) i0-!Y (1) i1-!Y (2) i2-!: :!:B such that each in is a trivial cofibration, and each map qn :Y (n) ! B is sharp. Then the induced map q :colimnY (n) ! B is sharp. Proof.Given a map f :A ! B, consider the pullbacks X(n) = Y (n) xB A. By the distributive law (3.7), colimnX(n) = colimnY (n)xB A. Since each map Y (n) ! B is sharp and each in is a weak equivalence, it follows that each X(n) ! X(n + 1) is a weak equivalence and thus a trivial cofibration. Thus the composite X(0) ! colimnX(n) is a trivial cofibration, and so base-change of q along f yields_a h* *omo- topy cartesian square. |__| 16 CHARLES REZK Proof of part 1 of (8.1).Let X0(n) = Y (n)xcolimY (n)colimX(n), whence we have that colimX0(n) colimX(n) by the distributive law (3.7). It suffices to show (1) that each map X0(n) ! Y (n) is sharp, and (2) that each map X(n) ! X0(n) is a weak equivalence. ` This is because (1), together with (4.1, P4) and the fact that nY (n) ! colim* *Y (n) is epi, implies that colimX(n) ! colimY (n) is sharp, and (2) then demonstrates that the appropriate squares are homotopy cartesian. Let X(n; m) = Y (n) xY (m)X(m) for m n. Then X0(n) colimmX(n; m) by the distributive law. We have that X(n; n) X(n), and each map X(n; m) ! X(n; m + 1) is a weak equivalence since X(m + 1) ! Y (m + 1) is sharp. Thus X(n) ! colimmX(n; m) X0(n) is a weak equivalence, proving (2). Claim (1) * *__ follows from (8.2) applied to the sequence X(n; m) over Y (n). |* *__| The following lemma describes conditions under which one may "glue" an object onto a sharp map and still obtain a sharp map. Lemma 8.3. Let X __i__//_Y f || g|| fflffl| fflffl| X0 _____//Y 0 p || q|| fflffl| fflffl| A _____//_B be a commutative diagram such that the top square is a push-out square, p, pf, * *and qg are sharp, f is a weak equivalence, and either i or f is a monomorphism. Then q is also sharp. Proof.It suffices by (2.7) to show that every base-change of q along a map U ! B produces a homotopy cartesian square. Since qg is sharp it suffices to show that U x g :U x Y ! U x Y 0 B B B is a weak equivalence. Via the pushout square U x X ______//U x Y B UxBi B UxBf|| || fflffl| fflffl| U x X0 ____//_U x Y 0 B B in which either the top or the left arrow is a cofibration, we see that it suff* *ices to show that U xB f is a weak equivalence, since sE is a left-proper model categor* *y. In fact, U xB f (U xB A) xA f; that is, U xB f is a base-change of f along a map into A. Thus since p and pf are sharp, this base-change of f is a weak_ equivalence, as desired. |__| We have need of the following peculiar lemma. FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 17 Lemma 8.4. In a Grothendieck topos E consider a diagram of the form A //___//_X |p| || fflffl| fflffl| A0 //___//X0 | |q | | fflffl| fflffl| B //___//_Y in which the horizontal arrows are mono, the top square is a pushout square, and the large rectangle is a pullback rectangle. Then the bottom square is also a p* *ullback square. Proof.It suffices to show that the lemma holds in a Boolean localization of E. In this case every subobject has a complement, so we may write X = A q C, X0 = A0q C0, and Y = B q D. To show that the lower square is a pullback, it suffices to show that q(C0) D. Since the top square is a pushout, p(C) = C0, a* *nd__ since the big rectangle is a pullback, qp(C) D, producing the desired result. * * |__| Proof of part 2 of (8.1).We have a diagram of the form X1 oo___X0 //i__//X2 (8.5) p1|| p0 || p2|| fflffl| fflffl|jfflffl| Y1 oo____Y0//___//_Y2 where pn is sharp for n = 0; 1; 2, each square is homotopy cartesian, and i and* * j are mono. We must show that the induced map X12 ! Y12 of pushouts is sharp, and that each square Xn ____//_X12 (8.6) pn|| p12|| fflffl| fflffl| Yn _____//_Y12 is homotopy cartesian for n = 0; 1; 2. We prove the claim by proving it for the following cases: (a) under the additional hypothesis that both of the squares in (8.5) are pull* *back squares, (b) under the additional hypothesis that the right-hand square in (8.5) is a p* *ull back square, and (c) under no additional hypotheses. In case (a), each square of the form (8.6) is necessarily a pullback square s* *ince i and j are mono; this can be seen by passing to a boolean localization, in whi* *ch case X2 X0 q X02and Y2 Y0 q Y20so that X12 X1 q X02and Y12 Y1 q Y20. Thus p12:X12! Y12must be sharp using (4.1, P4), since the pullback of p12along the epimorphism Y1 q Y2 ! Y12 is sharp. 18 CHARLES REZK In case (b), we let X00= Y0 x X1 and X02= X00[ X2, obtaining a diagram of Y1 X0 the form X0 //i__//X2 ~ || ~ || fflffl| fflffl| X1 oo___X00//___//X02 p1|| p00|| p02|| fflffl| fflffl|jfflffl| Y1 oo____Y0//___//_Y2 The map p00is sharp since it is a base-change of the sharp map p1. The map p02is sharp by (8.3) since i is mono. The lower right-hand square is a pullback square by (8.4). Then the claim reduces to case (a), since X1 [X00X02 X12. In case (c), let X00= Y0 x X2 and X01= X1 [ X00, obtaining a diagram of the Y2 X0 form X1fflfflX0oo_fflffl ~|| ~|| fflffl| fflffl|i0 X01oo___X00//___//X2 p01|| p00|| p2|| fflffl| fflffl|jfflffl| Y1 oo____Y0//___//_Y2 The map p00is the base-change of a sharp map p2 and hence is sharp; the map p01* *is sharp by (8.3) (note that X0 ! X00is mono). Thus the claim reduces to case_(b), since X01[X00X2 X12. |__| Proof of part 3 of (8.1).Let S = {1; : :;:n}; we prove the result by induction * *on n. The cases n = 0; 1 are trivial, and case n = 2 follows from (8.1, part 2). For a set T , as in (6.3) let PT denote the poset of proper subsets of T as i* *n (6.3). Then (6.4) provides a pushout square colimPS0X|S0_____//fflfflcolimPS0X0fflffl | | | | fflffl| fflffl| X(S0) ________//_colimPSX in which the vertical arrows are mono; here S0= {1; : :;:n - 1}. There is a sim* *ilar diagram for Y . One now deduces the result by induction on the size of S, apply* *ing (8.1, part 2) to the above square to carry out the induction step. Note that in order to apply the induction step, we need to know that the squa* *re colimPS0X|S0_____//colimPS0X0 | | | | fflffl| fflffl| colimPS0Y |S0____//colimPS0Y 0 FIBRATIONS AND HOMOTOPY COLIMITS OF SIMPLICIAL SHEAVES 19 is homotopy cartesian. This follows by induction from the fact that each of the squares in colimPS0X|S0_____//X(S0)____//X(S)oo___colimPS0X0 | | | | | | | | fflffl| fflffl| fflffl| fflffl| colimPS0Y |S0____//Y (S0)___//Y (S)oo__colimPS0Y 0 are homotopy cartesian. |___| 9.Sharp maps in a boolean localization In this section we go back to prove the results needed in the proof of (4.1) * *on sharp maps in a boolean localization. In the following B denotes a complete boo* *lean algebra. Proposition 9.1.Let f :X ! Y be a map in sShB. The following are equivalent. (1) f is sharp. (2) For all n 0 and all Sn ! Yn in ShB the induced pullback square P ________//_X | | | |f fflffl| fflffl| Sn x [n] _____//Y is homotopy cartesian. (3) For each n 0 there exists an epimorphism Sn ! Yn in ShB such that for each map ffi :[m] ! [n] of standard simplices, the induced diagram of pullback squares P _____h_____//_P_0_____//X | | | | | f| fflffl|1xffi fflffl| fflffl| Sn x [m] _____//Sn x [n]____//_Y is such that h is a weak equivalence. Let fl be an ordinal, viewed as a category. Given a functor X :fl ! sE such that colimff 2|B|and let fl be the smallest ordinal of car* *dinality c. Then for each b 2 B the object yb 2 ShB is small with respect to fl. That is, given a functor X :fl ! ShB, any map yb ! colimff