The homotopy type of a topological stack Johannes Ebert, 9 Mathematisches Institut der Universit¨at Bonn 0 0 Beringstraße 1 2 53115 Bonn, Germany n [email protected] a J January 21, 2009 1 2 ] Abstract T A The notion of the homotopy type of a topological stack has been around in the literature for some time. The basic idea is that an atlas X → X of a stack determines a topological . h groupoidXwithobjectspaceX. ThehomotopytypeofXshouldbetheclassifyingspaceBX. t a Thechoiceofan atlasisnotpart ofthedataof astack andhenceitisnotimmediately clear m why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this noteis to givean elementary construction of such a homotopy-typefunctor. [ 1 v 1 Introduction 5 9 The concept of a stack (which originated in algebraic geometry) plays an increasingly important 2 roleingeometrictopology,seeforexample[2],[5],[3],[4]. Inthisnoteweshowhowastackdefines 3 . an object of homotopy theory. 1 We assume that the reader is familiar with the terminology of stacks. Therefore we will not 0 spell out the basic definitions here. A stack over the site Top of topological spaces is a lax 9 0 sheaf of groupoids on the site of topological spaces; we refer the reader to the excellent [8] an : explanation of this definition. Stacks over Top form a 2-category all of whose 2-morphisms are v isomorphisms. Thereareseveralpossiblenotionsoftopological stacks. Ournotionismadeexplicit i X in2.3. Essentially,atopologicalstackisastackoverTopwhichcanberepresentedbyatopological r groupoid. Let TopStacks denote the 2-category of topological stacks. For any 2-category S, we a denote the underlying ordinary category by the symbol τ S. ≤1 We would like to construct a functor Ho : τ TopStacks → Top which assigns to a stack ≤1 its homotopy type. For set-theoretical reasons, we need to restrict to small subcategories of S ⊂ TopStacks. This level of sophistication is certainly sufficient for all applications of our construction to concrete mathematical problems. Furthermore, it turns out that we need to restrict to stacks which admit a presentation by a “paracompactgroupoid”(see belowfor details). Thisis arathermildcondition,whichissatisfied byvirtuallyallstacksofinterestingeometrictopology. Inthe sequel,weassumethatS isasmall 2-categoryand there is a fixed 2-functor J :S →TopStacks such that all stacks in the image of this functor admit a presentation by a paracompact groupoid. The first main result of this note is Theorem 1.1. There existsa functorHo:τ S →Top, which assigns tos∈Ob(S) a topological ≤1 space Ho(X) which is homotopy equivalent to BX, when X is a groupoid presenting the topological X = J(s). If f,g are two 1-morphisms with the same source and target, then Ho(f) and Ho(g) are homotopic if f and g are 2-isomorphic or if J(f) and J(g) are concordant (see Definition 2.10 below). 1 Let π (S) be the category which is obtained from τ S by identification of 2-isomorphic 1- 0 ≤1 morphisms;thereisaquotientfunctorτ S →π (S). Notethatthe fully faithfulYonedaembed- ≤1 0 ding st : Top → TopStacks defines a fully faithful functor Top → π TopStacks - homotopic 0 but different maps of spaces do not yield 2-isomorphic morphisms of stacks. Furthermore, let HoTopbethehomotopycategoryoftopologicalspaces. AsacorollaryofTheorem1.1,weobtain the existence of a homotopy type functor π (S)→HoTop. This homotopy type functor extends 0 both, the obvious functor Top → HoTop and the functor from topological groups to HoTop sending G to BG. ThereisanessentialfeatureofhomotopytypeswhichisabandonedinTheorem1.1. Letusdescribe what is missing. Let X be a space. Then we denote, as usual, the stack st(X) by the symbol X; there is no danger of confusion. The space Ho(X) should come with a map η : Ho(X) → X X which should be a universal weak equivalence, i.e. for any space Y and any Y →X, the pullback Y × Ho(X) → Y is a weak homotopy equivalence (the morphism Y → X is automatically X representable by [11], Corollary 7.3; thus Y × Ho(X) is a topological space). The map η is a X X generalization of the map BG→∗//G given by the universal principal G-bundle. Obviously, it is desirable that the maps η assemble to a natural transformation η : st◦Ho → X τ J offunctorsτ S →τ TopStacks. We werenotableto constructsucha naturaltransfor- ≤1 ≤1 ≤1 mation on the nose, but only up to contractible choice and up to 2-isomorphism. The following two definition make these notions precise. Definition 1.2. Let C be a (discrete, small) category. A functor definedupto contractiblechoice is a triple (C,p,F), where C is a topological category which has the same objects as C , p:C →C is a functorewhich is the ideentity on objects and a weak homotopy equivalence on morphism sepaces (i.e. C is a thickening of C in the sensethat the morphisms in C are replaced by contractible spaces e of morphisms) and F :C →Top is a continuous functor. e Definition 1.3. Let A be a topological category with discrete object set and B be a discrete 2- category all of whose 2-morphisms are isomorphisms. Let F ,F : A → B be two functors. A 0 1 pseudo-natural transformation η assigns to every object a ∈ A a 1-morphism η :F (a) →F (a) a 0 1 and to every morphism f : a → a′ of A a 2-isomorphism ηf : F1(f)◦ηa → ηa′ ◦F0(f) such that ηida =idηa and such that for any pair f,f′ of composable morphisms, the 2-isomorphisms ηf,ηf′ and ηf′◦f are compatible. For most (but not all) constructions of homotopy theory, a functor defined up to contractible choice is as good as an honest functor. Therefore the following theorem should be a satisfactory result for many purposes. Theorem 1.4. Let S be as above. Then there exists a functor defined up to contractible choice ] (τ S,p,Ho). The functor Ho is related to the functor Ho◦p from Theorem 1.1 by a zig-zag of ≤1 f f natural transformations which are weak homotopy equivalences on each object. ] Moreover, there exists a pseudo-natural transformation η : st◦Ho → J ◦p of functors τ S → ≤1 f TopStacks. Foranystack X∈S, themorphism η :Ho(X)→Xis a universalweakequivalence, X in the sense that for any space Y and any Y → X,fthe pullback Y × Ho(X) → Y is a weak X homotopy equivalence. The results of the present paper are very similar to those of Behrang Noohi’s recent paper [12] (in fact, they are slightly weaker). Proposition 11.2 in [12] implies 1.4. On the other hand our treatment is more elementary. Therefore we claim that the present paper should be useful for anyone whose main interest is in the applications of topological stacks to problems in geometry. However,thereisaflawinthetheory(alsoin[12])whichwewilldescribenow. Givenahomotopy invariant functor F from spaces to groups (or any other discrete category), we can extend F to stacks via Fˆ(X):=F(Ho(X)). (1.5) 2 ItfollowsthatFˆ(η )isanisomorphism. Formanyofthefunctorsofalgebraictopology,including X singular(co)homologyorhomotopygroups,thisisareasonabledefinition. Butthereareimportant homotopy-invariantfunctorsonspacesforwhich1.5isnot agooddefinition. Theprimeexampleis complexK-theory. AnyreasonabledefinitionoftheK-theoryofastackshouldsatisfyK0(∗//G)∼= RG for a compact Lie group G (RG is the representation ring). In fact the definition of the K- theoryofastackgivenin[5]satisfiesthiscondition. Ontheotherhand,thecelebratedAtiyah-Segal completiontheorem[1]shows,amongotherthings,thatthenaturalmapRG→K0(BG)isnotan isomorphism. Itfollowsthat, inthe presentsetup, K-theoryofastackis nothomotopy-invariant. Gepner and Henriques [6] developed a finer homotopy theory of stacks in which K-theory is homotopy invariant. On the other hand, their theory is much more involved than ours. Here is a brief outline of the paper. In section 2, we discuss the notion of a principal X-bundle for anarbitrarytopologicalgroupoidX anddefine the stack X //X ofprincipalX-bundles, which 0 1 is the prototype of a topological stack. Then we construct the universal principal X-bundle. The material in this section is a rather straightforward generalization of the classical theory of classifyingspacesfortopologicalgroups. However,weneedtobepreciseonthepoint-setlevel,and Theorem 2.7 is stronger than what is standard in the theory of fibre bundles. Section 3 contains theproofsofTheorem1.1and1.4. InanappendixA,weshowatechnicalresultwhichguarantees the paracompactness of the classifying space of a “paracompactgroupoid”. Acknowledgements The author was supported by a postdoctoralgrantfrom the German Academic Exchange Service (DAAD).HeenjoyedthehospitalityoftheMathematicalInstituteoftheUniversityofOxford. He is particularly grateful to Jeff Giansiracusa for numerous discussions about stacks. He also wants to thank the anonymous referee who pointed out a mistake at a crucial step of the argument in an earlier version of this paper. 2 Groupoids and stacks Principal bundles for groupoids Let X = (X ,X ,s,t,e,m,ι) be a topological groupoid: X is the object space, X the morphism 0 1 0 1 space, s,t : X1 → X0 are source and target maps, m : X1 ×X0 X1 → X1 is the multiplication, e : X → X is the unit map and ι : X → X sends a morphism to its inverse. The maps 0 1 1 1 s,t,e,m,ι are continuous and satisfy the usual identities. We use the following convention: the product xy =m(x,y) of x,y ∈X is defined if (and only if) t(x)=s(y). 1 WesaythatanX-spaceoveraspaceX consistsofaspaceE,twomapsp:E →X andq :E →X 0 and an ”action” α : E×X0 X1 = {(e,γ)∈ E×X1|q(e) = s(γ)} → E over X which is compatible with the projection to X and the multiplication in X. In other words, the diagrams E×X0 X1 α //E p1 p (cid:15)(cid:15) (cid:15)(cid:15) E p //X (P is the projection onto the first factor) and 1 E×X0 X1×X0 X1(id,m) // E×X0 X1 (α,id) α (cid:15)(cid:15) (cid:15)(cid:15) E×X0 X1 α //E 3 arerequiredtocommute. Isomorphisms,pullbacksandrestrictionsofX-spacesoverX aredefined in the obvious way. We will often write an X-space shortly as p:E →X, with the maps q and α understood. To define principal X-bundles, we start with trivial bundles. The diagram X t //X (2.1) 1 0 s (cid:15)(cid:15) X 0 defines an X-space over X (p:=s, q :=t, α:=m). This serves as the local model for a principal 0 X-bundle. Definition 2.2. Let X be a topological groupoid and let X be a topological space. A principal X-bundle on X is an X-space (E;p,q,α) over X which is locally trivial in the sense that each x∈X has an open neighborhood U ⊂X which admits a map h:U →X and an isomorphism of 0 the restriction E| with the pullback of 2.1 via h. U There is an equivalent notion which is more abstract but also more common in the theory of stacks. It is the notion of X-torsors. We will not use this notion, but we explain it briefly. Given any map T → X of spaces, we can form the groupoid X with object space T and morphism T space T × T. We say that a morphism of topologicalgroupoids X→Y is called cartesian if the X diagrams X //Y X // Y 1 1 1 1 s s t t (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) X // Y X // Y 0 0 0 0 arecartesian. AnX-torsor ona spaceX is a mapp:T →X whichadmitslocalsectionstogether with a cartesian morphism of groupoids φ:X →X. T Given such an X-torsor,a principal X-bundle is given as follows: The total space E is just T and φ0 gives a map E → X0. The action α is the composition T ×X0 X1 ∼= T ×X T → T (projection onto the first factor). Conversely,let (E,p,q,α) be a principal X-bundle. Put T :=E. The map p has local sections by the definitionofaprincipalbundle1. Lete ,e ∈E betwopointsinthe samefibre,p(e )=p(e ), 1 2 1 2 i.e. (e ,e )∈E× E. Thereexistsauniqueγ ∈X suchthatα(e ,γ )=e . Assigning 1 2 X (e1,e2) 1 1 (e1,e2) 2 (e ,e ) 7→γ defines a map φ :E× E → X which is continuous by the local triviality of 1 2 (e1,e2) 1 X 1 a principal bundle. This map fits into a commutative diagram T × T φ // X X 1 s,t s,t (cid:15)(cid:15) (cid:15)(cid:15) T q //X0, whichis easilyseento be cartesian. We will notuse the conceptofa torsoranymorein this note. One can also describe principal X-bundles in terms of transition functions, see [7]. It is worth to spell out what a principal X-bundle is for familiar groupoids. Let Y be a space, consideredasagroupoidXwithoutnontrivialmorphisms. Let(E,p,q,α)be aprincipalX-bundle on X. The maps in 2.1 are identities; thus p : E → X is a homeomorphism; q◦p−1 is a map X →Y, and under these identifications α is the canonical homeomorphism X × X →X. Thus X a principal bundle for the trivial groupoid is the same as a continuous map X →Y. 1Notethatι:X0→X1 isasectiontothemaps. 4 AsimilarsituationismetwhenX=Y isthegroupoidassociatedtoanopencoverU ofthespace U Y (see [13]). The mapE →X isthenanopencoverofX,q is acollectionoflocallydefinedmaps to Y (in fact, to the elements of the originalopen cover). The map α is precisely the information that all these locally defined maps fit together to form a globally defined map X →Y. If G is a topologicalgroup,consideredas a groupoidwith one object, then the notion of principal G-bundle from Definition 2.2 agrees with the traditional notion of a principal bundle, with the exception that we assume that G acts from the left on the total space. Similarly, if G acts on the space Y from the left, we form the groupoid G Y: the object space is R Y, the morphism space is G×Y and the action is used for the structural map of a groupoid. A principalG Y-bundleonaspaceX consistsofaprincipalG-bundleP →X andanG-equivariant R map q :P →Y. The stack of principal X-bundles The collection of all principal X-bundles on a space X, together with their isomorphisms, forms a groupoid which we denote by X //X (X). The functor X 7→ X //X (X) is a (lax) presheaf of 0 1 0 1 groupoids on the site of topological spaces. The local nature of principal bundles shows that this is actually a sheaf of groupoids, in other words, a stack, which we denote by X //X . 0 1 Clearly, morphisms of groupoids yield morphisms of stacks and natural transformations of mor- phismsgive2-morphismsofstacks. Thusthereisa2-functorfromtopologicalgroupoidstostacks, sending X to X //X . This 2-functor is far from being an equivalence of categories. There can be 0 1 a morphism φ:X→Y of groupoids such that the induced morphism φ :X //X →Y //Y is an ∗ 0 1 0 1 equivalence, but φ does not have an inverse. This is not particularily exotic: given a topological group G and a principal G-bundle P → X, then the obvious groupoid morphism G P → X R induces an equivalence of stacks P//G ∼= X, but there is no inverse unless P is trivial. A simi- lar situation is met when U is an open cover of the space X which defines a groupoid X and U an equivalence X → X. It is essential for the theory of stacks to allow for inverses of these U morphisms. It may seemthat the stacksX //X arequite special, but this is not the case. The data of a stack 0 1 X, a space X and a representable map ϕ:X →X determines a topological groupoid X. Namely, X = X, X = X × X; the structure maps for a groupoid are easy to find and the proof of the 0 1 X groupoid axioms is easy as well. Moreover,the map ϕ determines a morphism ϕˆ:X //X →X of stacks. It is easy to see that ϕ is 0 1 a chart (in the sense of [11], Def. 7.1) if and only if ϕˆ is an equivalence of stacks. Definition 2.3. A stack X over the site Top is a topological stack if there exists a topological groupoid X and an equivalence of stacks X //X →X. 0 1 This notion agrees with the notion of a ”pretopologicalstack” defined in [11], Def. 7.1. The universal principal bundle We now want to define the universal principal X-bundle. The topological category X ↓ X is the category of arrows in X; more precisely, an object is an arrow γ : x → y in X and a morphism from (γ :x →y) to (γ′ :x′ →y) is a morphism δ : x →x′ with2 δγ′ := m(δ,γ′) =γ; there is no morphism (γ :x→y)→(γ′ :x′ →y′) if y 6=y′. There is a topology on X ↓ X induced from the topology on X and the functor p : X ↓ X → X; p(γ : x → y) = x, p(δ) = δ is continuous. Similarly, the functor ζ : X ↓ X → X (the trivial 0 groupoid with object space X ) which sends γ :x→y to y is continuous. 0 In this note we say that the classifying space of a topological category C is the thick geometric realization kN Ck of the nerve N C of the category. The thick realization kX k of a simplicial • • • space X is the space • kX•k:= aXn×∆n/∼, n≥0 2Sic;rememberourconvention aboutmultiplicationinagroupoid. 5 where∼isthe equivalencerelationgeneratedby(ϕ∗x,t)∼(x,ϕ t)foranyinjective mapϕinthe ∗ simplex category △. We apply the classifying space construction to the functors p and ζ and obtain a diagram B(X↓X) Bζ //X (2.4) 0 Bp (cid:15)(cid:15) BX and, furthermore, a map B(X↓X)×X0 X1 →B(X↓X) which is given by multiplication. We will abbreviate EX := B(X ↓ X). In the appendix, we shall describe two other convenient models for the spaces BX and EX. Proposition 2.5. The diagram 2.4 is a principal X-bundle. The proof is a straightforward generalization of Milnor’s construction of classifying spaces for topologicalgroups. It is givenin[7]andalsoin[12]. The proofs usesa differentdescriptionofthe topological space BX. We say a few words about the latter point in the appendix. Proposition 2.6. There exists a section σ :X →EX of Bζ and a deformation retraction of EX 0 onto σ(X ) over X . Consequently, the space of sections of Bζ is contractible. 0 0 Proof. Let σ : X → X ↓ X be the functor which sends x to id : x → x. Clearly ζ ◦σ = id and 0 thereisanevidentnaturaltransformationT :idX↓X →σ◦ζ. ThecompositionT◦σ istheidentity transformation. So after realization, σ defines the desired section and T defines the deformation retraction. The well-known theorem that a continuous natural transformationbetween two functors of topo- logical categories defines a homotopy between the maps on classifying spaces still holds if the ge- ometric realizationofthe nerveis replacedby the thick geometricrealization,exceptthat we now get a contractible space of preferred homotopies instead of just one (combine the standard proof of the theoremin [13] with [14], Prop. A.1 (iii) and anexplicit computation ofkN (0→1)k). • Theorem 2.7. Let Euniv →Buniv be a principal X-bundle such that the space of sections to the structure map q : Euniv → X is contractible. Then for any principal X-bundle E → X on a 0 paracompact space X, the space of bundle morphisms E →Euniv is weakly contractible. In particular, this applies to the bundle EX constructed above. Proof. Let us begin with the trivial principal X-bundle X → X from 2.1. It is easy to see that 1 0 the space ofbundle morphismsfrom X →X to Euniv →Buniv is homeomorphic to the space of 1 0 sections s:X →Euniv of q. Thus, by assumption, the spaceof bundle morphisms is contractible 0 in this case. The same argument applies for a trivial principal bundle over a space different from X . 0 To achieve the global statement, we apply a trick, which ought to be standard in the theory of fibre bundles. Let p:E →X (plus the additional data) be a principal X-bundle. Choose an open covering (U ) of X, so that E| is trivial. For any finite nonempty S ⊂ I, let U := U . i i∈I Ui S Ti∈S i Clearly, E|US is trivial as well. Let FS := mapX(E|US,Euniv) be the space of bundle maps from E| toEuniv. We haveseenthatF isweaklycontractible. ForanyT ⊂S,thereis arestriction US S maprT :F →F . Let∆ bethe|S|−1-dimensionalsimplex{ t i∈RS| t =1;t ≥0}. S T S S Pi∈S i Pi i i We now claim that we can choose maps c :∆ →F S S S such that rT ◦c = c | whenever T ⊂ S. This is done by induction on |S|, using the con- S T S ∆T tractibility of F . S 6 Finally, let (λ ) be a locally finite partition of unity subordinate to (U ). For any point x∈X, i i∈I i let S(x)⊂I be the (finite) set of all i∈I with x∈supp(λ ). The formula i c(y)=cS(p(y))( X λi(p(y))i)(y) i∈S(p(y)) definesaglobalbundlemorphism. ThisshowsthatthespaceF ofbundlemorphismsisnonempty. A straightforwardversion of the preceding reasoning shows a relative version of it: if A⊂X is a cofibrant inclusion, then any bundle map E| → Euniv extends to X. Thus F is connected. A A parameterizedversionofthese argumentsshowsthatforanycompactK,map(K;F)isnonempty and connected. Thus F is weakly contractible. Proposition 2.8. The map νX :BX→X0//X1 given by the principal X-bundle 2.4 is a universal weak equivalence. For the proof of 2.8, we shall need a little lemma. Lemma 2.9. Let f : Z → Y be a map between topological spaces. Suppose that for any map p : U → Y, the space of sections to f : U × Z → U is weakly contractible. Then f is a U Y homotopy equivalence. Proof. We only need that the space is nonempty when p= id and path-connected when p =f. Y Theassumptionsimply thatthereexistsasections:Y →Z. Wehaveto showthats◦f :Z →Z is homotopic to the identity. The space of sections to the map f : Z × Z → Z; (z ,z ) 7→ z Z Y 1 2 1 is homeomorphic to the space of maps g : Z → Z with f ◦g = f. The maps id and s◦f both Z belong to that space and by assumption they are connected by a homotopy. Proof of Proposition 2.8. Let Y be a paracompact space and let P be a principal X-bundle on Y, which gives a map Y → X0//X1. We have to show that νX;Y : Y ×X0//X1 BX → Y is a weak homotopy equivalence. Note that the space of sections to νX;Y can be identified with the space of bundle maps P →EX, which is weakly contractible by Theorem 2.7. Likewise, let p:Z →Y be any map and let Q:=p∗P. The space of sections to Z ×X0//X1 BX∼=Z ×Y Y ×X0//X1 BX →Z is homeomorphicto the spaceof bundle maps fromQ to EX, whichis againcontractible. Therefore Lemma 2.9 can be applied. The classical theorem that for a topological group G, the set of isomorphism classes of principal G-bundles on a space Y is in bijective correspondance to the set of homotopy classes of maps Y →BG admits a generalization. Definition 2.10. Let X and Y be stacks. Let h : X → Y be two morphisms. A concordance i between h and h is a triple (h,β ,β ), where h : X×[0,1] → Y and β is a 2-isomorphism 0 1 0 1 i h →j∗h for i=0,1 (j denotes the inclusion X∼=X×{i}⊂X×[0,1]). i i i Let X[Y] be the set of concordance classes of elements in X(Y). The following is an immediate consequence of Proposition 2.8. Corollary 2.11. Let X berepresentedbythe groupoid X. Then thereis a naturalbijection X[Y]∼= [Y;BX]. An appropriate relative version is also true; we leave this to the reader. The last thing we want to showin this sectionis thatProposition2.8actually characterizesEX andBX up to homotopy. Let X be a groupoid presenting the stack X. Let p : X → X be a universal weak equivalence, which gives rise to a principal X-bundle E → X. We have seen that the space of bundle maps E → EX is contractible. Any such bundle map gives rise to a map X → BX. This map is a homotopy equivalences, which follows immediately from 2.8 and from the 2-commutativity of the diagram {{{{{{{{B== (cid:15)(cid:15)X X //X. 7 Paracompact groupoids Later on, we shall need a technical result. Slightly differing from standard terminology, we say that a topologicalspace X is paracompact if any open coveringof X admits a subordinate locally finite partition of unity. If X is Hausdorff and paracompact in the usual sense (i.e. any covering admits a locally finite refinement), then X is paracompact in the present sense, see [10], p. 427. We say that a topological groupoid X is paracompact and Hausdorff if all spaces X of the nerve n areparacompactandHausdorff. Ingeneral,theparacompactnessofX andX doesnotimplythe 0 1 paracompactnessofX. However,there arequite largeclassesofgroupoidswhichareparacompact andHausdorff. ExamplesofstackswhicharepresentablebyparacompactandHausdorffgroupoids include 1. All differentiable stacks modeled on finite-dimensional manifolds3: If X is a manifold and X →Xarepresentablesurjectivesubmersion,thenallfibredproductsX =X× X× ...X n X X are manifolds and hence they are paracompact. 2. Moregenerally,differentiablestacksmodeledonFr´echetmanifolds(undersomecountability conditions) 3. Alltopologicalstacksobtainedbytakingcomplexpointsofalgebraicstacks(somefiniteness condition is involved). 4. Quotient stacks X//G if G and X are metrizable4. Proposition A.3 says that the classifying space BX = kN Xk of a paracompact and Hausdorff • groupoid is paracompact. This is discussed in the appendix. 3 Proof of the main results We shall first prove Theorem 1.4 and then 1.1. Let us set up notation. First of all, let S be a small 2-category and J : S → TopStacks be a 2-functor, such that any stack in the image of J admits a presentation by a paracompact groupoid. To simplify notation, we assume that J is the inclusion of a small subcategory;the argument in the generalcase is the same. Stacks will be denotedbyGermanlettersX,Y,ZandtheyaretacitlyassumedtobeinS;likewiseformorphisms and 2-morphisms. By the corresponding symbols X,Y,Z we will denote paracompact groupoids representing the stacks. For any stack in S we choose a presentation by a paracompact groupoid5, denoted by ϕ : X X0//X1 →X. We canview ϕ−X1 as a principalX-bundle onthe stack X,denoted by UX. Insection 2,weconstructedamapνX :BX→X0//X1. WeconsiderthecompositionηX =ϕX◦νX :BX→X. Consider a morphism f :X→Y of stacks. There is a diagram BX BY νX νY (cid:15)(cid:15) (cid:15)(cid:15) X //X Y //Y 0 1 0 1 ϕX ϕY (cid:15)(cid:15) (cid:15)(cid:15) X f //Y, and the vertical maps are universal weak equivalences by Proposition 2.8. The pullback ηX∗f∗UY is a principal Y-bundle on BX. Let Ef be the space of bundle morphisms from ηX∗f∗UY to the 3Asusual,weassumemanifoldstobesecondcountable. 4It seems unlikely that the paracompactness of both G and X implies the paracompactness of the translation groupoid,becausetheproductoftwoparacompactspaces isingeneralnotparacompact. 5Hereweneedtheaxiomofchoice,henceweusethat S issmall. 8 universalY-bundleEY→BY. ByPropositionA.3BXisparacompactandtherefore,byTheorem 2.7, E is weakly contractible. Thus we get a 2-commutative diagram f Ef ×BX ǫf // BY (3.1) ηX◦p2 ηY (cid:15)(cid:15) (cid:15)(cid:15) X f //Y, in TopStacks; ǫ is the evaluation map and p denotes the projection onto the second factor. f 2 Iff :X→Yandg :Y→Zaremorphisms,thenthereisacompositionmapc :E ×E →E g,f g f g◦f which makes the diagram Egf ×OO BX ǫgf // BOOZ cg,f×id ǫg E ×E ×BX id×ǫf// E ×BY g f g commutative. The map cg,f arises in the following way. There is a specified map EY → UY of Y-bundles. Thus any element of Ef defines a map ηX∗f∗UY → ηY∗UY of Y-bundles and therefore, after application of the morphism g of stacks, a map ηXf∗g∗UZ → ηY∗g∗UZ of Z-bundles, which can be composed with any element of E . g Thecollectionofthesemapsisassociativeinthesensethatc ◦(c ×id)=c ◦(id×c ): h◦g,f h,g h,g◦f g,f E ×E ×E →E when h,g,f are composable morphisms. Also, id∈E . h g f h◦g◦f id Any 2-isomorphism φ : f → g yields an isomorphism ηX∗f∗UY ∼= ηX∗g∗UY and hence a homeomor- phism φ∗ :E →E . Moreover,the diagram g f Eg ×BX ǫf //BY (3.2) φ∗×id (cid:15)(cid:15) Ef ×BX ǫg // BY is commutative. ] Now we define an topological category τ S. It has the same objects as S (and the discrete ≤1 topology on the object set). The morphism spaces are ] τ≤1S(X;Y):= a Ef S(X;Y) with the composition described above. The morphism spaces have contractible components (one ] for each 1-morphism in S). There is an obvious functor p:τ S →τ (S). ≤1 ≤1 The classifying space construction determines a continuous functor ] Ho:τ S →Top, ≤1 f which sends a stack X to the space BX and which is the identity on morphism spaces. The universal weak equivalences ηX assemble to a pseudo-natural transformation of functors η :st◦Ho→J ◦p f ] offunctorsτ S →τ TopStacks (use3.1toverifythis). ThisfinishestheproofofTheorem1.4. ≤1 ≤1 Theorem 1.1 follows from Theorem 1.4 and a rectification procedure for homotopy-commutative diagrams which is proven in Nathalie Wahl’s paper [15], following ideas of Dwyer, Kan and G. Segal. Roughly, she proves that a functor which is defined up to contractible choices can be strictified. More precisely, Proposition2.1. of loc. cit can be reformulated as follows. 9 Proposition 3.3. [15] Let C be a discrete small6 category and let (C,p,F) be a functor to Top which is defined up to contractible choice. Then there exists a functorep F :C →Top and a zig- ∗ zag of natural transformations, which are weak homotopy equivalences on all objects, connecting p∗p F and F. ∗ To finish the proof of Theorem 1.1, take the functor Ho from Theorem 1.4 and put f Ho(X):=p Ho(X). ∗ f The additional assertions about 2-isomorphic and concordant morphisms of stacks follow from diagram 3.2 and 2.11, respectively. Uniqueness of the homotopy type functor Sofar,wehavenotaddressedthequestionwhetherourconstructionisunique. Hereisauniqueness result which seems to be satisfactory enough. Let S be a small category of stacks, all of whose objects can be presented by paracompact ] groupoids. Let (τ S,p,Ho) be the homotopy type functor and η :st◦Ho→J ◦p be the natural ≤1 transformationconstructefdinTheorem1.4. Let(τ]S′,p′,Ho′)beafunfctorτ S →Top,defined ≤1 ≤1 ′ f uptocontractiblechoiceandletη′ :st◦Ho →J ◦p′ be anaturaltransformationoffunctorssuch that η′ is a universal weak equivalence ffor any object X of S. These two functors together yield X a functor defined up to contractible choice on the category S×{0,1}. Using the arguments from the proof of Theorem 1.4, one can easily show: Proposition3.4. Underthecircumstances above, thereexists afunctorHo:S×(0→1)→Top, defineduptocontractiblechoicewhichagreeswith(τ]S,p,Ho)onS×{0}andwith(τ]S′,p′,Ho′) ≤1 ≤1 on S×{1}. The natural transformations η and η′ yield a nfatural transformation. f A Appendix: Point-set-topology of classifying spaces Let X be a topological groupoid. There are three different descriptions of the universal X-bundle each of which has some advantages. The first model, which was used in section 2 is the thick geometric realization of the categories X ↓ X and X. The use of this model makes the proofs of Lemma 2.6 and hence Theorem 2.7 particularly transparent. Another descriptiongoes backto Haefliger[7]; it is a generalizationofMilnor’s classicalconstruc- tion [9]. Let EMilX be the space consisting of sequences (t f ,t f ,...), where f ∈ X all have 0 0 1 1 i 1 the same target; t ∈ [0,1], t = 0 for all but finitely many i ∈ N, t = 1. Two sequences i i Pi i (t f ,t f ,...) and (t′f′,t′f′,...) are equivalent if t =t′ for each i and f =f′ whenever t 6=0. 0 0 1 1 0 0 1 1 i i i i i The topology on EMilX is the weakest topology such that the maps t : EMilX → [0,1] and i f :t−1(0,1]→X are continuous. i i 1 To obtain the space BMilX, we divide by the following equivalence relation. Two sequences (t f ,t f ,...) and (t′f′,t′f′,...) are identified if t =t′ for each i and if there exists an f ∈X 0 0 1 1 0 0 1 1 i i 1 such that f′ =f f for all i. i i A map EMilX → X is defined by sending (t f ,t f ,...) to the common target of the f ’s, and 0 0 0 1 1 i theactionofXisequallyeasytodefine. ThisdescriptionisconvenientfortheproofofProposition 2.5, see [12]. Lemma A.1. [13] There are natural homeomorphisms kN•Xk ∼= BMilX and kN•X ↓ Xk ∼= EMilX. There is a slightly differentdescriptionof kN Xk which helps to show PropositionA.3 below. Let • X be an arbitrary simplicial space. Let sk kX k:= X ×∆k/∼. Then • n • Sk≤n k kX•k∼=colimnsknkX•k 6Thisisanessentialassumption. 10