Homology, Homotopy and Appliations, vol. 7(1), 2005, pp.185 195 Introdution WHAT DOES THE CLASSIFYING SPACE OF A CATEGORY CLASSIFY? MICHAEL WEISS (ommuniated by Graham Ellis) Abstrat The lassifying spae of a small ategory lassifies sheaves whose values are ontravariant funtors from that ategory to sets and whose stalks are representable. Let C be a small ategory. Contravariant funtors from C to the ategory of sets, and natural transformations between them, will be alled C-sets and C-maps, respetively. The ategory of C-sets shares many good properties with the ategory of sets. (In short, it is a topos. See [4] or [7]. Here we will not make any expliit use of this fat.) The C-sets whih are of the form b mor C (b, ) for fixed C, and any isomorphi ones, are alled representable. By the Yoneda lemma, the representable C-sets form a full subategory of the ategory of all C-sets whih is equivalent to C. We will be onerned with sheaves of C-sets on a topologial spae X. For suh a sheaf, and x X, the stalk F x is again a C-set. It is the diret limit of the C-sets F (U) where U runs through the open neighborhoods of x. Theorem 0.1. The lassifying spae BC lassifies sheaves of C-sets with representable stalks. Notation, terminology, larifiations. Let F be any sheaf of C-sets on X. We may regard F as a ontravariant funtor (, U) F () (U) in two variables (where ob(c) and U is open in X). Speializing one of the variables, we obtain F (), a sheaf of sets on X, and F (U), a C-set. Let L be any sheaf of sets on X. The espae étalé of L, denoted Spé(L ), is the (disjoint) union of the stalks L x, suitably topologized. See [2, II.1, ex.1.13] for details. The sheaf L an be identified with the sheaf of ontinuous (partial) setions of the projetion Spé(L ) X. The projetion Spé(L ) X is an étale map alias loal homeomorphism. [But it happens frequently that X is Hausdorff while Spé(L ) is not.] The onstrution Spé(L ) leads to a good notion of pullbak of sheaves: for a map v : Y X, the Partially supported by the Royal Soiety. Reeived February 4, 2005, revised June 18, 2005; published on November 1, 2005. 2000 Mathematis Subjet Classifiation: 57T30, 18F20, 18G55 Key words and phrases: ategory, lassifying spae, sheaf, representable funtor 2005, Mihael Weiss. Permission to opy for private use granted.
Homology, Homotopy and Appliations, vol. 7(1), 2005 186 pullbak v L is defined in suh a way that Spé(v L ) = v Spé(L ). More generally, for a sheaf F of C-sets on X and v : Y X, the pullbak v F is defined in suh a way that Spé((v F ) () ) = v Spé(F () ). Let F and G be sheaves of C-sets on X, both with representable stalks. Let e 0, e 1 : X X [0, 1] be given by e 0 (x) = (x, 0) and e 1 (x) = (x, 1). The sheaves F and G are onordant if there exists a sheaf of C-sets H on X [0, 1], again with representable stalks, suh that e 0H = F and e 1H = G. The preise meaning of theorem 0.1 is as follows. Suppose that X has the homotopy type of a CW-spae. There is a natural bijetion from the homotopy set [X, BC] to the set of onordane lasses of sheaves of C-sets on X with representable stalks. Remark. Suppose that C is a group. To be more preise, suppose that C has just one objet and mor(, ) is a group. Let F be a sheaf of C-sets on a spae X. If the stalks of F are all representable, then the projetion Spé(F ) X is a prinipal mor(, )-bundle. Indeed any hoie of an open U and s F () (U) determines a bundle hart Spé(F U) = mor(, ) U. In this situation, onordant sheaves of C-sets on X (with representable stalks) are isomorphi, beause onordant implies isomorphi for prinipal mor(, )- bundles. Remark. The question in the title undoubtedly has many orret answers and a few have already been given elsewhere. Moerdijk [7, Introd.] has a result like theorem 0.1 in whih the representability ondition on stalks is replaed by a weaker ondition, that of being prinipal. To explain what a prinipal C-set is, we start with the following standard definitions: The transport ategory of a C-set S has objets (, x) where is an objet of C and x S(). A morphism from (, x) to (d, y) is a morphism g : d in C suh that the indued map S(g): S(d) S() takes y to x. (Some people would all this the opposite of the transport ategory of S.) A ategory D is filtered if it has at least one objet; for any two objets d 1, d 2 in D there exists another objet d 3 and morphisms d 1 d 3, d 2 d 3 ; for any two morphisms in D with the same soure and target, say f, g : a b, there exists a oequalizer (a morphism h: b in D suh that hf = hg). Now a C-set is prinipal if its transport ategory is filtered. (This is not exatly the terminology whih Moerdijk uses. He alls a sheaf of C-sets on X a prinipal C op -bundle if the transport ategory of eah stalk, as defined above, is filtered.) A representable C-set S is ertainly prinipal, sine the transport ategory of S has a terminal objet. The onverse does not hold. For example, suppose that C itself is a filtered ategory whih does not have a terminal objet. Define a C-set S in suh a way that S() has exatly one element, for every objet in C. Then the transport ategory of S is equivalent to C, so S is
Homology, Homotopy and Appliations, vol. 7(1), 2005 187 prinipal. But S is not representable, sine a representing objet would be a terminal objet for C. It follows that there exist sheaves of C-sets on some spaes X whih do not satisfy the ondition of theorem 0.1 but whih are prinipal C op -bundles aording to Moerdijk s definition. (Take X to be a point.) Moerdijk [7] takes the disussion muh further by onsidering topologial ategories, whih I have not attempted to do. Another preursor of theorem 0.1 is due to tom Diek (1972, unpublished). He used a notion of C-bundle defined in terms of a bundle atlas. His result was redisovered in [5, thm 4.1.2]. Theorem 0.1 is antiipated and illustrated to some extent in [6]. 1. The anonial sheaf on BC We are going to onstrut a sheaf E of C-sets on BC whih will eventually turn out to be universal. Reall to begin with that BC is the geometri realization of the simpliial set whose n-simplies are the ontravariant funtors [n] C, where [n] is the linearly ordered set {0, 1, 2,..., n}. (There are historial reasons for insisting on ontravariant funtors [n] C; the formulae for boundary operators look more familiar in the ase where C is a group or monoid.) Now suppose that U is open in BC and is an objet of C. Definition 1.1. An element of E () (U) is a funtion whih to every α: [n] op C and x B[n] op = n with α x U assigns a morphism s(α, x): α(0) in C. The funtion is required to be loally onstant in the seond variable, so that for y n suffiiently lose to x, with α y U, we have s(α, y) = s(α, x); natural in the first variable. That is, for an order-preserving g : [m] [n] and y m, we have s(α, g y) = α(0, g(0)) s(αg, y) where α(0, g(0)): α(g(0)) α(0) is the morphism in C indued by the unique morphism 0 g(0) in [n]. The ontravariant dependene of E () (U) on U and is obvious. The sheaf property is also obvious. Beause of the naturality ondition, an element s of E () (U) is determined by its values s(α, x) for nondegenerate α: [n] op C and x n n. Then α and x are determined by α x U; in partiular n is the dimension of the ell (in the anonial CW-deomposition of BC) to whih α x belongs. (Regarding ells, the onvention used here is that the ells of a CW-spae are pairwise disjoint, and eah ell is homeomorphi to some eulidean spae. This is in agreement with [1], for example.) Lemma 1.2. Fix a nondegenerate β : [m] op C and y m m. The stalk of E at β y is the ontravariant funtor mor(, β(0)). Proof. Any point of BC an be uniquely written as α z where α: [n] op C is nondegenerate and z n n. If α z is suffiiently lose to β y, then some
Homology, Homotopy and Appliations, vol. 7(1), 2005 188 degeneray of β will be a fae of α. That is, there are an order-preserving surjetion f : [k] [m] and an order-preserving injetion g : [k] [n] suh that αg = βf. And moreover, there will be w k suh that f w = y and z is lose to g w. For s in the stalk of E () at β y BC, we then have s(α, z) = s(α, g w) = α(0, g(0)) s(αg, w), s(αg, w) = s(βf, w) = s(β, f w) = s(β, y), so that s(α, z) = α(0, g(0)) s(β, y). Hene s is determined by s(β, y) mor(, β(0)). To establish the existene of a germ s with presribed value s(β, y), we proeed differently. Suppose indutively that the values of s at points near β y and in the (n 1)-skeleton of BC have already been determined onsistently, for some fixed n > m. For an n-simplex α: [n] op C we have an attahing map from n to the (n 1)-skeleton of BC. Hene s(α, x) is already determined for x in some open subset V of n. As a funtion on V, denoted informally s V, it satisfies the ontinuity and naturality onditions of definition 1.1 (mutatis mutandis). We now have to find an open W n suh that V = W n and an extension of s V from V to W. This is easy. For example, n an be identified with a one on n and W ould then be defined as the one on V minus the one point. Then s V has a unique extension from V to W. For an objet of C, let ( C) be the under ategory assoiated with. The objets of ( C) are the morphisms in C with soure, and the morphisms of ( C) are morphisms in C under. The lassifying spae B( C) is ontratible sine ( C) has an initial objet. The forgetful map B( C) BC has a anonial fatorization B( C) λ Spé(E () ) proj. BC. Indeed, any point of BC an be uniquely written as α z where α: [n] op C is nondegenerate and z n n. Lifting α z to B( C) amounts to speifying a morphism α(n) in C; lifting α z to Spé(E () ) amounts to speifying a morphism α(0) in C. Clearly a morphism α(n) determines a morphism α(0) by omposition with α(0, n): α(n) α(0). The map λ will be useful in the proof of Proposition 1.3. The spae Spé(E () ) is weakly ontratible. Proof. Let BC be the fat realization of the nerve of C, obtained by ignoring the degeneray operators. The quotient map q : BC BC is a quasifibration with ontratible fibers. To see this, note that the fat realization of any simpliial set Z an be desribed as the ordinary realization of another simpliial set Z whose n- simplies are triples (k, f, x) where x Z k and f : [n] [k] is an order-preserving surjetion. The forgetful simpliial map Z Z is a Kan fibration with ontratible fibers; hene the indued map of (lean) geometri realizations, Z Z, is a quasifibration with ontratible fibers. See [3]. Let E = Spé(E () ), let r : E BC be the projetion, and let Ē = q E. In the
Homology, Homotopy and Appliations, vol. 7(1), 2005 189 pullbak square r q r Ē B q r E B the map q is a quasifibration with ontratible fibers and r is a loal homeomorphism. It follows that r q is a quasifibration with ontratible fibers, and onsequently a weak homotopy equivalene. It remains to prove that Ē is ontratible, or equivalently, that the anonial map λ : B( C) Ē, the fat version of λ, is a weak homotopy equivalene. Suppose therefore that Y is any finitely generated -set (= simpliial set without degeneray operators ) and let f : Y Ē be any map. We want to show that, up to a homotopy, f lifts to B( C). The argument has two parts. (i) If rf : Y B is indued by a map of the underlying -sets, then f admits a unique fatorization through B( C). (ii) Modulo iterated baryentri subdivision of Y, and a homotopy of f, the omposition rf is indeed indued by a map of the underlying -sets. For the proof of (i), we may assume that Y is generated by a single n-simplex, so Y = n. Suppose that rf : n BC is the harateristi map of an n-simplex α: [n] op C in the nerve of C. The extra information ontained in f amounts to ompatible morphisms u i : α(i) for i = 0, 1,..., n; learly all u i are determined by u n. Together, u n and α determine an n-simplex in the nerve of ( C). For the proof of (ii), we note that the first baryentri subdivision of BC an be desribed as BC for another ategory C. An objet of C is a simplex of the nerve of C; a morphism from an m-simplex α to an n-simplex β is an injetive order-preserving v : [m] [n] with v β = α. The funtor C C given by α α(0) indues a -map from the nerve of C to the nerve of C, and then a map ϕ 1 : BC BC. This map is not a homeomorphism. There is of ourse another (well-known) map ϕ 0 : BC BC whih is a homeomorphism. What is important here is that ϕ 0 and ϕ 1 are homotopi in an obvious way, by a homotopy (ϕ t ) t [0,1]. (Eah trak of the homotopy is a straight line segment, or a single point, in a simplex of BC.) The homotopy (ϕ t ϕ 1 0 ) t [0,1], from the identity of BC to ϕ 1 ϕ 1 0, has a unique lift to a homotopy (ψ t : Ē [0, 1] Ē) t [0,1] with ψ 0 = id. (To verify this laim, ompare the pullbaks of Ē under the maps BC [0, 1] ϕϕ 1 0 BC, BC [0, 1] proj. BC. They are homeomorphi as spaes over BC [0, 1].) We an similarly look at iterated
Homology, Homotopy and Appliations, vol. 7(1), 2005 190 baryentri subdivisions of BC. They all have two anonial maps ϕ 0, ϕ 1 BC, one being a homeomorphism and the other being simpliial, and these two maps are homotopi by a homotopy (ϕ t ) t [0,1]. Again, the homotopy (ϕ t ϕ 1 0 ) t [0,1] has a unique lift to a homotopy (ψ t : Ē [0, 1] Ē) t [0,1] with ψ 0 = id. Coming bak now to maps Y Ē, any suh map is homotopi to a map f suh that rf is indued by a -map from some iterated baryentri subdivision of Y to some iterated baryentri subdivision of the nerve of C. Compose f with ψ 1 from the above homotopy. Then rψ 1 f is indued by a -map from Y to the nerve of C. 2. Resolutions The previous setion gives us a method to onvert a map f : X BC into a sheaf of C-sets on X with representable stalks, by f f E C. Going in the opposite diretion is more diffiult. From a sheaf F of C-sets on X, we shall onstrut a resolution p F : X F X and a map π F : X F BC. It turns out that p F is a homotopy equivalene if F has representable stalks and X is a CW-spae. Then we an hoose a homotopy inverse p 1 F and obtain a map π F p 1 : X BC, well F defined up to homotopy. Let O(X) be the poset of open subsets of a spae X, ordered by inlusion. Let F be a sheaf of C-sets on X. We an regard F as a ontravariant funtor from O(X) C to sets. The funtor F determines a transport ategory T F whose objets are the triples (U,, s) onsisting of an objet U in O(X), an objet in C, and s F () (U). A morphism from (U,, s) to (V, d, t) is a morphism U V in O(X) together with a morphism f : d in C suh that f (t) U = s. Let τ be the tautologial funtor (taking U O(X) to the spae U) from O(X) to spaes, and let ϕ: T F O(X) be the forgetful funtor. Put X F := τϕ. This omes with a anonial projetion p F : X F X, indued by the obvious natural inlusions τϕ(u,, s) X. There is also a projetion X F BT F whih we an ompose with the forgetful map BT F BC. This gives π F : X F BC. Proposition 2.1. If F has representable stalks, then the projetion p F : X F X is a weak homotopy equivalene. The proof relies on a few lemmas whih in turn rely on the notion of a mirofibration. Reall that a map p: E B is a Serre fibration if it has the homotopy lifting property for homotopies X [0, 1] B, with presribed initial lift X E, where X is a CW-spae. [It is enough to hek this in all ases where X is a disk.] A map p: E B is a Serre mirofibration if, for any homotopy h: X [0, 1] B with presribed initial lift h 0 : X E, there exist a neighborhood U of X {0} in X [0, 1] and a map h: U E suh that p h = h U and h(x, 0) = h 0 (x) for all x X. In that ase the map h is a mirolift of h. [Again it is enough to hek the miro-lifting property in all ases where X is a disk.]
Homology, Homotopy and Appliations, vol. 7(1), 2005 191 Lemma 2.2. Let p: E B be a Serre mirofibration. If p has weakly ontratible fibers, then it is a Serre fibration. Notes on the proof. This is essentially due to G. Segal [8, A.2]. The hypotheses here are slightly more general, though. There is a short indutive argument as follows. The indution step onsists in showing that if p: E B is a Serre mirofibration with ontratible fibers, then so is the projetion p I : E I B I. Here I = [0, 1], and the mapping spaes E I = map(i, E) and B I = map(i, B) ome with the ompatopen topology. The Serre mirofibration property for p I is obvious, so it is enough to establish the weak ontratibility of the fibers of p I. Suppose therefore given a map γ : I B and a map f : S n I E whih overs γ, so that pf(z, t) = γ(t) for z S n and t I. We must extend f to a map g : D n+1 I E whih overs γ. But that is easy: Use a suffiiently fine subdivision of I into subintervals [a r, a r+1 ] so that partial extensions g r : D n+1 [a r, a r+1 ] E of f an be onstruted, with pg r (z, t) = γ(t) for z D n+1 and t [a r, a r+1 ]. Then improve g r if neessary, on a small neighborhood of D n+1 {a r } in D n+1 [a r, a r+1 ], to ensure that g r (z, a r ) = g r 1 (z, a r ) for z D n+1. The indution beginning onsists in showing that p has the path lifting property. (That is, given a path γ : I B and a E with p(a) = γ(0), there exists a path ω : I E with pω = γ and ω(0) = a.) But that is also easy. Lemma 2.3. Let τ be the tautologial funtor from O(X) to spaes and let K be a ompat subset of τ. Then there exist a finite full sub-poset P O(X) and a subfuntor κ of τ P with ompat values suh that K κ. Remarks. The fullness assumption means that U, V P and U V imply U V in P. By a subfuntor κ of τ P is meant a seletion of subspaes κ(u) τ(u) = U, one for eah U P, suh that κ(v ) κ(u) if V U in P. The lemma is losely related to an observation for whih I am indebted to Larry Taylor: The mapping ylinder C of the inlusion of the open unit interval in the losed unit interval is not homeomorphi to a subset of [0, 1] 2. This is easy to verify, although surprising. The two endpoints of the losed unit interval, viewed as elements of the mapping ylinder C, don t have ountable neighborhood bases; hene C is not even metrizable. Equally surprising, and more to the point, is the following. Let K be a ompat subset of C. Then there exists a ompat subinterval L of the open unit interval suh that K is ontained in the mapping ylinder of the inlusion L [0, 1]. For the proof, exhaust the open unit interval by an asending sequene of ompat subintervals L i. Suppose if possible that for eah i there exists x i K whih is not ontained in the mapping ylinder of the inlusion L i [0, 1]. Then the x i form an infinite disrete losed subset of K, whih ontradits the ompatness of K. Proof of lemma 2.3. The lassifying spae BO(X) is a simpliial omplex. This has one n-simplex for eah subset of O(X) of the form {U 0, U 1,..., U n } where U i 1 is a proper subset of U i, for i {1, 2,..., n}. The image of C under the projetion
Homology, Homotopy and Appliations, vol. 7(1), 2005 192 τ BO(X) is ontained in a ompat simpliial subomplex BO(X), and without loss of generality we an assume that the subomplex has the form BP for a finite full sub-poset P of O(X). For eah simplex S of BP, let e(s) BP be the ell determined by S, so that e(s) is loally losed in BP and BP is the disjoint union (but not the oprodut in general) of the e(s) for the simplies S of BP. Let U(S) be the smallest of the open sets orresponding to the verties of S. The inverse image of e(s) for the projetion τ BO(X) is identified with e(s) U(S). Its intersetion with K is ontained in a subset of the form e(s) L(S), where L(S) U(S) is ompat. (This an be proved as in the remark just above.) Choose suh an L(S) for every simplex S in BP. For an element U of P let κ(u) := L(S). S with U(S) U Then κ(v ) κ(u) for V, U P with V U, and eah κ(u) is ompat. Corollary 2.4. The projetion p F : X F X is a Serre mirofibration. Proof. Write p = p F. Let q : X F BT F be the standard projetion (from the homotopy olimit to the lassifying spae of the indexing ategory). As we just disovered, the formula y (p(y), q(y)) need not define an embedding of X F in X BT F, but it ertainly defines an injetive map and we an use that to label elements of X F. In partiular, let h: D i [0, 1] X be a homotopy with an initial lift H 0 : D i X F, so that h(z, 0) = ph 0 (z). We need to find ε > 0 and a map H : D i [0, ε] X F suh that ph = h on D i [0, 1] and H(z, 0) = H 0 (z) for z D i. The plan is to define H in suh a way that (ph(z, t), qh(z, t)) = (h(z, t), qh 0 (z)), for (z, t) D i [0, ε], whih means that qh : D i [0, 1] BT F is a onstant homotopy. By lemma 2.3, the plan is sound, giving a well defined and ontinuous map D i [0, ε] X F for suffiiently small ε. Lemma 2.5. The projetion p F : X F X has ontratible fibers. Proof. The fiber over x X is identified with the homotopy olimit of the (ontravariant, set-valued) funtor (U, ) F () (U) where U runs through the open subsets of X ontaining x, and runs through the objets of C. By a well-known Fubini priniple for homotopy olimits, it is homotopy equivalent to the double homotopy olimit U x F () (U).
Homology, Homotopy and Appliations, vol. 7(1), 2005 193 In this expression the inside homotopy olimit is a homotopy olimit of sets (i.e., disrete spaes) taken over a direted poset, and therefore the anonial map U x is a homotopy equivalene. Therefore U x F () (U) olim U x F () (U) F () (U) F () x where F x () is the stalk of F () at x. But the stalk funtor F x is representable by assumption. The homotopy olimit of a representable funtor is ontratible. Proof of proposition 2.1. Apply lemma 2.2 and note that a Serre fibration with weakly ontratible fibers is a weak homotopy equivalene. 3. Classifiation of sheaves up to onordane Lemma 3.1. Let F 0 and F 1 be two sheaves of C-sets on X, both with representable stalks. Let g : F 0 F 1 be a binatural transformation. Then F 0 and F 1 are onordant. Proof. Let e 0 : X X I be given by e 0 (x) = (x, 0) and let p: X I X be the projetion. For an objet in C and an open subset U of X [0, 1], let U 0 = e 1 0 (U) and let G () (U) be the set of pairs (s, t) F 0 (U 0 ) p F 1 (U) suh that gs = e 0t. Now G is a sheaf of C-sets on X I with representable stalks. Its restritions to X {0} and X {1} are identified with F 0 and F 1, respetively. Corollary 3.2 (to proposition 2.1 and lemma 3.1). Let F be a sheaf of C-sets on X with representable stalks. Suppose that X is a CW-spae. Then π F p 1 F : X BC is a lassifying map for F. That is, (π F p 1 F ) E is onordant to F, with E as in definition 1.1. Proof. Abbreviate p = p F, π = π F. It is enough to show that the sheaves π E and p E on X F are onordant. By lemma 3.1, it is then also enough to make a map from π E to p F. That is what we will do, using the étale point of view. Therefore let z X F. We need to ompare the stalk of F at p(z) X with the stalk of E at π(z) BC. The point z maps to some ell in BT F whih orresponds to a nondegenerate diagram (U 0, 0 ) (U 1, 1 ) (U k 1, k 1 ) (U k, k ) in O(X) C, with p(z) U k, and an element s 0 F ( 0) (U 0 ). The stalk of E at π(z) is then represented by the objet 0. The germ of s 0 near p(z) amounts to a morphism from 0 to the objet whih represents the stalk of F at p(z); equivalently, by the Yoneda lemma, s 0 determines a C-map from the stalk of E at π(z) to the stalk of F at p(z). Letting z vary now, and seleting an objet in C, we obtain a map over X F from Spé(π E () ) to Spé(p F () ). This is ontinuous (verifiation left
Homology, Homotopy and Appliations, vol. 7(1), 2005 194 to the reader) and natural in, and therefore amounts to a map between sheaves of C-sets on X F, from π E to p F. Proof of theorem 0.1. Suppose that X is a CW-spae. Let g : X BC be any map and put F = g E. We have to show that g is homotopi to πp 1, where π = π F and p = p F. We note that X F also has the homotopy type of a CW-spae sine it is a homotopy olimit of open subsets of X (all of whih have the homotopy type of CW-spaes). Hene p is a homotopy equivalene. Therefore, showing g πp 1 amounts to showing that gp π. Now reall that X F was onstruted as the homotopy olimit of a funtor τϕ from a ertain ategory T F with objets (U,, s) to the ategory of spaes. The maps gp: X F BC and π : X F BC both have a fatorization of the following kind: (U,,s) U v Spé(E () ) w BC. Here v is (in both ases) indued by a natural transformation from the funtor (U,, s) U to the funtor (U,, s) Spé(E () ), given by the maps U Spé(E () ) ; x (x, s). In the ase of gp, the seond map w = w 0 in the fatorization is determined by the projetions Spé(E () ) BC. In the ase of π, the map w = w 1 is the omposition of the projetion to BT F and the forgetful map BT F BC. Consequently it is now suffiient to show that w 0 and w 1 are weakly homotopi, whih is to say, w 0 f w 1 f for any map f from a CW-spae to the ommon soure of w 0 and w 1. It is enough to hek this for a partiular f whih is a weak equivalene. A good hoie of suh an f is the map B( C) Spé(E () ) indued by the natural maps λ : B( C) Spé(E () ). By proposition 1.3, this f is indeed a weak homotopy equivalene. The maps w 0 f and w 1 f are easily seen to be homotopi. Indeed, eah ( C) has two obvious funtors two C, one given by ( d) d and the other by ( d). These are related by a natural transformation, whih determines a homotopy h () between the indued maps B( C) to BC. Integrating the homotopies h () one obtains a homotopy w 0 f w 1 f. Referenes [1] A. Dold, Letures on algebrai topology, Grundlehren, Springer, 1972. [2] R. Hartshorne, Algebrai geometry, Graduate Texts in Mathematis, Springer, 1977. [3] J.P.May, Simpliial objets in algebrai topology, Chiago Letures in Math., Univ. of Chiago Press, 1967. [4] S. Ma Lane and I. Moerdijk, Sheaves in geometry and logi, Springer, 1992.
Homology, Homotopy and Appliations, vol. 7(1), 2005 195 [5] I. Madsen and M. Weiss, The stable moduli spae of Riemann surfaes: Mumford s onjeture, preprint, arxiv:math.at/0212321, v3, 2002 2004. [6] I. Madsen and M. Weiss, The stable mapping lass group and stable homotopy theory, Proeedings of 2004 European Cong. of Maths. in Stokholm, European Math. So., 2005. [7] I. Moerdijk, Classifying spaes and lassifying topoi, Leture Notes in Math., vol. 1616, Springer, 1995. [8] G. B. Segal, Classifying spaes related to foliations, Topology 17 (1978), 367 382. This artile is available at http://intlpress.om/hha/v7/n1/a10/ Mihael Weiss Dept. of Mathematis, University of Aberdeen, AB24 3UE United Kingdom m.weiss@maths.abdn.a.uk