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An extension of Quillen's Theorem B

We prove a general version of Quillen's Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe's theorem on the stability of homotopy colimits in an infinity-topos, due to Rezk.


Introduction
Theorem B is one of the first results in Quillen's influential paper 'Higher K-Theory I' [13] and as such plays an important role in the foundations of algebraic K-theory. For a functor f : D → C between small categories, this theorem provides a way to identify the homotopy fibre of the induced map BD → BC between classifying spaces: it is the classifying space of the over-category D/x, provided that for each morphism x → y in C, the functor D/x → D/y induces a weak equivalence between the associated classifying spaces. This condition can also be phrased by saying that the classifying spaces of these various D/x form a diagram of spaces over C, on which C acts by weak equivalences. From this point of view, the theorem is very close to other results in the literature, such as Volker Puppe's theorem [12] on homotopy colimits of homotopy cartesian diagrams. A version of this theorem also holds for actions by homology equivalences, and this version yields the group completion theorem [9,14,10] and Bott periodicity [4].
These results all predate the development of Quillen model categories and their left Bousfield localizations, the homotopy theory of simplicial presheaves and sheaves, and the theory of ∞-categories and ∞-toposes. The purpose of this paper is to reconsider Quillen's Theorem B in the light of these developments. We will prove a very general version of Theorem B over an arbitrary site, for actions of a presheaf of simplicial categories on another simplicial presheaf. This general theorem states that if the action is by weak equivalences in some further left Bousfield localization of one of the standard model structures on simplicial presheaves, then the fibre and the homotopy fibre of the action become equivalent in this localization. See Theorem 5.1 below for a precise formulation.
This theorem has the expected applications, such as a version of the group completion theorem for actions of presheaves of simplicial monoids (such as the classifying space of a coproduct n BGL n (R) for a sheaf of rings R) (Examples 6.7 and 6.10 below) and a version of Puppe's theorem for homotopy cartesian morphisms between diagrams of simplicial presheaves over a site (Example 6.2 below). When applied to a left exact localization of simplicial presheaves, this result gives precisely what is sometimes referred to as (Rezk) descent for ∞-toposes [15]. When R is a sheaf of commutative rings on a site, the theorem shows that the associated projective space P ∞ is A 1 -homotopy equivalent to its group completion ΩB(P ∞ ) (Example 6.6). We expect our general version of Quillen's theorem to have further applications when applied to specific sites such as the Nisnevich topology for A 1 -homotopy theory [11].
The plan of this short paper is as follows. In Sections 2 and 3 we review the homotopy theory of simplicial and bisimplicial presheaves and sheaves. This material is largely standard, and can be found in many sources of which we will mention the main ones. In Section 4 we introduce the necessary notation and terminology for actions by categories on simplicial presheaves, so as to state and prove the main theorem in Section 5. Our proof closely follows the strategy of [10]. We conclude with some applications in Section 6.

Simplicial presheaves and sheaves
In this section we review some basic definitions and facts about the homotopy theory of simplicial presheaves and sheaves. Almost everything in this section traces back to [2,5,6].
Let (S, J) be a site, i.e. a small category S equipped with a Grothendieck topology J. Let PSh(S) and Sh(S, J) be the categories of presheaves (resp. sheaves) of sets on S and let be the adjoint pair given by the full embedding i * and the associated sheaf functor i * . By adjointness i * preserves all limits and i * preserves all colimits, while in addition i * preserves finite limits.
A point of the topos Sh(S, J) (or "of the site (S, J)") is such an adjoint pair p * : Sh(S, J) / / Sets : p * o o for which p * preserves finite limits (i.e. the pair forms a geometric morphism p : Sets → Sh(S, J)). The topos Sh(S, J) is said to have enough points if the collection of functors p * , for all points p of Sh(S, J), is jointly conservative (i.e. detects isomorphisms). Equivalently, Sh(S, J) has enough points if there exists a topological space X and a geometric morphism f : Sh(X) → Sh(S, J) for which f * is conservative (i.e. "f is surjective"). Many sites occurring in nature have enough points [3,8] and in some definitions and arguments we will assume that there are enough points, in order to help develop some intuition and to connect to the classical homotopy theory of simplicial sets. However, this assumption is never essential and can be circumvented by either working with a surjective "Boolean point" or by using the internal logic of Sh(S, J).
The adjoint pair (2.1) induces an adjoint pair between the categories of simplicial presheaves and sheaves. The category sPSh(S) can be endowed with the projective model structure, for which the fibrations and weak equivalences are defined levelwise: a map Y → X of simplicial presheaves on S is a fibration or weak equivalence if for each object S ∈ S, the map Y (S) → X(S) is a fibration or weak equivalence in the classical Kan-Quillen model structure on simplicial sets. All model categorical notions for presheaves will refer to the projective model structure, unless stated otherwise.
The category sSh(S, J) carries the Joyal or injective model structure, for which the cofibrations are the monomorphisms and the weak equivalences are the so-called local weak equivalences: a map Y → X of simplicial sheaves is called a local weak equivalence iff the map p * Y → p * X is a weak equivalence of simplicial sets for every point p. There are rather few fibrations in this model structure, but there is a wider class of so-called local fibrations, viz. the maps Y → X for which each p * Y → p * X is a Kan fibration. Equivalently, these are the maps for which the map is a surjection of sheaves of sets, for each n ≥ 1 and each 0 ≤ k ≤ n. Here we use that each simplicial set K and simplicial sheaf Y determine a sheaf (of sets) Y (K), determined by Alternatively, using that the Joyal model structure is a simplicial model structure, one can identify Y (K) with the sheaf of vertices of Y K . Similarly, a local trivial fibration is a map Y → X for which each p * Y → p * X is a trivial fibration of simplicial sets, or equivalently, for which each map is a surjection of sheaves of sets. One easily verifies that the adjoint pair is a Quillen pair. Among general Quillen pairs, it has some additional properties that are useful to keep in mind: (a) i * preserves weak equivalences between arbitrary objects (not just cofibrant ones). (b) Let us say that a map of simplicial presheaves Y → X is a local weak equivalence (resp. a local (trivial) fibration) if its image under i * is such.
Since i * preserves finite limits (as we already mentioned), it follows that any levelwise (trivial) fibration between simplicial presheaves is a local (trivial) fibration.
It follows from (a) and the fact that i * i * ∼ = id that Li * Ri * ≃ id, so that sSh(S, J) is a localization of sPSh(S). Since the weak equivalences in sSh(S, J) form an accessibly embedded accessible subcategory of the arrow category of sSh(S, J), it follows that there exists a left Bousfield localization sPSh(S) J of (the projective model structure on) sPSh(S) whose weak equivalences are the local weak equivalences. In this way one obtains a diagram of left Quillen functors As ordinary functors, j * and its right adjoint j * can be identified with i * and i * . The pair j * and j * forms a Quillen equivalence because Lj * Rj * ≃ id and j * preserves and detects weak equivalences. We will use the following simple observations. Lemma 2.2. In sSh(S, J), as well as in sPSh(S) J , the pullback along a local fibration is a homotopy pullback.
Proof. The two cases are proved in the same way. Let be a pullback in sSh(S, J) (or in sPSh(S) J ) in which f is a local fibration. The image of this pullback square under a point of Sh(S, J) is a homotopy pullback of simplicial sets, since the usual model structure on simplicial sets is right proper. In particular, pullbacks along local fibrations preserve local weak equivalences, so that sSh(S, J) is right proper as well. Now let Z ∼ ֒→ Z ′ → X be a factorization of g into a local weak equivalence, followed by a fibration. Then the pullback Z ′ × X Y computes the homotopy pullback of f and g and the map W → Z ′ × X Y is a local weak equivalence.
is a homotopy pullback in sPSh(S).
Proof. Let i * (Y ) → P → i * (X) be a factorization into a weak equivalence, followed by a fibration of simplicial presheaves. Then Q = P × i * (X) i * (Z) is the homotopy pullback in sPSh(S) since this model category is right proper. So i * (Q) ∼ = i * (P ) × X Z and i * (P ) fits into a sequence of a local weak equivalence, followed by a local fibration. The lemma now follows from Lemma 2.2.

Bisimplicial presheaves and sheaves
We will write bisSh(S, J) and bisPSh(S) for the categories of bisimplicial sheaves and presheaves on the site (S, J). These carry several model structures, but we will mostly be interested in the "diagonal" one, making the model categories Quillen equivalent to sPSh(S) and sSh(S, J), respectively. More precisely, write for the associated sheaf functor i * and its fully faithful right adjoint, and let be the diagonal functor. The functor δ * has a left adjoint δ ! and a right adjoint δ * . Using the same notation for sheaves, we obtain a diagram of adjoint pairs which are related by the following natural isomorphisms Proposition 3.1 (cf. [10]). The (projective, resp. Joyal) model structures can be transferred along the adjoint pair (δ ! , δ * ) and give model structures and Quillen equivalences We prove the second case; the other two cases are similar. To show that the transferred model structure exists, it suffices to verify that δ * δ ! maps generating trivial cofibrations to local weak equivalences that are monic. Indeed, these maps are stable under pushout and transfinite composition while δ * and δ ! both commute with colimits.
It is easy to check that δ * δ ! preserves monomorphisms. The fact that it preserves local weak equivalences follows immediately from the fact that the unit map X → δ * δ ! (X) is a levelwise weak equivalence of simplicial presheaves. Indeed, this just follows from the analogous statement for simplicial sets: by a standard skeletal induction it suffices to verify that for every simplex ∆[n], the map ∆ Similarly, the fact that X → δ * δ ! (X) is a levelwise weak equivalence shows that the Quillen pair is a Quillen equivalence (because δ * preserves and detects weak equivalences).
The proof of Proposition 3.1 applies equally well to further left Bousfield localizations of these model categories. More precisely, let λ be a set of maps (which one can always take to be cofibrations) in sPSh(S) and let us denote by i * : sPSh(S) J,λ / / sSh(S, J) λ : i * o o the associated Quillen equivalence between the left Bousfield localizations at λ and i * (λ), respectively. We will refer to the weak equivalences in these model structures as λ-equivalences (leaving the reference to J implicit when working with simplicial presheaves). The argument of Proposition 3.1 shows that these two model structures can be transferred to model structures on bisimplicial (pre)sheaves along (δ ! , δ * ), yielding two Quillen equivalences In fact, the transferred model structure bisPSh(S) J,λ is simply the left Bousfield localization of bisPSh(S) J at the set of maps δ ! (λ), and similarly for sheaves. Lemma 3.3. Let f : X → Y be a map of bisimplicial (pre)sheaves over S. If f induces a λ-equivalence X n,− → Y n,− of simplicial (pre)sheaves for each n ≥ 0, then the diagonal δ * X → δ * Y is a λ-equivalence as well.
Proof. This follows from the fact that δ * : bisSh(S, J) → sSh(S, J) λ is a left Quillen functor for the Reedy model structure on bisSh(S, J) = sSh(S, J) ∆ op .

Actions on simplicial presheaves and sheaves
We begin with some terminology and notation. Let C be a category object in one of the (model) categories sPSh(S) or sSh(S, J). Thus C is given by simplicial (pre)sheaves ob(C) and mor(C) of objects and morphisms, together with structure maps for source and target and two more structure maps for units and composition, all satisfying the usual identities. For any such category object C, its nerve N C is a bisimplicial (pre)sheaf whose diagonal we denote BC = δ * N C and call the classifying (pre)sheaf or "space" of C. Thus, BC is an object of sPSh(S) or sSh(S, J).
A left action of C on a simplicial presheaf X is given by maps π : X / / ob(C) and µ : s * (X) = mor(C) × ob(C) X / / X satisfying the usual identities (which express that for any S ∈ S, the components π S and µ S determine a covariant simplicial functor C(S) → sSet, natural in S).
The domain of the map µ is the pullback s * (X) of π along s. Such an action by C on X defines a new category object X C in sPSh(S) (or in sSh(S, J)) with while the new source and target map s * (X) ⇒ X are the projection and the action µ. For any object S ∈ S and any simplicial degree n, the category X C (S) n (in sets) can therefore be described as follows: the objects are n-simplices x ∈ X(S) n and a morphism x → y is a morphism φ : π(x) → π(y) in the category C(S) n such that µ(φ, x) = y. There is an obvious projection functor π : X C / / X which induces a map of classifying spaces For any n-simplex c ∈ ob(C)(S) n , i.e. a map S × ∆[n] → ob(C) of simplicial presheaves, we write X(c) for the pullback → BC and X(c) fits into a pullback of simplicial (pre)sheaves The action µ defines a map µ = (π 1 , µ) over mor(C) z z t t t t t t t t t mor(C).
If φ ∈ mor(C)(S) n is a morphism from c to d, i.e. φ : S × ∆[n] → mor(C) with sφ = c and tφ = d, then µ restricts to a map of simplicial presheaves Given a set of maps λ in sPSh(S) J or sSh(S, J), we can require these action maps to be weak equivalences in the resulting left Bousfield localization: Definition 4.1. Let C be a category acting on X in sPSh(S), as above. Then C is said to act by λ-equivalences if for any object S ∈ S and any morphism φ : There are some conditions closely related to this definition. Let us call a map of simplicial presheaves over a simplicial presheaf B Proof. Let φ : c → d be as in the definition and for any i = 0, . . . , n, consider the pullback where v i is the inclusion of the i-th vertex and u i and w i are its pullbacks. Each of these three maps is a local weak equivalence by Lemma 2.2, so that φ * is a λ-equivalence if and only if (φ i ) * is. Lemma 4.3. Let C be a category acting on X in sPSh(S), as above. Then C acts on X by λ-equivalences iff µ : s * (X) → t * (X) is a stable λ-equivalence over mor(C).
Proof. Since the maps φ * are pullbacks of µ over mor(C), the condition of the lemma is clearly sufficient. For the converse, consider a map A → mor(C) and let be the pullback of µ along A → mor(C). Consider the bisimplicial presheafX(s) ∈ bisPSh(S) whose value on an object S ∈ S has as (p, q)-simplices diagrams of the form In the same way, letX(t) be the bisimplicial presheaf obtained using t * (X) A instead of s * (X) A . For fixed S and p, the simplicial setX(s)(S) p is the nerve of a category whose objects are pairs consisting of a p-simplex of s * (X) A and a factorization of ∆[p] → s * (X) A (S) → A(S) through a simplex ∆[n] (as in the above diagram, for q = 0). For a fixed p-simplex of s * (X) A , there is an initial such factorization, so that there is a (natural) weak equivalence s * (X) A,p / /X (s)(S) p from a discrete simplicial set to the simplicial setX(s)(S) p . Taking diagonals, it follows that there is a (projective) weak equivalence of simplicial presheaves The same holds for t * (X) A → δ * X (t), of course.
On the other hand, in each fixed simplicial degree q, the mapX(s) −,q →X(t) −,q is a coproduct of maps φ * : X(c) → X(d), indexed by the composite maps These maps φ * are λ-equivalences by assumption, so the map δ * X (s) → δ * X (t) is a λ-equivalence as well, by Lemma 3.3. The commutative square now shows that s * (X) A → t * (X) A is a λ-equivalence, which finishes the proof.

The main theorem
In this section we will state and prove the main theorem. Some examples and applications have already been mentioned in the introduction and will be elaborated on in the next section. As before, we work over a fixed site (S, J) and consider the projective local model structure on sPSh(S) J , the injective one on sSh(S, J), as well as left Bousfield localizations of these at a set of maps λ.
Theorem 5.1. Let C be a category object acting on a simplicial presheaf X by λ-equivalences. Suppose π : X → ob(C) is a local fibration. Then for any object S ∈ S and any c ∈ C(S) 0 , the map from the pullback X(c) as in / / BC to the homotopy pullback is a λ-equivalence.
Remark 5.3. Note that the theorem refers to the homotopy pullback in the projective model structure and not in the λ-localized model structure. Of course, the two notions coincide in the case where the localization is (homotopy) left exact. This is the case where the model category sPSh(S) λ presents an ∞-topos.
It will be clear that our proof for presheaves applies to sheaves as well, but in fact the case of sheaves is also just a direct consequence: Corollary 5.4. Consider a left Bousfield localization sSh(S, J) λ of the Joyal model structure. If a category object C acts on a simplicial sheaf X by λ-equivalences and the map X → ob(C) is a local fibration, then the map is a λ-equivalence, where the homotopy pullback is computed in the Joyal model structure.
Proof. Form the homotopy pullback of simplicial presheaves The left Bousfield localization sSh(S, J) λ is Quillen equivalent to the left Bousfield localization sPSh(S) J,λ and the map i * X(c) → Q is a λ-equivalence of simplicial presheaves. It follows that X(c) → i * Q is a λ-equivalence of simplicial sheaves, so that the result follows from Lemma 2.3.
Proof (of Theorem 5.1). We follow the strategy from [10]. The square (5.2) in the theorem is obtained by applying the diagonal functor δ * to the pullback square of bisimplicial presheaves (5.5) Here X(c) and S ×∆[0] are considered as bisimplicial presheaves which are constant in one simplicial direction. It thus suffices to prove the theorem for the homotopy pullback of (5.5) in bisPSh(S). This homotopy pullback can be formed by factoring the map S × ∆[0] → N (C) as a trivial cofibration, followed by a fibration and then taking the pullback of N (X C ) → N (C) along that fibration. Such a factorization is obtained in the standard way from the small object argument, as a transfinite composition of pushouts of generating trivial cofibrations, i.e. maps T × δ ! Λ k [n] → T × δ ! ∆[n] for any object T ∈ S. Since pulling back along a map commutes with colimits in bisimplicial presheaves, it thus suffices to show that for any pullback diagram of the form / / N C (where i denotes the inclusion), the map X σi → X σ becomes a λ-equivalence after applying δ * . Indeed, then the map δ * (X σi ) → δ * (X σ ) becomes a trivial cofibration in the λ-localization of the injective model structure, and a transfinite composition of pushouts of these remains a λ-equivalence.
Let us explicitly spell out the bisimplicial presheaves X σ and X σi . The map σ : T × δ ! ∆[n] → N C is a string of morphisms and x ∈ X(R) q is an element whose image under π : X → ob(C) satisfies An object of X σi is a similar quadruple (f, α, β, x) satisfying the additional condition that there is some l = 0, . . . ,k, . . . , n such that α and β both miss l. Now consider the bisimplicial presheaves X 0 σ and X 0 σi whose (p, q)-simplices at R are quadruples (f, α, β, x) exactly as before, except that we require (so c 0 instead of c α(0) ). These bisimplicial presheaves fit into a commuting square where the vertical maps σ * = (σ α(0) • · · · • σ 1 ) * are induced by the action of C on X.
The top inclusion X 0 σi → X 0 σ fits into a pullback diagram of bisimplicial presheaves where all objects in the most right square are constant in one simplicial direction (the p-direction, in the above notation). Since the diagonal functor δ * preserves limits, it follows that δ * (X 0 σi ) → δ * (X 0 σ ) is the pullback of a (local) weak equivalence along the (local) fibration X → ob(C). Lemma 2.2 then implies that δ * (X 0 σi ) → δ * (X 0 σ ) is a (local) weak equivalence as well. To finish the proof, it remains to verify that the two vertical maps σ * induce λ-equivalences on the diagonals. But for a fixed p, the action map σ * : X 0 σ → X σ is a coproduct over α : [p] → [n] of maps of the form x x r r r r r r r r r r T × ∆[n].
These maps are all λ-equivalences of simplicial presheaves by assumption, so the induced map on diagonals is a λ-equivalence by Lemma 3.3.

Examples
Example 6.1 (Quillen's Theorem B). Let f : D → C be a functor between categories. Let X c = N (f /c) be the nerve of the comma category f /c for c ∈ C. These X c form a covariant diagram of simplicial sets indexed by C. The category C acts by weak equivalences on this diagram if for each α : c → c ′ in C, the functor f /c → f /c ′ induces a weak equivalence on nerves.
As a very special case of Theorem 5.1, we find that if this is the case, then X c is the homotopy fibre of hocolim X / / BC. The space hocolim X is the nerve of the category f /C and the spaces X c are the nerves of the fibres of the functor f /C → C [16].
There is an inclusion , which is left adjoint to the obvious projection f /C → D. This functor induces a homotopy equivalence on nerves, so that the map hocolim X → BC is homotopy equivalent to the map BD → BC. We therefore obtain Quillen's original Theorem B, identifying the homotopy fibre of BD → BC over c ∈ C with the nerve of f /c. Theorem 5.1 gives an extension to localizations (e.g., to the case where each X c → X c ′ is a homology isomorphism), as well as to functors D → C between (pre)sheaves of categories on a site (S, J). Example 6.2 (Homotopy colimits and Puppe's theorem). Let I be a small category and let X and Y be two I-indexed diagrams of simplicial sets. A natural transformation f : Y → X is called homotopy cartesian if for any morphism α : i → j in I, the naturality square is a homotopy pullback. Puppe's theorem [12] states that for any homotopy cartesian transformation f and any i 0 ∈ I, the square is a homotopy pullback. This theorem is in fact a special case of Theorem 5.1 (for the trivial site, so for simplicial sets rather than simplicial (pre)sheaves). Indeed, let C be the simplicial category X I with space of objectsX = i∈I X i and space of morphisms The natural transformation f defines an action of X I onỸ = i∈I Y i . After replacing Y → X by a fibration in the projective model structure on sSet I , the hypothesis on the squares (6.3) mean precisely that X I acts by weak homotopy equivalences. The space BX I is a model for hocolim i X i , and Theorem 5.1 gives for this special case that (6.4) is a homotopy pullback.
Still working on the trivial site, Theorem 5.1 gives variations of Puppe's theorem for left Bousfield localizations. For example, suppose that all the squares (6.3) are "homology cartesian", in the sense that for each vertex x ∈ X i , the map from the homotopy fibre of f i over x to the one of f j over α * (x) is a homology equivalence. Then the map from Y i0 to the homotopy pullback inscribed in (6.4) is also a homology equivalence.
For a left Bousfield localization λ of the model category sPSh(S) J or sSh(S, J), we obtain a similar result for a map f : Y → X between I-diagrams of simplicial (pre)sheaves: Theorem 5.1 states that the map (homotopy pullback in the non-localized model structure) is a λ-weak equivalence whenever the map between homotopy fibres is a λ-weak equivalence for each i ∈ I and each vertex x ∈ X i (S). If the localization λ is left exact, then Theorem 5.1 translates into the statement that if each square (6.3) is homotopy cartesian in the λ-localized model structure, then so is each pullback square (6.4). This is a version of Puppe's theorem for ∞-toposes, which is also referred to as descent, cf. [15] or [7, Chapter 6.1.3]. To state this criterion, let us assume that for any λ-equivalence X → X ′ between simplicial presheaves and any S ∈ S, the map X × S → X ′ × S is a λ-equivalence. This holds in various cases, e.g. for ∞-toposes (cf. Example 6.2) and for A 1 -model structures [11] (cf. Example 6.6). It follows that f × g : This is unital and associative up to homotopy, so that homotopy classes of maps into M ′ form a monoid and π λ 0 (M ) is a sheaf of monoids. The map M → ΩBM is a λ-equivalence whenever π λ 0 (M ) is a sheaf of groups.
To see this, take S ∈ S and m ∈ M 0 (S), with image m ′ in M ′ 0 (S). To see that m * : M /S → M /S is a λ-equivalence, it suffices to verify that m ′ * : M ′ /S → M ′ /S is a λ-equivalence. Because π λ 0 (M ) is a sheaf of groups, there is a cover α i : is a homotopy equivalence. Similarly, the restriction of m ′ to an iterated pullback S i0...in admits a homotopy inverse and (m ′ i0...in ) * is a homotopy equivalence as well. These weak equivalences assemble into a natural weak equivalence of bisimplicial presheaves i0...in then endows A ∞ * and its quotient P ∞ with the structure of a commutative monoid. Let us use the criterion of Example 6.5 to verify that is an A 1 -weak equivalence. In fact, π A 1 0 (P ∞ ) is the terminal sheaf: for any S ∈ S and any point Identifying A 1 -homotopic elements in π 0 (P ∞ ) therefore yields the terminal sheaf, which implies that π A 1 0 (P ∞ ) is terminal as well (by [11,Corollary 3.22]). Examples 6.7 and 6.10 and Variant 6.11 generalize the classical argument of the group completion theorem (see [9,14]) to Bousfield localizations of simplicial (pre)sheaves. We only describe the case of simplicial sheaves, the case of simplicial presheaves being completely analogous. sending each simplicial sheaf X to its homology sheaves, i.e. the associated sheaves of the presheaves H * (X(−); Z). This functor has the following properties: (1) It sends local weak equivalences to isomorphisms of sheaves of graded abelian groups. (2) If X : I → sSh(S) is a filtered diagram of simplicial sheaves, then the natural map colim h * (X i ) / / h * (hocolim X i ) is an isomorphism.
(3) Let X and Y be two I-indexed diagrams of simplicial sheaves and let f : X → Y be a natural transformation between them. If each h * (X i ) → h * (Y i ) is an isomorphism, then the map h * (hocolim X) → h * (hocolim Y ) is an isomorphism. (4) It is lax symmetric monoidal, i.e. there are natural maps where ⊗ denotes the usual tensor product of sheaves of graded abelian groups. In particular, h * sends simplicial monoids to graded rings. (5) h * is part of an indexed functor in the following sense. For every sheaf (of sets) S, its category of elements S/S inherits a natural Grothendieck topology from (S, J). As in (6.8), there is a functor (h * ) /S taking the homology of simplicial sheaves over S/S, which satisfies conditions (1) - (4). For any map of sheaves f : S → T , these functors fit into a square which commutes up to natural isomorphism sSh(S/T ) Here f * restricts a (simplicial) sheaf along the functor S/S → S/T . because π 0 (M ) ∼ = N is generated by a single element 1. The group completion theorem of Example 6.7 now asserts that the map induces an isomorphism on homology sheaves. which takes values in sheaves of graded modules over a sheaf of graded-commutative rings E * ( * ) and satisfies conditions (1) -(5) above.