HYPERCOVERS IN TOPOLOGY DANIEL DUGGER AND DANIEL C. ISAKSEN Abstract.We show that if U* is a hypercover of a topological space X then the natural map hocolimU* ! X is a weak equivalence. This fact is used to construct topological realization functors for the A1-homotopy theory* * of schemes over real and complex fields. 1.Introduction Let X be a topological space, and let U = {Ua} be an open cover of X. From this data one may build the ~Cech complex ~C(U)*, which is the simplicial space ` oo___` oo___` Ua0oo___ Ua0a1 oo___ooUa0a1a2._. . Here Ua0...an= Ua0\. .\.Uan, and the face maps are obtained by omitting indices_ we have chosen not to draw the degeneracies for typographical reasons. Segal [S1] proved that if X has a partition of unity subordinate to U then the map |C~(U)*| ! X is a homotopy equivalence, where |- | denotes geometric realizatio* *n. Our first goal in this paper is to generalize this result to the following theo* *rem. Theorem 1.1. For every open cover U of X, the natural map hocolim~C(U)* ! X is a weak equivalence. There are two steps in the argument. First, we prove that |C~(U)*| ! X is a weak equivalence for arbitrary open covers. It is possible to deduce this from * *Segal's result, making use of the fact that weak equivalences are detected by spheres, * *and spheres always have partitions of unity. But instead of going this route we giv* *e a proof that avoids Segal's theorem completely, and is quite elementary. The second step is to deal with the difference between |C~(U)*| and hocolim~C* *(U)*. For any simplicial object W* in a model category, there are general criteria fo* *r when its geometric realization agrees with its homotopy colimit (cf. [H , Th. 19.6.4* *]); un- fortunately these criteria apply only when the objects Wn are all cofibrant, and we are definitely not assuming that the open sets Ua and their intersections are cofibrant. To get around this we prove a curious theorem (given in Appendix A) that when computing homotopy colimits for topological spaces one never has to worry about this cofibrancy issue. Strange, but true. The main goal of this paper is generalizing Theorem 1.1 so that it applies to `hypercovers', rather than just ~Cech covers. These are defined in detail in Se* *ction 4, but for now we will just give an intuitive definition. An open hypercover of* * a space X is a simplicial space U* such that ____________ Date: November 25, 2001. 1991 Mathematics Subject Classification. 55U35, 14F20, 14F42. Key words and phrases. hypercover, homotopy colimit, geometric realization, * *motivic homo- topy theory. The second author was supported by an NSF Postdoctoral Research Fellowship. 1 2 DANIEL DUGGER AND DANIEL C. ISAKSEN (1)Each Un is a disjoint union of open subsets of X, (2)The spaces appearing in U0 are an open cover of X, (3)The spaces in U1 cover the double intersections of those in level 0, (4)The spaces in U2 cover the triple intersections of those in level 1, and so * *on. Of course making sense of (4)_especially the `and so on' part_requires a certain amount of bookkeeping, which is why we are postponing the formal definition. But the essence is that hypercovers are like ~Cech complexes except that instead of* * taking the double intersections at level 1 we may refine them further, and we may cont* *inue this refining process at each level. Our second main result is then Theorem 1.2. If U* is an open hypercover of a space X, then the natural map hocolimU* ! X is a weak equivalence. This result could almost be considered folklore since everyone immediately ag* *rees it's true, but a proof seems to be missing from the literature. One might consi* *der tackling it by appealing to the Whitehead theorem, proving an isomorphism on fundamental groupoids and homology with local coefficients. This is the approach taken in [F , Prop. 8.1] in the related context of 'etale hypercovers, but this* * is messy and obscures in computation the underlying geometric explanation of the theorem. In the case of topological spaces, the isomorphism on fundamental groupoids was the subject of the paper [RT ] (although they only dealt with ~Cech complexes, * *not hypercovers). The approach we take here, on the other hand, is very elementary. The idea is to reduce to the case of ~Cech covers in a clever way. Our interest in these results arose from attempts to understand topological r* *e- alization functors in the A1-homotopy theory of schemes [MV ]. Given an algebra* *ic variety X defined over C, there is an associated topological space X(C) obtaine* *d by giving X the analytic topology. Of course this should extend to a map of `homot* *opy theories' from the Morel-Voevodsky category Spc(C) to the category of topologic* *al spaces. In [MV ] this extension is only provided at the level of homotopy categ* *ories, but we are interested in extending it to the model category level. The key fact needed to make this work is precisely Theorem 1.2. This is worked out in detail* * in Section 5, following the basic program of [I] (also outlined in [D2 , Rem. 8.2]* *). We also prove that taking analytic spaces for schemes defined over R induces a Qui* *llen map from Spc(R) to Z2-equivariant topological spaces. Finally, we give in this paper several interesting corollaries to Theorem 1.2* *. On the whole these seem too disparate to recount in the introduction, but as an ex* *ample let us mention two of them. We refer the reader to Sections 3 and 4 for more re* *sults like these. Corollary 1.3. Let E ! B be any map which is locally split (for example, a covering space), and form the associated ~Cech complex ~C(E)* given by C~(E)n := En+1B= E xB E xB . .x.BE. Then the natural map hocolim~C(E)* ! B is a weak equivalence. Corollary 1.4. Let U be an open cover of a space X with the property that every finite intersection Ua0...anis covered by other elements of U. Form the diagram consisting of all the Ua's and all the inclusions between them. Then the homoto* *py colimit of this diagram is weakly equivalent to X. The first corollary is an immediate consequence of Proposition 4.10, and the second is restated and proved as Proposition 4.6(c). HYPERCOVERS IN TOPOLOGY 3 Using open covers to give homotopy decompositions for spaces, or to detect we* *ak equivalences, is of course a classical topic. In addition to [S1] it is worthwh* *ile to mention [Mc1 ], [Mc2 ], and [Dk ]. Hypercovers were invented by Verdier in [SGA* *4 , Expose V, Sec. 7], where they were used as a way of computing sheaf cohomology in arbitrary Grothendieck topologies. We would like to express our thanks to Bill Dwyer, Phil Hirschhorn, Michael Mandell, and Jeff Smith for several useful conversations about these results. 1.5. Notation, terminology, and other annoyances. We assume that the reader is familiar with homotopy colimits, and in a few places also with the th* *eory of model categories. The original reference for the latter is [Q ], but we gene* *rally follow [H ] in notation and terminology ([Ho ] is also a good reference). Regar* *ding homotopy colimits, [H ] uses `hocolimD' to denote the result of applying a cert* *ain explicit formula to any diagram D. This has the disadvantage that the resulting object has the correct homotopy type only when the diagram consists entirely of cofibrant objects. We instead adopt the position that `hocolimD' should always denote the correct homotopy-invariant construction: it is obtained by first app* *lying cofibrant-replacement to the objects in the diagram, and only then using the us* *ual explicit formulas. In model-theoretic terms, homotopy colimit is the left deri* *ved functor of the ordinary colimit functor, when the category of diagrams is given* * the projective model structure (see below). Having made the previous point, we now get to say that for topological spaces* * it isn't really necessary. This is definitely a non-standard fact, but we've banis* *hed it to Appendix A so it won't distract the reader from the general theme of the pap* *er. On the other hand, it is a useful result and we'd like to call the reader's att* *ention to it: when taking homotopy colimits for diagrams of topological spaces one doesn't first have to make all the spaces involved cofibrant. The usual formulas are al* *ready homotopy-invariant. We review one last piece of machinery, used often in the body of the paper. Given a small category I, recall that there is a model structure on the categor* *y of diagrams sSetIsuch that a map is a weak equivalence (resp., fibration) if it is* * so in every spot of the diagram [H , Sec. 13.8]. We call this the projective model st* *ructure on sSetI, and the cofibrant diagrams have the property that the homotopy colimit and ordinary colimit are weakly equivalent. Finally, some notation: Throughout this paper our open covers U = {Ua} are always indexed by a set A. In particular, we are allowing the possibility that Ua = Ua0for different values a 6= a0. For every finite set oe = {a0, . .,.an} * *in A, we'll write Uoeor Ua0...anfor Ua0\ . .\.Uan. Also, once and for all we fix our * *model for n as the subset of Rn+1 consisting of (n + 1)-tuples t = (t0, . .,.tn) suc* *h that 0 ti 1 for all i and ni=0ti = 1. The symbol Top denotes the category of all topological spaces_we don't assume any hypotheses like compactly-generated. 2. ~Cech complexes The purpose of this section is to prove the following: Theorem 2.1. For any open cover U of a topological space X, the natural map ß : |C~(U)*| ! X is a weak equivalence. We start by recalling the following result and its corollary: 4 DANIEL DUGGER AND DANIEL C. ISAKSEN Proposition 2.2 (Gray). Let f :X ! Y be a map of spaces and let U and V form an open cover of Y . Suppose that the induced maps f-1 U ! U, f-1 V ! V, and f-1 (U \ V ) ! U \ V are all weak equivalences. Then X ! Y is also a weak equivalence. This is proven (in more generality) in [Gr , 16.24], using an elegant small-s* *implices argument. With enough technology it can also be done by a Whitehead-type theo- rem: it's easy to see that X ! Y is an isomorphism on ß0, a souped-up van Kampen theorem yields the isomorphism on ß1, and for homology with local coefficients * *one uses the Mayer-Vietoris exact sequence. Gray's argument is much nicer, though. Corollary 2.3 (May). Let f :X ! Y be a map of spaces and let U = {Ua} be an open cover of Y . Suppose that f-1 Uoe! Uoeis a weak equivalence for every fini* *te set oe of indices. Then X ! Y is also a weak equivalence. May deduces the generalization by a quick application of Zorn's Lemma [M2 , Cor. 1.4]: look at the set of all opens W such that f-1 (W \ Uoe) ! W \ Uoeis a weak equivalence for all oe, including oe = ;. This set has a maximal element, * *and Gray's result shows it must be X. In an earlier paper McCord proved a more gene* *ral version of this result [Mc1 , Th. 6], but the proof is quite a bit more complic* *ated. Proof of Theorem 2.1.Given any open set V in X, the space ß-1(V ) is homeomor- phic to the space |C~(U0)*|, where U0 is the open cover {Ua \ V } of the space * *V . This definitely uses the fact that V is open. We want to consider the maps ß-1(Uoe) ! Uoe, but in this case the cover U0of Uoeactually contains the whole space Uoeas one of its elements. From the follow* *ing lemma we know that under this condition |C~(U0)*| ! Uoeis a weak equivalence; so by Corollary 2.3 the map |C~(U)*| ! X is a weak equivalence as well. Lemma 2.4. Let U be an open cover of X such that Ub = X for some index b. Then the natural map |C~(U)*| ! X is a weak equivalence (in fact, a homotopy equivalence). Proof.There is a section Ø: X ! |C~(U)*| obtained from the map Ub 0 ! |C~(U)*| and the identification Ub = X. We only need to show that Øß is homotop* *ic to the identity. Let ~C(U)*xI be the simplicial space obtained by crossing all the levels of ~* *C(U)* with the unit interval. Then |C~(U)* x I| is the quotient " # a Ua0...anx n x I = ~ a0...an where the relations are the usual ones, not affecting the I factor at all. Defi* *ne a map |C~(U)* x I| ! |C~(U)*| in the following way. Take an element (x, t0, . .,.* *tn, s) where x belongs to Ua0...anand (t0, . .,.tn) belongs to n, and send it to the * *element (x, 1 - s, st0, . .,.stn) in the factor Uba0...an n+1. This definition respec* *ts the various identifications. Now, there is also an obvious map f :|C~(U)* x I| ! |C~(U)*| x I induced by sending (x, t, s) to ((x, t), s). We claim that this is a homeomorphism, thereb* *y giving us a homotopy |C~(U)*| x I ! |C~(U)*| between Øß and the identity. The reason f is a homeomorphism is just because geometric realization and crossing with I are both left adjoints, and the right adjoints are easily seen to commute. It is im* *portant HYPERCOVERS IN TOPOLOGY 5 that I and n are locally compact Hausdorff so that the relevant mapping spaces with compact-open topologies have the correct adjointness properties. 2.5. Connection with Segal's results. To close this section we make the con- nection between our Theorem 2.1 and the result proven in [S1]. Segal doesn't explicitly deal with ~Cech complexes, but the objects he deals with turn out to* * be homeomorphic to them. This connection will be needed later on. Let A be the indexing set for a cover U. We have already introduced the ~Cech complex ~C(U)*, but if A is given an ordering we may also consider the ordered ~Cech complex ~Co(U)* which is often easier to work with. This is the simplicial ` space given by C~o(U)n = a0...anUa0...an, where the coproduct ranges over all ordered multi-indices in A. That is, we only consider multi-indices for which a0 a1 . . .an. Note that there is an inclusion of simplicial spaces ~Co(U)* ! ~C* *(U)*. Proposition 2.6. The map C~o(U)* ! C~(U)* induces a homotopy equivalence |C~o(U)*| ! |C~(U)*|. Proof.For any (not necessarily ordered) multi-index a0. .a.n, there is a canoni* *cal reordering aoe0. .a.oensuch that aoe0 . . .aoen. If ai = aj for some i < j, th* *en always choose oei < oej. This allows us to define an inverse map |C~(U)*| ! |C~* *o(U)*|. If (x, t) is an element of Ua0...an n, then send (x, t) to the element (x, oe* *t) of Uaoe0...aoen n, where oet is defined by (oet)i= toei. One composition is the equal to the identity. It remains to construct a homot* *opy H : |C~(U)*| x I ! |C~(U)*| between the other composition and the identity. As in the proof of Lemma 2.4, we use the space |C~(U)* x I| rather than |C~(U)*| x* * I. We define H as follows: An element (x, t) of Ua0...an n is equivalent in |C~(* *U)*| to the element (x, t0, . .,.tn, 0, . .,.0) of Ua0...anaoe0...aoen 2n+1. Also* *, (x, oet) is equivalent in |C~(U)*| to the element (x, 0, . .,.0, toe0, . .,.toen) of Ua0...* *anaoe0...aoen 2n+1. Define H((x, t), s) to be the element (x, st0, . .,.stn, (1 - s)toe0, . .,.(1 - s)toen) of Ua0...anaoe0...aoen 2n+1. Proposition 2.7. Let U be an open cover of a space X indexed by a set A. Consid* *er the realization of the simplicial space a [n] 7! Uoen, oe0 ... oen where the coproduct is indexed by chains of nonempty, finite subsets of A. This realization is homeomorphic to the realization |C~o(U)*| of the ordered ~Cech c* *omplex and is homotopy equivalent to |C~(U)*|. The realization in the above proposition is the object considered in [S1]. T* *he ordered ~Cech complex is another construction of the same space, which for us s* *eems somewhat easier to work with. One disadvantage, of course, is that it is not na* *tural: a total ordering on A must be chosen to begin with. Proof.The second claim follows from the first claim and Proposition 2.6. For the first claim, it is convenient to use a slightly unusual construction * *of |C~o(U)*|. When forming the geometric realization, instead of forming Cartesian products with k we instead form products with sd k; since they are homeomor- phic it doesn't matter which one we use. Given this, the key observation is tha* *t we 6 DANIEL DUGGER AND DANIEL C. ISAKSEN can coordinatize sd k in the following way: assuming that the vertices of k are labelled by the numbers 0, . .,.k in the usual way, a point on sd k is represen* *ted uniquely by a chain of proper inclusions oe0 . . .oej of subsets of {0, . .,.* *k} together with an element t of j. Essentially, the chain of subsets determines * *in which sub-simplex the point lies, and then t gives local coordinates inside that sub-simplex. Using this coordinate scheme, we can write down maps in both directions be- tween"the two realizations# " # a a Uoenx n = ~ and Ua0...akx sd k = ~ . oe0 ... oen a0 ... ak For instance, let's give the map from left to right. Using degeneracy relation* *s, a point p in the left space can be represented by a chain of proper inclusions oe0 . . .oen, a point x of Uoen, and an element t of n. Let a0, a1, . .,.ak* * be the ordered list of elements of oen. The chain oe* together with t defines a p* *oint s in sd k, and so we map p to the pair (x, s). It is easy to see that this map * *is well-defined and continuous, and just as easy to write down its inverse. In the case that {Ua} admits a partition of unity {_a} it is fairly easy to s* *ee that the map ß :|C~o(U)*| ! X admits a section: First, a point x of X has a neighborhood which intersects the support of _a only for finitely many indices a = a0, . .,.an. The section Ø sends x to the point of |C~o(U)*| represented by* * (x, t) in Ua0...an n where ti = _ai(x). One has to check that Ø is continuous (use the local-finiteness of the partition of unity), and that Øß ' id via a straigh* *t-line homotopy. See Proposition 4.1 of [S1]. 3.Passing to homotopy colimits The results of the previous section all concerned geometric realizations. In * *this section we translate these into results about various homotopy colimits. In ge* *n- eral, there is a `Reedy cofibrancy' condition on simplicial spaces which guaran* *tees that geometric realization and homotopy colimit agree. Unfortunately our ~Cech complexes are not Reedy cofibrant, due to the fact that the open sets appearing in them are not necessarily cofibrant spaces. However, Theorem A.8 shows that in the category of topological spaces this cofibrancy issue is unimportant: homoto* *py colimits can be computed naively, without first making things cofibrant. This f* *act saves the day. Theorem 3.1. If U is an open cover of a space X, then the natural map hocolim~C(U)* ! X is a weak equivalence. Proof.By Theorem A.8, we can compute the homotopy colimit in the Strom model category. In this model structure the ~Cech complex is Reedy cofibrant (it has * *free degeneracies in the sense of Definition A.4), and so the realization already ha* *s the correct homotopy type. Theorem 2.1 now gives the result. Here are several alternative formulations: Proposition 3.2. Let A be an indexing set for the cover U, and let PA denote the partially ordered set consisting of all nonempty finite subsets of A. Let de* *note HYPERCOVERS IN TOPOLOGY 7 the functor PopA! Top which sends oe to Uoe. Then the natural map hocolim ! X is a weak equivalence. Proof.To construct hocolim we can take the realization of the simplicial repla* *ce- ment for (by Theorem A.8 we don't need to first make the spaces cofibrant). That is, we take the realization of the simplicial space a [n] 7! Uoen, oe0 ... oen where the coproduct is indexed by chains of nonempty, finite subsets of A. Now Proposition 2.7 tells us that this realization is homotopy equivalent to |C~(U)* **|, so Theorem 2.1 finishes the proof. Corollary 3.3. Let PU denote the subcategory of Top whose objects are the open sets Ua belonging to U together with their finite intersections; the morphisms * *are the inclusions of open subsets of X. Let denote the inclusion functor PU ! To* *p. Then the natural map hocolim ! X is a weak equivalence. Proof.Consider the obvious functor F :PopA! PU sending oe to Uoe. We will show that it is homotopy cofinal, so pick an object V in PU and look at the undercat* *egory (V # F ). It suffices to show that any map K ! N(V # F ) can be extended over the cone on K, as K ranges over all finite simplicial sets. Every n-simplex s * *in K maps to a chain of open sets V ! Uoe0! Uoe1! . .!.Uoenin (V # F ). Since K has only finitely-many non-degenerate simplices, only finitely-many of * *the Uoewill ever appear. Define ~ to be the union of all the oei arising from the m* *ap K ! N(V # F ). To extend the map over CK, we send the cone on s to the (n+1)- simplex corresponding to the chain V ! U~ ! Uoe0! Uoe1! . .!.Uoen. The following corollary was shown to us by Bill Dwyer. Let (Top # X)U denote the full subcategory`of (Top # X) consisting of all maps Z ! X that factor thro* *ugh the space E = aUa. Let : (Top # X)U ! Top be the canonical functor sending Z ! X to Z. We would like to claim that the homotopy colimit of the diagram is weakly equivalent to X, but (Top # X)U is not a small category. So we choose* * an infinite cardinal ~ larger than the size of E and restrict to the spaces Z that* * have at most ~ elements. As the proof of the corollary indicates, the weak homotopy type of hocolim is independent of the choice of ~, as long as ~ is sufficientl* *y large. Corollary 3.4. For the functor : (Top # X)U ! Top defined above, the natural map hocolim ! X is a weak equivalence. Proof.The nth level of the ~Cech complex is EnX:= E xX E xX . .x.XE (n factors). Let's write C = (Top # X)U , for brevity. So we have the functor F : op ! C giv* *en by [n] 7! EnX. The composition op ! C ! Topis just ~C(U)*. Because of Theorem 3.1, it will be enough to show that F is homotopy cofinal. For this we pick an object z : Z ! X in C and show that (z # F ) is contracti* *ble. This undercategory is isomorphic to the category of simplices of K, where K is * *the simplicial set sending [n] to Hom C(z, EnX). But observe that Hom C(z, EnX) is * *equal to T nwhere T = Hom C(z, EX ). So K is the simplicial set [n] 7! T n, which is contractible because T is nonempty (using the fact that z :Z ! X factors through E). Thus (z # F ) is isomorphic to the category of simplices of a contractible simplicial set, and therefore has a contractible nerve. 8 DANIEL DUGGER AND DANIEL C. ISAKSEN Corollary 3.5 (Small simplices theorem). Let Sing UX denote the simplicial set whose n-simplices are the maps n ! X that factor through some Ua. Then Sing UX ! Sing X is a weak equivalence. Proof.Let PA be the category defined in Proposition 3.2, where A is the indexing set for the cover. Consider the diagram : PopA! sSetdefined by (oe) = Sing (U* *oe). By general nonsense |hocolim | ' hocolim| |. Also, there is a commutative dia- gram hocolimPopA|Sing Uoe|__//|Sing X| | | | | fflffl| fflffl| hocolimPopAUoe_________//X in which the vertical maps are weak equivalences because the natural map |Sing Y| ! Y is a weak equivalence for every space Y . We know from Proposition 3.2 that the bottom horizontal map is a weak equivalence, so the top horizontal map is also a weak equivalence. We conclude that the map hocolim ! Sing X is a weak equivalence of simplicial sets. Therefore, we shall compare hocolim and Sing UX. For the moment, assume that A is finite. Notice that PopAis a Reedy category * *[Ho , Def. 5.2.1], where we think of all the maps as being directed upward. Since the* *re are no non-identity downwardomaps,pthe fibrations are objectwise in the Reedy model structure on sSetPA (see [Ho , Th. 5.2.5]). So in this case the Reedy and proje* *ctive model structures (cf. Section 1.5) are the same. In particular, a Reedy-cofibra* *nt diagram is also projective-cofibrant, which guarantees that the homotopy colimit and the ordinary colimit are weakly equivalent. The functor may be checked to be Reedy cofibrant: at the spot indexed by oe = {a0, . .,.an}, the latching object is the subobject of Sing Uoeconsisting * *of all simplices which are contained in some other Ub. The fact that it is actually a subobject says that the latching map is a cofibration. So we know that hocolim and colim are weakly equivalent. It is easy to check that colim ~=Sing UX. We have shown that if U is a finite cover, then Sing UX is weakly equivalent to Si* *ng X. Now let A be arbitrarily large. For any finite subcollection U0, let [U0 deno* *te the union of the open sets in U0. Then we know the map Sing U0([U0) ! Sing ([U0) is a weak equivalence. But Sing UX ! Sing X is the filtered colimit of these maps, where the indexing category is the poset of all finite subcollections U0. This * *uses that each space n is compact. Our result now follows from the fact that filter* *ed colimits of simplicial sets preserve weak equivalences. 4.Hypercovering Theorems In this section we define hypercovers, and then prove our main result, Theo- rem 1.2. We go on to deduce various corollaries. Before giving a rigorous definition of hypercovers, we need to recall a few p* *ieces of machinery related to simplicial objects. For any category C, let sC denote t* *he category of simplicial objects in C. Likewise, let s n C denote the category of truncated simplicial objects of dimension n. There is the obvious forgetful fun* *ctor skn:sC ! s n C, and if C has all finite limits then skn has a right adjoint cal* *led coskn; these are the skeleton and coskeleton functors. If U* belongs to sC then HYPERCOVERS IN TOPOLOGY 9 we'll often abbreviate coskn(sknU)* as just cosknU*. Finally, the nth matching object MnU is defined to be the nth object of coskn-1U*. There is a canonical map of simplicial spaces U* ! coskn-1U*, and in level n it gives Un ! MnU. In levels less than n, this map is the identity. We write coskXnfor the nth coskel* *eton functor for s(Top # X). These definitions have somewhat easier interpretations when C is the category* * of topological spaces. To describe these, note that any simplicial set may be rega* *rded as a simplicial space which is discrete in every dimension, and if U* and W* are simplicial spaces then the set of maps from U* to W* has a natural topology com* *ing from the compact-open topology on function spaces. Using these observations, one checks that (i)Un ~=Map ( n, U*), (ii)[cosknU]k ~=Map (skn k, U*), and (iii)MnU ~=Map (@ n, U*). The first property is immediate from the Yoneda lemma. The second property follows from the first and the adjunction between sknand coskn. The third prope* *rty is a special case of the second. Finally, say that a map of spaces`Z ! X is an open covering map if it is isomorphic to a map of the form aUa ! X where {Ua} is an open cover of X. Definition 4.1. A hypercover of a space X is an augmented simplicial space U* ! X such that the maps Un ! MXnU are open covering maps for all n 0. Here MXnU denotes the nth matching object of U* computed in the category s(Top # X) of simplicial spaces over X. Note that MX0U ~= X, so the condition for n = 0 says that U0 ! X is an open covering map. Also MX1U ~= U0 xX U0, so when n = 1 we are requiring U1 ! U0 xX U0 to be an open covering map. The reader should be aware that when n > 1 the objects MnU and MXnU turn out to be isomorphic, so one can forget about the extra complication of the overcategory. Using properties (i)-(iii) above, it can be checked that if U* ! X is a hyper* *cover and K ! L is an inclusion of finite simplicial sets, then the map Map (L, U*) ! Map (K, U*) is also an open covering map. From this, it follows that coskXnU* !* * X is a hypercover whenever U* ! X is a hypercover. Also, each map Uk ! [coskXnU]k is an open covering map. We leave it to the reader to check that in a hypercover each Un must be a dis* *joint union of open subsets of X, and that ~Cech complexes are the hypercovers for which the maps Un ! MXnU are all isomorphisms. Generalizing this, a hypercover U* ! X is called bounded if there exists an N such that the maps Un ! MXnU are isomorphisms for all n > N. The smallest such N for which this happens is called the dimension of the hypercover. Said intuitively, the bounded hypercove* *rs of dimension N are the hypercovers for which the refinement process stops after the Nth level. A hypercover U* ! X has dimension at most N if and only if U* ~=coskXNU*. Lemma 4.2. If U* ! X is a bounded hypercover, then hocolimU* ! X is a weak equivalence. A more detailed version of the following proof, given in the context of an ar* *bitrary Grothendieck topology, appears in [DHI ]. 10 DANIEL DUGGER AND DANIEL C. ISAKSEN Proof.We proceed by induction, starting from the fact that bounded hypercovers of dimension 0 are just ~Cech covers and therefore are handled by Theorem 3.1. Suppose that U* ! X is a bounded hypercover of dimension n + 1. Define V* to be cosknU*, so V* is a bounded hypercover of dimension at most n. Therefore, we may assume by induction that hocolimV* ! X is a weak equivalence. The canonical map U* ! V* gives an open covering map Un+1 ! Vn+1, by the very definition of what it means for U* to be a hypercover (since Vn+1 = Mn+1U). In fact, one can check that Uk ! Vk is an open covering map for all k. Consider the following bisimplicial object, augmented horizontally by V*: V*oo___U* oo___U*oxV*U*o_oo__.o.o._oo_ The kth row is the (augmented) ~Cech complex for the open covering map Uk ! Vk. Note that for 0 k n the kth row is the constant simplicial object with value Uk because Uk ! Vk is the identity. Call this bisimplicial object (without the horizontal augmentation) W**. Let D* denote the diagonal of W**. Standard homotopy theory tells us that hocolimD* may be computed (up to weak equivalence) by first taking the homotopy colimits of the rows of W**, and then taking the homotopy colimits of the resul* *ting simplicial object. But the homotopy colimit of the kth row is just Vk by Theorem 3.1. Since V* is a bounded hypercover of dimension at most n, we have assumed t* *hat hocolimV*is weakly equivalent to X. So hocolimD* ! X is a weak equivalence. We claim that U* is a retract, over X, of D*. Note first that one has, in com* *plete generality, a map U* ! D*; in dimension k it is the unique horizontal degeneracy W0k ! Wkk. To produce a map D* ! U* it is enough to give skn+1D* ! skn+1U*, because U* = coskn+1U*. Notice that sknD* = sknU*. Choosing any face map [0] ! [n+1] gives a map Wn+1,n+1! W0,n+1, which is just Dn+1 ! Un+1. This induces a corresponding map skn+1D* ! skn+1U* as desired. It is straightforward to check that U* ! D* ! U* is the identity (because U* = coskn+1U* one only has to check it on (n + 1)-skeleta), and all the maps commute with the augmentations down to X. We have already shown that hocolimD* ! X is a weak equivalence. Since hocolimU* ! X is a retract of hocolimD* ! X, it must also be a weak equivalence. Theorem 4.3. If U* ! X is a hypercover then the maps hocolimU* ! |U*| ! X are all weak equivalences. Proof.The fact that hocolimU* ! |U*| is a weak equivalence follows just as in Theorem 3.1 for the case of ~Cech complexes: we may compute the homotopy colimit in the Strom model category, where the simplicial object U* is Reedy cofibrant * *since it has free degeneracies (Definition A.4). To show that |U*| ! X is a weak equivalence, note first that we have an iso- morphism ßk|U*| ! ßk|coskk+1U*|. (This is true for any map of simplicial spaces X* ! Y* which is an isomorphism on (k + 1)-skeleta_an easy proof is to apply the singular functor everywhere to get into bisimplicial sets, then use the diagona* *l in place of realization.) But coskk+1U* is a bounded hypercover, so Lemma 4.2 tells ~= us that ßk|coskk+1U*| -! ßkX. HYPERCOVERS IN TOPOLOGY 11 4.4. Complete covers. In this section we don't quite consider hypercovers, but rather a related concept which captures the same phenomena. This second approach was suggested to us by Jeff Smith. Definition 4.5. An open cover U = {Ua} of a space X is called complete if for all finite sets oe of indices, the intersection Uoeis covered by elements of U.* * It is called a ~Cech cover if every Uoeis again an element of the cover. Complete covers appear in [DT , Satz 2.2], where they were used in the context of identifying quasi-fibrations. The paper [Mc1 ] then used them to detect weak equivalences. We blur the distinction between a cover and the full subcategory that it spans inside the category of open sets of X. Given a cover U, we can construct an associated simplicial space in the following way: For any n 0, let Pn denote the category of nonempty subsets of {0, . .,.n}, where the maps are the inclusi* *ons. Note that the assignment [n] 7! Pn defines a cosimplicial category in the obvio* *us way. (Application of the nerve functor everywhere gives the cosimplicial space [n] ! sd n.) Define * to be the simplicial space a [n] 7! F ({0, . .,.n}), F :Popn!U where the coproduct runs over all functors Pnop! U. The faces and degeneracies are induced by those in P in the expected way. To give a point in 3, for example, is to give the following data: (1)A sequence of opens U0, . .,.U3 in U, (2)6 open subsets U01, U02, . .,.U23 in U such that Uij Ui\ Uj; (3)4 open subsets U012, . .,.U123in U such that Uijk Uij\ Ujk\ Uik; (4)An open subset U0123in U which is contained in all the Uijk; (5)A point on U0123. It is usually helpful to think of these open sets as indexed by the faces of a * *3-simplex. In forming the ~Cech complex of a cover U we are throwing in all the finite intersections Uoeinto the higher levels of the simplicial object, and these are* * typically objects which are not in U itself. The simplicial object * is in some sense th* *e closest thing we can get to a ~Cech complex while requiring all the open sets to belong* * to U. Proposition 4.6. (a)If the cover U is complete then * is a hypercover of X. (b)Regarding U as a category, let : U ! Top be the obvious inclusion. Then hocolim ' | *|. (c)If the cover U is complete then the natural map hocolim ! X is a weak equivalence. Proof.For part (a), consider the full subcategory ~Pnof Pn consisting of all ob* *jects except for {0, 1, . .,.n}. Then the matching space Mn is equal to a h " i F~(oe) . ~F:~Popn!Uoe2P~n For example, a point in M3 is determined by the data in (1)-(3) above, together with a point in U012\ U013\ U023\ U123. 12 DANIEL DUGGER AND DANIEL C. ISAKSEN Since the cover is complete, for each functor F~:P~opn! U and each element x of \oe2P~n~F(oe), there exists an extension F of ~Fto Pnopsuch that x belongs* * to F ({0, . .,.n}). This shows that n ! Mn is an open covering map, which finish* *es part (a). Part (b) is almost trivial, given the right machinery. To form hocolim we can work in the Strom model structure on Top (see Appendix A), where we first take the simplicial replacement a [n] 7! U0 U0!...!Un and then form the realization. Here the coproduct is indexed over all functors n ! U, where n denotes the category of n composable maps. Note that * was formed in almost the same way as the simplicial replacement of , except we ind* *exed the coproduct by functors Pnop! U. Each Pn is essentially just a subdivision of n, so it's not surprising that | *| is another model of the homotopy colimit. In somewhat more detail: Let sd0denote the `opposite' of the usual subdivision functor on sSet, in which the orientations of all the simplices have been chang* *ed so that they point away from the barycentres, rather than towards them. (We need this because we are using Pnoprather than Pn.) The functor sd0has a right adjoint Ex0. There is a natural `first vertex map' sd0K ! K, inducing K ! Ex0K. Given our diagram : U ! Top, the realization of the simplicial replacement is isomorphic to the coend U B, where B :U ! sSet sends Ua to the classifying space B(Ua # U). Likewise, one checks that the realization of * is isomorphic * *to the coend U Ex0B, where Ex0B is the obvious composite functor. The natural map B ! Ex 0B is an objectwise weak equivalence. The object B of sSetU is cofibrant (see [H , Cor. 15.8.8]), where this diagram category has the projecti* *ve model structure described in Section 1.5. The exact same arguments show that Ex0B is also cofibrant in this structure. So we have an objectwise weak equival* *ence between two cofibrant diagrams. The diagram : U ! Top is objectwise cofibrant (since we are working with the Strom model structure on Top), and so by [H , Cor. 19.3.5] it follows that U B ! U Ex0B is a weak equivalence. Finally, part (c) is an immediate consequence of (a), (b), and Theorem 4.3. The following corollary was originally proven by McCord [Mc1 , Th. 6], but is* * an easy consequence of our hypercovering theorem. It generalizes May's result from Corollary 2.3, which handled the case of ~Cech covers. For the proof we will ne* *ed the following observations: (1) If U ! X is an open covering map and f :Y ! X is any map, there is an induced open covering map Y xX U ! Y . (2) If U* ! X is a hypercover and f :Y ! X is a map of spaces, one gets a hypercover f-1 U* ! Y whose space in level n is Y xX Un. Corollary 4.7. Let f :X ! Y be a map of spaces. Suppose there is a complete cover U = {Ua} of Y such that each f-1 (Ua) ! Ua is a weak equivalence. Then f itself is a weak equivalence. Proof.From U form the associated hypercover Y*as described in the paragraph preceding Proposition 4.6. Pulling this back to X gives a hypercover X*:= f-1 * * Y*, as described above (note that this is not the hypercover associated to the cove* *ring {f-1 Ua}). Now f induces a map X*! Y*compatible with the augmentations. This map of simplicial spaces is a levelwise weak equivalence, by assumption. U* *pon HYPERCOVERS IN TOPOLOGY 13 taking homotopy colimits we get hocolim X*__~__//hocolim Y* ~ || |~| fflffl| fflffl| X _____________//Y, and so we conclude that X ! Y is also a weak equivalence. 4.8. Generalized hypercovers for topological spaces. Up until now we have only considered open covers, but now we turn to a broader notion. We'll say that a map p: E ! B of spaces is a generalized cover if it is locally split: that is, every element of B has a neighborhood U such that p-1(U) ! U admits a section. Observe that covering spaces, and in fact fibre bundles in general, are general* *ized covers. The point for us is that generalized covers and open covers generate t* *he same Grothendieck topology on topological spaces. Definition 4.9. An augmented simplicial space U* ! X is a generalized hyper- cover of X if the maps Un ! MXnU are generalized covers. Proposition 4.10. If U* is a generalized hypercover of X then hocolimU* ! X is a weak equivalence. Proof.Results from [DHI ], in the context of an arbitrary Grothendieck topology, show that this is a consequence of Theorem 4.3. The essential point is that gen* *er- alized hypercovers can all be refined by open hypercovers. To deduce this from the results of [DHI ] we do the following: Pick a regular cardinal ~ larger than the size of all the sets in U*. Let Top~ denote the cate* *gory of topological spaces of size less than ~, and make it into a Grothendieck site* * via the usual notion of open cover. Form the universal model category U(Top~) (see the paper [D2 ]) and localize it with respect to the set S consisting of all ma* *ps hocolimV* ! X, where V* ! X is an open hypercover. Theorem 4.3 implies that that there is a `realization map' U(Top~)=S ! Top. The results of [DHI ] say th* *at in U(Top~)=S one actually knows that hocolimU* ! X is a weak equivalence for all generalized hypercovers U*, and then applying our realization functor tells* * us this must hold in Top as well. Corollary 1.3 is an immediate consequence of the above proposition. Example 4.11. Let G be a topological group and consider the usual covering space , :EG ! BG. Form the ~Cech complex ~C(,)*, which is a generalized hypercover of BG. Using only the fact that EG has a free G-action, one can see that the nth l* *evel of ~C(,)* is homeomorphic to Gn x EG, and the face and degeneracy maps are the familiar ones of the two-sided bar construction B(*, G, EG). Now using that EG is contractible, we find that ~C(,)* is levelwise weakly equivalent to the simp* *licial space * oo___Goo_oo_GoxoG_.o...o_ The above proposition tells us that |C~(,)*| ' BG, and so in this way we recover the usual bar construction for BG. 14 DANIEL DUGGER AND DANIEL C. ISAKSEN 5.Topological realization functors for A1-homotopy theory Let k be a field. Morel and Voevodsky [MV ] produced a model category Spc(k) which captures the `motivic homotopy theory' of smooth schemes over k. Here Spc(k) stands for `spaces over k'. It is the category of simplicial presheaves * *on the Nisnevich site of smooth schemes over Speck. When k comes with an embedding k ,! C, then any k-scheme X gives rise to a topological space X(C) consisting of its C-valued points with the analytic topology. A natural expectation is to use this functor to relate Spc(k) to the * *usual model category Top of topological spaces. Morel and Voevodsky showed how to extend this functor on the level of homotopy categories (by somewhat awkward methods), but they didn't produce functors at the model category level. In this section we use Proposition 4.10 to produce such functors, with the small provis* *ion that we have to replace Spc(k) with a Quillen-equivalent variant. We also addre* *ss the situation when k ,! R, in which case one can construct topological realizat* *ion functors into Z2-equivariant spaces. As in [D2 ], a Quillen pair L: M Æ N: R will be called a Quillen map M ! N. 5.1. The Complex case. Let T denote either the Zariski, Nisnevich, or 'etale Grothendieck topology on the category Sm=k of smooth k-schemes. In the termi- nology of [D2 ], let Spc0(k)T denote the universal model category built from Sm* *=k subject to the following relations: (1)X q Y -~! (X [ Y ) (here q denotes the coproduct in our model category, whereas [ denotes disjoint union of schemes); (2)hocolimU* -~!X for any T-hypercover U* of a smooth scheme X (called `basal hypercovers' in [DHI ]); (3)X x A1 -~!X. (Relation (1) is morally a special case of (2), but must be included separately* * for technical reasons_see [DHI ]). The model categories Spc(k)T and Spc0(k)T have the same underlying category and the same class of weak equivalences, but differ in their notions of cofibra* *tion and fibration. They are injective and projective versions of the same homotopy theory. Theorem 5.2. There are Quillen maps Spc0(k)et! Top and Spc0(k)Nis ! Top sending a smooth k-scheme X to X(C). Proof.By general nonsense from [D2 ], to give a Quillen map Spc0(k)T ! Top we just need to give a functor Sm=k ! Top which respects the above relations. The functor we're interested in is X 7! X(C), and this clearly preserves relations * *(1) and (3). In the case of the 'etale topology, the fact that it preserves relati* *on (2) is just Proposition 4.10; the point is that if p: E ! B is an 'etale cover, then p(C): E(C) ! B(C) satisfies the hypotheses of the inverse function theorem and hence is locally split. Since the 'etale topology is finer than the Nisnevich topology, there is an o* *bvious map Spc0(k)Nis ! Spc0(k)et(in essence, there are more relations of type (2) for* * the 'etale topology). So one also gets a topological realization map Spc0(k)Nis ! T* *op by composition. HYPERCOVERS IN TOPOLOGY 15 It is possible to show that the functor X 7! X(C) takes elementary distinguis* *hed squares [MV ] to homotopy pushout squares of topological spaces. Together with results of [B ], this can be used to give an alternative proof of the above the* *orem for the Nisnevich topology. 5.3. The Real case. If we have a Real field k ,! R, then the space X(C) comes equipped with an action of the group Gal(C=R) = Z2. So we might hope to compare Spc0(k) to a model category of Z2-equivariant spaces. Recall that if G is a finite group then there are two notions of weak equival* *ence for G-spaces, called the non-equivariant and G-equivariant equivalences. An equivariant map X ! Y is a non-equivariant equivalence if it is a weak equivale* *nce after forgetting the equivariant structure, and it is a G-equivariant equivalen* *ce if XH ! Y H is a non-equivariant weak equivalence for every subgroup H G. There are associated G-equivariant and non-equivariant model structures on the catego* *ry of G-spaces, which we will denote Top(G) and Top(G)non. If p: E ! B is an equivariant map which is also a covering space (non- equivariantly), the map hocolim~C(E)* ! B is a non-equivariant equivalence but not necessarily a G-equivariant equivalence. For instance, if p is G ! * then t* *he map hocolim~C(E)* ! B is equal to EG ! *. So when we have a subfield k ,! R the arguments given above show that the functor X 7! X(C) induces a Quillen map Spc0(k)et! Top(Z2)non, but not a Quillen map Spc0(k)et! Top(Z2). However, when we use the Nisnevich topology something special happens. Lemma 5.4. If E ! B is a Nisnevich cover of k-schemes, then E(C)Z2 ! B(C)Z2 is locally split. For a counterexample to this in the case of 'etale covers, try SpecC ! SpecR. Proof.First note that X(C)Z2 is homeomorphic to X(R) for any scheme X over k. The map p(R) : E(R) ! B(R) is surjective by the defining property of Nisnevich covers; every R-point in B lifts to E. By definition of 'etale covers, p(R) satisfies the hypothesis of the inverse * *function theorem. Since p(R) is surjective, it is locally split. Theorem 5.5. There is a Quillen map Spc0(k)Nis ! Top(Z2) sending a smooth k-scheme X to X(C). Proof.The argument exactly parallels the non-equivariant case in Theorem 5.2, so the only nontrivial part is to show that if U* ! X is a Nisnevich hypercover then the map hocolimU*(C) ! X(C) is a Z2-equivariant weak equivalence of Z2- spaces. The fact that it is a non-equivariant equivalence has already been disc* *ussed in Theorem 5.2, because U* ! X is in particular an 'etale hypercover. So we must consider what happens when we take Z2-fixed points. It is a fact that for any diagram D of G-spaces (G any finite group) and any subgroup H of G, one has (hocolimD)H ' (hocolimDH ) (see Remark 5.6 below). So we just need to convince ourselves that hocolim{U*(C)Z2} ! X(C)Z2 is a non- equivariant weak equivalence. But by the above lemma one sees that U*(C)Z2 is a generalized hypercover of X(C)Z2, and so the result is an instance of Proposi- tion 4.10. Remark 5.6. In the above proof we needed the fact that (hocolimD)H is weakly equivalent to hocolim(DH ). This is well-known in equivariant topology, but it* *'s 16 DANIEL DUGGER AND DANIEL C. ISAKSEN hard to find an actual reference. We give a brief sketch, for which we are grat* *eful to Michael Mandell. First of all, it clearly suffices to consider the case where all the Diare co* *fibrant. This means in particular that they are Hausdorff. We form hocolimD by first writing down the simplicial replacement of the diagram, and then taking geometr* *ic realization. Taking H-fixed points obviously commutes with the simplicial repla* *ce- ment functor, so it suffices to worry about the geometric realization part. But one can check that if X* is a simplicial space in which all Xn are Hausdorff, t* *hen |X* |H is homeomorphic to |XH*|. To do this, use the skeletal filtration on |X** * | and the fact that |Skn X*| is obtained from |Skn-1 X*| by pushing out along a closed inclusion (this is one of the places where the Hausdorff condition is needed). * *Check that taking fixed-points commutes with filtered colimits, and for Hausdorff spa* *ces it also commutes with pushouts along closed inclusions. Appendix A. Homotopy colimits for diagrams of non-cofibrant spaces Let Top denote the category of all topological spaces, with its usual model c* *ate- gory structure. Given a diagram D :I ! Top, the usual instructions for computing the homotopy colimit of D are (1) to apply a cofibrant-replacement functor to every object in the diagram, and (2) to then use an explicit formula like that * *of Bousfield-Kan [BK , Sec. XII.2]. This is the situation in an arbitrary model ca* *tegory. In this section we show that for the special case of Top, the first step of cof* *ibrant- replacement is actually not needed. What we show is that no matter what formula one uses for computing homotopy colimits_whether it is the Bousfield-Kan for- mula or your favorite alternative_that formula always gives a homotopy invariant construction in Top, even without the cofibrant-replacement step. This fact see* *ms not to be well known, although it could be argued that the seeds lie there in t* *he collective subconscious of algebraic topologists. In any case, for our purposes* * here we need to bring it into the light of day. The most useful way to formulate this result seems to be in model category terms, as a comparison between the usual model structure on Top and the Strom model structure, where everything is cofibrant. See Theorem A.8. We would like to thank Phil Hirschhorn for helpful conversations about the results in this section, in particular for his ideas on removing an annoying T1* * sep- aration condition. The final form of Lemmas A.2 and A.3 is something we owe to him. To begin with, we need the following Lemma A.1. Let A ! B and X ! Y be weak equivalences. Given a diagram A x Dn oo___A x Sn-1 ____//_X | | | | | | fflffl| fflffl| |fflffl B x Dn oo___B x Sn-1 ____//_Y, where the maps in the left-hand-square are the obvious ones, the induced map fr* *om the pushout of the top row to the pushout of the bottom row is also a weak equi* *va- lence. Note that if A and B are cofibrant then this is an easy consequence of left- properness for Top, but we claim the result in greater generality. HYPERCOVERS IN TOPOLOGY 17 Proof.Let XA and YB be the pushouts of the top and bottom rows respectively, and write f :XA ! YB for the map between them. We will produce a suitable cover of these spaces and use Proposition 2.2. Let UB be the pushout of B x (Dn - {0})oo___B x Sn-1 ____//_Y. Write Dfflfor {x 2 Dn :|x| < ffl} (where 0 < ffl < 1), and let VB = B x Dffl. T* *he spaces UB and VB clearly form an open cover of YB , and notice that UB deformat* *ion- retracts down to Y . The intersection UB \ VB is equal to B x (Dffl- {0}). The same definitions give us a cover {UA , VA } of XA , and it is easy to che* *ck that f-1 (UB ) = UA and f-1 (VB ) = VA . So the map f-1 (VB ) ! VB is the map A x Dffl! B x Dffl, which is a weak equivalence. Similar reasoning shows that f-1 (UB \ VB ) ! UB \ VB is a weak equivalence. Finally, one argues that f-1 (UB ) ! UB is a weak equivalence because it deformation-retracts down to X ! Y . Proposition 2.2 now shows that XA ! YB is a weak equivalence. We'll say that an inclusion Y ,! Z is relatively T1 if given any open set U i* *n Y and any point z of Z\U, there is an open set W of Z such that U W and z =2W (compare the similar definition from [Ho , p. 50]). It follows that if E is any* * finite subset of Z\U, one can find an open set W Z which contains U and doesn't intersect E. Note that a space X is T1 precisely if all the inclusions {x} ,! X* * are relatively T1. Lemma A.2. Given a pushout diagram of the form A xfSnflffl_//Yfflffl ____ | _____ fflffl| fflffl___ A x Dn+1 ____//_________Z, the inclusion Y ,! Z is relatively T1. Proof.Suppose given a point z in Z and an open U in Y . Either z is in Y or else it is represented by a pair (a, t) where t is in the interior of Dn+1. The argu* *ment works the same for the two cases, and so for convenience we'll assume the latte* *r. Pull back U to A x Sn and express it as a union of rectangles Vix Wi, where Vi is open in A and Wi is open in Sn. Each Wi can be fattened into an open subset Wi0of Dn+1 with the properties that Wi0\ Sn = Wi and Wi0does not contain t. Let M be the union of the Vix Wi0; it is an open subset of A x Dn+1. Let N be the union of the images of M and U in Z. One checks that N \ Y = U, and the pullback of N to A x Dn+1 is M. So N is open in Z and N contains U, but N does not contain z. The following lemma is well-known for closed inclusions of T1-spaces (see also [Ho , Prop. 2.4.2]). The usual proof still works in our case. Lemma A.3. Suppose that Y1 ,! Y2 ,! . .i.s a sequence of relatively T1 inclusio* *ns and that K is a compact space. Then any map f :K ! colimY factors through some Yk. Proof.Suppose the map does not factor through any Yk. By taking a subsequence of Y if necessary, we can find a sequence of points k1, k2, . .i.n K with the p* *roperty that f(ki) lies in Yi\Yi-1. 18 DANIEL DUGGER AND DANIEL C. ISAKSEN Pick an n and set Vn = Yn. Next, choose an open set Vn+1 in Yn+1 which contai* *ns Vn but doesn't contain f(kn+1). Then pick an open set Vn+2 in Yn+2 which contai* *ns Vn+1 but neither f(kn+1) nor f(kn+2). Continuing this process gives an infinite sequence of opens, so their colimit Wn is an open subset of colimY . As n varies, the open subspaces Wn form a cover of colimY . But f(K) is a compact subspace of colimY , and it is not covered by any finite subcover. This* * is a contradiction. We now need some machinery related to simplicial spaces. Definition A.4. A simplicial space X* is said to be split, or to have free dege* *n- eracies, if there exist subspaces Nk ,! Xk such that the canonical map a Noe! Xk oe is an isomorphism. Here the variable oe ranges over all surjective maps in of* * the form [k] ! [n], Noedenotes a copy of Nn, and the map Noe! Xk is the one induced by oe*: Xn ! Xk (see [AM , Def. 8.1]). The idea is that the spaces Nk represent the `non-degenerate' part of Xk, sit* *ting inside of Xk as a direct summand. It is an easy exercise to check that if X* has free degeneracies and all the Nk are cofibrant spaces, then X* is Reedy cofibra* *nt in sTop. If X* is any simplicial space, let Skn X* be the simplicial space equaling X* through dimension n and equaling the degenerate subspaces of X* in larger di- mensions. This is slightly different than the n-truncated simplicial space sknX* **. There are maps Sk0X* ! Sk1X* ! . .a.nd the colimit is X*. It follows that |X* | is equal to colimn|Skn X*|, using that geometric realization is a left ad* *joint (and this doesn't require any assumptions on X, only hinging upon the fact that the spaces n are locally compact Hausdorff). An important point is that when X* has free degeneracies the space |Skn X*| is obtained from |Skn-1 X*| via the pushout diagram (A.1) Nn x @ n _____//|Skn-1_X*| | ______ | ____ fflffl| fflffl__ Nn x n _____//___________|Skn X*|. Proposition A.5. Let X* be a simplicial space with free degeneracies. If K is a compact space then any map K ! |X* | factors through some |Skn X*|. Proof.This is a direct application of Lemmas A.2 and A.3, using the skeletal fi* *l- tration of |X* | and the pushout square (A.1). The following corollary is the crucial ingredient for Theorem A.8. It is very similar to things in the literature, notably [M1 , Th. 11.13] and [S2, Lem. A.5* *]. May's result assumes the spaces are compactly-generated and Hausdorff, and also that the realizations are simply-connected. Segal's result is more similar to o* *urs, and the proofs follow the same pattern, but he works with homotopy equivalences rather than weak equivalences. Corollary A.6. If X* ! Y* is a map of simplicial spaces with free degeneracies such that Xn ! Yn is a weak equivalence for each n, then |X*| ! |Y*| is also a weak equivalence. HYPERCOVERS IN TOPOLOGY 19 Proof.For every k and every basepoint * of X0, there is an isomorphism colimnßk(| SknX*|, *) ! ßk(|X*|, *) (and the same statement holds with X* replaced by Y*). This follows from Proposition A.5, taking K to be a sphere. Therefore, it suffices to show that |Skn X*| ! |Skn Y*| is a weak equivalence. Using induction, this follows from t* *he pushout square (A.1) and Lemma A.1. Recall that the Strom model category is a model structure for topological spa* *ces, denoted TopS, in which the weak equivalences are homotopy equivalences and the cofibrations (resp., fibrations) are the Hurewicz cofibrations (resp., fibratio* *ns). Note that all objects are cofibrant in this structure. Proposition A.7. The Strom model category is left proper and simplicial. Proof.Left properness is automatic when all objects are cofibrant [H , Cor. 11.* *1.3]. The simplicial action is of course given by A K ~= A x |K|. To establish the simplicial structure we use the reductions outlined in [D1 , Sec. 3]. If A ! B * *is a Hurewicz cofibration and K ! L is a cofibration of simplicial sets, then |K| ! * *|L| is a closed cofibration and therefore the map a A L B K ! B L A K ~ is a cofibration by [L, Cor. 1]. If A æ B is a Hurewicz cofibration and a homot* *opy equivalence,~then certainly A K ! B K is still a homotopy equivalence. And if K æ L is a trivial cofibration of simplicial sets then |K| ! |L| is actuall* *y a homotopy equivalence, hence A K ! A L is also a homotopy equivalence. Theorem A.8. Let D :I ! Top be a diagram of spaces. Then the homotopy colimits of D as computed in Top and TopS have the same weak homotopy type. Proof.In TopS, since all objects are cofibrant, we can compute hocolimD by first taking the simplicial replacement of D and then applying the realization functo* *r. In Top we first apply a cofibrant-replacement functor to all the objects in the diagram, and only then do we take simplicial replacement and realize. Simplicial replacements always have free degeneracies (see [D2 , Proof of Lem. 2.7]), hence Corollary A.6 applies. Remark A.9. Theorem A.8 also holds if one uses the category of compactly- generated, weak Hausdorff spaces with its usual model structure. The same proofs work, with some extra caution that the various colimits are what they're suppos* *ed to be. References [SGA4]M. Artin, A. Grothendieck, and J.L. Verdier, Th'eorie des Topos et Cohomo* *logie Etale des Sch'emas, Springer Lecture Notes in Math. 270, Springer-Verlag, Berli* *n, 1972. [AM] M. Artin and B. Mazur, 'Etale homotopy, Springer Lecture Notes in Math. 1* *00, Springer- Verlag, Berlin, 1969. [B] B. Blander, Local projective model structures on simplicial presheaves, K* *-theory, to ap- pear. [BK] A.K. Bousfield and D.M. Kan, Homotopy limits, completions, and localizati* *ons, Springer Lecture Notes in Math. 304, Springer-Verlag, New York, 1972. 20 DANIEL DUGGER AND DANIEL C. ISAKSEN [Dk] T. tom Dieck, Partitions of unity in homotopy theory, Comp. Math. 23, Fas* *c. 2 (1972), 159-167. [DT] A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische produkte* *, Ann. Math. 67, no. 2, (1958), 239-281. [D1] D. Dugger, Replacing model categories by simplicial ones, Trans. Amer. Ma* *th. Soc. 353 (2001), 5003-5027. [D2] D. Dugger, Universal homotopy theories, Adv. Math., to appear. [DHI] D. Dugger, S. Hollander, and D. C. Isaksen, Hypercovers and simplicial pr* *esheaves, preprint. [F] E. M. Friedlander, Simplicial schemes and 'etale homotopy type, Annals of* * Mathematics Studies 104, Princeton University, 1982. [Gr] B. Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic* * Press, 1975. [H] P. S. Hirschhorn, Localization of Model Categories, preprint, version dat* *ed April 12, 2000. (Available at http://www-math.mit.edu/~psh). [Ho] M. Hovey, Model Categories, Mathematical Surveys and Monographs vol. 63, * *Amer. Math. Soc., 1999. [I] D. C. Isaksen, Etale realization on the A1-homotopy theory of schemes, pr* *eprint. [L] J. Lillig, A union theorem for cofibrations, Arch. Math. XXIV (1973), pp.* * 410-414. [M1] J.P. May, Geometry of iterated loop spaces, Springer Lecture Notes in Mat* *h. 271, Springer-Verlag, Berlin-New York, 1972. [M2] J.P. May, Weak equivalences and quasifibrations, Groups of self-equivalen* *ces and related topics (Proc. Conf. Universit'e de Montr'eal, Montreal, Quebec, 1988), Le* *cture Notes in Math. 1425, Springer-Verlag, Berlin, 1990, pp. 91-101. [Mc1] M.C. McCord, Singular homology groups and homotopy groups of finite topol* *ogical spaces, Duke Math. J. 33 (1966), 465-474. [Mc2] M.C. McCord, Homotopy type comparison of a space with complexes associate* *d with its open covers, Proc. Amer. Math. Soc. 18 (1967), 705-708. [MV] F. Morel and V. Voevodsky, A1-homotopy theory of schemes, Inst. Hautes Et* *udes Sci. Publ. Math. 90 (2001), 45-143. [Q] D. Quillen, Homotopical Algebra, Springer Lecture Notes in Math. 43, Spri* *nger-Verlag, Berlin, 1969. [RT] A. Razak Salleh and J. Taylor, On the relation between fundamental groupo* *ids of the classifying space of the nerve of an open cover, J. Pure Appl. Algebra 37* * (1985), no. 1, 81-93. [S1] G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes * *Sci. Publ. Math. 34 (1968), 105-112. [S2] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312. Department of Mathematics, Purdue University, West Lafayette, IN 47907 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: ddugger@math.purdue.edu E-mail address: isaksen.1@nd.edu