HYPERCOVERS AND SIMPLICIAL PRESHEAVES
DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
Abstract.We use hypercovers to study the homotopy theory of simpli
cial presheaves. The main result says that model structures for simplic*
*ial
presheaves involving local weak equivalences can be constructed by local*
*iz
ing at the hypercovers. One consequence is that the fibrant objects can *
*be
explicitly described in terms of a hypercover descent condition. These i*
*deas
are central to constructing realization functors on the homotopy theory *
*of
schemes [DI1, Is]. We give a few other applications for this new descrip*
*tion 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. Other applications 20
8. Verdier sites 25
9. Internal hypercovers 28
Appendix A. ~Cech localizations 30
References 38
1.Introduction
This paper is concerned with the subject of homotopical sheaf theory, as it h*
*as
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
localtoglobal 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 alg*
*ebraic
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 nonabelian example of algebraic Ktheory.
Here the site C is a category of schemes, and the functor F assigns to each sch*
*eme
X its algebraic Ktheory spectrum K(X). Thomason's paper [Th ] (building on
1
2 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
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 pictu*
*re.
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 EilenbergMacLane objects, and the whole
theory can in some sense be regarded as the study of nonadditive sheaf cohomol*
*ogy.
Jardine's model structure has recently served as the foundation from which Morel
and Voevodsky built their A1homotopy theory for schemes [MV ].
One important ingredient missing from Jardine's work is a description of the
fibrant objects. They can be characterized in terms of a certain lifting proper*
*ty,
but this is not so enlightening and not very useful in practice. Intuitively i*
*t has
always been clear that the fibrant objects should be the simplicial presheaves *
*that
satisfy some kind of descent. Our main goal in this paper is to clarify this i*
*ssue
and give an explicit interpretation of fibrancy for simplicial presheaves.
To explain the basic ideas, let's assume our site is the category of topologi*
*cal
spaces equipped withQthe usual openQcovers. A presheaf of sets F is a sheaf if *
*F (X)
is the equalizer of aF (Ua) ' a,bF (Ua\ Ub) whenever {Ua} is an open cover *
*of
X. This equalizer is in fact the same as the inverse limit of the entire cosimp*
*licial
diagram
Q ____//_Q _____////
a F (Ua)____//_a,bF (Uab)___//_. . .
where we have abbreviated Ua0...anfor Ua0 \ . .\.Uan and have refrained from
drawing the codegeneracies for typographical reasons. For a presheaf of simplic*
*ial
sets (or taking values in some other homotopical objects like spectra), it is n*
*atural
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 proper*
*ty,
when it holds for all open covers, is called ~Cech descent. It can also be expr*
*essed
in a slightly more compact way, if one recalls that the`~Cech complex ~CU assoc*
*iated
to a cover {Ua} of X is the simplicial object [n] 7! a0,...,anUa0...an. Then*
* F
satisfies ~Cech descent if the natural map
F (X) ! holimnF (C~Un)
is a weak equivalence.
A motivating example is given by the functor Topop! Spectrataking X to EX ,
where E is a fixed spectrum and EX denotes the function spectrum. This functor
has ~Cech descent, because X is weakly equivalent to the homotopy colimit of the
~Cech complex for any open cover_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 ~Cech descent (although this erroneous *
*claim
has appeared in a couple of preprints). 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, whi*
*ch
is similar to a ~Cech complex except in level n we only need to have a cover of*
* the
nfold intersections Ua0...an. A precise definition requires a morass of machi*
*nery
(see Section 4). A simplicial presheaf F satisfies descent for the hypercover
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 3
U ! X if the natural map
F (X) ! holimnF (Un)
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:
Theorem 1.1. The fibrant objects in Jardine's model category sP re(C)L are tho*
*se
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 stru*
*cture
for the discrete topology on C (see Section 2 for more about this). The fibran*
*cy
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 c*
*om
plicated conditions of a similar `diagrammatic nature'. In practice such condit*
*ions
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 b*
*elow
and the discussion in Section 2.
We'd like to point out that the above theorem can be reinterpreted in terms *
*of
giving `generators' and `relations' for the homotopy theory of simplicial presh*
*eaves,
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 u*
*ni
versal homotopy theory UC=S constructed by
(1)Formally adding homotopy colimits to the category C, to create UC; and then
(2)Imposing relations requiring that for every hypercover U ! X, the map
hocolimnUn ! X is a weak equivalence.
In other words, the result says that everything special about the homotopy th*
*eory
of simplicial presheaves can be derived from the basic fact that one can recons*
*truct
X as the homotopy colimit of any of its hypercovers. The above theorem is cruci*
*al
to the construction of 'etale realization functors for A1homotopy 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 fibran*
*cy
conditions of the injective model structure are replaced by a much simpler cond*
*ition.
Compare the following result to Theorem 1.1.
Theorem 1.3. The fibrant objects in the model category UC=S are those simplici*
*al
presheaves that
(1)are objectwise fibrant (i.e., each F (X) is a fibrant simplicial set), and
(2)satisfy descent for all hypercovers U ! X.
The main ideas we use to prove these results are very simple, and worth sum
marizing. They exactly parallel classical facts about CWcomplexes. The two key
ingredients are:
4 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
(i)In the category of simplicial presheaves one can construct objects analogous
to CWcomplexes, the only difference being that one has different kinds of
0simplices corresponding to the different representable presheaves rX. (And
as a consequence, there are different kinds of nsimplices corresponding to
the objects 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 CW
approximation, 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 con*
*struct
ing a hypercover. These ideas are explored in detail in Section 5.
This paper came into existence because we needed to use Theorems 6.2, 7.6,
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 careful*
*ly
setting down the machinery of hypercovers, hopefully in a way that will be usab*
*le
by other people. For this reason the paper sometimes takes on an expository ton*
*e.
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 wh*
*ich
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 8) and internal hypercovers (Section 9*
*) one
can implement for sites like those encountered in algebraic geometry. This subj*
*ect
of homotopical sheaf theory is rapidly finding applications in many contexts, s*
*o 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 Quill*
*en
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 no*
*tation and
terminology. [Ho ] is also a good reference.
Section 3 reviews material from [DI2] on lifting properties for sim*
*plicial
presheaves, and how these can be used to characterize local weak equivalences.
Section 4 introduces the machinery needed for defining and working with hyper
covers. 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 probl*
*ems
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 Section 6, where they appear as Theorem 6.2 and Corollary 6.3.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 5
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 7
we give a few more applications. One of the most interesting, given in Section *
*7.1,
is a much simpler approach to the changeofsite functors of [MV ]. We also dis*
*cuss
a generalization of the Verdier hypercovering theorem in Theorem 7.6.
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 8 and*
* 9.
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 sporad*
*ically
in Sections 6 and 7, 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
~Cech descent and hypercover descent. Again, the principal motivation comes from
the fact that ~Cech descent is more easily dealt with in practice. We show, amo*
*ng
other things, that having descent for ~Cech complexes is equivalent to having d*
*escent
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 sim
plicial object of C, then rU is the simplicial presheaf given by the formula
rU(Y )n = Hom C(Y, Un)_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 finitetype o*
*ver S.
The full subcategory of schemes which are smooth over S is denoted Sm=S. Finall*
*y,
in a simplicial model category we write Map (A, B) for the simplicial mapping s*
*pace.
2.Model structures on simplicial presheaves
We start by recalling that for any small category C there are two Quillen equ*
*iva
lent model structures on the category of diagrams sSetC. In each case a map D !*
* E
is a weak equivalence if D(c) ! E(c) is a weak equivalence of simplicial sets f*
*or
each c in C. Such a map is usually called an objectwise weak equivalence. In
the projective model structure on sSetC 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), and
(2)a cofibration if it has the leftliftingproperty with respect to the acycli*
*c fibra
tions.
Dually, in the injective model structure the cofibrations are objectwise and the
fibrations have the rightliftingproperty with respect to acyclic cofibrations*
*. The
names `projective' and `injective' come from the analogy between the two usual
model structures on chain complexes of Rmodules. 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).
6 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
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 isomor
phisms on all sheaves of homotopy groups [I, Jo, J3]. 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 loca*
*liza
tion of sP re(C) at the class L of local weak equivalences; we denote this loca*
*lization
as sP re(C)L . Of course since L is a class of maps there is no a priori guaran*
*tee that
the Bousfield localization exists, but Jardine was able to construct it directl*
*y_it is
only after the fact that one can identify it as a localization.
Similarly, one can define a model structure UCL by localizing UC at the same
class L (cf. [Bl, Thm. 1.5]). The identity maps induce a Quillen equivalence
UCL ! 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 ar*
*e 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 UCL 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 simpl*
*icial
presheaf and K is a simplicial set then K F and F K are defined objectwise, by
(K F )(X) = K x 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 DF : op!
sP re(C) by sending [n] to Fn. Here Fn is just a presheaf of sets, but we can r*
*egard
it as a discrete simplicial presheaf in the obvious way. The realization of th*
*is
simplicial diagram is precisely F . The BousfieldKan map hocolimDF ! DF 
is a weak equivalence in this case, by some basic model category theory. So any
simplicial presheaf F is weakly equivalent to hocolimDF . 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 p*
*ur
poses. See [DI2] for the proof that the two definitions agree and for more deta*
*ils
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
n,k X _____//F
 
 
fflffl fflffl
n X _____//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 n U ! F . These liftings are not required to be compatible for the
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 7
different U's. A map F ! G is called a local fibration if it has local liftings*
* with
respect to the maps n,k X ! 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 simpli
cial presheaves is a local weak equivalence if and only if every square
@ n X ____//_F
 
 
fflffl fflffl
n X _____//G
has local relative homotopyliftings, in the following sense: after restricting*
* to
the pieces U ! X of some covering sieve, one has maps n U ! F making the
upper triangle commute on the nose and the lower triangle commute up to simplic*
*ial
homotopy relative to @ n U.
The reader may consult [DI2] for a detailed discussion of this kind of relati*
*ve
homotopylifting 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 equiva*
*lence.
Proposition 3.2 ([DI2, Prop. 7.2]).A map F ! G admits local liftings in every
square
@ n X ____//_F::____
 _________
 _______
fflffl___fflffl_
n X _____//G
if and only if it is a local acyclic fibration.
One consequence of the above result is that local acyclic fibrations are clos*
*ed
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 simplicia*
*l sets,
then the induced map
F L! F KxGK GL
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 Ea of representables, f is a generalized cover precisely when the s*
*ieve
generated by the maps {Ea ! B} is a covering sieve of B. n
For a simplicial presheaf F , let M~nF denote the 0th object of F @ (the `ti*
*lde'
is to distinguish this from a slightly different construction used later in the*
* paper).
This is the presheaf of sets whose value ~MnF (X) is the set of all maps @ n ! *
*F (X).
8 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
There is a natural map Fn ! M~nF induced by F n! F @ n. Proposition 3.2 can
be rephrased as saying that F ! G is a local acyclic fibration if and only if t*
*he
maps
(3.4) Fn ! M~nF xM~nGGn
are generalized covers, for all n 0. Using this observation, most properties*
* of
generalized covers can automatically be seen to hold for local acyclic fibratio*
*ns.
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 t*
*he
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 context*
*s.
4.1. The definition.
Definition 4.2.Let X belong to C and suppose that U is a simplicial presheaf wi*
*th
an augmentation U ! X. This map is called a hypercover of X if each Un 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 s*
*ays
that the maps U0 ! X, U1 ! U0 xX U0, and Un ! M~nU (for n 1) are all
generalized covers. This is not particularly enlightening, but it's easy to pr*
*ovide
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 ~Cech complexes, which have the form
` _____//` ____//_`
. . .Ua0a1a2____////_Ua0a1__//_Ua0_____//X
for some chosen covering family {Ua ! X}. Here Ua0...anis the fibreproduct
Ua0xX . .x.XUan. The ~Cech complexes are the hypercovers for which the maps
U1 ! U0 xX U0 and Un ! M~nU are all isomorphisms. In an arbitrary hypercover
one takes the iterated fibreproducts at each level but then is allowed to refi*
*ne that
object further, by taking a generalized cover of it. We refer the reader to [A*
*M ,
Section 8] for further discussion of hypercovers.
Next is the formal definition of hypercover descent:
Definition 4.3.An objectwisefibrant simplicial presheaf F satisfies descent for
a hypercover U ! X if F (X) is weakly equivalent to the homotopy limit of the
diagram
Q a ____//_Q a ____//_//
aF (U0)____//_aF (U1)_____//_.,. .
where the products range over the representable summands of each Un. If F is not
objectwisefibrant, we say it satisfies descent if some objectwisefibrant repl*
*acement
for F does.
The definition has been arranged so that if F ! G is an objectwise weak equiv*
*a
lence, then F satisfies descent for U ! X if and only if G does. While the defi*
*nition
reflects our intuitive notion of descent, the next lemma gives a more concise r*
*efor
mulation in terms of simplicial mapping spaces.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 9
Lemma 4.4.
(i)A simplicial presheaf F satisfies descent for a hypercover U ! X if and only
if Map (X, ^F) ! Map (U, ^F) is a weak equivalence of simplicial sets, wher*
*e ^F
is an injectivefibrant replacement for F .
(ii)Let U0 be a cofibrant replacement for U in UC. Then F satisfies descent for
U ! X if and only if Map (X, ^F) ! Map (U0, ^F) is a weak equivalence of
simplicial sets, where ^Fis an objectwisefibrant replacement for F .
Note that any split hypercover (see Definition 4.13) is cofibrant in UC, in w*
*hich
case one can apply (ii) with U0 = U.
Proof.This is by general nonsense. Consider the diagram op! sP re(C) given by
[n] ! Un, and let ~Ube its homotopy colimit. This is not the same as U, but the*
*re
is a map ~U! U which is an objectwise weak equivalence (see Remark 2.1). Let ^F
be an injectivefibrant replacement for F , which a fortiori is an objectwisef*
*ibrant
replacement as well. Then Map (U~, ^F) is weakly equivalent to Map (U, ^F) sin*
*ce
~U! U is a weak equivalence between injectivecofibrant objects. But Map (U~, ^*
*F)
is
Q a
Map (hocolimnUn, ^F) ' holimnMap(Un, ^F) ' holimn a^F(Un).
Since Map (X, ^F) is equal to ^F(X), the condition that Map (X, F ) ! Map (U, F*
* )
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 Un is cofibrant in UC and so ~U= hocolimnUn is also
cofibrant. In other words ~Uis a cofibrant replacement for U, and so ~U' U0. If
^Fis an objectwisereplacement for F then Map (U0, F ) ' Map (U~, F ), and as i*
*n (i)
Q *
* __
the latter is equivalent to holimn aF (Uan). The rest of the proof is the same*
*. __
A more elegant way to phrase the above result is to say that F satisfies desc*
*ent for
U ! X if and only if hMap (X, F ) ! hMap (U, F ) is a weak equivalence of simpl*
*icial
sets, where hMap (, ) denotes a homotopy function complex [H , Ch. 17] in eit*
*her
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 cocomplete_in our applications M
is P re(C), but for the moment let us work in the more general setting. Let +
denote the augmented cosimplicial indexing category: it is obtained by adjoining
an initial object [1] to . Let s+ M denote the category of functors op+! M,
i.e. the category of augmented simplicial objects. We regard a simplicial set K*
* as
belonging to s+ Setby letting K1 consist of a single point.
If S is a set and X belongs to M, let X.Sdenote a product of copies of X inde*
*xed
by the elements of S. Given a simplicial set K and an object W of s+ M, we rega*
*rd
these as functors K : op+! Setand W : op+! M and then form the resulting
10 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
end, denoted hom+ (K, W ):
hY Y i
hom+ (K, W ) := eq Wn.Kn' Wm.Kn.
n [n]![m]
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 bi
jectively 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 Kn, 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 te*
*n
soring 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 ) is isomorphic to homX (K, W ), where homX (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+ ( n, W ) ~=Wn.
(v)hom+ (, W ) takes colimits of simplicial sets to limits in M # X. In other
words, if K = colimiKi, then hom+ (K, W ) ~=limXihom+(Ki, W ).
Definition 4.8.The object hom+ (@ n, W ) is the nth augmented matching
space MnW . The induced map hom+ ( n, W ) ! hom+ (@ n, W ), which we
may now write as Wn ! MnW , is the nth matching map for W . Note that
W0 ! M0W is just the augmentation since @ 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 , Ch. 16]. For n 2, MnW is
isomorphic to M~nU = hom (@ n, W ) because @ 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 Un
is a coproduct of repesentables and the maps Un ! MnU are all generalized cover*
*s.
Definition 4.10.A hypercover U ! X is bounded if there exists an n 0 such
that the maps Uk ! MkU are isomorphisms for all k > n. The smallest such n for
which this is true is called the height of the hypercover, and denoted htU.
We have already remarked that the hypercovers of height 0 are precisely the ~*
*Cech
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.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 11
Lemma 4.11. If U ! X is a bounded hypercover of height at most n, then the
induced maps hom+ ( k, U) ! hom+ (skn k, U) are isomorphisms for all k.
Proof.When k n, the result is easy because k equals skn k. In general, k
is obtained from skn k by gluing on finitely many simplices of dimension at lea*
*st
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
hom+ (L, U)______//hom+(K, U)
 
 
fflffl fflffl
hom+ ( i, U)____//_hom+(@ i, U).
The bottom map is the matching map Ui ! MiU, 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 ntruncated simplicial objects and augmented n
truncated simplicial objects over M. There is an obvious forgetful functor s+ M*
* !
s+ M n called skn, and this has a right adjoint called coskn. These are the ske*
*leta
and coskeleta functors for augmented simplicial objects. If W belongs to s+ M,
we abbreviate cosknsknW to just cosknW .
The kth object of cosknU is
[cosknU]k ~=hom+ ( k, cosknU) ~=hom+ (skn k, U)
(use Remark 4.6 for the second isomorphism). In particular, the (n + 1)st object
of cosknU is what we have been calling Mn+1U. Observe also, using Lemma 4.11,
that a hypercover U has height at most n if and only if U ~=cosknU.
Now, the functor sknalso has a left adjoint dgnn:s+ M n ! s+ M, called the n
degeneration functor. The simplicial object dgnnU 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 [dgnn U]n+1*
* is
called the (n + 1)st latching object for U and is denoted Ln+1 U. This latching
object is the one that arises when considering Reedy model structures of simpli*
*cial
diagram categories [H , Ch. 16]. Note that U, dgnn U, and cosknU all have the
same nskeleton, so there are canonical maps dgnnU ! U ! cosknU; looking in
level n + 1 gives Ln+1U ! U ! Mn+1U.
Definition 4.13.An object W of s+ M is said to be split, or to have free de
generacies, if there exist subobjects Nk ,! Wk such that the canonical maps
Nk q LkW ! Wk are isomorphisms for all k 0. This is equivalent to requir
ing that the canonical map
a
Noe! Wk
oe
is an isomorphism, where the variable oe ranges over all surjective maps in of
the form [k] ! [n], Noedenotes a copy of Nn, and the map Noe! Wk is the one
induced by oe*: Wn ! Wk (see [AM , Def. 8.1]).
12 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
The idea is that the objects Nk represent the nondegenerate 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 forg*
*ets the
augmentation.
We are particularly interested in split hypercovers. If U ! X is a split hy
percover then LkU is a summand of Uk, each LkU is a coproduct of representables,
and each representable summand of LkU is the image under some degeneracy of a
representable from Uk1 (but not uniquely). It follows from [D , Cor. 9.4] that*
* split
hypercovers are cofibrant in UCL , 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 Un into a copro*
*duct
of representables is unique up to permutations of the summands, and (2) to give
a map qiAi ! qjBj between coproducts of representables corresponds to giving,
for each index i, an index j(i) and a map Ai ! Bj(i). Because of these remarks,
one can construct a simplicial set K by taking Kn to be the set of representable
summands of Un. We'll refer to K as the indexing simplicial set for U.
Now suppose a: L ! K is a map of simplicial sets. If opL denotes the opposite
category of simplices of L [H , Def. 16.1.15], there is an obvious diagram opL*
* !
sP re(C) # X which sends a ksimplex oe to the representable which is the summa*
*nd
of Uk corresponding to a(oe). 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
a
hom+ (L, U) ~= U(a).
a:L!K `
In particular, the matching object MnU is isomorphic to a:@ n!K U(a).
Note that opL is an infinite category. If L has the property that every non
degenerate simplex has nondegenerate faces (e.g. L = @ n), then one can use a
smaller version. Let opndL be the subcategory whose objects are the nondegene*
*rate
simplices, and where the maps correspond to face maps. Under the above assump
tion on L, it is an easy exercise to check that opndL ,! opL 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 opndL
in practice.
5. Hypercovers and lifting problems
In Proposition 3.1 we saw how local weak equivalences relate to solutions of
homotopylifting problems after passing from a representable to the elements of
a covering sieve. These liftings could not necessarily be made compatible on the
different pieces of the sieve, however. In this section we show that one can ar*
*range
for this kind of compatibility by using hypercovers.
The proof of our main result (Theorem 6.2) is mostly formal, except for the
key ingredient provided by the next proposition. Recall that, just as for ordin*
*ary
covering families, a refinement of a hypercover U ! X is another hypercover V !
X that factors through U.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 13
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
(5.2) K U _____//F
 
 
fflffl fflffl
L U _____//G
in which U ! X is a hypercover, there exists another hypercover V ! X refining
U and liftings as in the following diagram:
(5.3) K V ____//_K U____//F44__________
 _______________
 ____________ 
fflffl_________ fflffl
L V_____//L U____//_G.
To summarize the basic idea of the proof of Proposition 5.1, 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 locallifting
property (3.1) that there is a covering family {Va ! U0} with liftings sa: Va !*
* F .
In general, saVaband sbVabare not equal, but the two liftings become homotopic
after projecting down to G. We can lift this homotopy to F by passing to a suit*
*able
covering family of Vab, again using the fact that F ! G is a local weak equival*
*ence.
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 general*
*ized
cover Z ! J such that Z is a coproduct of representables and such that the diag*
*ram
__7F7____
___________
________ 
_________ fflffl
Z ____//_J___//_G
has a lifting.
Proof.For each map f :X ! J from a representable, choose a covering sieve Rf
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
_ !
a a
Z = U .
Xf!J iU!XnRf
The obvious map Z ! J is a generalized cover, and the composite Z ! J ! G__
lifts to F . __
14 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
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 ntruncated
hypercover V ! X refining sknU such that the diagram
(5.6) o7F7o
ooooo 
ooo 
oooo fflffl
V ____//_U___//_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
nskeleta the diagram is equal to (5.6).
Proof.The core of the proof is just an ArtinMazur argument [AM , Ch. 8]. First
form the pullback F 0= U xG F . The map F 0! U is still a local acyclic fibrati*
*on,
and we need only produce an (n + 1)truncated hypercover W and a lifting into F*
* 0.
In other words, we can reduce to the case where U = G (and F = F 0). Note that
in this case G is locally fibrant_the representable X is locally fibrant for tr*
*ivial
reasons, and U ! X is a local fibration. Moreover, since F ! G is a local fibra*
*tion,
F is locallynfibrant+as1well.n+1
Now F ! F @ 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
Fn+1 ! Mn+1F (the n = 0 case being only slightly different). Our initial diagram
gives a map V @ n+1! F @ n+1, and the 0th level has the form Mn+1V ! Mn+1F .
So Lemma 5.4 says that there is a generalized cover Z ! Mn+1V , where Z is a
coproduct of representables, such that the composite Z ! Mn+1F lifts through
Fn+1. We take W to be the (n + 1)truncated hypercover with sknW = sknV_and
Wn+1 = Z q Ln+1V . __
Proof of Proposition 5.1.Given a square as in the statement of the proposition,*
* it
may be interpreted as a map
U ! F KxGK GL.
We are trying to produce a hypercover V ! X refining U ! X and a lifting
___F5L5_____
_____________
__________ 
______________ fflffl
V ____//_U___//_F KxGK GL.
The vertical map is a local acyclic fibration by Proposition 3.3, so the hyperc*
*over_
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 hyper*
*cover
V ! X refining U ! X and relativehomotopyliftings in the diagram (5.3).
Recall that relativehomotopyliftings were defined in Proposition 3.1, and d*
*is
cussed extensively in [DI2].
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 15
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
K V ___//_F55__________
ssss______________
sss___________ 
yyss_____fflffl__fflffl_
L V L V ___//_G55________
ss ____________
i0 i1sssss_____________
fflfflyyss________
RH V.
Here RH denotes the pushout of L x 1  K x 1 ß!K, and the maps i0 and
i1 are the obvious inclusions L ,! RH.
Consider the square
F RH ____//_F LxGL GRH
i*1 
fflffl fflffl
F L_____//_F KxGK GL.
By [DI2, Cor. 7.5], the fact that F ! G is a local weak equivalence between loc*
*ally
fibrant objects implies that the horizontal maps are also local weak equivalenc*
*es.
By the same result, the fact that i1: L ! RH is a weak equivalence of simplicial
sets implies that the left vertical map is a local weak equivalence. So we conc*
*lude
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 KxGK GL, 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 LxGL GRH . This provides the__
necessary relativehomotopylifting. __
Lemma 5.8. Let F ! G be a map between locally fibrant simplicial presheaves.
Assume we have a square of simplicial sets
K _____//M
 
 
fflfflfflffl
L _____//N
such that both K ! M and M qK L ! N are cofibrations. Then the induced map
F M xGM GN ! F KxGK GL
is a local fibration.
Proof.The hypotheses imply that both F M ! F K and GN ! G(MqK L) =
GM xGK GL are local fibrations, using [J2, Cor. 1.5]. Now we observe that there
are pullback squares
F M xGM GN ________//_GN F M xGK GL _____//F M
   
   
fflffl fflffl fflffl fflffl
F M xGK GL ____//_GM xGK GL F KxGK GL _____//F K,
16 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
and the pullback of a local fibration is again a local fibration. Finally,_the *
*map we
want is just the composite F M xGM GN ! F M xGK GL ! F KxGK GL. __
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. Thi*
*s lets
us identify the fibrant objects in the model structure. Similar results are pro*
*ven
for the projective version UCL .
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. The
collection is called split if every hypercover in S can be refined by a split h*
*ypercover
which also belongs to S.
For instance, Theorem 8.6 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. Then the localization sP re(C)=S exists and coincides with Jardine's mod*
*el
structure sP re(C)L . Similarly, the localization UC=S exists and coincides wi*
*th
UCL .
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 ] f*
*or a
summary treatment or [H ] for complete details. The fibrations and weak equiva
lences in M=S are called Sfibrations and Sequivalences, while the cofibration*
*s 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 theor*
*em in
the case where S is the collection of all hypercovers, and this is not a set: i*
*n our
definition of hypercover one can have arbitrarily large coproducts of represent*
*ables
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 re*
*gular
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 6.6.
The corollary below follows easily from the theorem, and in fact the two are
equivalent.
Corollary 6.3.Let S be a collection of hypercovers which contains a set that is
dense.
(i)A simplicial presheaf F is fibrant in sP re(C)L if and only if F is injecti*
*ve
fibrant and satisfies descent for all hypercovers in S.
(ii)F is fibrant in UCL if and only if it is objectwise fibrant and satisfies *
*descent
for all hypercovers in S.
In particular, a simplicial presheaf F satisfies descent for all hypercovers if*
* and only
if it satisfies descent for all elements of S.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 17
Proof.First observe that the fibrant objects in sP re(C)=S are the injectivefi*
*brant
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 cofibrantreplacements 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 (i).
The last statement in the corollary now follows as well. The point is that Th*
*e
orem 6.2 applies not only to S, but also to the collection of all hypercovers_so
these two collections give the same localization. Thus, an injectivefibrant ob*
*ject
satisfies descent for the hypercovers in S if and only if it satisfies descent *
*for all
hypercovers. Now use that a simplicial presheaf F satisfies descent for a hyper*
*cover
U precisely when an injectivefibrantreplacement for F does.
The proof of (ii) is the same as (i), but any hypercover U must be replaced by
a cofibrant object before it appears in a mapping space, and Lemma 4.4(ii)_is u*
*sed
instead of Lemma 4.4(i). __
To prove Theorem 6.2, we need a general criterion for checking whether two
localizations are identical:
Lemma 6.4. 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 Sfibration X i Y between Sfibr*
*ant
objects is a T equivalence, then it is an Sequivalence.
Proof.We must show the hypothesis implies that every T equivalence A ! B is an
Sequivalence. Let L denote a fibrantreplacement functor in M=S, and consider
the square
A ______//B
~ S ~ S
fflffl fflffl
LA _____//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 Sacyclic cofibration followed by an Sfibration:
LA //~S//_X__////_LB.
Note that X is Sfibrant, 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 Sfibration, and the
domain and codomain are Sfibrant. Our hypothesis then says that X ! LB is an
Sequivalence. Applying the twooutofthree property (twice) shows that A_! B
is an Sequivalence. __
For the moment let S be a set of hypercovers that is dense. Because S is a se*
*t,
we know that the model structure sP re(C)=S exists (by [H ], using that sP re(C)
is left proper and cellular). The fibrant objects in sP re(C)=S (called Sfibr*
*ant
objects) are the injectivefibrant objects which satisfy descent for all hyperc*
*overs
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 n*
*ow
check the criterion from the above lemma:
18 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
Lemma 6.5. Let F and G be Sfibrant objects, and let f : F ! G be an Sfibrati*
*on
that is also a local weak equivalence. If X is a representable, then every squa*
*re
@ n X ____//_F
 
 
fflffl fflffl
n X _____//G
has a lifting. In particular, f is actually an objectwise acyclic fibration and*
* therefore
an Sequivalence.
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 r*
*espect
to the maps @ n ! n.
Now we prove the first claim. First, f is an objectwise fibration since every
Sfibration is an injectivefibration and also a projectivefibration. This im*
*plies
that f is also a local fibration. Because f is both a local fibration and a lo*
*cal
weak equivalence, Proposition 5.1 guarantees us a hypercover U ! X such that
the diagram
@ n U ____//_@ n X____//F44____________
 _________________
 ______________ 
fflffl_________ fflffl
n U ______// n X_____//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:
Map (X, F n)_______~________//Map(U, F n)
 
fflfflfflffl fflfflfflffl
Map (X, G n xG@ n F @ n)_~__//Map(U, G n xG@ n F @ n).
All the model categories wenhavenbeen consideringnare simplicial model categori*
*es,
and this implies that F ! G xG@ n F @ is an Sfibration between Sfibrant
objects. It is a local weak equivalence by Proposition 3.3. Therefore, the vert*
*ical
maps above are fibrations of simplicial sets because both X and U are Scofibra*
*nt.
Likewise, the horizontal maps are weak equivalences becausenUn! X is an Sn
equivalence between Scofibrant objects and both F and G xG@ n F @ are
Sfibrant.
We are given a 0simplex 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 im*
*age
of x in the lower right corner lifts to the upper right corner. Since the horiz*
*ontal
maps are weak equivalences, there is another 0simplex y belonging to the conne*
*cted
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 h*
*as a
lift. __
Proof of Theorem 6.2.We first consider the claim for sP re(C)L . For the case w*
*hen
our collection of hypercovers S is itself a set, we have already done all the w*
*ork.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 19
Since hypercovers are local weak equivalences we know S L, and so we are in t*
*he
situation of Lemma 6.4. The necessary condition was verified in Lemma 6.5.
In the general case, let S0 be a dense set of hypercovers contained in S. As
shown in the previous paragraph, sP re(C)=S0 is equal to sP re(C)L . So every l*
*ocal
weak equivalence is a weak equivalence in sP re(C)=S0, and in particular every
hypercover in S is an S0equivalence. This shows that sP re(C)=S exists and is
equal to sP re(C)=S0.
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 6.5 for the
projective model structure, assuming that F and G are fibrant objects of UC=S
and that F ! G is an objectwise fibration. The only difference in the proof is *
*that
one replaces U by a cofibrant object before dealing with simplicial mapping spa*
*ces.
The rest of the argument is exactly the same, as is the generalization to_the_c*
*ase
where S need not be a set. __
6.6. 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 (4.14) that to any hypercover U ! X one can
attach an indexing simplicial set K, where Kn is the set`of representable summa*
*nds
of Un. The size of the hypercover is the cardinality of nKn, 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 6.7.The class of all hypercovers has a subset which is dense.
Proof.Choose a regular cardinal ~ sufficiently large compared to the cardinalit*
*y of
the set of morphisms in C, and let S denote the set of all hypercovers of size *
*less
than ~. We will show that any hypercover U ! X can be refined by one in S.
Since U0 ! X is a generalized cover, there`is a covering sieve R of X such th*
*at
every W ! X in R lifts through U0. Let V0 = W!X W , where the coproduct
ranges over all maps W ! X in R. The number of summands in V0 is clearly
bounded by ~.
Now assume by induction that we have constructed an ntruncated hypercover
V ! X which refines U, and such that the number of summands in V is less than ~.
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 fr*
*om_
a representable into Mn+1V . This can be deduced from Proposition 4.15. _*
*_
6.8. A short example about fibrant replacement. We end this section with a
simple (and wellknown) example demonstrating the use of Corollary 6.3. Let A be
a presheaf of abelian groups on the site C, and let I* denote an injective reso*
*lution
of the sheafification ~Ain 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 Ik in dimension n  k
when k < n, and has the presheaf of nboundaries Bn in dimension 0. The Dold
Kan correspondence lets us identify presheaves of (nonnegatively 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.
20 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
The map A ! I0 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 fib*
*rant
object in UCL . This shows that one can identify weak homotopy classes of maps
Ho ( k=@ k X, K(A, n)) with Hnk(I(X)), which is just the sheaf cohomology
group HnkC(X, ~A). 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 Un. It is
known that after sheafification Z[U] becomes a resolution of Z[X] (this is basi
cally 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) ! Iq(Up). By running the
spectral sequence for the homology of this bicomplex, making use of the fact th*
*at
the Ik'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.
7.Other applications
Our main application for studying hypercovers is to produce realization funct*
*ors
on A1homotopy theory [DI1, Is]. In this section we consider a few other applic*
*ations
to the homotopy theory of simplicial presheaves.
7.1. 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 well
behaved in relation to the homotopy theory of simplicial presheaves. Here is a
general result which is now easy to prove:
Proposition 7.2.Suppose that there is a dense set S of hypercovers in C such th*
*at
f* takes elements of S to hypercovers in D. Then the adjoint functors (f*, f*) *
*give
a Quillen map UCL ! UDL . (Recall that a Quillen map is just a Quillen pair
[H , Defn. 8.6.1] regarded as a map of model categories in the direction of the*
* left
adjoint.)
In this result one cannot replace UCL by sP re(C)L . The functor f* usually d*
*oes
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 collect*
*ion
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 we have already seen in Theorem 6.2 that these localizations_are just
UCL and UDL . __
Suppose that f* preserves finite limits; then Mn(f*U) ~=f*(MnU). Suppose
also that f is continuous, in the sense that {f(Ua) ! f(X)} generates a covering
sieve of f(X) if {Ua ! X} is a covering sieve. Then f* preserves generalized
covers. So under these two hypotheses one finds that f* preserves hypercovers.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 21
The hypothesis that f* preserves finite limits is not always satisfied in examp*
*les of
interest (see [MV , Ex. 1.19, p. 103]), and so here is a slightly different cri*
*terion
which is useful:
Corollary 7.3.Suppose that C and D are Verdier sites (see Section 8). Assume
the functor f :C ! D preserves finite limits for diagrams of basal maps, and ta*
*kes
covering families {Ua ! X} in C to covering families {f(Ua) ! f(X)} in D. Then
(f*, f*) give a Quillen map UCL ! UDL .
Proof.Preserving finite limits implies that f* preserves matching objects (use
Proposition 4.15 and the material in section 8). So the condition about preserv*
*ing
covering families shows that f* takes basal hypercovers in C to basal hypercove*
*rs_
in D. Thus, Proposition 7.2 applies. __
As an example, let S ! T be a map of schemes and consider the basechange
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, 'etale, or Nis*
*nevich)
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.
7.4. Computing homotopy classes of maps.
Given a simplicial presheaf F , we will use HF to denote a fibrantreplacement
in sP re(C)L (or in UCL , depending on the context). In some sense the ultimate
goal of sheaf theory is to compute the simplicial sets HF (X). For instance, if
A is a presheaf of abelian groups and F = K(A, n) is the associated Eilenberg
MacLane simplicial presheaf, then ßiHF (X) = Hni(X, ~A) (see Section 6.8) . 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 this
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 th*
*is
section we give analogs of the Verdier hypercovering theorem, which show how to
compute some invariants of HF (X) using hypercovers.
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, ß(F, G) denotes the set
sP re(C)(F, G)=~, where the equivalence relation is generated by simplicial hom*
*o
topy.
Given an object X in C, let HCX denote the full subcategory of sP re(C) con
sisting of all hypercovers of X. We let ßHCX denote the category with the same
objects, but where ßHCX (U, V ) equals ß(U, V ).
Proposition 7.5.The category ßHCX is filtered.
This proposition is proved in [SGA4 , Expos'e V, 7.3.2] and also in [AM , Sec*
*tion 8]
with a slightly different notion of hypercover (see Section 9). We prove it aga*
*in here
because it is straightforward with the techniques that we have already develope*
*d.
Proof.If U ! X and V ! X are both hypercovers, then so is U xX V ! X. Thus,
we only need show that two parallel arrows V ' U in ßHCX can be equalized.
22 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
The two maps from V ! X to U ! X can be assembled into the square
@ 1 V _____//U
 
 
fflffl fflffl
1 V _____//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
@ 1 W ____//_@ 1 V____//U44iiii
 iiiiiii 
 iiiii 
fflffliiii fflffl
1 W ______// 1 V____//X.
The two compositions W ! U are simplicially homotopic and hence equal_in
ßHCX . __
The following is a generalization of the Verdier hypercovering theorem [SGA4 ,
Expos'e 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 fr*
*om
general considerations about the category of locally fibrant simplicial preshea*
*ves
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 considerat*
*ions,
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 handl*
*es
all cases at once.
Theorem 7.6. Let F be a locallyfibrant simplicial presheaf and let X belong t*
*o C.
Let F ! HF be a fibrant replacement for F in sP re(C)L . Then
(a)Given a 0simplex 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:
V ___v__//F
 
 
fflfflpfflffl
X ____//_HF.
We say that `p is represented by the map v'.
(b)Given a finite simplicial set K, there is an isomorphism
Ho (K X, F ) ~=colimU!Xß(K U, F )
where the colimit is taken over (the opposite category of) ßHCX .
(c)Given p and V ! F as in (a), there is an isomorphism
ßn(HF (X), p) ~=colimU!Vßn(Map (U, HF ), pU ) ~=colimU!Vß( n=@ n U, F )v*
*U.
Here vU denotes the map U ! V ! F , and pU denotes the map U !
V ! X ! HF . The colimits are taken over the overcategory ßHCX # V of
hypercovers refining V , and ß( n=@ n U, F )vUdenotes the set of all maps
f : n=@ n U ! F such that f * U is the given map vU :U ! V ! F ,
modulo simplicial homotopy relative to * U.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 23
Proof.Part (a) is a direct consequence of Proposition 5.7 because F and HF are
both locallyfibrant.
For surjectivity in (b), note that any element ff of Ho (K X, F ) is repres*
*ented
by an actual map K X ! HF . From Proposition 5.7 again, we get a hypercover
U ! X and a diagram
f
K U _____//_F
 
 
fflffl fflffl
K X ____//_HF
commuting up to simplicial homotopy. The map f has image ff in Ho (K X, F ).
For injectivity, 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
(@ 1 x K) V ____//_(@ 1 x K) _U____//_F33________________
 _______________________
 _________________ 
fflffl__________ fflffl
( 1 x K) V _____//_( 1 x K) _U___//HF
for some refinement V of U, where the lift is a relativehomotopylifting. In p*
*artic
ular, the upper left triangle commutes on the nose, so the two maps K U ! F
are equal in colimU!X ß(K U, F ).
For (c), note that the natural map Map (X, HF ) ! Map (U, HF ) is a weak equi*
*v
~=
alence. So it induces an isomorphism ßn(HF (X), p) ! ßn(Map (U, HF ), pU ), a*
*nd
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 ß( n=@ n U, F )vU! ßn(Map (U, HF ), pU ). As in the proof of part (b)
above, the fact that these maps give an isomorphism after passing to the colimi*
*t_is
a direct consequence of Proposition 5.7. __
Note that if S is a dense set of hypercovers then the colimits in the above r*
*esults
can just as well be taken over the full subcategory of ßHCX whose objects belong
to S. 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.
7.7. The coconnected case.
Definition 7.8.A locallyfibrant simplicial presheaf F is said to be locally n
coconnected if it has the following property: for any X in C and any 0simplex x
in F (X), the homotopy group sheaves ßk(F, x) on C # X vanish for all k n.
Using techniques from [DI2], a locallyfibrant simplicial presheaf is locally*
* n
coconnected if and only if it has the local lifting property with respect to th*
*e maps
@ k X ! k X for k > n.
Not surprisingly, for ncoconnected presheaves one can calculate homotopy
classes of maps by only using bounded hypercovers. This is what we'll prove nex*
*t.
If n 0, let HCX (n) denote the category of bounded hypercovers U ! X
of height at most n (see (4.10)). Let ßHCX (n) denote the category with the
24 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
same objects but with simplicial homotopy classes of maps. Arguments similar to
Proposition (7.5) show that ßHCX (n) is filtered.
Proposition 7.9.Suppose that F is locally fibrant and locally ncoconnected. Th*
*en
given a finite simplicial set K, there is an isomorphism
Ho (K X, F ) ~=colimU!Xß(K U, cosknF )
where the colimit is taken over the category ßHCX (n).
Proof.First, the map F ! cosknF is a local weak equivalence between locally
fibrant objects. So we can say that
Ho (K X, F ) ~=Ho (K X, cosknF ) ~=colimU!Xß(K U, cosknF )
where the colimit runs over the full category ßHCX ; the second isomorphism com*
*es
from Theorem 7.6. We need to show that
colim ß(K U, cosknF ) ! colim ß(K U, cosknF )
U2ßHCX(n) U2ßHCX
is an isomorphism. Observe that for any simplicial set L, a map L U ! cosknF
factors through coskn(L U), and the map L U ! coskn(L U) factors as
L U ! L cosknU ! coskn(L U). Applying this when L = K shows surjectivity, __
and from L = K x 1 one can deduce injectivity. __
Proposition 7.10.Let S be a Noetherian scheme, and let C be Sm=S (or Sch=S)
with either the `etale 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 boun*
*ded
hypercover of X can be refined by a ~Cech complex.
Proof.In the case of the 'etale topology, this is essentially the content of [A*
*r,
Thm. 4.1]. Since the result is trivial for hypercovers of height 0, we'll supp*
*ose
by induction that it works for hypercovers of height at most n. Let U ! X be a
hypercover of height n+1. By Theorem 8.6, U can be refined by a basal hypercover
U0 ! X (see Section 8 below). Let V = cosknU0, which is a hypercover of height *
*at
most n. By induction,`there is an 'etale covering family {Wi! X} such that ~CW *
*re
fines V (where W = Wi). Consider the induced map ~CWn+1 ! Vn+1 = Mn+1U0.
The map U0n+1! Mn+1U0 is an 'etale cover, which pulls back to an 'etale cover
E ! ~CWn+1. Theorem 4.1 of [Ar] (applied to the case of no geometric points) sa*
*ys
that there is a refinement Z of W such that the map ~CZn+1 ! ~CWn+1 factors
through E. In particular, this means that C~Z refines U0 (and therefore U) up
through dimension n + 1; since the height of U is n + 1, this means it automati*
*cally
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 7.11.Let C and X be as above, and suppose F is a locally fibrant sim
plicial presheaf which is locally ncoconnected. If K is a finite simplicial se*
*t, there
is an isomorphism
Ho (K X, F ) ~=colimU!Xß(K U, cosknF )
where the colimit is taken over the category ßHCX (0) consisting of the ~Cech c*
*om
plexes.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 25
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 ~Cech 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 subcat*
*_
egory ßHCX (0) is final in ßHCX (n + 1). __
Remark 7.12. The above proposition and its corollary are not true for the Zari*
*ski
topology, and therefore not for the open covering topology on an arbitrary topo*
*log
ical space. We repeat the example of [MV , Ex. 1.10, p. 99]: Let X = SpecR be t*
*he
semilocalization of A2kat the points (0, 0) and (0, 1). As a topological space*
* X has
exactly two closed points x1 and x2 (of codimension 2), infinitely many points *
*of
codimension 1 (corresponding to the irreducible closed curves in A2 passing thr*
*ough
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 x1 and x2, respectively.
Let U1 = X  {x1} and U2 = X  {x2}. Pick two of the codimension 1 points
f and g which specialize to both x1 and x2. Let W1 = (U1 \ U2)  {f} and
W2 = (U1\ U2)  {g}. Let 0 = U1q U2 and 1 = (U1q U2) q W1q W2 (the first
part is degenerate). Consider the hypercover cosk1 . This hypercover cannot be
refined by a ~Cech complex.
8.Verdier sites
The definition of hypercover we've adopted so far in this paper is extremely
broad. It 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 sub*
*ject
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 hy
percover of a space X basically corresponds to giving a simplicial space U* such
that each matching map Un ! MnU is locally split. This allows for an incredible
amount of freedom in what a hypercover can look like, so much so that it's very
difficult to say anything concrete about it. To make things easier, it is reaso*
*nable
that one should be able to look just at the `open hypercovers', where the maps
Un ! MnU all have the form qaWa ! MnU for some open covering {Wa} 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 ex*
*tra
data that one can talk about a special kind of `basal hypercover' rather than t*
*he
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 8.1.A Verdier site is a category C together with a given collection
of covering families {Ua ! 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.
26 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
(ii)If {Ua ! X} is a covering family and Y ! X is a map, then the pullbacks
Y xX Ua all exist, and {Y xX Ua ! Y } is a covering family.
(iii)If {Ua ! X} is a covering family and one is given a collection of covering
families {Vab! Ua}, then the collection of compositions {Vab! Ua ! X} is
also a covering family.
(iv)If U ! X is a basal map then the diagonal U ! U xX 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 geometr*
*ic
Grothendieck sites satisfy the above axioms: these include topological spaces, *
*where
the covering families are open covers, as well as the Zariski, Nisnevich, and '*
*etale
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 a*
*re
used in A1homotopy 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 {Va !*
* X} is
a finite collection ofQbasal mapsQand {Ua ! Va} is another collection of basal *
*maps,
then the induced map X Ua ! XVa of fibreproducts is again`basal.`
For the following definition, note that to give a map f : irXi ! jrYj be
tween coproducts of representables one must choose, for every index i, a prescr*
*ibed
value of j and a map Xi! Yj.
Definition 8.2. `
(a)A map f :X ! Y is basal`if X is a coproduct irXi of representables, Y
is also a coproduct jrYj of representables, and the various maps Xi ! Yj
determining f are all basal, in the sense of Definition 8.1.
(b)A basal hypercover U ! X is a hypercover such that the matching maps
Un ! MnU are all basal.
The second part of this definition only makes sense if one knows that the mat*
*ch
ing objects MnU are all coproducts of representables, but we will see in Propos*
*i
tion 8.5 that this is the case. First, an easy lemma:
Lemma 8.3. 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 fibreproducts. The necessary pullbacks in C exist_
because pullbacks along basal maps always exist. __
Lemma 8.4. Let U ! X be an ntruncated basal hypercover, and let K be a
finite simplicial set of dimension at most n. Then hom +(K, U) is a coproduct of
representables.
Proof.We proceed by induction on the dimension of K. When K is empty,
hom+ (K, U) is just X, which is a coproduct of representables 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 ksimplices. By repeating the following argument, we may
assume that only one ksimplex was attached. It follows that hom +(K, U) is the
pullback of the diagram
hom +( k, U) ! hom+ (@ k, U) hom+ (L, U).
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 27
All three objects are coproducts of representables, the first because Uk is a c*
*oprod
uct of representables and the last two by the induction hypothesis. Since the l*
*eft
map above is basal, Lemma 8.3 tells us that hom +(K, U) is also a coproduct_of
representables. __
Let us return momentarily to Definition 8.2(b). If U ! X is a hypercover and
U0 ! X is basal, then the above proposition specialized to @ 1 shows that M1U is
a coproduct of representables. So we may ask that U1 ! M1U be basal, which in
turn forces M2U to be a coproduct of representables. This shows that our defini*
*tion
of basal hypercover makes sense, in a recursive sort of way.
Proposition 8.5.Let K ! L be any map of finite simplicial sets whose dimen
sions are at most k, and let U ! X be a ktruncated 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 pro*
*perty
stated in the lemma. By definition of basal hypercovers, C contains the generat*
*ing
cofibrations @ n ! n. Cobase changes preserve C by Lemmas 4.7(v), 8.3, and
8.4. Also, finite compositions preserve C because basal maps are closed under f*
*inite
composition. This shows that C contains all inclusions of finite simplicial set*
*s.
In particular, ; ! n belongs to C. This means that Un ! X is basal for
every basal hypercover U ! X. By the definition of Verdier sites, the map Un !
Un xX Un is also basal. In other words, C contains the codiagonal n q n ! 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. Unfortunat*
*ely,
dealing with these basal maps definitely increases the technical complications.
Theorem 8.6. 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 U0 ! 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 {Wa ! X} for which every element belongs to R. Setting V0 = arWa, we
have that V0 ! X is basal and refines U0 ! X.
Continuing by induction, we may assume we have built a split, basal, ntrunca*
*ted
hypercover V which refines U (up through dimension n). Our job is to define Vn+*
*1.
We consider the maps
Un+1


fflffl
Mn+1V _____//Mn+1U,
where all the objects are coproducts of representables. Using the same reasoning
as in the first paragraph, there is a map W ! Mn+1V that is basal, that is a
generalized cover, and that fits in the upper left corner of this diagram, i.e.*
*, it
28 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
refines the pullback generalized cover Un+1xMn+1U Mn+1V ! Mn+1V . Set Vn+1 =
W qLn+1V . Now V is a split, (n+1)truncated hypercover; the question is whether
Vn+1 ! Mn+1V is basal. Because of the way W was constructed, we need only
show that the map Ln+1V ! Mn+1V is basal.
Recall from Section 4.12 that there is a natural map dgnn V ! cosknV . In
dimension n this is the identity map on Vn, and in dimension n + 1 it's the map
Ln+1V ! Mn+1V . Picking any degeneracy si from level n to n + 1, we get a
diagram
Ln+1VO____//_Mn+1VOOO
si si
 
Vn __________Vn.
Every representable summand of Ln+1V is of the form si(rU) for some i and some
representable summand rU of Vn, so it suffices to show that the righthand map
si : Vn ! Mn+1V is basal. But this degeneracy is induced by the corresponding
collapse map @ n+1 ! n, i.e., the composition s: @ n+1 ,! n+1 si! n. In
other words, si coincides with hom+ ( n, V ) ! hom+ (@ n+1, V ). The fact that_
this is basal follows from Proposition 8.5. __
Remark 8.7. Suppose there is a regular cardinal ~ with the property that every
covering family in C has size less than ~. By tracing through the above proof,
following similar observations to those in Proposition 6.7, one can show that t*
*he
split, basal hypercover can be constructed so that in each level it has fewer t*
*han ~
summands. This is needed in the next section.
9. Internal hypercovers
In this final section we give a slight modification of Theorem 6.2 which is u*
*seful
in applications_for instance, it is needed in [Is]. This involves once again tw*
*eaking
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 q rY .
Throughout this section we work with a Verdier site for which there exists a
regular cardinal ~ such that:
(1)Every covering family {Ui! X} has cardinality less than ~.
(2)Coproducts of size less than ~ exist in C.
(3)If {Xi} is a`set of objectsSwhose cardinality is less than ~, then the map of
presheaves irXi! r( iXi) becomes an isomorphism after sheafification.
For example, if Smk denotes the category of smooth schemes of finite type ove*
*r a
fixed ground field k, we may give it the structure of a Verdier`site by saying *
*that the
covering families are finite collections {Ui ! X} such that Ui ! X is an 'eta*
*le
(or Zariski or Nisnevich) cover. This generates the usual Grothendieck topology,
and satisfies the above properties with ~ = @0.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 29
Definition 9.1.Given an object X of C, an internal hypercover of X is a sim
plicial object U in sC which is augmented by X, with the property`that each mat*
*ching
map Un ! MnU is isomorphic over MnU to a map of the form iVi! MnU, for
some basal maps {Vi! MnU} which generate a covering sieve.
Of course one has to worry about whether the matching object MnU exists, since
the site C need not have arbitrary limits. But we shall see that the condition *
*on
Uk ! MkU for k n  1 guarantees that MnU 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 qK M, W ) exists and is isomorphic to the above pullb*
*ack.
Proposition 9.2.If U ! X is an ntruncated 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 MnU = hom+ (@ n, U) exists.
Proof.The proof follows the same lines as the proof of Lemma 8.4. _*
*__
We continue our notational convention of writing U for a simplicial object of*
* C
and also for the simplicial presheaf that it represents.
Theorem 9.3. The model category sP re(C)L of simplicial presheaves may be ob
tained as the localization`ofSsP re(C) at the following collection of maps I:
(i)Maps of the form Wi! ( Wi), for collections {Wi} in C of size less than
~.
(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 Theor*
*em 6.2.
For maps of type (i) it follows from assumption (3) at the beginning of this se*
*ction,
because every simplicial presheaf is locally weakly equivalent to its sheafific*
*ation.
So sP re(C)L is a stronger localization than sP re(C)I. To see that the localiz*
*ations
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 ~, then V ! X is a weak
equivalence in sP re(C)I. This is by virtue of Theorem 6.2, Theorem 8.6, and
Remark 8.7.
Each presheaf Vn may be decomposed`as a coproduct of representables in anS
essentially unique way: Vn = ffVnff. We define an object U of sC by Un = ffV*
*nff,
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 th*
*at
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 mo
ment, relation (i) in our definition of sP re(C)I shows that Vn ! rUn is an Iw*
*eak
equivalence for each n. Since every simplicial presheaf F is the homotopy colim*
*it
hocolimnFn (see Remark 2.1), it follows that V ! rU is also an Iweak equivalen*
*ce.
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
30 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
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 nondegenerate simplices in K, using the
same pullback square from Lemma 8.4 and the fact that sheafification preserves
finite limits.
So taking K = @ n we have that MnV ! Mn(rU) induces an isomorphism on
sheafifications, and in particular is a generalized cover. This, together with *
*the fact
that Vn ! MnV is a generalized cover, shows immediately that the same is true of
rUn ! Mn(rU). So rU is a hypercover of X.
Finally, to see that U is an internal hypercover one just uses that the Yoned*
*a__
embedding preserves all limits that exist: so Mn(rU) is isomorphic to r(MnU). *
*__
Appendix A. ~Cech localizations
This appendix is a bit of an aside from the main body of the paper. Here we
investigate how descent for ~Cech complexes compares to descent for all hyperco*
*vers.
These are not equivalent notions in general_see Example A.10_although in some
cases they turn out to agree. Unlike hypercover descent, ~Cech 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 ~Cech descent actually implies desce*
*nt for
all bounded hypercovers, and we give some related results of interest.
If , :F ! G is a map of presheaves of sets, the ~Cech complex of , is the
simplicial presheaf ~C, (often denoted ~CF by abuse) given by
[n] 7! F xG F xG . .x.GF (n + 1 factors).
A simplicial presheaf F is said to have ~Cech descent if it satisfies descent f*
*or ~CU
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 {Ua ! X} be any set of`maps in C, and let R ,! X be
the sieve generated by these maps. Let U = arUa. Then there is a natural map
~CU ! R, and this map is an objectwise weak equivalence.
Proof.If , :K ! L is any map of simplicial sets, then the ~Cech complex C~, is
fibrant and homotopy discrete. This shows that the natural map ~CU ! ß0C~U is
an objectwise weak equivalence. The presheaf R is equal to the presheaf ß0C~U,_
i.e., R(Y ) = ß0C~U(Y ) for all Y in C. __
Let ~Cdenote 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)~*
*Cdenote
the Bousfield localization of sP re(C) at the set ~C. We'll refer to this model*
* cate
gory as the ~Cech localization of sP re(C), for reasons which will shortly beco*
*me
apparent (see Corollary A.3).
Given a covering`sieve R ,! X, let ~C(R) denote the ~Cech complex correspondi*
*ng
to the cover Ua ! X where the coproduct ranges over all maps Ua ! X in
the sieve. The above proposition implies that ~CR ! X factors through R, and
~CR ! R is an objectwise weak equivalence. So localizing at the set {R ,! X} is
equivalent to localizing at {C~R ! X}. We will see in a minute that this is act*
*ually
equivalent to localizing at {C~U ! X}, for all generalized covers U ! X, and so
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 31
our ~Cech localization is analogous to the hypercover localization of Theorem 6*
*.2.
The advantage of starting with just the sieves rather than the generalized cove*
*rs is
that these form a set, and so the localization automatically exists.
Proposition A.2. Given any simplicial presheaf F , the map F ! ~Ffrom F to
its (levelwise) sheafification is a weak equivalence in sP re(C)~C.
Unfortunately the proof of this result is somewhat involved, so we'll postpon*
*e it
until the end of the section.
Corollary A.3. Let F ! G be any generalized cover of presheaves of sets. Then
the map ~CF ! G is a weak equivalence in sP re(C)~C.
Proof.The map C~F ! G factors as C~F ! ß0C~F ! 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)~C, and so we can conclude the same for the composite ~CF ! G. _*
*_
We now derive the connection with hypercovers. Recall from Definition 4.10 th*
*at
a hypercover U ! X is bounded if U = cosknU 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)~C.
Proof.We proceed by induction, starting from the fact that bounded hypercovers
of height 0 are just ~Cech 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 cosknU, 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)~C. The canonic*
*al
map U ! V gives a generalized cover Un+1 ! Vn+1, by the very definition of what
it means for U to be a hypercover. Lemma A.5 below shows that in fact Uk ! Vk
is a generalized cover for all k.
Consider the following bisimplicial object, augmented horizontally by V :
V oo___U oo___UoxVoU_ oo___.o.o._oo_
The kth row is the (augmented) ~Cech complex for the generalized cover Uk ! Vk.
Note that for 0 k n the kth row is the constant simplicial object with value
Uk because Uk ! Vk is the identity. Call this bisimplicial object (without the
horizontal augmentation) W**. There is an obvious map hocolimW**! X.
One may compute hocolimW**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)~Cthe homotopy colimit of the kth row is just Vk by Corollary A.3. Also,
we have assumed by induction that V ' hocolimkVk is weakly equivalent to X. So
hocolimW**! X is a weak equivalence.
Let D denote the diagonal of W**. Standard homotopy theory tells us that
D = hocolimkDk is weakly equivalent to hocolimW**. 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 W0k ! Wkk.
To produce a map D ! U it is enough to give skn+1D ! skn+1U, because U =
coskn+1U. But note that sknD = sknU, and choosing any face map [0] ! [n + 1]
32 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
gives a map Dn+1 ! Un+1, inducing a corresponding map skn+1D ! skn+1U as
desired.
It is straightforward to check that U ! D ! U is the identity (because U =
coskn+1U one only has to check it on (n + 1)skeleta), and all the maps commute
with the augmentations down to X. We have already shown that D ! X is a weak
equivalence in sP re(C)~C. 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 ! cosknU is
a generalized cover in every dimension.
Proof.First note that for k n, the map Uk ! [cosknU]k is the identity, so it *
*is
a generalized cover. For any k, the map Uk ! [cosknU]k may be rewritten as
hom+ ( k, U) ! hom+ (skn k, U).
But U = coskn+1U, so the domain may be written as
hom+ ( k, U) = hom+ ( k, coskn+1U) = hom+ (skn+1 k, U).
So we are interested in the map hom+ (skn+1 k, U) ! hom+ (skn k, U) induced
by the inclusion skn k ! skn+1 k. Recall from Lemma 4.11 the pullback square
Q n+1
hom+ (skn+1 k, U)_____//_X hom+ ( , U)
 
 
fflffl Q fflffl
hom+ (skn k, U)______//X hom+ (@ n+1, U).
The map hom+ ( n+1, U) ! hom+ (@ n+1, U) is just the matching map Un+1 !
MnU, and is therefore a generalized cover. So the right vertical map in the abo*
*ve
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 IR denote the full subcategory of C # X
consisting of all maps in R. Consider the diagram IR ! sP re(C) sending (U ! X)
to U. The colimit of this diagram is R, and we will write the homotopy colimit
as hocolimRU. The natural map from the homotopy colimit to the colimit gives
hocolimRU ! 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 hocolimRU ! X, where R ,! X is a covering sieve;
(c)The class of all hypercovers of height 0, i.e., the ~Cech complexes ~CU ! X;
(d)The class of all bounded hypercovers U ! X;
(e)The class of maps F ! ~Ffrom simplicial presheaves to their sheafifications.
If the topology on C is given by a basis of covering families, then one can als*
*o add
(a')The set of all covering sieves RU ,! X where RU is the sieve generated by t*
*he
covering family {Ua ! X}.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 33
It may seem surprising that the localization in (e) does not give the usual n*
*otion
of local weak equivalence, but only the weaker ~Cech 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 hocolimRU ! R is an objectwise weak equivalence immedi
ately 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 pr*
*oof
that the localizations in (a') and (e) coincide follows the proof of Propositio*
*n_A.2
more or less verbatim. __
It would be interesting to know more about sP re(C)~C, 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)~Cis that it is sometimes a more co*
*nvenient
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
~Cech complexes if and only if it satisfies descent for all bounded hypercovers.
Proof.Let F 0be a fibrantreplacement for F in sP re(C). The statement of the
corollary for F 0requires that F 0be local with respect to the ~Cech complexes *
*~CV !
X if and only if it is local with respect to the bounded hypercovers U ! X. Thi*
*s 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 0has descent for that same class,_b*
*ecause
F ! F 0is 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 ~Cech descent if and *
*only if
it satisfies`descent for all the ~Cech complexes ~CU ! X in which X is a repres*
*entable
and U = aUa for some covering family {Ua ! 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.
Corollary A.9. Let F be an objectwisefibrant simplicial presheaf with the prop*
*erty
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 a*
*ll ~Cech
complexes.
Proof.First we need to consider the localization UC=S, where S is the set of ma*
*ps
{@ n+1 X ! n+1 XX 2 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 localization LSG 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) ! LSG(X) are isomorphisms up through simplicial dimension n. So
34 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
LSG(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*
*SG
is objectwise weakly equivalent to LS(Ex 1 G), and so it is still true that G(X*
*) and
LSG(X) have the same (n  1)type for all X 2 C.
General localization theory says that a map G ! H is an Sequivalence if and
only if LSG ! LSH is an objectwise equivalence, and so this is the same as sayi*
*ng
that G(X) ! H(X) induces isomorphisms on all homotopy groups up through
dimension n  1, for every X. In particular, the map G ! cosknG is an S
equivalence.
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
if it is objectwise fibrant, has ~Cech 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 ~Cech complexes. Then we know that F is fibrant in UC=T . To verify des*
*cent
for all hypercovers, it suffices by Corollary 6.3 to verify it for all split hy*
*percovers
U ! X. But note that U ! cosknU is necessarily a T equivalence (because it is *
*an
Sequivalence). Yet cosknU is a bounded hypercover of X, and hence cosknU ! 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 and U are cofibrant *
*in
UC=T , the morphism Map (X, F ) ! Map (U, F ) is a weak equivalence; so_F_satis*
*fies
descent for U ! X by Lemma 4.4. __
Here is an example showing that the ~Cech localization can be strictly weaker
than the localization at all hypercovers. In other words, we exhibit a simplic*
*ial
presheaf which has descent for all ~Cech complexes but does not have descent fo*
*r all
hypercovers. The example is a slight modification of one suggested to us by Car*
*los
Simpson.
Example A.10. Let X = X0 be the open interval (0, 1). Now let U0 = (0, 2_3),
V0 = (1_3, 1), and X1 = U0\V0. Note that X1 ~=X, and let U1 = (1_3, 5_9) , V1 =*
* (4_9, 2_3),
and X2 = U1 \ V1. Again one has X2 ~=X, and we define U2, V2, and X3 in the
expected way. Continue. Our site C consists of the spaces {Xi, Ui, Vi i 0} w*
*ith
the inclusions between them, and equipped with the usual notion of open cover.
The category C is depicted as
U0 U1 . . .
_ ``BBB __ ``BBB zzz
___ BBB ___ BBB zzz
~~___ B ~~__ B __zz
X0 X1 X2
``BBB _ ``BBB __ bbDDD
BBB ___ BBB ___ DDD
B ~~___ B ~~__ DD
V0 V1 . ...
Define a presheaf of spaces on our site in the following way:
F (X0) = ;, F (Un) = Dn+, F (Vn) = Dn, and F (Xn+1) = Sn (n 0).
Here Dn+and Dndenote the upper and lower hemispheres of Sn. The restric
tion maps F (Un) ! F (Xn+1) and F (Vn) ! F (Xn+1) are the inclusions of the
hemispheres in Sn, while the maps F (Xn) ! F (Un) and F (Xn) ! F (Vn) are the
inclusions of the boundaries of the hemispheres. Define the simplicial presheaf*
* G
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 35
by G(W ) = ZSing F (W )_that is, G(W ) is the result of applying the free abeli*
*an
group functor to the singular complex of F (W ). Using the DoldKan correspon
dence, one can regard G as a presheaf of chain complexes; then G(W ) is the usu*
*al
complex for computing the singular homology of F (W ).
Now G has ~Cech descent: this can be checked using Corollary A.8, and so the
main point is that for every n the square
G(Un [ Vn)_______//G(Un)
 
 
fflffl fflffl
G(Vn) _______//G(Un \ Vn)
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 sligh*
*tly
complicated, but the idea is this: Start with 0 = U0 q V0, then consider cosk0
except replace each nondegenerate occurrence of X1 with U1 q V1. Next take
cosk1 , replace each nondegenerate occurrence of X2 with U2q V2, and continue.
This gives the hypercover ! X.
Now we will be more precise. Let Pn be the category of all nonempty subsets
of {0, 1, . .,.n}, with inclusions. Note that the objects of Pn can be identifi*
*ed with
the subsimplices of n, and that [n] 7! Pn forms a simplicial category. Let Sn
denote the set of all functors J :Pnop! C with the following properties:
(1)All the values of J belong to {Ui, Vi i 0};T
(2)Given a subset oe = {i0, . .,.ik} in Pn, if j J({i0, . .,.^ij, . .,.ik}) = *
*Um (resp.
Vm )Tthen J(oe) = Um (resp. Vm ).
(3)If jJ({i0, . .,.^ij, . .,.ik}) = Xm then J(oe) is either Um or Vm (and *
*this
includes the case k = 0).
Let denote the simplicial presheaf defined by
a
n := J({0, 1, . .,.n}),
J2Sn
with simplicial structure induced by that of P . Intuitively, each summand of n
corresponds to an nsimplex 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 is a
hypercover of X0 (use Proposition 4.15). Check that 0 = U0 q V0 and 1 =
U0 q (U1 q V1) q (U1 q V1) q V0.
We claim that holimnG( n) is not connected, whereas G(X) = 0. To calculate
ß0(holimnG( n)) we can just work in the category of chain complexes. The cosim
plicial object [n] 7! G( n) corresponds to a double complex, and we are trying *
*to
compute the 0th homology of the total complex (the one called Tot in [W ], rat*
*her
than Tot ). But observe that each G( n) has homology only in dimension 0 be
cause each G(Ui) and G(Vi) is contractible. Therefore, the E1term of the spect*
*ral
sequence for the homology of the bicomplex is concentrated in a line. Thus, the*
* bi
complex's 0th homology is the kernel of d0 d1: H0G( 0) ! H0G( 1), which is Z.
This completes the verification that G does not satisfy descent for the hyperco*
*ver
.
36 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
Example A.11. Sometimes the localizations UC~Cand UCL do coincide. Let S
be a Noetherian scheme of finite dimension and let C be the site Sm=S with eith*
*er
the Zariski or Nisnevich topology (one can also take Sch=S here). For these sit*
*es
the localizations UC~Cand UCL agree. For the Zariski topology this is a direct
consequence of the `BrownGersten Theorem', which identifies the fibrant objects
in UCL with the objectwisefibrant simplicial presheaves satisfying ~Cech desce*
*nt for
all twofold Zariski covers {U, V ! X}. This is essentially proven in [BG ], al*
*though
one has to translate their proof into our more modern setting. See also [Bl, Le*
*mmas
4.1, 4.3].
For the Nisnevich topology we have to explain a little more. Given an element*
*ary
distinguished square {U ,! X, p: V ! X} [MV , Def. 1.3, p.96], let P (U, V )
denote the simplicial presheaf which has U q V in dimension 0, U q p1(U) q V in
dimension 1, and is degenerate in higher dimensions. The BrownGersten Theorem
in this context is [MV , Lemma 1.6, p.98]; together with [Bl, Lemma 4.3], it im*
*plies
that UCL is the localization of UC at the maps P (U, V ) ! X for all elementary
distinguished squares. We already know that UCL is a stronger localization than
UC~C, so we just need to show that the maps P (U, V ) ! X are weak equivalences
in UC~C.
To see that P (U, V ) ! X is a weak equivalence in UC~C, first note that the
sections P (U, V )(Z) are simplicial sets with nondegenerate simplices only in*
* di
mensions 0 and 1. Each component is a star, centered at a 0simplex correspondi*
*ng
to a map Z ! U (because every map Z ! V can be an endpoint of at most one
1simplex). Therefore each component is contractible, so P (U, V ) ! ß0P (U, V *
*) is
an objectwise weak equivalence. The codomain is just the presheaf U qp1U V , so
we are reduced to showing that the map U qp1U V ! X is a weak equivalence
in UC~C. 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 s*
*talks,
or look at [MV , Lemma 3.1.6]).
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 ! ~Fis a
weak equivalence in sP re(C)~C. 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 homoto*
*py
colimit of the discrete presheaves Fn (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 sU = tU for ev*
*ery
U ! X in R. Secondly, BF is defined to be the pushout
`
R _____//_F
 
 
` fflffl fflffl
X ____//_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 ~F.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 37
We will show that the maps F ! BF and F ! AF are ~Cech weak`equivalences,`
for any presheaf F . The first claim is very easy: Since R ! X is an acyclic
cofibration in sP re(C)~C, its cobase change F ! BF is also an acyclic cofibrat*
*ion.
Unfortunately the second claim is much more difficult. The idea is to build up a
~Cech weak equivalence F ! L1 F by brute force, in such a way that there is an
objectwise weak equivalence L1 F ! AF .
Given a covering sieve R ,! X, let JnR be the pushout
@ n R _____//@ n X
 
 
fflffl fflffl
n R ______//_JnR.
The natural map JnR ! n X is a ~Cech acyclic cofibration, by SM7. Note that
to give a map JnR ! G is the same as giving a map @ n ! G(X) together with
a compatible family of extensions n ! G(U) for all maps U ! X in R.
Let L0F = F , and let Ln+1F be obtained from LnF as the pushout
`
(A.13) Jn+1R _______//LnF
fflffl fflffl
~  ~
` fflffl fflffl
n+1 X ____//_Ln+1F
where the coproducts run over all X in C, all covering sieves R ,! X, and all m*
*aps
Jn+1R ! LnF . Let L1 F be the colimit of the chain L0F ! L1F ! L2F ! . ...
Since each LnF ! Ln+1F is a ~Cech acyclic cofibration, the composite F ! L1 F
is also a ~Cech acyclic cofibration.
To get a feel for what's happening here, let's look just at L1F . A map J1R !*
* F
corresponds to giving a map @ 1 ! F (X) together with a compatible family of
extensions 1 ! F (U) for all U ! X in the sieve. Since F is discrete, this mea*
*ns
that we are giving two elements of F (X) which agree when restricted to pieces
of the sieve. When we form the pushout 1 X J1R ! F we are adding a
1simplex into F (X) which will identify these elements in ß0. So it follows t*
*hat
ß0L1F (Y ) = AF (Y ), for all Y .
When we pass from LnF to Ln+1F 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 L1 F (X) is fibrant and homotopy discrete, and th*
*at
ß0L1 F (X) = AF (X). This will imply that the natural map L1 F ! ß0L1 F is
an objectwise weak equivalence, and the target is identified with AF . We will *
*then
have F ~! L1 F ~! AF in sP re(C)~C.
The argument will proceed by establishing the following properties: Given any
Y in C,
(i)The map of simplicial sets LnF (Y ) ! Ln+1F (Y ) is an isomorphism on n
skeleta.
(ii)The simplicial set LnF (Y ) has dimension at most n, i.e., it is degenerat*
*e in
degrees greater than n.
(iii)Given any nsimplex oe in LnF (Y ), there is a covering sieve R ,! Y such
that oeU is in the image of Ln1F (U) ! LnF (U) for every U ! Y in R. In
particular, oeU is a degenerate nsimplex.
38 DANIEL DUGGER, SHARON HOLLANDER, AND DANIEL C. ISAKSEN
(iv)Given any nsimplex oe in LnF (Y ), there is a covering sieve R ,! Y such t*
*hat
oeU is in the image of F (U) ! LnF (U) for every U ! Y in R.
(v)For n 2, any map @ n ! Ln1F (Y ) extends to a map n ! LnF (Y ).
(vi)Any map 2,k! L1F (Y ) extends to @ 2 ! L2F (Y ).
Granting these for the moment, let us show they imply the desired result. To
show that L1 F (X) is fibrant and homotopy discrete, it is enough to verify tha*
*t it
has the extension property with respect to the maps @ n ! n (n 2) and the
maps 2,k! 2. These are easy consequences of parts (i), (v), and (vi). Also, p*
*art
(i) tells us that ß0L1F ! ß0L1 F is an isomorphism, and we have already remarked
that ß0L1F ~=AF . This finishes the proof, granting the statements outlined abo*
*ve.
Claim (i) follows from the fact that Jn+1R(Y ) ! ( n+1 X)(Y ) is an isomor
phism on nskeleta. Part (ii) follows from an induction, using that ( 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 oe :@ n ! Ln1F (Y ). By (iv), for each face dioe there
is a covering sieve Ri,! Y such that dioeU is in the image of F (U) ! Ln1F (U*
*),
for every U ! Y in Ri. There is of course a covering sieve R which refines all *
*the
Ri. So for each U ! Y in R and each i, there is (n1)simplex ffU,iin F (U) whi*
*ch
maps to (dioe)U .
Now, it is not clear that as i varies the (n1)simplices ffU,ifit together t*
*o give a
map ffU :@ n ! F (U). However, we know they fit together in Ln1F (U), and the
map F (U) ! Ln1F (U) is a cofibration of simplicial sets, hence a monomorphism.
So the ffU,imust fit together in F (U) as well.
Secondly, it is not immediately clear that the ffU patch together over the co*
*vering
sieve R: that is, we must check that given maps U ! V ! X where V ! X is in
R, then ffU coincides with the restriction of ffV to U. Again, this follows fro*
*m the
fact that everything patches together in Ln1F and the fact that F ! Ln1F is
an objectwise cofibration.
So we have constructed a map ff: @ n R ! F such that the composite map
@ n R ! F ! Ln1F coincides with oeR . Now we use the fact that F is a
discrete simplicial presheaf, from which it follows that ff can be extended to *
*a map
~ff: n R ! F . Composing this with F ! Ln1F and patching with oe gives a
map JnR ! Ln1F , and this will extend to n Y once we pass to LnF . The
upshot is that we've shown oe extends to n under the map Ln1F (Y ) ! LnF (Y ).
References
[SGA4]M. Artin, A. Grothendieck, J.L Verdier, Th'eorie des Topos et Cohomologie*
* 'Etale des
Schemas, Lecture Notes in Math. 270, Springer, 1972.
[AM] M. Artin and B. Mazur, 'Etale Homotopy, Lecture Notes in Math. 100, Spring*
*er, 1969.
[Ar] M. Artin, On the joins of Hensel rings, Adv. Math. 7 (1971), 282296.
[Bl] B. Blander, Local projective model structures on simplicial presheaves, K*
*theory 24 (2001),
no. 3, 283301.
[Bo] M. Boardman, Conditionally convergent spectral sequences, in Homotopy Inva*
*riant Alge
braic Structures (Baltimore, MD, 1998), 4984, Amer. Math. Soc., 1999.
[B] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Tr*
*ans. Amer.
Math. Soc. 186 (1973), 419458.
[BK] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions, and Localizat*
*ions, Lec
ture Notes in Math. 304, Springer, 1972.
HYPERCOVERS AND SIMPLICIAL PRESHEAVES 39
[BG] K. Brown and S. Gersten, Algebraic Ktheory as generalized sheaf cohomolog*
*y, in Higher
Ktheory I, 266292, 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, Hypercovers in topology, preprint, 2001.
[DI2]D. Dugger and D. C. Isaksen, Weak equivalences for simplicial presheaves, *
*preprint, 2002.
[DS] W. G. Dwyer and J. Spali'nski, Homotopy theories and model categories, in *
*Handbook of
algebraic topology, 73126, NorthHolland, 1995.
[He] A. Heller, Homotopy Theories, Mem. Amer. Math. Soc. 71 (1988), no. 383.
[H] P. S. Hirschhorn, Localization of Model Categories, 2001 preprint.
(Available at http://wwwmath.mit.edu/~psh).
[Ho] M. Hovey, Model Categories, Mathematical Surveys and Monographs vol. 63, A*
*mer. Math.
Soc., 1999.
[I] L. Illusie, Complexe cotangent et d'eformations I, Lecture Notes in Math. *
*239, Springer,
1971.
[Is] D. C. Isaksen, 'Etale realization on the A1homotopy theory of schemes, pr*
*eprint, 2001.
[J1] J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemporary Mat*
*hematics 55,
Part I (1986), 193239.
[J2] J.F. Jardine, Simplicial presheaves, J. Pure and Appl. Algebra 47 (1987), *
*3587.
[J3] J.F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. M*
*ath. 39 (1987),
no. 3, 733747.
[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, A1homotopy theory, Inst. Hautes Etudes Sci. Pu*
*bl. Math.
90 (2001), 45143.
[Q] D. Quillen, Homotopical Algebra, Lecture Notes in Math. 43, Springer, 1969.
[Sp] E. H. Spanier, Algebraic Topology, Springer, 1966.
[Th] R. Thomason, Algebraic Ktheory and 'etale cohomology, Ann. Sci. Ec. Norm.*
* Sup. (4) 18
(1985), no. 3, 437552.
[W] C. Weibel, An Introduction to Homological Algebra, Cambridge University Pr*
*ess, 1994.
Department of Mathematics, Purdue University, West Lafayette, IN 47907
Department of Mathematics, University of Chicago, Chicago, IL 60637
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
Email address: ddugger@math.purdue.edu
Email address: sjh@math.uchicago.edu
Email address: isaksen.1@nd.edu