HYPERCOVERS AND SIMPLICIAL PRESHEAVES

DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN

Abstract. We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

Contents

1. Introduction 1 2. Model structures on simplicial presheaves 5 3. Local weak equivalences and local lifting properties 6 4. Background on hypercovers 8 5. Hypercovers and lifting problems 12 6. Hypercovers and localizations 16 7. Fibrations and descent conditions 19 8. Other applications 23 9. Verdier sites 29 10. Internal hypercovers 32 Appendix A. Čech localizations 33 References 42

1. Introduction

This paper is concerned with the subject of homotopical sheaf theory, as it has developed over time in the articles [I, B, BG, Th, Jo, J1, J2, J3, J4]. Given a fixed Grothendieck site 𝒞, one wants to consider contravariant functors F defined on 𝒞 whose values have a homotopy type associated to them. The most basic question is: what should it mean for F to be a sheaf? The desire is for some kind of local-to-global property—also called a descent property—where the value of F on an object X can be recovered by homotopical methods from the values on a cover. Perhaps the earliest instance where such a concept had to be tackled was in algebraic geometry, where people had to deal with presheaves of chain complexes defined on a space X. Because of its abelian nature this could be handled by classical homological algebra, and led to the Grothendieck definition of hypercohomology. Much later, people encountered the non-abelian example of algebraic K-theory. Here the site 𝒞 is a category of schemes, and the functor F assigns to each scheme X its algebraic K-theory spectrum K(X). Thomason’s paper [Th] (building on earlier work from [B, BG]) combined homotopy theory and sheaf theory to study the descent properties of this functor.

The work of [BG, Jo, J2] brought the use of model categories into this picture. In the most modern of these [J2], Jardine defined a model category structure on presheaves of simplicial sets with the property that the weak equivalences are local in nature. Classical invariants such as sheaf cohomology arise in this setting as homotopy classes of maps into certain Eilenberg-Mac Lane objects, and the whole theory can in some sense be regarded as the study of non-additive sheaf cohomology. Jardine’s model structure has recently served as the foundation from which Morel and Voevodsky built their 𝔸^{1}-homotopy theory for schemes [MV].

One important ingredient missing from Jardine’s work is a description of the fibrant objects and the fibrations. They can be characterized in terms of a certain lifting property, but this is not so enlightening and not very useful in practice. Our main goal in this paper is to give explicit, simple characterizations of the fibrations and fibrant objects in terms of descent conditions. This is equivalent to describing Jardine’s model category as a very explicit Bousfield localization.

To explain the basic ideas, let’s assume our site is the category of topological spaces equipped with the usual open covers. A presheaf of sets F is a sheaf if F (X) is the equalizer of ∏_{a} F (U_{a}) ⇉ ∏_{a,b} F (U_{a} ∩ U_{b}) whenever {U_{a}} is an open cover of X. This equalizer is in fact the same as the inverse limit of the entire cosimplicial diagram

\xymatrix{∏_{a} F (U_{a}) \ar[r] \ar[r] & ∏_{a,b} F (U_{ab}) \ar[r] \ar[r] \ar[r] & ⋯}

where we have abbreviated U_{a_{0}...a_{n}} for U_{a_{0}} ∩ ⋯ ∩ U_{a_{n}} and have refrained from drawing the codegeneracies for typographical reasons. For a presheaf of simplicial sets (or taking values in some other homotopical objects like spectra), it is natural to replace the limit by a homotopy limit. So one requires that F (X) be weakly equivalent to the homotopy limit of the above cosimplicial diagram. This property, when it holds for all open covers, is called Čech descent. It can also be expressed in a slightly more compact way, if one recalls that the Čech complex ČU associated to a cover {U_{a}} of X is the simplicial object [n] ↦ ∐_{a_{0}···a_{n}} U_{a_{0}···a_{n}}. Then F satisfies Čech descent if the natural map

F (X) → holim_{n} F (ČU_{n})

is a weak equivalence.

A motivating example is given by the functor 𝒯op^{op} → 𝒮pectra taking X to E^{X}, where E is a fixed spectrum and E^{X} denotes the function spectrum. This functor has Čech descent, because X is weakly equivalent to the homotopy colimit of the Čech complex associated to any open cover of X—this was shown in [DI1, Thm. 1.1].

Now, it is not true that the fibrant objects in Jardine’s model category are just the simplicial presheaves which satisfy Čech descent (although this erroneous claim has appeared in a couple of preprints, for instance [HS]). See the appendix, Example A.10, for an example. What we show in this paper is that one has to instead consider descent for all hypercovers. A hypercover is a simplicial object U, augmented by X, which is similar to a Čech complex except in level n we only need to have a cover of the n-fold intersections U_{a_{0}···a_{n}}. A precise definition requires a morass of machinery (see Section 4). A simplicial presheaf F satisfies descent for the hypercover U → X if the natural map

F (X) → holim_{n} F (U_{n})

is a weak equivalence (see Definition 4.3).

What we will show is that the fibrant objects in Jardine’s model category are essentially the simplicial presheaves which satisfy descent for all hypercovers (there is an analagous criterion for fibrations in terms of a relative descent property, given in Section 7):

Theorem 1.1. The fibrant objects in Jardine’s model category sP re(𝒞)_{ℒ} are those simplicial presheaves that:

(1) are fibrant in the injective model structure sP re(𝒞), and

(2) satisfy descent for all hypercovers U → X.

The injective model structure on sP re(𝒞) just refers to Jardine’s model structure for the discrete topology on 𝒞 (see Section 2 for more about this). The fibrancy conditions for this model structure are awkward to describe, but they also aren’t very interesting—they have no dependence on the Grothendieck topology, only on the shape of the underlying category 𝒞. The conditions require that each F (X) be a fibrant simplicial set, certain maps F (X) → F (Y) be fibrations, and more complicated conditions of a similar ‘diagrammatic nature’. In practice such conditions are not very important, and in fact there’s a way to get around them completely by using the projective version of Jardine’s model structure; see Theorem 1.3 below and the discussion in Section 2.

We’d like to point out that the above theorem can be re-interpreted in terms of giving ‘generators’ and ‘relations’ for the homotopy theory of simplicial presheaves, in the manner introduced by [D]. Using the language of that paper, we prove

Theorem 1.2. Jardine’s model category sP re(𝒞)_{ℒ} is Quillen equivalent to the universal homotopy theory U𝒞/S constructed by

(1) formally adding homotopy colimits to the category 𝒞, to create U𝒞;

(2) imposing relations requiring that for every hypercover U → X, the map hocolim_{n} U_{n} → X is a weak equivalence.

In other words, the result says that everything special about the homotopy theory of simplicial presheaves can be derived from the basic fact that one can reconstruct X as the homotopy colimit of any of its hypercovers. The above theorem is crucial to the construction of étale realization functors for 𝔸^{1}-homotopy theory [Is], as well as the analogous question about topological realization functors [DI1].

One advantage of the model structure U𝒞/S over the model structure sP re(𝒞)_{ℒ} is that the fibrant objects are much easier to describe. The inexplicit fibrancy conditions of the injective model structure are replaced by a much simpler condition. Compare the following result to Theorem 1.1.

Theorem 1.3. The fibrant objects in the model category U𝒞/S are those simplicial presheaves that:

(1) are objectwise-fibrant (i.e., each F (X) is a fibrant simplicial set),

(2) satisfy descent for all hypercovers U → X.

Again, an analagous criterion for all fibrations is given in Section 7.

The main ideas we use to prove these results are very simple, and worth summarizing. They exactly parallel classical facts about CW-complexes. The two key ingredients are:

(i) In the category of simplicial presheaves one can construct objects analogous to CW-complexes, the only difference being that one has different kinds of 0-simplices corresponding to the different representable presheaves rX. (And as a consequence, there are different kinds of n-simplices corresponding to the objects Δ^{n} ⊗ rX.) Every simplicial presheaf has a cellular approximation built up out of representables in this way (see [D, Section 2.6]).

(ii) Weak equivalences for simplicial presheaves are characterized by a certain ‘local lifting criterion’, where lifting problems can be solved by passing from a representable object to the pieces of a cover. See Proposition 3.1 and the paper [DI2].

From these two basic principles, it’s inevitable that hypercovers will arise in the solution of lifting problems. One starts building a lift inductively on a CWapproximation, and the obstructions to extending the lift are made to vanish by passing to a finer cover at each stage. Thus, one finds oneself inductively constructing a hypercover. These ideas are explored in detail in Section 5.

This paper came into existence because we needed to use Theorems 6.2, 8.6, and A.6(c,d) in other work. We originally hoped for a very short paper, but to actually write down complete proofs one has to be able to manipulate hypercovers with a certain amount of ease—and the literature on this subject is not the most helpful. So in the end a large portion of the paper has been devoted to carefully setting down the machinery of hypercovers, hopefully in a way that will be usable by other people. For this reason the paper sometimes takes on an expository tone. We have tried to be clear and thorough, and for good or bad this has come at the expense of brevity. Also, one of our goals has been to adopt definitions which can be applied to any Grothendieck site, not just the classical ones which get used most often. The result is theorems which are simple enough to state and prove, but sometimes hard to apply in practice. To complement this, we have included the reductions to Verdier sites (Section 9) and internal hypercovers (Section 10) that one can implement for sites like those encountered in algebraic geometry. This subject of homotopical sheaf theory is rapidly finding applications in many contexts, so we have tried to give a presentation that is clear enough, and general enough, to be useful to a variety of practitioners.

1.4. Organization of the paper.

In Section 2 we review the basic model categories that will be used throughout the paper. One of these is Jardine’s model structure, and the other is a Quillen equivalent version which has fewer cofibrations and more fibrations. We assume throughout that the reader is familiar with the theory of model categories—the original reference for this subject is [Q], but we generally follow [H] in notation and terminology. [Ho] is also a good reference.

Section 3 reviews material from [DI2] on lifting properties for simplicial presheaves, and how these can be used to characterize local weak equivalences. Section 4 introduces the machinery needed for defining and working with hypercovers. The section is a bit long, and serves mostly as a reference section for the rest of the paper—it can comfortably be skimmed the first time through.

In Section 5 we show how hypercovers enter into the solution of lifting problems in the homotopy theory of simplicial presheaves. These are the key observations which are needed for the main results. The proofs of these main results are then given in Sections 6 and 7, where they appear as Theorem 6.2, Corollary 7.1, and Theorem 7.8.

One application of the results on hypercovers is to realization functors from the homotopy theory of schemes—this is treated in the papers [DI1, Is]. In Section 8 we give a few more applications. One of the most interesting, given in Section 8.1, is a much simpler approach to the change-of-site functors of [MV]. We also discuss a generalization of the Verdier hypercovering theorem in Theorem 8.6.

In applications one rarely wants to work with all hypercovers, because this is just too broad a class of objects. In the case of the ‘geometric’ sites which are most commonly used, one can adopt more restrictive definitions and have all the above results still go through. These reductions are explored in Sections 9 and 10. We axiomatize what is necessary into the notion of a Verdier site, which comes equipped with a special class of ‘basal hypercovers’. These ideas appear sporadically in Section 8, but hopefully the reader can just refer back to the later sections as necessary.

Finally, the paper contains an appendix which explores the difference between Čech descent and hypercover descent. Again, the principal motivation comes from the fact that Čech descent is more easily dealt with in practice. We show, among other things, that having descent for Čech complexes is equivalent to having descent for all bounded hypercovers (the ones where the refinement process stops at some finite level). This is an important ingredient in [DI1].

1.5. Notation and Terminology. If X is an object of a site 𝒞, then the representable simplicial presheaf rX on 𝒞 is given by the formula rX(Y) = Hom_{𝒞} (Y, X). Note that each simplicial set rX(Y) is discrete. If U is a simplicial object of 𝒞, then rU is the simplicial presheaf given by the formula rU(Y)_{n} = Hom_{𝒞} (Y, U_{n})—these, of course, are usually not discrete. We frequently abuse notation and write simply X (or U) for the presheaf rX (or rU).

If S is a scheme, then 𝒮ch/S denotes the category of schemes of finite-type over S. The full subcategory of schemes which are smooth over S is denoted Sm/S. Finally, in a simplicial model category we write Map(A, B) for the simplicial mapping space.

2. Model structures on simplicial presheaves

We start by recalling that for any small category 𝒞 there are two Quillen equivalent model structures on the category of diagrams s𝒮et^{𝒞}. In each case a map D → E is a weak equivalence if D(c) → E(c) is a weak equivalence of simplicial sets for each c in 𝒞. Such a map is usually called an objectwise weak equivalence. In the projective model structure on s𝒮et^{𝒞} one defines a map D → E to be

(1) A fibration if every D(c) → E(c) is a fibration of simplicial sets (i.e., D → E is an objectwise fibration).

(2) A cofibration if it has the left-lifting-property with respect to the acyclic fibrations.

Dually, in the injective model structure the cofibrations are objectwise and the fibrations have the right-lifting-property with respect to acyclic cofibrations. The names ‘projective’ and ‘injective’ come from the analogy between the two usual model structures on chain complexes of R-modules. For notational convenience, the projective model structure is denoted U𝒞 (as was done in [D], where it was pointed out that U𝒞 has a certain universal property) and the injective model structure is denoted sP re(𝒞).

When 𝒞 comes equipped with a Grothendieck topology, then one can construct refinements of these model structures which reflect the topology on 𝒞. A map of simplicial presheaves F → G is a local weak equivalence if it induces isomorphisms on all sheaves of homotopy groups [I, Jo, J2]. In this paper we will use an alternative characterization in terms of homotopy liftings, described below.

Jardine’s model structure on simplicial presheaves is the left Bousfield localization of sP re(𝒞) at the class ℒ of local weak equivalences; we denote this localization as sP re(𝒞)_{ℒ} . Of course since ℒ is a class of maps there is no a priori guarantee that the Bousfield localization exists, but Jardine was able to construct it directly—it is only after the fact that one can identify it as a localization.

Similarly, one can define a model structure U𝒞_{ℒ} by localizing U𝒞 at the same class ℒ (cf. [Bl, Thm. 1.5]). The identity maps induce a Quillen equivalence U𝒞_{ℒ} → sP re(𝒞)_{ℒ} , so once again these are projective and injective versions of the same underlying homotopy theory. The injective version has the advantage that every object is cofibrant, but in the projective version the fibrant objects are easier to understand and the representable presheaves are still cofibrant. Also, it is usually easier to construct functors out of the projective version [D]. We state most of our results only in terms of sP re(𝒞)_{ℒ} , but analogous statements for U𝒞_{ℒ} are also true with only minor differences between the proofs.

Both U𝒞 and sP re(𝒞) are proper, simplicial model categories: if F is a simplicial presheaf and K is a simplicial set then K ⊗ F and F^{K} are defined objectwise, by

(K ⊗ F)(X) = K × F (X)  and  (F^{K})(X) = F (X)^{K}.

From general considerations [H, Thm. 4.1.1], all localizations of U𝒞 and sP re(𝒞) that we consider are also left proper, simplicial model categories.

Remark 2.1. If F is a simplicial presheaf, then one obtains a diagram D_{F} : Δ^{op} → sP re(𝒞) by sending [n] to F_{n}. Here F_{n} is just a presheaf of sets, but we can regard it as a discrete simplicial presheaf in the obvious way. The realization of this simplicial diagram is precisely F. The Bousfield-Kan map hocolim D_{F} → |D_{F} | is a weak equivalence in this case, by some basic model category theory. So any simplicial presheaf F is weakly equivalent to hocolim D_{F}. This observation will be needed often.

3. Local weak equivalences and local lifting properties

Local weak equivalences are usually defined in terms of sheaves of homotopy groups. Here we recall a different description which is more suitable for our purposes. See [DI2] for the proof that the two definitions agree and for more details on the results in this section.

First, recall that if X is in 𝒞 and F and G are simplicial presheaves, then a diagram such as

\xymatrix{Λ^{n,k} ⊗ X \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ X \ar[r] & G}

has local liftings if there exists a covering sieve R of X such that for every map U → X in the sieve, the diagram one obtains by restricting from X to U has a lifting Δ^{n} ⊗ U → F. These liftings are not required to be compatible for the different U’s. A map F → G is called a local fibration if it has local liftings with respect to the maps Λ^{n,k} ⊗ X → Δ^{n} ⊗ X, for all X in 𝒞. A simplicial presheaf is called locally fibrant if F → ∗ is a local fibration.

Proposition 3.1 ([DI2, Th. 6.15]). A map F → G between locally fibrant simplicial presheaves is a local weak equivalence if and only if every square

\xymatrix{∂Δ^{n} ⊗ X \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ X \ar[r] & G}

has local relative homotopy-liftings, in the following sense: after restricting to the pieces U → X of some covering sieve, one has maps Δ^{n} ⊗ U → F making the upper triangle commute on the nose and the lower triangle commute up to simplicial homotopy relative to ∂Δ^{n} ⊗ U.

The reader may consult [DI2] for a detailed discussion of this kind of relativehomotopy-lifting property.

The following two results from [DI2] will be used later. Recall that a map is a local acyclic fibration if it is both a local fibration and a local weak equivalence.

Proposition 3.2 ([DI2, Prop. 7.2]). A map F → G admits local liftings in every square

\xymatrix{∂Δ^{n} ⊗ X \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ X \ar@{.>}[ur] \ar[r] & G}

if and only if it is a local acyclic fibration.

One consequence of the above result is that local acyclic fibrations are closed under pullbacks (in [J2] this was proven only when the domain and codomain are locally fibrant).

Proposition 3.3 ([J4, Lemma 19],[DI2, Cor. 7.4]). Let F → G be a local fibration (resp. local acyclic fibration). If K ↪ L is an inclusion of finite simplicial sets, then the induced map

F^{L} → F^{K} ×_{G^{K}} G^{L}

is a local fibration (resp. local acyclic fibration).

Let f: E → B be a map between presheaves of sets. One says that f is a generalized cover (or local epimorphism) if it has the following property: given any map rX → B, there is a covering sieve R ↪ X such that for every element U → X in R, the composite rU → rX → B lifts through f. The ‘generalized’ adjective is there to remind us that we are looking at a map between presheaves, not actual objects of the site. In the case where B is representable and E is a coproduct ∐ E_{a} of representables, f is a generalized cover precisely when the sieve generated by the maps {E_{a} → B} is a covering sieve of B.

For a simplicial presheaf F, let M̃_{n}F denote the 0th object of F^{∂Δ^{n}} (the ‘tilde’ is to distinguish this from a slightly different construction used later in the paper).

This is the presheaf of sets whose value M̃_{n}F (X) is the set of all maps ∂Δ^{n} → F (X). There is a natural map F_{n} → M̃_{n}F induced by F^{Δ^{n}} → F^{∂Δ^{n}}. Proposition 3.2 can be rephrased as saying that F → G is a local acyclic fibration if and only if the maps

F_{n} → M̃_{n}F ×_{M̃_{n}G} G_{n} (3.4)

are generalized covers, for all n ≥ 0. Using this observation, most properties of generalized covers can automatically be seen to hold for local acyclic fibrations.

4. Background on hypercovers

This section contains the necessary machinery for defining and working with hypercovers. Unfortunately there is quite a bit of annoying category theory, and some readers may wish to only skim this section their first time through. This should be enough to understand the basic definitions that are used throughout the paper. In Section 4.12 we recall the coskeleton and degeneration functors, which appear when passing between simplicial objects and truncated simplicial objects. These notions are used later in the paper, but only in fairly technical contexts.

4.1. The definition.

Definition 4.2. Let X belong to 𝒞 and suppose that U is a simplicial presheaf with an augmentation U → X. This map is called a hypercover of X if each U_{n} is a coproduct of representables, and U → X is a local acyclic fibration.

Using (3.4) one can rewrite the second condition in a more explicit way: it says that the maps U_{0} → X, U_{1} → U_{0} ×_{X} U_{0}, and U_{n} → M̃_{n}U (for n ≥ 1) are all generalized covers. This is not particularly enlightening, but it’s easy to provide some intuition behind it. For convenience we assume our Grothendieck topology is given by a basis of covering families. Then the easiest examples of hypercovers are the Čech complexes, which have the form

\xymatrix{⋯ ∐ U_{a_{0}a_{1}a_{2}} \ar[r] \ar[r] \ar[r] & ∐ U_{a_{0}a_{1}} \ar[r] \ar[r] & ∐ U_{a_{0}} \ar[r] & X}

for some chosen covering family {U_{a} → X}. Here U_{a_{0}···a_{n}} is the fibre-product U_{a_{0}} ×_{X} ⋯ ×_{X} U_{a_{n}}. The Čech complexes are the hypercovers for which the maps U_{1} → U_{0} ×_{X} U_{0} and U_{n} → M̃_{n}U are all isomorphisms. In an arbitrary hypercover one takes the iterated fibre-products at each level but then is allowed to refine that object further, by taking a generalized cover of it. We refer the reader to [AM, Section 8] for further discussion of hypercovers.

Next is the formal definition of hypercover descent:

Definition 4.3. An objectwise-fibrant simplicial presheaf F satisfies descent for a hypercover U → X if the natural map from F (X) to the homotopy limit of the diagram

\xymatrix{∏_{a} F (U_{0}^{a}) \ar[r] \ar[r] & ∏_{a} F (U_{1}^{a}) \ar[r] \ar[r] \ar[r] & ⋯}

is a weak equivalence. Here the products range over the representable summands of each U_{n}. If F is not objectwise-fibrant, we say it satisfies descent if some objectwisefibrant replacement for F does.

The definition has been arranged so that if F → G is an objectwise weak equivalence, then F satisfies descent for U → X if and only if G does. While the definition reflects our intuitive notion of descent, the next lemma gives a more concise reformulation in terms of simplicial mapping spaces.

Lemma 4.4.

(i) A simplicial presheaf F satisfies descent for a hypercover U → X if and only if Map(X, F̂) → Map(U, F̂) is a weak equivalence of simplicial sets, where F̂ is an injective-fibrant replacement for F.

(ii) Let U′ be a cofibrant replacement for U in U𝒞. Then F satisfies descent for U → X if and only if Map(X, F̂) → Map(U′, F̂) is a weak equivalence of simplicial sets, where F̂ is an objectwise-fibrant replacement for F.

Note that any split hypercover (see Definition 4.13) is cofibrant in U𝒞, in which case one can apply (ii) with U′ = U.

Proof. This is by general nonsense. Consider the diagram Δ^{op} → sP re(𝒞) given by [n] → U_{n}, and let Ũ be its homotopy colimit. This is not the same as U, but there is a map Ũ → U which is an objectwise weak equivalence (see Remark 2.1). Let F̂ be an injective-fibrant replacement for F, which a fortiori is an objectwise-fibrant replacement as well. Then Map(Ũ, F̂) is weakly equivalent to Map(U, F̂) since Ũ → U is a weak equivalence between injective-cofibrant objects. But Map(Ũ, F̂) is

Map(hocolim_{n} U_{n}, F̂) ≃ holim_{n} Map(U_{n}, F̂) ≃ holim_{n}_{∏a}F̂(U_{n}^{a}).

Since Map(X, F̂) is equal to F̂(X), the condition that Map(X, F̂) → Map(U, F̂) be a weak equivalence is a direct translation of the homotopy limit formulation in Definition 4.3. This proves (i).

For (ii), note that each U_{n} is cofibrant in U𝒞 and so Ũ = hocolim_{n} U_{n} is also cofibrant. In other words Ũ is a cofibrant replacement for U, and so Ũ ≃ U′. If F̂ is an objectwise replacement for F then Map(U′, F̂) ≃ Map(Ũ, F̂), and as in (i) the latter is equivalent to holim_{n} ∏_{a} F̂(U_{n}^{a}). The rest of the proof is the same. □

A more elegant way to phrase the above result is to say that F satisfies descent for U → X if and only if hMap(X, F) → hMap(U, F) is a weak equivalence of simplicial sets, where hMap(−, −) denotes a homotopy function complex [H, Ch. 17] in either U𝒞 or sP re(𝒞).

4.5. Machinery.

Definition 4.2 is very compact, but it’s not always such an easy thing to work with. For the rest of this section we will set down more convenient techniques for constructing and working with hypercovers. This material is used throughout the paper, but many readers will want to skip ahead and refer back to this section only when needed.

Let ℳ be a category which is complete and co-complete—in our applications ℳ is P re(𝒞), but for the moment let us work in the more general setting. Let Δ_{+} denote the augmented cosimplicial indexing category: it is obtained by adjoining an initial object [−1] to Δ. Let s_{+}ℳ denote the category of functors Δ_{+}^{op} → ℳ, i.e. the category of augmented simplicial objects. We regard a simplicial set K as belonging to s_{+}𝒮et by letting K_{−1} consist of a single point.

If S is a set and X belongs to ℳ, let X^{S} denote a product of copies of X indexed by the elements of S. Given a simplicial set K and an object W of s_{+}ℳ, we regard these as functors K: Δ_{+}^{op} → 𝒮et and W: Δ_{+}^{op} → ℳ and then form the resulting end, denoted hom_{+}(K, W):

hom_{+}(K, W) := eq[∏_{n} W_{n}^{K_{n}} ⇉ ∏_{[n]→[m]} W_{m}^{K_{n}}].

The + subscript is to remind us of the augmentations.

Remark 4.6. As with any end, this construction exhibits a useful adjointness property. If Z is in ℳ, then the maps Z → hom_{+}(K, W) in ℳ correspond bijectively with the maps Z ⊗K → W in s_{+}ℳ. Here Z ⊗K is the augmented simplicial object which in dimension n is a coproduct, indexed by the set K_{n}, of copies of Z.

In the unaugmented simplicial category sℳ, we can compute unaugmented ends hom(K, W) in an analogous way. Again, this construction is right adjoint to tensoring with K.

The following lemma can be proved with the above adjointness property and the Yoneda lemma.

Lemma 4.7. Let W → X be an augmented simplicial object (that is, X is the augmentation).

(i) hom_{+}(K, W) ≅ hom^{X}(K, W), where hom^{X}(K, W) is computed in the unaugmented simplicial overcategory s(ℳ ↓ X).

(ii) hom_{+}(K, W) ≅ hom(K, W) if K is connected. (iii) hom_{+}(∅, W) ≅ X, and so for any simplicial set K there is a canonical map

hom_{+}(K, W) → X. (iv) hom_{+}(Δ^{n}, W) ≅ W_{n}.

(v) hom_{+}(−, W) takes colimits of simplicial sets to limits in ℳ ↓ X. In other words, if K = colim_{i} K_{i}, then hom_{+}(K, W) ≅ lim_{i}^{X} hom_{+}(K_{i}, W).

Definition 4.8. The object hom_{+}(∂Δ^{n}, W) is the nth augmented matching space M_{n}W. The induced map hom_{+}(Δ^{n}, W) → hom_{+}(∂Δ^{n}, W), which we may now write as W_{n} → M_{n}W, is the nth matching map for W. Note that W_{0} → M_{0}W is just the augmentation since ∂Δ^{0} = ∅.

We have chosen to work with these augmented constructions only because they seem to make for the most compact and intuitive proofs. Note that the augmented matching objects and maps are the ones that arise when considering Reedy model structures of simplicial objects in (ℳ ↓ X) [H, 15.2.2]. For n ≥ 2, M_{n}W is isomorphic to M̃_{n}U = hom(∂Δ^{n}, W) because ∂Δ^{n} is connected. The following lemma is a reformulation of (3.4).

Lemma 4.9. An augmented simplicial presheaf U → X is a hypercover iff each U_{n} is a coproduct of repesentables and the maps U_{n} → M_{n}U are all generalized covers.

Definition 4.10. A hypercover U → X is bounded if there exists an n ≥ 0 such that the maps U_{k} → M_{k}U are isomorphisms for all k > n. The smallest such n for which this is true is called the height of the hypercover, and denoted ht U.

We have already remarked that the hypercovers of height 0 are precisely the Čech complexes. If one thinks of the nth level of a hypercover as refining the (n + 1)-fold ‘intersections’ of the objects in previous levels, then a bounded hypercover is one where the refinement process stops at some point. The following lemma is a minor ingredient in the discussion of coskeleta in Section 4.12 below, but the ideas from the proof reappear several times throughout the paper.

Lemma 4.11. If U → X is a bounded hypercover of height at most n, then the induced maps hom_{+}(Δ^{k}, U) → hom_{+}(sk_{n} Δ^{k}, U) are isomorphisms for all k.

Proof. When k ≤ n, the result is easy because Δ^{k} equals sk_{n} Δ^{k}. In general, Δ^{k} is obtained from sk_{n} Δ^{k} by gluing on finitely many simplices of dimension at least n + 1. It suffices to show that hom_{+}(L, U) → hom_{+}(K, U) is an isomorphism if L is obtained from K by attaching a simplex of dimension i, where i > n. Using Lemma 4.7 we obtain a pullback square

\xymatrix{hom_{+}(L, U) \ar[r] \ar[d] & hom_{+}(K, U) \ar[d] \\ hom_{+}(Δ^{i}, U) \ar[r] & hom_{+}(∂Δ^{i}, U).}

The bottom map is the matching map U_{i} → M_{i}U, which is an isomorphism since i > n. Hence the top map is also an isomorphism. □

4.12. Skeleta, coskeleta, and split objects.

We continue to assume that ℳ is complete and cocomplete. Let sℳ_{≤n} and s_{+}ℳ_{≤n} denote the categories of n-truncated simplicial objects and augmented ntruncated simplicial objects over ℳ. There is an obvious forgetful functor s_{+}ℳ → s_{+}ℳ_{≤n} called sk_{n}, and this has a right adjoint called cosk_{n}. These are the skeleta and coskeleta functors for augmented simplicial objects. If W belongs to s_{+}ℳ, we abbreviate cosk_{n} sk_{n} W to just cosk_{n} W.

The kth object of cosk_{n} U is

[cosk_{n} U]_{k} ≅ hom_{+}(Δ^{k}, cosk_{n} U) ≅ hom_{+}(sk_{n} Δ^{k}, U)

(use Remark 4.6 for the second isomorphism). In particular, the (n + 1)st object of cosk_{n} U is what we have been calling M_{n+1}U. Observe also, using Lemma 4.11, that a hypercover U has height at most n if and only if U ≅ cosk_{n} U.

Now, the functor sk_{n} also has a left adjoint dgn_{n}: s_{+}ℳ_{≤n} → s_{+}ℳ, called the ndegeneration functor. The simplicial object dgn_{n} U is obtained from U by freely adding the images of the degeneracies in dimensions higher than n (and so, in particular, note that the augmentations are irrelevant). The object [dgn_{n} U]_{n+1} is called the (n + 1)st latching object for U and is denoted L_{n+1}U. This latching object is the one that arises when considering Reedy model structures of simplicial diagram categories [H, 15.2.2]. Note that U, dgn_{n} U, and cosk_{n} U all have the same n-skeleton, so there are canonical maps dgn_{n} U → U → cosk_{n} U; looking in level n + 1 gives L_{n+1}U → U → M_{n+1}U.

Definition 4.13. An object W of sℳ is said to be split, or to have free degeneracies, if there exist subobjects N_{k} ↪ W_{k} such that the canonical maps N_{k} ⨿ L_{k}W → W_{k} are isomorphisms for all k ≥ 0. This is equivalent to requiring that the canonical map

∐_{σ} N_{σ} → W_{k}

is an isomorphism, where the variable σ ranges over all surjective maps in Δ of the form [k] → [n], N_{σ} denotes a copy of N_{n}, and the map N_{σ} → W_{k} is the one induced by σ^{∗}: W_{n} → W_{k} (see [AM, Def. 8.1]).

The idea is that the objects N_{k} represent the non-degenerate part of W in dimension k, and that the leftover degenerate part is as free as possible. The same definition as above can be applied to augmented simplicial objects, and the result is that such an object is split if and only if it is split when one forgets the augmentation.

We are particularly interested in split hypercovers. If U → X is a split hypercover then L_{k}U is a summand of U_{k}, each L_{k}U is a coproduct of representables, and each representable summand of L_{k}U is the image under some degeneracy of a representable from U_{k−1} (but not uniquely). It follows from [D, Cor. 9.4] that split hypercovers are cofibrant in U𝒞_{ℒ} , which is why we care about them.

4.14. Computing matching objects.

Suppose that U → X is an augmented simplicial presheaf which in each level is a coproduct of representables. Note that (1) the decomposition of U_{n} into a coproduct of representables is unique up to permutations of the summands, and (2) to give a map ⨿_{i}A_{i} → ⨿_{j}B_{j} between coproducts of representables corresponds to giving, for each index i, an index j(i) and a map A_{i} → B_{j(i)}. Because of these remarks, one can construct a simplicial set K by taking K_{n} to be the set of representable summands of U_{n}. We’ll refer to K as the indexing simplicial set for U.

Now suppose a: L → K is a map of simplicial sets. If Δ^{op}L denotes the opposite category of simplices of L [H, Def. 15.1.16], there is an obvious diagram Δ^{op}L → sP re(𝒞) ↓ X which sends a k-simplex σ to the representable which is the summand of U_{k} corresponding to a(σ). We’ll write U(a) for the limit of this diagram.

The following observation is straightforward (use Remark 4.6):

Proposition 4.15. There is an isomorphism of presheaves

hom_{+}(L, U) ≅ ∐_{a: L→K} U(a).

In particular, the matching object M_{n}U is isomorphic to ∐_{a: ∂Δ^{n}→K} U(a).

Note that Δ^{op}L is an infinite category. If L has the property that every nondegenerate simplex has nondegenerate faces (e.g. L = ∂Δ^{n}), then one can use a smaller version. Let Δ_{nd}^{op}L be the subcategory whose objects are the non-degenerate simplices, and where the maps correspond to face maps. Under the above assumption on L, it is an easy exercise to check that Δ_{nd}^{op}L ↪ Δ^{op}L is final (use the fact that in any simplicial set a degenerate simplex is an iterated degeneracy of a unique nondegenerate simplex). Hence the limit U(a) can be computed over Δ_{nd}^{op}L in practice.

5. Hypercovers and lifting problems

In Proposition 3.1 we saw how local weak equivalences relate to solutions of homotopy-lifting problems—one gets lifts after passing from a representable to the elements of a covering sieve. Typically, these liftings can’t be made compatible on the different pieces of the sieve. In this section we find that one can arrange for this kind of compatibility by using hypercovers.

The following proposition is the key ingredient in the proof of our main result, Theorem 6.2. Recall that, just as for ordinary covering families, a refinement of a hypercover U → X is another hypercover V → X that factors through U.

Proposition 5.1. Let F → G be a local acyclic fibration and let K → L be a cofibration of finite simplicial sets. For any square

\xymatrix{K ⊗ U \ar[r] \ar[d] & F \ar[d] \\ L ⊗ U \ar[r] & G} (5.2)

in which U → X is a hypercover, there exists another hypercover V → X refining U and liftings as in the following diagram:

\xymatrix{K ⊗ V \ar[r] \ar[d] & K ⊗ U \ar[r] & F \ar[d] \\ L ⊗ V \ar[r] \ar@{.>}[urr] & L ⊗ U \ar[r] & G.} (5.3)

To summarize the basic idea of the proof, let’s assume that K → L is ∅ → ∗ and that the Grothendieck topology comes with a specified basis of covering families. Starting with a map U → G, we know by the local-lifting property (3.1) that there is a covering family {V_{a} → U_{0}} with liftings s_{a}: V_{a} → F. In general, s_{a} |_{V_{ab}} and s_{b} |_{V_{ab}} are not equal, but the two liftings become homotopic after projecting down to G. We can lift this homotopy to F by passing to a suitable covering family of V_{ab}, again using the fact that F → G is a local weak equivalence. Next we move on to consider patching on the triple intersections. Once again, we can patch up to homotopy after refining the triple intersections by a covering family. In this way we build a hypercover V over which a lifting is defined. The work in this section is just a precise way of saying all this.

The proof involves an inductively constructed hypercover, and the following lemma is the core of the induction step:

Lemma 5.4. Let F and G be presheaves of sets, and let F → G be a generalized cover. If J is a presheaf of sets with a map J → G, then there exists a generalized cover Z → J such that Z is a coproduct of representables and such that the diagram

\xymatrix{ &  & F \ar[d] \\ Z \ar@{.>}[urr] \ar[r] & J \ar[r] & G}

has a lifting.

Proof. For each map f: X → J from a representable, choose a covering sieve R_{f} of X so that the composites U → X → J → G lift to F for every U → X in the sieve. Let Z denote the coproduct

Z = ∐_{X⟶^{f}J}( ∐_{in R_{f}^{U→X}} U).

The obvious map Z → J is a generalized cover, and the composite Z → J → G lifts to F. □

Proposition 5.5. Let F → G be a local acyclic fibration, and let U → G be a map where U → X is a hypercover. Let n ≥ 0, and suppose that there is an n-truncated hypercover V → X refining sk_{n} U and a map V → F such that the diagram

(5.6)  \xymatrix{ &  & F \ar[d] \\ V \ar[r] \ar[urr] & U \ar[r] & G}

commutes. Then there is an (n + 1)-truncated hypercover W → X refining U and a map W → F making the corresponding diagram commute, and such that on n-skeleta the diagram is equal to (5.6).

Proof. The core of the proof is just an Artin-Mazur argument [AM, Ch. 8]. First form the pullback F′ = U ×_{G} F. The map F′ → U is still a local acyclic fibration, and we need only produce an (n + 1)-truncated hypercover W and a lifting into F′. In other words, we can reduce to the case where U = G (and F = F′). Note that in this case G is locally fibrant—the representable X is locally fibrant for trivial reasons, and U → X is a local fibration. Moreover, since F → G is a local fibration, F is locally fibrant as well.

Now F^{Δ^{n+1}} → F^{∂Δ^{n+1}} is a local fibration by Proposition 3.3, so the map in the 0th level is a generalized cover by (3.4). When n > 0 this map is precisely F_{n+1} → M_{n+1}F (the n = 0 case being only slightly different). Our initial diagram gives a map V^{∂Δ^{n+1}} → F^{∂Δ^{n+1}}, and the 0th level has the form M_{n+1}V → M_{n+1}F. So Lemma 5.4 says that there is a generalized cover Z → M_{n+1}V, where Z is a coproduct of representables, such that the composite Z → M_{n+1}F lifts through F_{n+1}. We take W to be the (n + 1)-truncated hypercover with sk_{n} W = sk_{n} V and W_{n+1} = Z ⨿ L_{n+1}V. □

Proof of Proposition 5.1. Given a square as in the statement of the proposition, it may be interpreted as a map

U → F^{K} ×_{G^{K}} G^{L}.

We are trying to produce a hypercover V → X refining U → X and a lifting

\xymatrix{ &  & F^{L} \ar[d] \\ V \ar[r] \ar@{.>}[urr] & U \ar[r] & F^{K} ×_{G^{K}} G^{L}.}

The vertical map is a local acyclic fibration by Proposition 3.3, so the hypercover can be produced inductively using Proposition 5.5. □

If we have a map F → G which is a local weak equivalence but not necessarily a fibration, we can say the following:

Proposition 5.7. Let F → G be a local weak equivalence between locally fibrant simplicial presheaves. Then given any diagram as in (5.2), there exists a hypercover V → X refining U → X and relative-homotopy-liftings in the diagram (5.3).

Recall that relative-homotopy-liftings were defined in Proposition 3.1, and discussed extensively in [DI2].

Proof. Given a diagram as in (5.2), we need to produce a hypercover V → X refining U → X together with liftings in the diagram

\xymatrix{ & K ⊗ V \ar[r] \ar[d] \ar[dl] & F \ar[d] \\ L ⊗ V \ar[d]_{i_{0}} \ar@{.>}[urr] & L ⊗ V \ar[r] \ar[dl]_{i_{1}} & G \\ RH ⊗ V. \ar@{.>}[urr]}

Here RH denotes the pushout of L × Δ^{1} ⟵ K × Δ^{1} ⟶^{π} K, and the maps i_{0} and i_{1} are the obvious inclusions L ↪ RH.

Consider the square

\xymatrix{F^{RH} \ar[d]_{i_{1}^{∗}} \ar[r] & F^{L} ×_{G^{L}} G^{RH} \ar[d] \\ F^{L} \ar[r] & F^{K} ×_{G^{K}} G^{L}.}

By [DI2, Cor. 7.5], the fact that F → G is a local weak equivalence between locally fibrant objects implies that the horizontal maps are also local weak equivalences. By the same result, the fact that i_{1} : L → RH is a weak equivalence of simplicial sets implies that the left vertical map is a local weak equivalence. So we conclude that the same is true of the right vertical map. Even more, the right vertical map is a local fibration by Lemma 5.8 below.

Our initial data from (5.2) was a map U → F^{K} ×_{G^{K}} G^{L}, so by Proposition 5.1 (for n = 0 and K → L equal to ∅ → ∗) there is a hypercover V → X refining U → X for which the composite lifts through F^{L} ×_{G^{L}} G^{RH}. This provides the necessary relative-homotopy-lifting. □

Lemma 5.8. Let F → G be a map between locally fibrant simplicial presheaves. Assume we have a square of finite simplicial sets

\xymatrix{K \ar[d] \ar[r] & M \ar[d] \\ L \ar[r] & N}

such that both K → M and M ⨿_{K} L → N are cofibrations. Then the induced map

F^{M} ×_{G^{M}} G^{N} → F^{K} ×_{G^{K}} G^{L}

is a local fibration.

Proof. The hypotheses imply that both F^{M} → F^{K} and G^{N} → G^{(M⨿_{K}L)} = G^{M} ×_{G^{K}} G^{L} are local fibrations, using [J2, Cor. 1.5]. Now we observe that there are pullback squares

\xymatrix{F^{M} ×_{G^{M}} G^{N} \ar[r] \ar[d] & G^{N} \ar[d] & F^{M} ×_{G^{K}} G^{L} \ar[r] \ar[d] & F^{M} \ar[d] \\ F^{M} ×_{G^{K}} G^{L} \ar[r] & G^{M} ×_{G^{K}} G^{L} & F^{K} ×_{G^{K}} G^{L} \ar[r] & F^{K},}

and the pullback of a local fibration is again a local fibration. Finally, the map we want is just the composite F^{M} ×_{G^{M}} G^{N} → F^{M} ×_{G^{K}} G^{L} → F^{K} ×_{G^{K}} G^{L}. □

6. Hypercovers and localizations

In this section we prove the main theorem, that Jardine’s model category can be obtained by localizing the injective structure sP re(𝒞) at the hypercovers. This lets us identify the fibrant objects in the model structure. Similar results are proven for the projective version U𝒞_{ℒ} .

We start with a definition:

Definition 6.1. A collection of hypercovers S is called dense if every hypercover U → X in sP re(𝒞) can be refined by a hypercover V → X which belongs to S.

For instance, Theorem 9.6 shows that when 𝒞 is a Verdier site the collection of basal hypercovers is both split and dense. The following is our main goal.

Theorem 6.2. Let S be a collection of hypercovers which contains a set that is dense (e.g., the collection of all hypercovers). Then the localization sP re(𝒞)/S exists and coincides with Jardine’s model structure sP re(𝒞)_{ℒ} . Similarly, the localization U𝒞/S exists and coincides with U𝒞_{ℒ} .

Our notation is that if ℳ is a model category and S is a collection of maps, then ℳ/S denotes the left Bousfield localization of ℳ at S (if it exists)—see [D] for a summary treatment or [H] for complete details. The fibrations and weak equivalences in ℳ/S are called S-fibrations and S-equivalences, while the cofibrations are the same as those in ℳ.

The hypothesis of the theorem is a little stronger than just assuming that S is dense, because S itself may not be a set. For the same reason, the existence of the localization is not automatic. One of the things we will do is apply this theorem in the case where S is the collection of all hypercovers, and this is not a set: in our definition of hypercover one can have arbitrarily large coproducts of representables appearing. So we’ll need to verify that S contains a dense set, and this can be done by making use of the fact that our site is small. We choose a suitably large regular cardinal, and then we only consider hypercovers in which the number of summands in each level is bounded by our cardinal. For now we can ignore this point, but see Section 6.5.

To prove Theorem 6.2, we need a general criterion for checking whether two localizations are identical:

Lemma 6.3. Let ℳ be a model category, and let S ⊆ T be two classes of maps for which the localizations ℳ/S and ℳ/T exist. For the two localizations to be the same it suffices to check the following: if an S-fibration X ↠ Y between S-fibrant objects is a T -equivalence, then it is an S-equivalence.

Proof. We must show the hypothesis implies that every T -equivalence A → B is an S-equivalence. Let L denote a fibrant replacement functor in ℳ/S, and consider the square

\xymatrix{A \ar[r] \ar[d]_{∼}^{S} & B \ar[d]_{∼}^{S} \\ LA \ar[r] & LB.}

Since S ⊆ T the two vertical maps are T -equivalences, and the top map is a T - equivalence by assumption—so the bottom map is one as well. Now factor the bottom map in ℳ/S as an S-acyclic cofibration followed by an S-fibration:

[figure: LA; ∼; S; X; LB.]

Note that X is S-fibrant, because LB is. Also, since both the first map and the composite are T -equivalences, so is the second map.

Therefore the map X → LB is a T -equivalence and an S-fibration, and the domain and codomain are S-fibrant. Our hypothesis then says that X → LB is an S-equivalence. Applying the two-out-of-three property (twice) shows that A → B is an S-equivalence. □

For the moment let S be a set of hypercovers that is dense. Because S is a set, we know that the model structure sP re(𝒞)/S exists (by [H, Thm. 4.1.1], using that sP re(𝒞) is left proper and cellular). The fibrant objects in sP re(𝒞)/S (called S-fibrant objects) are the injective-fibrant objects which satisfy descent for all hypercovers in S. Since every hypercover is a local weak equivalence by definition, sP re(𝒞)_{ℒ} is a localization of sP re(𝒞)/S. To show that the two structures coincide, we now check the criterion from the above lemma:

Lemma 6.4. Let F and G be S-fibrant objects, and let f : F → G be an S-fibration that is also a local weak equivalence. If X is a representable, then every square

\xymatrix{∂Δ^{n} ⊗ X \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ X \ar[r] & G}

has a lifting. In particular, f is actually an objectwise acyclic fibration and therefore an S-equivalence.

Proof. The second claim follows from the first by adjointness and because acyclic fibrations of simplicial sets are detected by the right lifting property with respect to the maps ∂Δ^{n} → Δ^{n}.

Now we prove the first claim. First, f is an objectwise fibration since every S-fibration is an injective-fibration and also a projective-fibration. This implies that f is also a local fibration. Because f is both a local fibration and a local weak equivalence, Proposition 5.1 guarantees us a hypercover U → X such that the diagram

\xymatrix{∂Δ^{n} ⊗ U \ar[r] \ar[d] & ∂Δ^{n} ⊗ X \ar[r] & F \ar[d] \\ Δ^{n} ⊗ U \ar[r] \ar@{.>}[urr] & Δ^{n} ⊗ X \ar[r] & G}

has a lifting. In applying Proposition 5.1, we have used that X is (trivially) a hypercover of itself. Since S is dense, we may refine U and assume that U → X belongs to S. We now write down the following diagram of simplicial mapping spaces:

\xymatrix{Map(X, F^{Δ^{n}}) \ar[r]^{∼} \ar@{->>}[d] & Map(U, F^{Δ^{n}}) \ar@{->>}[d] \\ Map(X, G^{Δ^{n}} ×_{G^{∂Δ^{n}}} F^{∂Δ^{n}}) \ar[r]^{∼} & Map(U, G^{Δ^{n}} ×_{G^{∂Δ^{n}}} F^{∂Δ^{n}}).}

All the model categories we have been considering are simplicial model categories, and this implies that F^{Δ^{n}} → G^{Δ^{n}} ×_{G^{∂Δ^{n}}} F^{∂Δ^{n}} is an S-fibration between S-fibrant objects. Therefore, the vertical maps above are fibrations of simplicial sets because both X and U are S-cofibrant. Likewise, the horizontal maps are weak equivalences because U → X is an S-equivalence between S-cofibrant objects and both F^{Δ^{n}} and G^{Δ^{n}} ×_{G^{∂Δ^{n}}} F^{∂Δ^{n}} are S-fibrant.

We are given a 0-simplex x in the lower left corner in the above diagram, and we want to find a lift in the upper left corner. We have already shown that the image of x in the lower right corner lifts to the upper right corner. Since the horizontal maps are weak equivalences, there is another 0-simplex y belonging to the connected component of x such that y has a lift in the upper left corner. But fibrations of simplicial sets are surjective onto the components in their images, so x also has a lift. □

Proof of Theorem 6.2. We first consider the claim for sP re(𝒞)_{ℒ} . For the case when our collection of hypercovers S is itself a set, we have already done all the work. Since hypercovers are local weak equivalences we know S ⊆ ℒ, and so we are in the situation of Lemma 6.3. The necessary condition was verified in Lemma 6.4.

In the general case, let S′ be a dense set of hypercovers contained in S. As shown in the previous paragraph, sP re(𝒞)/S′ is equal to sP re(𝒞)_{ℒ} . So every local weak equivalence is a weak equivalence in sP re(𝒞)/S′, and in particular every hypercover in S is an S′-equivalence. This shows that sP re(𝒞)/S exists and is equal to sP re(𝒞)/S′.

The argument for U𝒞/S is basically the same. Assume first that S is a set of hypercovers which is dense. One reproves the analog of Lemma 6.4 for the projective model structure; the only difference in the proof is that one replaces U by a cofibrant object before dealing with simplicial mapping spaces. The rest of the argument is exactly the same, as is the generalization to the case where S need not be a set. □

6.5. Cardinality considerations. Early in this section we mentioned that the collection of all hypercovers is not a set, but contains a subset that is dense. We will now give the proof. Recall from Section 4.14 that to any hypercover U → X one can attach an indexing simplicial set K, where K_{n} is the set of representable summands of U_{n}. The size of the hypercover is the cardinality of ∐_{n} K_{n}, i.e., the number of representable summands that appear in U. The main point is that in the arguments from Proposition 5.5 and Lemma 5.4, one can control the size of the constructed hypercover.

Proposition 6.6. The class of all hypercovers has a subset which is dense.

Proof. Choose a regular cardinal λ sufficiently large compared to the cardinality of the set of morphisms in 𝒞, and let S denote the set of all hypercovers of size less than λ. We will show that any hypercover U → X can be refined by one in S.

Since U_{0} → X is a generalized cover, there is a covering sieve R of X such that every W → X in R lifts through U_{0}. Let V_{0} = ∐_{W →X} W, where the coproduct ranges over all maps W → X in R. The number of summands in V_{0} is clearly bounded by λ.

Now assume by induction that we have constructed an n-truncated hypercover V → X which refines U, and such that the number of summands in V is less than λ. To extend V we use the argument from Proposition 5.5, where we must show that Z does not have too many representable summands. Inspecting the construction of Z given in Lemma 5.4, it suffices to show that there aren’t too many maps from a representable into M_{n+1}V. This can be deduced from Proposition 4.15. □

7. Fibrations and descent conditions

This section identifies the fibrations and fibrant objects in sP re(𝒞)_{ℒ} and U𝒞_{ℒ} . We start with the fibrant objects, where the result follows from Theorem 6.2:

Corollary 7.1. Let S be a collection of hypercovers which contains a set that is dense. A simplicial presheaf F is fibrant in sP re(𝒞)_{ℒ} (resp. in U𝒞_{ℒ} ) if and only if F is injective-fibrant (resp. objectwise-fibrant) and satisfies descent for all hypercovers in S.

Note that an immediate consequence is that a simplicial presheaf F satisfies descent for all hypercovers if and only if it satisfies descent for all elements of S.

Proof. First observe that the fibrant objects in sP re(𝒞)/S are the injective-fibrant objects F such that Map(X, F) → Map(U, F) is a weak equivalence for every U → X in S [H, Thm. 4.1.1(2)]. (Since everything is cofibrant in sP re(𝒞), one doesn’t have to take cofibrant replacements for U and X.) Lemma 4.4(i) says the latter condition is the same as F satisfying descent for U. By Theorem 6.2 the model structure sP re(𝒞)_{ℒ} is the same as sP re(𝒞)/S, so this proves one case.

The proof of the second case is the same, but any hypercover U must be replaced by a cofibrant object before it appears in a mapping space, and Lemma 4.4(ii) is used instead of Lemma 4.4(i). □

Using the above corollary, we can actually identify the fibrations in both sP re(𝒞)_{ℒ} and U𝒞_{ℒ} in terms of a relative descent condition. The idea for this proof is due to Blander [Bl, Prop. 4.1]. However, the interpretation in terms of descent conditions requires extra care with various homotopy-limit constructions. We start by recalling the relevant definitions.

Let P be the cosimplicial simplicial set B(Δ ↓ −) described in [H, 14.7.7]. Then hom^{Δ}(P, X) is defined in [H, 18.3.6] to be the homotopy limit of a cosimplicial diagram X. However, this construction only has good properties when X is objectwisefibrant. Therefore, we will define holim_{Δ} X to be hom^{Δ}(P, X̂), where X̂ is an objectwise-fibrant replacement for X. If X is not objectwise-fibrant, there is no obvious guarantee that hom^{Δ}(P, X) and holim_{Δ} X are weakly equivalent; we will need to be careful about this below.

In the following definition, F (U) is the cosimplicial simplicial set which appears in Definition 4.3. If Z is any simplicial set, then cZ denotes the corresponding constant cosimplicial simplicial set.

Definition 7.2. An objectwise fibration F → G satisfies descent for the hypercover U → X if the natural map F (X) → holim_{Δ}[cG(X) ×_{G(U)} F (U)] is a weak equivalence.

Note that this reduces to our previous definition when G = ∗.

Remark 7.3. By manipulating homotopy limits, one can see that the above definition is equivalent to requiring that F (X) be the homotopy limit of the diagram

\xymatrix{ & ∏_{a} F (U_{0}^{a}) \ar[r] \ar[r] \ar[d] & ∏_{a} F (U_{1}^{a}) \ar[r] \ar[r] \ar[r] \ar[d] & ⋯ \\ G(X) \ar[r] & ∏_{a} G(U_{0}^{a}) \ar[r] \ar[r] & ∏_{a} G(U_{1}^{a}) \ar[r] \ar[r] \ar[r] & ⋯}

or alternatively of the diagram

G(X) → holim_{Δ} G(U) ← holim_{Δ} F (U).

We will not need either of these criteria, however.

Our goal is to show that the fibrations in sP re(𝒞)_{ℒ} or in U𝒞_{ℒ} can be characterized using the above descent condition. The proof is more complex than one might imagine, and proceeds in a few steps.

Lemma 7.4.

(i) Let Z be a simplicial set, let W^{∗} be a cosimplicial simplicial set, and suppose there is an objectwise fibration W^{∗} → cZ. Then hom_{Δ}(P, W) is weakly equivalent to the homotopy limit of W^{∗}.

(ii) An objectwise fibration of simplicial presheaves F → G satisfies descent for U → X if and only if the natural map F (X) → hom^{Δ}(P, cG(X) ×_{G(U)} F (U)) is a weak equivalence.

Proof. For (i), pick a fibrant replacement Z → Ẑ, and factor the composite W → cZ → cẐ into an objectwise acyclic cofibration W ↣^{∼} Ŵ followed by an objectwise fibration Ŵ ↠ cẐ. Let E = (cZ)×_{cẐ}Ŵ, and consider the two maps hom_{Δ}(P, W) → hom_{Δ}(P, E) → hom_{Δ}(P, Ŵ). The last object is the homotopy limit of W by definition, so we will show that both maps are equivalences.

First consider the diagram

\xymatrix{Z \ar[r]^{∼} \ar[d]_{∼} & hom_{Δ}(P, cZ) \ar[d] & hom_{Δ}(P, E) \ar[d] \ar[l] \\ Ẑ \ar[r]^{∼} & hom_{Δ}(P, cẐ) & hom_{Δ}(P, Ŵ) \ar@{->>}[l]}

Note that P_{0} = B(Δ ↓ [0]) is actually simplicially contractible, since [0] is the terminal object of Δ. Because hom^{Δ}(P, cZ) = Map(P_{0}, Z) it follows that the canonical map Z = hom^{Δ}(∗, cZ) → hom^{Δ}(P, cZ) is a simplicial homotopy equivalence (and similarly for Ẑ). So the left horizontal maps in the diagram are simplicial homotopy equivalences.

The right square is a pullback square. It follows that the right-most vertical map is the pullback of a weak equivalence along a fibration, hence a weak equivalence (by right properness of s𝒮et).

Next we consider the diagram

\xymatrix{W \ar[rr]^{∼} \ar@{->>}[dr] &  & E \ar@{->>}[dl] \\  & cZ.}

It is a general fact that a right Quillen functor preserves weak equivalences between fibrations over a given object (this is Ken Brown’s Lemma [H, 7.7.2] in the overcategory). Since hom_{Δ}(P, −) is a right Quillen functor, we therefore have hom_{Δ}(P, W) ⟶^{∼} hom_{Δ}(P, E). This finishes the proof of (i).

Part (ii) is an application of (i), noting that cG(X) ×_{G(U)} F (U) → cG(X) is an objectwise fibration. □

Our next task is to reinterpret descent in yet another way, as a lifting condition. In U𝒞, consider the maps hocolim_{n} U_{n} → X and factor them as hocolim_{n} U_{n} ↣ B(U) −↠^{∼} X, where the first map is a projective cofibration and the second is an objectwise acyclic fibration. From now on we’ll denote the object hocolim_{n} U_{n} by A(U), for short. Note that since each U_{n} is projective-cofibrant, so is hocolim U_{n} by [H, 18.4.2]—so both A(U) and B(U) are projective-cofibrant. Also, X is both cofibrant and fibrant in U𝒞, therefore B(U) is cofibrant-fibrant and B(U) → X is a simplicial homotopy equivalence.

Fix a collection of hypercovers S which contains a dense set. Let J_{𝒞} be the collection of all maps Λ^{n,k} ⊗ Z → Δ^{n} ⊗ Z (for all Z ∈ 𝒞) and also of all the maps

[A(U) ⊗ Δ^{n}] ⨿_{A(U)⊗∂Δ^{n}} [B(U) ⊗ ∂Δ^{n}] → B(U) ⊗ Δ^{n}

where U → X ranges over the hypercovers in S.

Lemma 7.5. A map F → G is an objectwise fibration satisfying descent with respect to all hypercovers in S if and only if it has the right-lifting-property with respect to the maps in J_{𝒞}.

Note that, as a consequence, objectwise fibrations satisfying descent are closed under pullbacks.

Proof. A map F → G is an objectwise fibration if and only if it has the right-lifting-property with respect to the maps Λ^{n,k} ⊗ Z → Δ^{n} ⊗ Z. If F → G is an objectwise fibration then the map

Map(B(U), F) → Map(A(U), F) ×_{Map(A(U),G)} Map(B(U), G)

is a fibration because U𝒞 is a simplicial model category, and it’s a weak equivalence if and only if F → G has the right-lifting-property with respect to the maps

[A(U) ⊗ Δ^{n}] ⨿_{A(U)⊗∂Δ^{n}} [B(U) ⊗ ∂Δ^{n}] → B(U) ⊗ Δ^{n},

for all n. Because B(U) → X is a simplicial homotopy equivalence, it follows that both Map(X, F) → Map(B(U), F) and Map(X, G) → Map(B(U), G) are simplicial homotopy equivalences. From this one sees that

Map(B(U), F) → Map(A(U), F) ×_{Map(A(U),G)} Map(B(U), G)

is a weak equivalence if and only if

Map(X, F) → Map(A(U), F) ×_{Map(A(U),G)} Map(X, G)

is one.

We have A(U) = hocolim_{Δ^{op}} U = U ⊗_{Δ} P, which gives the canonical identification of Map(A(U), F) with hom^{Δ}(P, F (U)) [H, 18.1.10]. A diagram chase now shows that an objectwise fibration F → G has the lifting property in question if and only if the natural maps

hom^{Δ}(P, cF (X)) → hom^{Δ}(P, F (U)) ×_{hom^{Δ}(P,G(U))} hom^{Δ}(P, cG(X))

are weak equivalences. The object on the left is simplicially homotopy equivalent to F (X), as in the proof of Lemma 7.4. Using that hom^{Δ}(P, −) is a right adjoint, the object on the right can be identified with hom^{Δ}(P, cG(X) ×_{G(U)} F (U)). Now Lemma 7.4(ii) tells us that the above map is a weak equivalence if and only if F → G satisfies descent for U → X. □

Lemma 7.6. Let ℳ be a model category, and let J be a set of acyclic cofibrations which permits the small object argument. Suppose every map that is both a J-injective and a weak equivalence is also a fibration. Then the J-injectives are precisely the fibrations.

Proof. The small object argument shows that any acyclic cofibration A → B may be factored as A → X → B where the first map is a relative J-cell complex (therefore an acyclic cofibration) and the second is a J-injective. The two-out-of-three property says that X → B is a weak equivalence, and then our assumption implies it is a fibration. The retract argument then shows that A → B is a retract of A → X. In other words, every acyclic cofibration is a retract of a relative J-cell complex. From this it follows that J-injectives have the right-lifting-property with respect to all acyclic cofibrations, hence J-injectives are fibrations. □

Proposition 7.7. The set J_{𝒞} is a set of generating acyclic cofibrations in U𝒞_{ℒ} .

Proof. The maps in J_{𝒞} are acyclic cofibrations in U𝒞_{ℒ} because U𝒞_{ℒ} is a simplicial model category and because the maps A(U) → B(U) are acyclic cofibrations in U𝒞_{ℒ} . It suffices to show that if F → G is a J_{𝒞} -injective which is also a local weak equivalence, then F → G is an objectwise acyclic fibration. This will imply that it is also an acyclic fibration in U𝒞_{ℒ} , and in particular a fibration in U𝒞_{ℒ} . Then we apply Lemma 7.6.

We need to show that for every X ∈ 𝒞 and every point x ∈ G(X), the fiber of F (X) → G(X) over x is contractible. Let us replace F and G by their restrictions to the site 𝒞 ↓ X. The map of restricted presheaves is still a local weak equivalence, and still satisfies descent with respect to a dense set of hypercovers (because every hypercover in 𝒞 ↓ X is essentially a hypercover in 𝒞). By Lemma 7.5 F → G has the right-lifting-property with respect to the corresponding set J_{(𝒞↓X)} in U(𝒞 ↓ X).

Our point x ∈ G(X) now corresponds to a map ∗ → G in U(𝒞 ↓ X). Consider the pullback square

\xymatrix{H \ar[r] \ar[d] & F \ar[d] \\ ∗ \ar[r] & G.}

The map H → ∗ still has the right-lifting-property with respect to J_{(𝒞↓X)} , and so it is an objectwise fibration satisfying descent by Lemma 7.5. Therefore H is fibrant in U(𝒞 ↓ X)_{ℒ} by Corollary 7.1. Moreover, since F → G is a local acyclic fibration, so is H → ∗. Thus H → ∗ is an acyclic fibration in U(𝒞 ↓ X)_{ℒ} , hence an objectwise acyclic fibration. This implies that H(∗) is contractible. But H(∗) is another name for the fiber of our original map F (X) → G(X), so we are done. □

Theorem 7.8. Let S be a collection of hypercovers which contains a set that is dense. A map of simplicial presheaves F → G is a fibration in sP re(𝒞)_{ℒ} (resp. in U𝒞_{ℒ} ) if and only if it is an injective fibration (resp. an objectwise fibration) and satisfies descent for all hypercovers in S.

Proof. The statement for U𝒞_{ℒ} follows from Proposition 7.7 and Lemma 7.5. For sP re(𝒞)_{ℒ} one repeats all the above arguments, but in the definition of J_{𝒞} the maps Λ^{n,k} ⊗ Z → Δ^{n} ⊗ Z are replaced with a set of generating acyclic cofibrations for sP re(𝒞). □

7.9. A short example about fibrant replacement. We end this section with a simple (and well-known) example demonstrating the use of Corollary 7.1. Let A be a presheaf of abelian groups on the site 𝒞, and let I_{∗} denote an injective resolution of the sheafification Ã in the category of sheaves. We will explain how to use I_{∗} to construct a fibrant replacement for the simplicial presheaf K(A, n).

Let ℐ denote the chain complex of presheaves which has I_{k} in dimension n − k when k < n, and has the presheaf of n-boundaries B_{n} in dimension 0. The Dold-Kan correspondence lets us identify presheaves of (non-negatively graded) chain complexes with the abelian group objects in sP re(𝒞), and so ℐ can be regarded as a simplicial presheaf. Since right now we are only dealing with abelian things, it’s easier just to think about chain complexes, though.

The map A → I_{0} induces a map K(A, n) → ℐ (and recall that as a chain complex, K(A, n) has A in dimension n and 0 everywhere else). This map is a local weak equivalence because it induces isomorphisms on homology group sheaves. We claim that ℐ satisfies descent for all hypercovers, and so is a fibrant object in U𝒞_{ℒ} . This shows that one can identify weak homotopy classes of maps Ho (Δ^{k}/∂Δ^{k} ⊗ X, K(A, n)) with H_{k}(ℐ(X)), which is just the sheaf cohomology group H_{𝒞}^{n−k}(X, Ã). The connection with sheaf cohomology is also explained in [J1, Section 2].

If U → X is a hypercover of X, let ℤ[U] denote the chain complex of presheaves obtained by applying the free abelian group functor to the presheaves U_{n}. It is known that after sheafification ℤ[U] becomes a resolution of ℤ[X] (this is basically the ‘Illusie Conjecture’—see [J1, Thm. 2.5] for a proof). The mapping space Map(U, ℐ) may be identified with Map(ℤ[U], ℐ) using adjointness, and this is just the total complex associated to the bicomplex (p, q) → ℐ_{q}(U_{p}). By running the spectral sequence for the homology of this bicomplex, making use of the fact that the ℐ_{k}’s are injective sheaves (k ≥ 1) and ℤ[U]^{∼} is a resolution of ℤ[X]^{∼}, one finds that the spectral sequence collapses and the homology is just that of Map(X, ℐ). In other words, Map(X, ℐ) → Map(U, ℐ) is a weak equivalence.

8. Other applications

Our main application for studying hypercovers is to produce realization functors on 𝔸^{1}-homotopy theory [DI1, Is]. In this section we consider a few other applications to the homotopy theory of simplicial presheaves.

8.1. Change of site.

Suppose that 𝒞 and 𝒟 are Grothendieck sites, and f: 𝒞 → 𝒟 is a functor. The direct image functor f_{∗}: sP re(𝒟) → sP re(𝒞) has a left adjoint f^{∗}. One is interested in conditions on f which imply that these adjoint functors are wellbehaved in relation to the homotopy theory of simplicial presheaves. Here is a general result which is now easy to prove:

Proposition 8.2. Suppose that there is a dense set S of hypercovers in 𝒞 such that f^{∗} takes elements of S to hypercovers in 𝒟. Then the adjoint functors (f^{∗}, f_{∗}) give a Quillen map U𝒞_{ℒ} → U𝒟_{ℒ} . (Recall that a Quillen map is just a Quillen pair [H, Defn. 8.5.2] regarded as a map of model categories in the direction of the left adjoint.)

In this result one cannot replace U𝒞_{ℒ} by sP re(𝒞)_{ℒ} . The functor f^{∗} usually does not preserve monomorphisms, which are the cofibrations in sP re(𝒞)_{ℒ} .

Proof. Using general facts about the universal model category U𝒞 [D, Prop. 2.3], the functors (f^{∗}, f_{∗}) are a Quillen map from U𝒞 to U𝒟. If T denotes the collection of hypercovers in 𝒟, then we have assumed that f^{∗} maps S into T. Therefore, by general considerations [D, Section 5] one gets a Quillen pair between U𝒞/S and U𝒟/T. But by Theorem 6.2 these localizations are just U𝒞_{ℒ} and U𝒟_{ℒ} . □

Suppose that f is continuous, in the sense that {f(U_{a}) → f(X)} generates a covering sieve of f(X) if {U_{a} → X} is a covering sieve. It follows that f^{∗} preserves generalized covers: this is easy for maps whose target is a representable, and the general case can be deduced using that the target is a colimit of representables. If one also supposes that f^{∗} preserves finite limits, then M_{n}(f^{∗}U) ≅ f^{∗}(M_{n}U) and hence f^{∗} preserves hypercovers. Unfortunately there are examples of interest in which f^{∗} does not preserve finite limits (see [MV, Ex. 1.19, p. 103]), and so here is a slightly different criterion which is useful:

Corollary 8.3. Suppose that 𝒞 and 𝒟 are Verdier sites (see Section 9). Assume the functor f: 𝒞 → 𝒟 preserves finite limits for diagrams of basal maps, and takes covering families {U_{a} → X} in 𝒞 to covering families {f(U_{a}) → f(X)} in 𝒟. Then (f^{∗}, f_{∗}) give a Quillen map U𝒞_{ℒ} → U𝒟_{ℒ} .

Proof. The assumptions imply that f^{∗} preserves matching objects of basal hypercovers (use Proposition 4.15 and the material in Section 9). So the condition about preserving covering families shows that f^{∗} takes basal hypercovers in 𝒞 to basal hypercovers in 𝒟. Thus, Proposition 8.2 applies. □

As an example, let S → T be a map of schemes and consider the base-change functor f: Sm/T → Sm/S from the category of smooth schemes over T to the category of smooth schemes over S. This functor satisfies the properties of the above proposition for any of the standard topologies (such as Zariski, étale, or Nisnevich) on Sm/S and Sm/T. So one gets a Quillen pair U(Sm/T)_{ℒ} → U(Sm/S)_{ℒ} , by the above corollary. Compared to the discussion in [MV], this approach is much simpler.

8.4. Computing homotopy classes of maps.

Given a simplicial presheaf F, we will use ℍF to denote a fibrant replacement in sP re(𝒞)_{ℒ} (or in U𝒞_{ℒ} , depending on the context). In some sense the ultimate goal of sheaf theory is to compute the homotopy types of the simplicial sets ℍF (X).

For instance, if A is a presheaf of abelian groups and F = K(A, n) is the associated Eilenberg-Mac Lane simplicial presheaf, then π_{i}ℍF (X) = H^{n−i}(X, Ã) (see Section 7.9). If F is a presheaf of chain complexes then ℍF (X) computes the hypercohomology of X with coefficients in F, and this is where the notation ℍF comes from (in the present context it goes back to [Th]).

There is no known method for computing ℍF in general—one can use the small object argument, but this is not very computable. For ‘nice’ sites one can use the Godement resolution [J2, Prop. 3.3], but this is also not so computable. In this section we give analogs of the Verdier hypercovering theorem, which show how to compute some invariants of ℍF (X) using hypercovers. It would be interesting to construct an explicit model for the simplicial set ℍF (X) using hypercovers, but we haven’t been able to do this.

We’ll write Ho (F, G) for the set of weak homotopy classes of maps from F to G in the homotopy category of sP re(𝒞)_{ℒ} . Likewise, π(F, G) denotes the set sP re(𝒞)(F, G)/∼, where the equivalence relation is generated by simplicial homotopy.

Given an object X in 𝒞, let HC_{X} denote the full subcategory of sP re(𝒞) consisting of all hypercovers of X. We let πHC_{X} denote the category with the same objects, but where πHC_{X}(U, V) equals π(U, V).

Proposition 8.5. The category πHC_{X} is filtered.

This proposition is proved in [SGA4, Exposé V, 7.3.2] and also in [AM, Section 8] with a slightly different notion of hypercover (see Section 10). We prove it again here because it is straightforward with the techniques that we have already developed.

Proof. If U → X and V → X are both hypercovers, then so is U ×_{X} V → X. Thus, we only need show that two parallel arrows V ⇉ U in πHC_{X} can be equalized.

The two maps from V → X to U → X can be assembled into the square

\xymatrix{∂Δ^{1} ⊗ V \ar[r] \ar[d] & U \ar[d] \\ Δ^{1} ⊗ V \ar[r] & X}

in which the bottom map factors through V → X. The right vertical arrow is a local acyclic fibration by definition. Therefore, we apply Proposition 5.1 and obtain another hypercover W → X that refines V, together with a diagram

\xymatrix{∂Δ^{1} ⊗ W \ar[r] \ar[d] & ∂Δ^{1} ⊗ V \ar[r] & U \ar[d] \\ Δ^{1} ⊗ W \ar[r] \ar[urr] & Δ^{1} ⊗ V \ar[r] & X.}

The two compositions W → U are simplicially homotopic and hence equal in πHC_{X}. □

The following is a generalization of the Verdier hypercovering theorem [SGA4, Exposé V, 7.4.1(4)]. The case K = ∗ of part (b) appeared in [B], and is cited several times in Jardine’s papers (see [J2, p. 83], for instance). It can be deduced from general considerations about the category of locally fibrant simplicial presheaves being a ‘category with fibrant objects’. The generalization to arbitrary K, as well as to the relative setting in (c), doesn’t seem to follow from these considerations, however. The case of arbitrary K can be deduced from K = ∗ using [DI2, Cor. 7.5], but the material in Section 5 makes it just as easy to give a proof which handles all cases at once.

Theorem 8.6. Let F be a locally fibrant simplicial presheaf and let X belong to 𝒞. Let F → ℍF be a fibrant replacement for F in sP re(𝒞)_{ℒ} . Then

(a) Given a 0-simplex p of ℍF (X), there is a hypercover V → X and a map v : V → F such that the following square commutes up to simplicial homotopy:

\xymatrix{V \ar[r]^{v} \ar[d] & F \ar[d] \\ X \ar[r]^{p} & ℍF.}

We say that ‘p is represented by the map v’.

(b) Given a finite simplicial set K, there is an isomorphism

colim_{U→X} π(K ⊗ U, F) → Ho (K ⊗ X, F)

where the colimit is taken over (the opposite category of) πHC_{X}.

(c) Given p and v as in (a), there is an isomorphism

π_{n}(ℍF (X), p) ≅ colim_{U→V} π_{n}(Map(U, ℍF), p|_{U}) ≅ colim_{U→V} π(Δ^{n}/∂Δ^{n} ⊗ U, F)_{v|_{U}} .

Here v|_{U} denotes the map U → V → F, and p|_{U} denotes the map U → V → X → ℍF. The colimits are taken over the overcategory πHC_{X} ↓ V of hypercovers refining V, and π(Δ^{n}/∂Δ^{n} ⊗ U, F)_{v|_{U}} denotes the set of all maps f: Δ^{n}/∂Δ^{n} ⊗ U → F such that f |_{∗⊗U} is the given map v|_{U}: U → V → F, modulo simplicial homotopy relative to ∗ ⊗ U.

Proof. Part (a) is a direct consequence of Proposition 5.7 because F and ℍF are both locally fibrant.

For surjectivity in (b), note that any element α of Ho (K ⊗ X, F) is represented by an actual map K ⊗ X → ℍF. From Proposition 5.7 again, we get a hypercover U → X and a diagram

\xymatrix{K ⊗ U \ar[r]^{f} \ar[d] & F \ar[d] \\ K ⊗ X \ar[r] & ℍF}

commuting up to simplicial homotopy. The map f has image α in Ho (K ⊗ X, F).

For injectivity in (b), suppose given two maps K ⊗ U → F that have the same image in Ho (K ⊗ X, F). Since K ⊗ U → K ⊗ X is a local weak equivalence, this means that the two compositions K ⊗ U → ℍF are simplicially homotopic. Hence we have a diagram

\xymatrix{(∂Δ^{1} × K) ⊗ V \ar[r] \ar[d] & (∂Δ^{1} × K) ⊗ U \ar[r] & F \ar[d] \\ (Δ^{1} × K) ⊗ V \ar[r] \ar@{.>}[urr] & (Δ^{1} × K) ⊗ U \ar[r] & ℍF}

for some refinement V of U, where the lift is a relative-homotopy-lifting. In particular, the upper left triangle commutes on the nose, so the two maps K ⊗ U → F are equal in colim_{U→X} π(K ⊗ U, F).

For (c), note that the natural map Map(X, ℍF) → Map(U, ℍF) is a weak equivalence. So it induces an isomorphism π_{n}(ℍF (X), p) ⟶^{≅} π_{n}(Map(U, ℍF), p|_{U}), and after taking the colimit over all U we get the first isomorphism in the theorem.

For the second isomorphism, observe that composing with F → ℍF induces maps π(Δ^{n}/∂Δ^{n} ⊗ U, F)_{v|_{U}} → π_{n}(Map(U, ℍF), p|_{U}). As in the proof of part (b) above, the fact that these maps give an isomorphism after passing to the colimit is a direct consequence of Proposition 5.7. □

Note that if S is a dense set of hypercovers then the colimits in the above results can just as well be taken over the full subcategory of πHC_{X} whose objects belong to S.

8.7. The coconnected case.

Definition 8.8. A locally fibrant simplicial presheaf F is said to be locally ncoconnected if it has the following property: for any X in 𝒞 and any 0-simplex x in F (X), the homotopy group sheaves π_{k}(F, x) on 𝒞 ↓ X vanish for all k ≥ n.

Using techniques from [DI2], a locally fibrant simplicial presheaf is locally ncoconnected if and only if it has the local lifting property with respect to the maps ∂Δ^{k} ⊗ X → Δ^{k} ⊗ X for k > n.

Not surprisingly, for n-coconnected presheaves one can calculate homotopy classes of maps by only using bounded hypercovers. This is what we’ll prove next.

If n ≥ 0, let HC_{X}(n) denote the category of bounded hypercovers U → X of height at most n (see Definition 4.10). Let πHC_{X}(n) denote the category with the same objects but with simplicial homotopy classes of maps. Arguments similar to Proposition 8.5 show that πHC_{X}(n) is filtered.

Proposition 8.9. Suppose that F is locally fibrant and locally n-coconnected. Then given a finite simplicial set K, there is an isomorphism

Ho (K ⊗ X, F) ≅ colim_{U→X} π(K ⊗ U, cosk_{n} F)

where the colimit is taken over the category πHC_{X}(n).

Proof. First, the map F → cosk_{n} F is a local weak equivalence between locally fibrant objects. So we can say that

Ho (K ⊗ X, F) ≅ Ho (K ⊗ X, cosk_{n} F) ≅ colim_{U→X} π(K ⊗ U, cosk_{n} F)

where the colimit runs over the full category πHC_{X}; the second isomorphism comes from Theorem 8.6. We need to show that

colim_{U∈πHC_{X}(n)} π(K ⊗ U, cosk_{n} F) → colim_{U∈πHC_{X}} π(K ⊗ U, cosk_{n} F)

is an isomorphism. Observe that for any simplicial set L, a map L ⊗ U → cosk_{n} F factors through cosk_{n}(L⊗U), and the map L⊗U → cosk_{n}(L⊗U) factors as L⊗U → L⊗cosk_{n} U → cosk_{n}(L⊗U). Applying this when L = K shows surjectivity because cosk_{n} U belongs to HC_{X}(n), and from L = K × Δ^{1} one can deduce injectivity. □

Proposition 8.10. Let S be a Noetherian scheme, and let 𝒞 be Sm/S (or 𝒮ch/S) with either the étale or Nisnevich topology. Let X be an object in 𝒞 with the property that every finite set of points is contained in an affine open. Then every bounded hypercover of X can be refined by a Čech complex.

Proof. In the case of the étale topology, this is essentially the content of [Ar, Thm. 4.1]. Since the result is trivial for hypercovers of height 0, we’ll suppose by induction that it works for hypercovers of height at most n. Let U → X be a hypercover of height n+1. By Theorem 9.6, U can be refined by a basal hypercover U′ → X (see Section 9 below). Let V = cosk_{n} U′, which is a hypercover of height at most n. By induction, there is an étale covering family {W_{i} → X} such that ČW refines V (where W = ∐ W_{i}). Consider the induced map ČW_{n+1} → V_{n+1} = M_{n+1}U′. The map U′_{n+1} → M_{n+1}U′ is an étale cover, which pulls back to an étale cover E → ČW_{n+1}. Theorem 4.1 of [Ar] (applied to the case of no geometric points) says that there is a refinement Z of W such that the map ČZ_{n+1} → ČW_{n+1} factors through E. In particular, this means that ČZ refines U′ (and therefore U) up through dimension n + 1; since the height of U is n + 1, this means it automatically refines U in all dimensions. This completes the proof.

For the Nisnevich topology it is essentially the same argument, only using a revised version of [Ar, Thm 4.1]—see [MV, Prop. 1.9, p. 99]. □

Corollary 8.11. Let 𝒞 and X be as in Proposition 8.10, and suppose F is a locally fibrant simplicial presheaf which is locally n-coconnected. If K is a finite simplicial set, there is an isomorphism

Ho (K ⊗ X, F) ≅ colim_{U→X} π(K ⊗ U, cosk_{n} F)

where the colimit is taken over the category πHC_{X}(0) consisting of the Čech complexes.

In particular, if A is a presheaf of abelian groups and we take K = ∗ and F = K(A, n), then the above corollary gives the isomorphism between Čech cohomology and sheaf cohomology established in [Ar] and [MV, Prop. 1.9, p. 99].

Proof. This is a direct consequence of the previous two propositions. The subcategory πHC_{X}(0) is final in πHC_{X}(n). □

Remark 8.12. The above proposition and its corollary are not true for the Zariski topology, and therefore not for the open covering topology on an arbitrary topological space. We repeat the example of [MV, Ex. 1.10, p. 99]: Let X = Spec R be the semi-localization of 𝔸_{k}^{2} at the points (0, 0) and (0, 1). As a topological space X has exactly two closed points x_{1} and x_{2} (of codimension 2), infinitely many points of codimension 1 (corresponding to the irreducible closed curves in 𝔸^{2} passing through both (0, 0) and (0, 1)), and a generic point of codimension 0. Any open cover of X can be refined by a cover with exactly two elements: take any of the pieces containing x_{1} and x_{2}, respectively.

Let U_{1} = X − {x_{1}} and U_{2} = X − {x_{2}}. Pick two of the codimension 1 points f and g which specialize to both x_{1} and x_{2}. Let W_{1} = (U_{1} ∩ U_{2}) − {f} and W_{2} = (U_{1} ∩ U_{2}) − {g}. Let Ω_{0} = U_{1} ⨿ U_{2} and Ω_{1} = (U_{1} ⨿ U_{2}) ⨿ W_{1} ⨿ W_{2} (the first part is degenerate). Consider the hypercover cosk_{1} Ω. This hypercover cannot be refined by a Čech complex.

9. Verdier sites

The definition of hypercover we’ve adopted so far in this paper has the advantage of working for any Grothendieck site; but it is so broad that it can sometimes be cumbersome. One is often in the position of having to check that something works for all hypercovers, and so it is important to have—whenever possible—a smaller collection of objects to deal with. This is the subject of the present section.

To see the basic problem, look at the site of topological spaces with the Grothendieck topology given by open covers. Under Definition 4.2, to give a hypercover of a space X basically corresponds to giving a simplicial space U_{∗} such that each matching map U_{n} → M_{n}U is locally split. This allows for an incredible amount of freedom in what a hypercover can look like, so much that it’s very difficult to say anything concrete about it. To make things easier, it is reasonable that one should be able to look just at the ‘open hypercovers’, where the maps U_{n} → M_{n}U all have the form ⨿_{a}W_{a} → M_{n}U for some open covering {W_{a}} of the target. These are much more manageable objects.

The notion of a Verdier site—introduced in the following definition—is just an axiomatization of the above situation. It is a Grothendieck site with enough extra data that one can talk about a special kind of ‘basal hypercover’ rather than the more general notion we have been working with. A Verdier site is almost just a Grothendieck site with a basis, but we need to throw in one extra property.

Definition 9.1. A Verdier site is a category 𝒞 together with a given collection of covering families {U_{a} → X} satisfying the properties below. A map U → X in 𝒞 is basal if it belongs to one of these covering families. With this terminology, the properties can be stated as follows:

(i) Any single isomorphism {Z → X} forms a covering family. (ii) If {U_{a} → X} is a covering family and Y → X is a map, then the pullbacks

Y ×_{X} U_{a} all exist, and {Y ×_{X} U_{a} → Y } is a covering family. (iii) If {U_{a} → X} is a covering family and one is given a collection of covering

families {V_{ab} → U_{a}}, then the collection of compositions {V_{ab} → U_{a} → X} is also a covering family.

(iv) If U → X is a basal map then the diagonal U → U ×_{X} U is also basal.

Conditions (i)–(iii) say that the collection of covering families serves as a basis for a Grothendieck topology on 𝒞 in the usual way. Most of the familiar geometric Grothendieck sites satisfy the above axioms, including topological spaces, where the covering families are open covers, as well as the Zariski, Nisnevich, and étale topologies on schemes. The reason for not assuming that 𝒞 has all pullbacks is so that our results apply to the Grothendieck topologies on smooth schemes which are used in 𝔸^{1}-homotopy theory [MV].

Observe that pullbacks along any basal map always exist (part (ii)), and that any composition of basal maps is again basal (part (iii)). It follows that if {V_{a} → X} is a finite collection of basal maps and {U_{a} → V_{a}} is another collection of basal maps, then the induced map ∏_{X} U_{a} → ∏_{X} V_{a} of fibre-products is again basal.

For the following definition, note that to give a map f: ∐_{i} rX_{i} → ∐_{j} rY_{j} between coproducts of representables one must choose, for every index i, a prescribed value of j and a map X_{i} → Y_{j}.

Definition 9.2.

(a) A map f: W → Y in sP re(𝒞) is basal if W is a coproduct ∐_{i} rW_{i} of representables, Y is also a coproduct ∐_{j} rY_{j} of representables, and the various maps W_{i} → Y_{j} determining f are all basal, in the sense of Definition 9.1.

(b) A basal hypercover U → X is a hypercover such that the matching maps U_{n} → M_{n}U are all basal.

The second part of this definition only makes sense if one knows that the matching objects M_{n}U are all coproducts of representables, but we will see in Lemma 9.4 that this is the case. First, an easy lemma:

Lemma 9.3. Let F → H ← G be maps between coproducts of representables, where G → H is basal. Then the pullback is also a coproduct of representables, and the map from the pullback to F is basal.

Proof. Use the fact that the Yoneda embedding preserves whatever limits exist, and that coproducts commute with fibre-products in sP re(𝒞). The necessary pullbacks in 𝒞 exist because pullbacks along basal maps always exist. □

Lemma 9.4. Let U → X be an n-truncated basal hypercover, and let K be a finite simplicial set of dimension at most n. Then hom_{+}(K, U) is a coproduct of representables. In particular, this is true for M_{n+1}U = hom_{+}(∂Δ^{n+1}, U).

Proof. We proceed by induction on the dimension of K. When K is empty hom_{+}(K, U) is just X, which is a representable by assumption.

Now assume that the lemma has been proven for simplicial sets of dimension at most k − 1, and let K be obtained from a (k − 1)-dimensional simplicial set L by attaching finitely many k-simplices. By repeating the following argument, we may assume that only one k-simplex is attached. It follows that hom_{+}(K, U) is the pullback of the diagram

hom_{+}(Δ^{k}, U) → hom_{+}(∂Δ^{k}, U) ← hom_{+}(L, U).

All three objects are coproducts of representables, the first because U_{k} is a coproduct of representables and the last two by the induction hypothesis. Since the left map above is basal (being the matching map in a basal hypercover), Lemma 9.3 tells us that hom_{+}(K, U) is also a coproduct of representables. □

Let us return momentarily to Definition 9.2(b). If U → X is a hypercover and U_{0} → X is basal, then the above proposition specialized to ∂Δ^{1} shows that M_{1}U is a coproduct of representables. So we may ask that U_{1} → M_{1}U be basal, which in turn forces M_{2}U to be a coproduct of representables. This shows that our definition of basal hypercover makes sense, in a recursive sort of way.

Proposition 9.5. Let K → L be any map of finite simplicial sets whose dimensions are at most k, and let U → X be a k-truncated basal hypercover. Then the map hom_{+}(L, U) → hom_{+}(K, U) is basal.

Proof. Consider the class C of all maps of finite simplicial sets having the property stated in the proposition. By definition of basal hypercovers, C contains the generating cofibrations ∂Δ^{n} → Δ^{n}. Cobase changes preserve C by Lemmas 4.7(v), 9.3, and 9.4. Also, finite compositions preserve C because basal maps are closed under finite composition. This shows that C contains all inclusions of finite simplicial sets.

In particular, ∅ → Δ^{n} belongs to C. This means that U_{n} → X is basal for every basal hypercover U → X. Using axiom (iv) of Verdier sites one can deduce that U_{n} → U_{n} ×_{X} U_{n} is also basal (note that in the present context these objects are coproducts of representables, unlike in the axiom). In other words, C contains the codiagonal Δ^{n} ⨿ Δ^{n} → Δ^{n} for every n.

Every surjection can be built from the above codiagonals with finitely many compositions and cobase changes. Thus, every surjection belongs to C. But every map is a composition of a surjection with an inclusion, so every map belongs to C. □

The proposition below is the main thing we need about basal hypercovers. See [AM, Lem. 8.8] for the same result without reference to basal maps. Unfortunately, dealing with these basal maps definitely increases the technical complications.

Theorem 9.6. In a Verdier site, any hypercover may be refined by a split, basal hypercover. In particular, the basal hypercovers are dense.

Proof. Let U → X be any hypercover. The fact that U_{0} → X is a generalized cover means there is a covering sieve R of X such that every map in R lifts through U_{0}. But our Grothendieck topology was generated by a basis, so there is a covering family {W_{a} → X} for which every element belongs to R. Setting V_{0} = ∐_{a} rW_{a}, we have that V_{0} → X is basal and refines U_{0} → X.

Continuing by induction, we may assume we have built a split, basal, n-truncated hypercover V which refines U (up through dimension n). Our job is to define V_{n+1}. We consider the maps

\xymatrix{ & U_{n+1} \ar[d] \\ M_{n+1}V \ar[r] & M_{n+1}U,}

where all the objects are coproducts of representables by Lemma 9.4. Using the same reasoning as in the first paragraph, there is a map W → M_{n+1}V that is basal, that is a generalized cover, and that fits in the upper left corner of this diagram, i.e., it refines the pullback generalized cover U_{n+1} ×_{M_{n+1}U} M_{n+1}V → M_{n+1}V. Set V_{n+1} = W ⨿ L_{n+1}V. Now V is a split, (n + 1)-truncated hypercover; the question is whether V_{n+1} → M_{n+1}V is basal. Because of the way W was constructed, we need only show that the map L_{n+1}V → M_{n+1}V is basal.

Recall from Section 4.12 that there is a natural map dgn_{n} V → cosk_{n} V. In dimension n this is the identity map on V_{n}, and in dimension n + 1 it’s the map L_{n+1}V → M_{n+1}V. Picking any degeneracy s_{i} from level n to n + 1, we get a diagram

[figure: L_{n+1}V; M_{n+1}V; V_{n}; s_{i}; V_{n}.; s_{i}]

Every representable summand of L_{n+1}V is of the form s_{i}(rU) for some i and some representable summand rU of V_{n}, so it suffices to show that the right-hand map s_{i} : V_{n} → M_{n+1}V is basal. But this degeneracy is induced by the corresponding collapse map ∂Δ^{n+1} → Δ^{n}, i.e., the composition s: ∂Δ^{n+1} ↪ Δ^{n+1} ⟶^{s_{i}} Δ^{n}. In other words, s_{i} coincides with hom_{+}(Δ^{n}, V) → hom_{+}(∂Δ^{n+1}, V). The fact that this is basal follows from Proposition 9.5. □

Remark 9.7. Suppose there is a regular cardinal λ with the property that every covering family in 𝒞 has size less than λ. Following similar observations to those in Proposition 6.6, the split, basal hypercover of the above proposition can be constructed so that in each level it has fewer than λ summands. This is needed in the next section.

10. Internal hypercovers

In this final section we give a slight modification of Theorem 6.2 which is useful in applications—for instance, it is needed in [Is]. This involves once again tweaking the definition of hypercover in a certain way.

What sometimes happens is that the Grothendieck site 𝒞 is rich enough that one can talk about hypercovers as elements of s𝒞 rather than sP re(𝒞), and this is usually a convenience. For example this is the approach taken in [AM], and it is also used in [DI1] in the context of simplicial spaces. Handling this involves only a slight difference from what we have done, mostly caused by the fact that the coproduct in 𝒞 (which we will denote by ∪) is not the same as the coproduct of presheaves: i.e., r(X ∪ Y) is not the same as rX ⨿ rY.

Throughout this section we work with a Verdier site for which there exists a regular cardinal λ such that:

(1) Every covering family {U_{i} → X} has cardinality less than λ.

(2) Coproducts of size less than λ exist in 𝒞.

(3) If {X_{i}} is a set of objects whose cardinality is less than λ, then the map of presheaves ∐_{i} rX_{i} → r(⋃_{i} X_{i}) becomes an isomorphism after sheafification. For example, if Sm/k denotes the category of smooth schemes of finite type over

a fixed ground field k, we may give it the structure of a Verdier site by saying that the covering families are finite collections {U_{i} → X} such that ∐ U_{i} → X is an étale (or Zariski or Nisnevich) cover. This generates the usual Grothendieck topology, and satisfies the above properties with λ = ℵ_{0}.

Definition 10.1. Given an object X of 𝒞, an internal hypercover of X is a simplicial object U in s𝒞 which is augmented by X, with the property that each matching map U_{n} → M_{n}U is isomorphic over M_{n}U to a map of the form ∐_{i} V_{i} → M_{n}U, for some basal maps {V_{i} → M_{n}U} which generate a covering sieve.

Of course one has to worry about whether the matching object M_{n}U exists, since the site 𝒞 need not have arbitrary limits. But we shall see that the condition on U_{k} → M_{k}U for k ≤ n − 1 guarantees that M_{n}U does in fact exist. Even though 𝒞 is not necessarily complete, the conclusions of Lemma 4.7 are still valid when the limits hom_{+}(K, W) do exist in 𝒞. For example, if hom_{+}(L, W), hom_{+}(K, W), and hom_{+}(M, W) all exist, and the pullback of

hom_{+}(L, W) → hom_{+}(K, W) ← hom_{+}(M, W)

also exists, then hom_{+}(L ⨿_{K} M, W) exists and is isomorphic to the above pullback.

Lemma 10.2. If U → X is an n-truncated internal hypercover then the object hom_{+}(K, U) exists whenever K is a simplicial set of dimension at most n. In particular, the matching object M_{n+1}U = hom_{+}(∂Δ^{n+1}, U) exists.

Proof. The proof follows the same lines as the proof of Lemma 9.4. □

We continue our notational convention of writing U for a simplicial object of 𝒞 and also for the simplicial presheaf that it represents.

Theorem 10.3. The model category sP re(𝒞)_{ℒ} of simplicial presheaves may be obtained as the localization of sP re(𝒞) at the following collection of maps ℐ:

(i) Maps of the form ∐ W_{i} → (⋃ W_{i}), for collections {W_{i}} in 𝒞 of size less than λ.

(ii) The maps rU → rX, for all internal hypercovers U → X.

Proof. Let sP re(𝒞)_{ℐ} denote the localization we’re considering. First note that all the maps in ℐ are local weak equivalences. For maps of type (ii), this is Theorem 6.2. For maps of type (i) it follows from assumption (3) at the beginning of this section, because every simplicial presheaf is locally weakly equivalent to its sheafification. So sP re(𝒞)_{ℒ} is a stronger localization than sP re(𝒞)_{ℐ}. To see that the localizations coincide, it will suffice to show that if V → X is a basal hypercover in which the number of summands in each level is smaller than λ, then V → X is a weak equivalence in sP re(𝒞)_{ℐ}. This is by virtue of Theorem 6.2, Theorem 9.6, and Remark 9.7.

Each presheaf V_{n} may be decomposed as a coproduct of representables in an essentially unique way: V_{n} = ∐_{α} V_{nα}. We define an object U of s𝒞 by U_{n} = ⋃_{α} V_{nα}, and with face and degeneracy maps lifted from those in V. For the rest of the proof we will be careful to distinguish U from the simplicial presheaf rU. Observe that there is a canonical map V → rU, commuting with the augmentations down to X.

We claim that U is an internal hypercover of X. Assuming this for the moment, relation (i) in our definition of sP re(𝒞)_{ℐ} shows that V_{n} → rU_{n} is an ℐ-weak equivalence for each n. Since every simplicial presheaf F is the homotopy colimit hocolim_{n} F_{n} (see Remark 2.1), it follows that V → rU is also an ℐ-weak equivalence. Using that U → X is an internal hypercover, relation (ii) gives that rU ≃ X; so one concludes that V ≃ X as well.

It remains only to verify that U is an internal hypercover. First note that hom_{+}(K, V) → hom_{+}(K, rU) induces an isomorphism on sheafifications for every finite simplicial set K. When K has dimension 0 this follows from property (3) that we assumed at the beginning of the section. For higher dimensional K one proceeds by induction on the number of non-degenerate simplices in K, using the same pullback square from Lemma 9.4 and the fact that sheafification preserves finite limits.

So taking K = ∂Δ^{n} we have that M_{n}V → M_{n}(rU) induces an isomorphism on sheafifications, and in particular is a generalized cover. This, together with the fact that V_{n} → M_{n}V is a generalized cover, shows immediately that the same is true of rU_{n} → M_{n}(rU). So rU is a hypercover of X.

Finally, to see that U is an internal hypercover one just uses that the Yoneda embedding preserves all limits that exist: so M_{n}(rU) is isomorphic to r(M_{n}U). □

Appendix A. Čech localizations

This appendix is a bit of an aside from the main body of the paper. Here we investigate how descent for Čech complexes compares to descent for all hypercovers. These are not equivalent notions in general—see Example A.10—although in some cases they turn out to agree. Unlike hypercover descent, Čech descent is often a reasonably straightforward thing to verify; so it’s useful to know how strong a notion it is. In this section we show that Čech descent actually implies descent for all bounded hypercovers, and we give some related results of interest. Čech descent has also been explored in papers of Simpson, for instance in [HS].

If ξ: F → G is a map of presheaves of sets, the Čech complex of ξ is the simplicial presheaf Čξ (often denoted ČF by abuse) given by

[n] ↦ F ×_{G} F ×_{G} ⋯ ×_{G} F (n + 1 factors).

A simplicial presheaf F is said to have Čech descent if it satisfies descent for ČU → X whenever U → X is a generalized cover in which X is representable and U is a coproduct of representables.

Here is a short proposition we will need to use often:

Proposition A.1. Let {U_{a} → X} be any set of maps in 𝒞, and let R ↪ X be the sieve generated by these maps. Let U = ∐_{a} rU_{a}. Then there is a natural map ČU → R, and this map is an objectwise weak equivalence.

Proof. If ξ: K → L is any map of simplicial sets, then the Čech complex Čξ is fibrant and homotopy discrete. This shows that the natural map ČU → π_{0}ČU is an objectwise weak equivalence. The presheaf R is equal to the presheaf π_{0}ČU, i.e., R(Y) = π_{0}ČU(Y) for all Y in 𝒞. □

Let 𝒞̌ denote the set of maps {R ↪ X}, where X runs over all objects in 𝒞 and R runs over all covering sieves (this is a set because 𝒞 is small). Let sP re(𝒞)𝒞̌ denote the Bousfield localization of sP re(𝒞) at the set 𝒞̌. We’ll refer to this model category as the Čech localization of sP re(𝒞), for reasons which will shortly become apparent (see Corollary A.3).

Given a covering sieve R ↪ X, let Č(R) denote the Čech complex corresponding to the cover ∐ U_{a} → X where the coproduct ranges over all maps U_{a} → X in the sieve. The above proposition implies that ČR → X factors through R, and ČR → R is an objectwise weak equivalence. So localizing at the set {R ↪ X} is equivalent to localizing at {ČR → X}. We will see in a minute that this is actually equivalent to localizing at {ČU → X}, for all generalized covers U → X, and so our Čech localization is analogous to the hypercover localization of Theorem 6.2. The advantage of starting with just the sieves rather than the generalized covers is that these form a set, and so the localization automatically exists.

Proposition A.2. Given any simplicial presheaf F, the map F → F̃ from F to its (levelwise) sheafification is a weak equivalence in sP re(𝒞)𝒞̌.

Unfortunately the proof of this result is somewhat involved, so we’ll postpone it until the end of the section.

Corollary A.3. Let F → G be any generalized cover of presheaves of sets. Then the map ČF → G is a weak equivalence in sP re(𝒞)𝒞̌.

Proof. The map ČF → G factors as ČF → π_{0}ČF → G. As in the proof of Proposition A.1, the first map is an objectwise weak equivalence. The second map is a monomorphism of presheaves, and the fact that F → G is a generalized cover shows that it is a local epimorphism. Hence, the map becomes an isomorphism upon sheafification. Proposition A.2 then shows that it is a weak equivalence in sP re(𝒞)𝒞̌, and so we can conclude the same for the composite ČF → G. □

We now derive the connection with hypercovers. Recall from Definition 4.10 that a hypercover U → X is bounded if U = cosk_{n} U for some n.

Proposition A.4. Given a bounded hypercover U of X, the map U → X is a weak equivalence in sP re(𝒞)𝒞̌.

The following proof was the inspiration for the proof of [DI1, Lem. 4.2].

Proof. We proceed by induction, starting from the fact that bounded hypercovers of height 0 are just Čech complexes and therefore are handled by Corollary A.3.

Suppose that U → X is a bounded hypercover of height n + 1. Define V to be cosk_{n} U, so V is a bounded hypercover of height at most n. Therefore, we may assume by induction that V → X is a weak equivalence in sP re(𝒞)𝒞̌. The canonical map U → V gives a generalized cover U_{n+1} → V_{n+1}, by the very definition of what it means for U to be a hypercover. Lemma A.5 below shows that in fact U_{k} → V_{k} is a generalized cover for all k.

Consider the following bisimplicial object, augmented horizontally by V:

\xymatrix{V & U \ar[l] & U ×_{V} U \ar[l] \ar[l] & ⋯ \ar[l] \ar[l] \ar[l]}

The kth row is the (augmented) Čech complex for the generalized cover U_{k} → V_{k}. Note that for 0 ≤ k ≤ n the kth row is the constant simplicial object with value U_{k} because U_{k} → V_{k} is the identity. Call this bisimplicial object (without the horizontal augmentation) W_{∗∗}. There is an obvious map hocolim W_{∗∗} → X.

One may compute hocolim W_{∗∗} by first taking the homotopy colimit of the rows, and then taking the homotopy colimit of the resulting simplicial object. But in sP re(𝒞)𝒞̌ the homotopy colimit of the kth row is just V_{k} by Corollary A.3. Also, we have assumed by induction that V ≃ hocolim_{k} V_{k} is weakly equivalent to X. So hocolim W_{∗∗} → X is a weak equivalence.

Let D denote the diagonal of W_{∗∗}. Standard homotopy theory tells us that D = hocolim_{k} D_{k} is weakly equivalent to hocolim W_{∗∗}. We claim that U is a retract over X of D. Note first that one has, in complete generality, a map U → D; in dimension k it is the unique horizontal degeneracy W_{0k} → W_{kk}.

To produce a map D → U it is enough to give sk_{n+1} D → sk_{n+1} U, because U = cosk_{n+1} U. But note that sk_{n} D = sk_{n} U, and choosing any face map [0] → [n + 1] gives a map D_{n+1} → U_{n+1}, inducing a corresponding map sk_{n+1} D → sk_{n+1} U as desired.

It is straightforward to check that U → D → U is the identity (because U = cosk_{n+1} U 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 D → X is a weak equivalence in sP re(𝒞)𝒞̌. Since U → X is a retract of D → X, it must also be a weak equivalence. □

Lemma A.5. If U is a hypercover of height n + 1, then the map U → cosk_{n} U is a generalized cover in every dimension.

Proof. First note that for k ≤ n, the map U_{k} → [cosk_{n} U]_{k} is the identity, so it is a generalized cover. For any k, the map U_{k} → [cosk_{n} U]_{k} may be rewritten as

hom_{+}(Δ^{k}, U) → hom_{+}(sk_{n} Δ^{k}, U).

But U = cosk_{n+1} U, so the domain may be written as

hom_{+}(Δ^{k}, U) = hom_{+}(Δ^{k}, cosk_{n+1} U) = hom_{+}(sk_{n+1} Δ^{k}, U).

So we are interested in the map hom_{+}(sk_{n+1} Δ^{k}, U) → hom_{+}(sk_{n} Δ^{k}, U) induced by the inclusion sk_{n} Δ^{k} → sk_{n+1} Δ^{k}. Recall from Lemma 4.11 the pullback square

\xymatrix{hom_{+}(sk_{n+1} Δ^{k}, U) \ar[d] \ar[r] & ∏_{X} hom_{+}(Δ^{n+1}, U) \ar[d] \\ hom_{+}(sk_{n} Δ^{k}, U) \ar[r] & ∏_{X} hom_{+}(∂Δ^{n+1}, U).}

The map hom_{+}(Δ^{n+1}, U) → hom_{+}(∂Δ^{n+1}, U) is just the matching map U_{n+1} → M_{n}U, and is therefore a generalized cover. So the right vertical map in the above square is a finite product of generalized covers, which is again a generalized cover. Finally, we see that the left vertical map is a pullback of a generalized cover, hence also a generalized cover. □

If R ↪ X is a covering sieve, let ℐ_{R} denote the full subcategory of 𝒞 ↓ X consisting of all maps in R. Consider the diagram ℐ_{R} → sP re(𝒞) sending (U → X) to U. The colimit of this diagram is R, and we will write the homotopy colimit as hocolim_{R} U. The natural map from the homotopy colimit to the colimit gives hocolim_{R} U → R, and this turns out to be an objectwise weak equivalence by [D, Lemma 2.7]. This fact has nothing to do with sieves, and is true in a slightly generalized form for arbitrary simplicial presheaves.

Theorem A.6. The following classes of maps give the same localization of sP re(𝒞):

(a) The set of all covering sieves R ↪ X.

(b) The set of all maps hocolim_{R} U → X, where R ↪ X is a covering sieve.

(c) The class of all hypercovers of height 0, i.e., the Čech complexes ČU → X.

(d) The class of all bounded hypercovers U → X.

(e) The class of maps F → F̃ from simplicial presheaves to their sheafifications. If the topology on 𝒞 is given by a basis of covering families, then one can also add (a’) The set of all covering sieves R_{U} ↪ X where R_{U} is the sieve generated by the

covering family {U_{a} → X}.

It may seem surprising that the localization in (e) does not give the usual notion of local weak equivalence, but only the weaker Čech version. Example A.10 shows that it really is weaker. Also, note that the above theorem could just as well have been stated for U𝒞 rather than sP re(𝒞)—the proofs are essentially the same.

Proof. The fact that hocolim_{R} U → R is an objectwise weak equivalence immediately shows that the localizations in (a) and (b) are the same. And we have seen in Proposition A.1 that the localizations in (a) and (c) are the same.

The localization in (d) is a priori stronger than that in (a); Proposition A.4 shows that the two localizations actually agree. Likewise, the localization in (e) is stronger than the one in (a), because R ↪ X becomes an isomorphism upon sheafification. Proposition A.2 shows that they agree.

Finally, if our topology is given by a basis of covering families then the proof that the localizations in (a’) and (e) coincide follows the proof of Proposition A.2 more or less verbatim. □

It would be interesting to know more about sP re(𝒞)𝒞̌, for instance to have an explicit characterization of the weak equivalences. Perhaps this wouldn’t be so useful, since the chief interest in sP re(𝒞)𝒞̌ is that it is sometimes a more convenient version of sP re(𝒞)_{ℒ} (see Example A.11).

Corollary A.7. Let F be a simplicial presheaf. Then F satisfies descent for all Čech complexes if and only if it satisfies descent for all bounded hypercovers.

Proof. Let F′ be a fibrant replacement for F in sP re(𝒞). The statement of the corollary for F′ requires that F′ be local with respect to the Čech complexes ČV → X if and only if it is local with respect to the bounded hypercovers U → X. This is true by the above proposition (parts (c) and (d)). But of course F has descent for a certain class of objects precisely when F′ has descent for that same class, because F → F′ is an objectwise weak equivalence. □

Corollary A.8. Suppose the Grothendieck topology on 𝒞 is given by a basis of covering families. Then a simplicial presheaf F satisfies Čech descent if and only if it satisfies descent for all the Čech complexes ČU → X in which X is a representable and U = ∐_{a} U_{a} for some covering family {U_{a} → X} in the basis.

Proof. This is similar to the proof of Corollary A.7, using Theorem A.6 (parts (a’) and (c)) and Proposition A.1. □

The following result can be useful for verifying hypercover descent. We deduce it from our general analysis of hypercovers, but the statement also follows from results of [HS] (see especially Proposition 6.1) and should be credited to that paper.

Corollary A.9. Let F be an objectwise-fibrant simplicial presheaf with the property that F (X) has no homotopy in dimension n or higher, for every X in 𝒞. Then F satisfies descent for all hypercovers if and only if it satisfies descent for all Čech complexes.

Proof. First we need to consider the localization U𝒞/S, where S is the set of maps {∂Δ^{n+1} ⊗ X → Δ^{n+1} ⊗ X|X ∈ 𝒞}. It is easy to check that the fibrant objects in U𝒞/S are the simplicial presheaves G such that each G(X) is fibrant and has no homotopy above dimension n − 1. Given an objectwise-fibrant simplicial presheaf G, one can construct the fibrant replacement L_{S}G via the small object argument applied to the maps in S. By thinking about this, one sees that the maps of simplicial sets G(X) → L_{S}G(X) are isomorphisms up through simplicial dimension n. So L_{S}G(X) has the same homotopy groups as G(X) up through dimension n−1, but no homotopy groups in higher dimensions. Even if G is not objectwise-fibrant, L_{S}G is objectwise weakly equivalent to L_{S}(Ex^{∞} G), and so it is still true that G(X) and L_{S}G(X) have the same (n − 1)-type for all X ∈ 𝒞.

General localization theory says that a map G → H is an S-equivalence if and only if L_{S}G → L_{S}H is an objectwise equivalence, and so this is the same as saying that G(X) → H(X) induces isomorphisms on all homotopy groups up through dimension n − 1, for every X. In particular, the map G → cosk_{n} G is an Sequivalence.

Now consider the localization U𝒞/T, where T is the union of S and the set of all covering sieves R ↪ X. A simplicial presheaf F is fibrant in U𝒞/T precisely when it is objectwise-fibrant, has Čech descent, and each F (X) has no homotopy in dimension n or higher.

Suppose that F is as in the statement of the corollary, and that F satisfies descent for all Čech complexes. Then we know that F is fibrant in U𝒞/T. But note that U → cosk_{n} U is necessarily a T -equivalence (because it is an S-equivalence). Yet cosk_{n} U is a bounded hypercover of X, and hence cosk_{n} U → X is a T -equivalence as well (using the U𝒞 version of Proposition A.4). Hence U → X is also a T - equivalence. Since F is fibrant in U𝒞/T and X is cofibrant in U𝒞/T, the morphism Map(X, F) → Map(U′, F) is a weak equivalence, where U′ is a cofibrant replacement for U in U𝒞. Thus F satisfies descent for U → X by Lemma 4.4(ii). □

Here is an example showing that the Čech localization can be strictly weaker than the localization at all hypercovers. In other words, we exhibit a simplicial presheaf which has descent for all Čech complexes but does not have descent for all hypercovers. The example is a slight modification of one suggested to us by Carlos Simpson.

Example A.10. Let X = X_{0} be the open interval (0, 1). Now let U_{0} = (0, \frac{2}{3}), V_{0} = (\frac{1}{3}, 1), and X_{1} = U_{0}∩V_{0}. Note that X_{1} ≅ X, and let U_{1} = (\frac{1}{3}, \frac{5}{9}) , V_{1} = (\frac{4}{9}, \frac{2}{3}), and X_{2} = U_{1} ∩ V_{1}. Again one has X_{2} ≅ X, and we define U_{2}, V_{2}, and X_{3} in the expected way. Continue. Our site 𝒞 consists of the spaces {X_{i}, U_{i}, V_{i} | i ≥ 0} with the inclusions between them, and equipped with the usual notion of open cover. The category 𝒞 is depicted as

\xymatrix{ & U_{0} \ar[dl] &  & U_{1} \ar[dl] &  & ⋯ \ar[dl] \\ X_{0} &  & X_{1} \ar@{-->}[ul] \ar[dl] &  & X_{2} \ar@{-->}[ul] \ar[dl] \\  & V_{0} \ar@{-->}[ul] &  & V_{1} \ar@{-->}[ul] &  & · · · . \ar[ul]}

Define a presheaf of topological spaces on our site in the following way:

F (X_{0}) = ∅, F (U_{n}) = D_{+}^{n}, F (V_{n}) = D_{−}^{n}, and F (X_{n+1}) = S^{n} (n ≥ 0).

Here D_{+}^{n} and D_{−}^{n} denote the upper and lower hemispheres of S^{n}. The restriction maps F (U_{n}) → F (X_{n+1}) and F (V_{n}) → F (X_{n+1}) are the inclusions of the hemispheres in S^{n}, while the maps F (X_{n}) → F (U_{n}) and F (X_{n}) → F (V_{n}) are the inclusions of the boundaries of the hemispheres. Define the simplicial presheaf G by G(W) = ℤSing F (W)—that is, G(W) is the result of applying the free abelian group functor to the singular complex of F (W). Using the Dold-Kan correspondence, one can regard G as a presheaf of chain complexes; then G(W) is the usual complex for computing the singular homology of F (W).

Now G has Čech descent: this can be checked using Corollary A.8, and so the main point is that for every n the square

\xymatrix{G(U_{n} ∪ V_{n}) \ar[r] \ar[d] & G(U_{n}) \ar[d] \\ G(V_{n}) \ar[r] & G(U_{n} ∩ V_{n})}

is a homotopy pullback. On the other hand, we will construct a hypercover for which G does not have descent. The combinatorics of this construction are slightly complicated, but the idea is this: Start with Ω_{0} = U_{0} ⨿ V_{0}, then consider cosk_{0} Ω except replace each non-degenerate occurrence of X_{1} with U_{1} ⨿ V_{1}. Next take cosk_{1} Ω, replace each non-degenerate occurrence of X_{2} with U_{2} ⨿ V_{2}, and continue. This gives the hypercover Ω → X.

Now we will be more precise. Let P_{n} be the category of all nonempty subsets of {0, 1, …, n}, with inclusions. Note that the objects of P_{n} can be identified with the sub-simplices of Δ^{n}, and that [n] ↦ P_{n} forms a simplicial category. Let S_{n} denote the set of all functors J: P_{n}^{op} → 𝒞 with the following properties:

(1) All the values of J belong to {U_{i}, V_{i} | i ≥ 0}.

(2) Given a subset σ = {i_{0}, …, i_{k}} in P_{n}, if ⋂_{j} J({i_{0}, …, î_{j}, …, i_{k}}) = U_{m} (resp. V_{m}) then J(σ) = U_{m} (resp. V_{m}).

(3) If ⋂_{j} J({i_{0}, …, î_{j}, …, i_{k}}) = X_{m} then J(σ) is either U_{m} or V_{m} (and this includes the case k = 0). Let Ω denote the simplicial presheaf defined by

Ω_{n} := ∐_{J∈S_{n}} J({0, 1, …, n}),

with simplicial structure induced by that of P. Intuitively, each summand of Ω_{n} corresponds to an n-simplex together with a certain labelling of its simplices given by J: the labelling is such that smaller simplices are labelled by larger opens, and such that the above properties are satisfied. The reader is encouraged to work out what these properties say for small values of n, and to verify that Ω is a hypercover of X_{0} (use Proposition 4.15). Check that Ω_{0} = U_{0} ⨿ V_{0} and Ω_{1} = U_{0} ⨿ (U_{1} ⨿ V_{1}) ⨿ (U_{1} ⨿ V_{1}) ⨿ V_{0}.

We claim that holim_{n} G(Ω_{n}) is not connected, whereas G(X) = 0. To calculate π_{0}(holim_{n} G(Ω_{n})) we can just work in the category of chain complexes. The cosimplicial object [n] ↦ G(Ω_{n}) corresponds to a double complex, and we are trying to compute the 0th homology of the total complex (the one called Tot^{Π} in [W], rather than Tot^{⊕}). But observe that each G(Ω_{n}) has homology only in dimension 0 because each G(U_{i}) and G(V_{i}) is contractible. Therefore, the E_{1}-term of the spectral sequence for the homology of the bicomplex is concentrated in a line. Thus, the bicomplex’s 0th homology is the kernel of d_{0} − d_{1} : H_{0}G(Ω_{0}) → H_{0}G(Ω_{1}), which is ℤ. This completes the verification that G does not satisfy descent for the hypercover Ω.

Example A.11. Sometimes the localizations U𝒞𝒞̌ and U𝒞_{ℒ} do coincide. Let S be a Noetherian scheme of finite dimension and let 𝒞 be the site Sm/S with either the Zariski or Nisnevich topology (one can also take 𝒮ch/S here). For these sites the localizations U𝒞𝒞̌ and U𝒞_{ℒ} agree. For the Zariski topology this is a direct consequence of the ‘Brown-Gersten Theorem’, which identifies the fibrant objects in U𝒞_{ℒ} with the objectwise-fibrant simplicial presheaves satisfying Čech descent for all two-fold Zariski covers {U, V → X}. This is essentially proven in [BG], although one has to translate their proof into our more modern setting. See also [Bl, Lem. 4.1, 4.3].

For the Nisnevich topology we have to explain a little more. Given an elementary distinguished square {U ↪ X, p: V → X} [MV, Def. 1.3, p. 96], let P (U, V) denote the simplicial presheaf which has U ⨿ V in dimension 0, U ⨿ p^{−1}(U) ⨿ V in dimension 1, and is degenerate in higher dimensions. The Brown-Gersten Theorem in this context is [MV, Lem. 1.6, p. 98]; together with [Bl, Lem. 4.3], it implies that U𝒞_{ℒ} is the localization of U𝒞 at the maps P (U, V) → X for all elementary distinguished squares. We already know that U𝒞_{ℒ} is a stronger localization than U𝒞𝒞̌, so we just need to show that the maps P (U, V) → X are weak equivalences in U𝒞𝒞̌.

To see that P (U, V) → X is a weak equivalence in U𝒞𝒞̌, first note that for any Z the simplicial set P (U, V)(Z) has non-degenerate simplices only in dimensions 0 and 1. Each component is a star, centered at a 0-simplex corresponding to a map Z → U (because every map Z → V can be an endpoint of at most one 1-simplex). Therefore each component is contractible, so P (U, V) → π_{0}P (U, V) is an objectwise weak equivalence. The codomain is just the presheaf U ⨿_{p^{−1}U} V, so we are reduced to showing that the map U ⨿_{p^{−1}U} V → X is a weak equivalence in U𝒞𝒞̌. By the U𝒞 version of Theorem A.6(e) it suffices to show that this map induces an isomorphism on sheafifications, and this is routine (use Nisnevich stalks, or look at [MV, Lem. 1.6, p. 98]).

A.12. A leftover proof. The final goal of this section is to give the proof of Proposition A.2: if F is a simplicial presheaf we need to show that F → F̃ is a weak equivalence in sP re(𝒞)𝒞̌. In fact it will suffice to do this when F is a discrete simplicial presheaf, since a simplicial presheaf F can be recovered as a homotopy colimit of the discrete presheaves F_{n} (Remark 2.1). Unfortunately, even to prove our claim for discrete simplicial presheaves seems to require a wrestling match with the small object argument.

So from now on F is just a presheaf of sets. We introduce two constructions: First, 𝒜F is the presheaf defined by 𝒜F (X) = F (X)/ ∼, where two sections s and t are equivalent if there is a covering sieve R ↪ X such that s|_{U} = t|_{U} for every U → X in R. Secondly, ℬF is defined to be the pushout

\xymatrix{∐ R \ar[r] \ar[d] & F \ar[d] \\ ∐ X \ar[r] & ℬF}

where the coproduct is indexed over all objects X in 𝒞, all covering sieves R ↪ rX, and all maps R → F. One may check that 𝒜ℬF is what is usually denoted F^{+}, and so 𝒜ℬ𝒜ℬF is the sheafification F̃.

We will show that the maps F → ℬF and F → 𝒜F are Čech weak equivalences, for any presheaf F. The first claim is very easy: Since ∐ R → ∐ X is an acyclic cofibration in sP re(𝒞)𝒞̌, its cobase change F → ℬF is also an acyclic cofibration. Unfortunately the second claim is much more difficult. The idea is to build up a Čech weak equivalence F → L_{∞}F by brute force, in such a way that there is an objectwise weak equivalence L_{∞}F → 𝒜F.

Given a covering sieve R ↪ X, let J^{n}R be the pushout

\xymatrix{∂Δ^{n} ⊗ R \ar[d] \ar[r] & ∂Δ^{n} ⊗ X \ar[d] \\ Δ^{n} ⊗ R \ar[r] & J^{n}R.}

The natural map J^{n}R → Δ^{n} ⊗ X is a Čech acyclic cofibration because sP re(𝒞)𝒞̌ is a simplicial model category. Note that a map J^{n}R → G is the same as a map ∂Δ^{n} → G(X) together with a compatible family of extensions Δ^{n} → G(U) for all maps U → X in R.

Let L_{0}F = F, and let L_{n+1}F be obtained from L_{n}F as the pushout

[figure: ∐ J^{n+1}R; ∼; L_{n}F; ∼; ∐ Δ^{n+1} ⊗ X; L_{n+1}F] (A.13)

where the coproducts run over all X in 𝒞, all covering sieves R ↪ X, and all maps J^{n+1}R → L_{n}F. Let L_{∞}F be the colimit of the chain L_{0}F → L_{1}F → L_{2}F → · · ·. Since each L_{n}F → L_{n+1}F is a Čech acyclic cofibration, the composite F → L_{∞}F is also a Čech acyclic cofibration.

To get a feel for what’s happening here, let’s look just at L_{1}F. A map J^{1}R → F corresponds to giving a map ∂Δ^{1} → F (X) together with a compatible family of extensions Δ^{1} → F (U) for all U → X in the sieve. Since F is discrete, this means that we are giving two elements of F (X) which agree when restricted to pieces of the sieve. When we form the pushout Δ^{1} ⊗ X ← J^{1}R → F we are adding a 1-simplex into F (X) which identifies these elements in π_{0}. So it follows that π_{0}L_{1}F (Y) = 𝒜F (Y), for all Y.

When we pass from L_{n}F to L_{n+1}F something similar is happening—we will see it boils down to killing off all the higher homotopy, in the end because the objects F (X) were all discrete and so had no higher homotopy to begin with. So the goal is to show that each L_{∞}F (X) is fibrant and homotopy discrete, and that π_{0}L_{∞}F (X) = 𝒜F (X). This will imply that the natural map L_{∞}F → π_{0}L_{∞}F is an objectwise weak equivalence, and the target is identified with 𝒜F. We will then

have F ⟶^{∼} L_{∞}F ⟶^{∼} 𝒜F in sP re(𝒞)𝒞̌.

The argument will proceed by establishing the following properties: Given any Y in 𝒞,

(i) The map of simplicial sets L_{n}F (Y) → L_{n+1}F (Y) is an isomorphism on nskeleta.

(ii) The simplicial set L_{n}F (Y) has dimension at most n, i.e., it is degenerate in

degrees greater than n. (iii) Given any n-simplex σ in L_{n}F (Y), there is a covering sieve R ↪ Y such

that σ|_{U} is in the image of L_{n−1}F (U) → L_{n}F (U) for every U → Y in R. In particular, σ|_{U} is a degenerate n-simplex.

(iv) Given any n-simplex σ in L_{n}F (Y), there is a covering sieve R ↪ Y such that

σ|_{U} is in the image of F (U) → L_{n}F (U) for every U → Y in R.

(v) For n ≥ 2, any map ∂Δ^{n} → L_{n−1}F (Y) extends to a map Δ^{n} → L_{n}F (Y). (vi) Any map Λ^{2,k} → L_{1}F (Y) extends to ∂Δ^{2} → L_{2}F (Y).

Granting these for the moment, let us show they imply the desired result. To show that L_{∞}F (X) is fibrant and homotopy discrete, it is enough to verify that it has the extension property with respect to the maps ∂Δ^{n} → Δ^{n} (n ≥ 2) and the maps Λ^{2,k} → Δ^{2}. These are easy consequences of parts (i), (v), and (vi). Also, part (i) tells us that π_{0}L_{1}F → π_{0}L_{∞}F is an isomorphism, and we have already remarked that π_{0}L_{1}F ≅ 𝒜F. This finishes the proof, granting the statements outlined above.

Claim (i) follows from the fact that J^{n+1}R(Y) → (Δ^{n+1} ⊗ X)(Y) is an isomorphism on n-skeleta. Part (ii) follows from an induction, using that (Δ^{n} ⊗ X)(Y) has dimension n and that F (Y) has dimension 0 (since we assumed that F is a presheaf of sets). Part (iii) is a straightforward analysis of diagram (A.13), and (iv) follows from (iii) by induction. We will show that (v) is a consequence of (iv), and a similar argument proves (vi).

Suppose we have a map σ: ∂Δ^{n} → L_{n−1}F (Y). By (iv), for each face d_{i}σ there is a covering sieve R_{i} ↪ Y such that d_{i}σ|_{U} is in the image of F (U) → L_{n−1}F (U), for every U → Y in R_{i}. There is of course a covering sieve R which refines all the R_{i}. So for each U → Y in R and each i, there is (n−1)-simplex α_{U,i} in F (U) which maps to (d_{i}σ)|_{U}.

Now, it is not clear that as i varies the (n−1)-simplices α_{U,i} fit together to give a map α_{U}: ∂Δ^{n} → F (U). However, we know they fit together in L_{n−1}F (U), and the map F (U) → L_{n−1}F (U) is a cofibration of simplicial sets, hence a monomorphism. So the α_{U,i} must fit together in F (U) as well.

Secondly, it is not immediately clear that the α_{U} patch together over the covering sieve R: that is, we must check that given maps U → V → X where V → X is in R, then α_{U} coincides with the restriction of α_{V} to U. Again, this follows from the fact that everything patches together in L_{n−1}F and the fact that F → L_{n−1}F is an objectwise cofibration.

So we have constructed a map α: ∂Δ^{n} ⊗ R → F such that the composite map ∂Δ^{n} ⊗ R → F → L_{n−1}F coincides with σ|_{R}. Now we use the fact that F is a discrete simplicial presheaf, from which it follows that α can be extended to a map ᾱ: Δ^{n} ⊗ R → F. Composing this with F → L_{n−1}F and patching with σ gives a map J^{n}R → L_{n−1}F, and this extends to Δ^{n} ⊗Y once we pass to L_{n}F. The upshot is that we’ve shown σ extends to Δ^{n} under the map L_{n−1}F (Y) → L_{n}F (Y).

References

[SGA4] M. Artin, A. Grothendieck, J.L Verdier, Théorie des Topos et Cohomologie Étale des Schemas, Lecture Notes in Math. 270, Springer, 1972.

[AM] M. Artin and B. Mazur, Étale Homotopy, Lecture Notes in Math. 100, Springer, 1969.

[Ar] M. Artin, On the joins of Hensel rings, Adv. Math. 7 (1971), 282–296.

[Bl] B. Blander, Local projective model structures on simplicial presheaves, K-theory 24 (2001), no. 3, 283–301.

[B] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419–458.

[BG] K. Brown and S. Gersten, Algebraic K-theory as generalized sheaf cohomology, in Higher K-theory I, 266–292, Lecture Notes in Math. 341, Springer, 1973.

[D] D. Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176.

[DI1] D. Dugger and D. C. Isaksen, Topological hypercovers and 𝔸^{1}-realizations, Math. Zeit., to appear.

[DI2] D. Dugger and D. C. Isaksen, Weak equivalences of simplicial presheaves, Proc. Conf. in Algebraic Topology (Evanston, 2002), to appear.

[H] P. S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs vol. 99, Amer. Math. Soc., 2003.

[HS] A. Hirschowitz and C. Simpson, Descente pour les n-champs, preprint, 2001.

[Ho] M. Hovey, Model Categories, Mathematical Surveys and Monographs vol. 63, Amer. Math. Soc., 1999.

[I] L. Illusie, Complexe cotangent et déformations I, Lecture Notes in Math. 239, Springer, 1971.

[Is] D. C. Isaksen, Étale realization on the 𝔸^{1}-homotopy theory of schemes, Adv. Math., to appear.

[J1] J. F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Mathematics 55, Part I (1986), 193–239.

[J2] J. F. Jardine, Simplicial presheaves, J. Pure and Appl. Algebra 47 (1987), 35–87.

[J3] J. F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), no. 3, 733–747.

[J4] J. F. Jardine, Boolean localization, in practice, Doc. Math. 1 (1996), 245–275.

[Jo] A. Joyal, unpublished letter to A. Grothendieck.

[MV] F. Morel and V. Voevodsky, 𝔸^{1}-homotopy theory, Inst. Hautes Etudes Sci. Publ. Math. 90 (2001), 45–143.

[Q] D. Quillen, Homotopical Algebra, Lecture Notes in Math. 43, Springer, 1969.

[Th] R. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. Ec. Norm. Sup. (4) 18 (1985), no. 3, 437–552.

[W] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1994.

Department of Mathematics, University of Oregon, Eugene, OR 97403

Department of Mathematics, University of Chicago, Chicago, IL 60637

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

   E-mail address: ddugger@math.uoregon.edu
   E-mail address: sjh@math.uchicago.edu
   E-mail address: isaksen.1@nd.edu