FLASQUE MODEL STRUCTURES FOR PRESHEAVES
DANIEL C. ISAKSEN
Abstract.By now it is well known that there are two useful (objectwise or
local) families of model structures on presheaves: the injective and pro*
*jective.
In fact, there is at least one more: the flasque. For some purposes, bot*
*h the
projective and the injective structure run into technical and annoying (*
*but
surmountable) difficulties for different reasons. The flasque model stru*
*cture,
which possesses a combination of the convenient properties of both struc*
*tures,
sometimes avoids these difficulties.
1.Introduction
It is well known that there are two useful classes of model structures on sim
plicial presheaves: the projective and injective. This paper is about a third c*
*lass
of intermediate model structures, called the flasque structures. Flasque preshe*
*aves
arise on a regular basis in homotopical sheaf theory [BG ] [J1] [DI], so it is *
*surprising
that these model structures have not been described previously, especially sinc*
*e the
proofs (as will be seen) are pretty simple if one takes the right perspective. *
*The
goal of this paper is to show that the usefulness of flasque presheaves arises *
*from
the fact that there are flasque model structures with convenient properties.
The point of the flasque model structures is that they share some of the adva*
*n
tageous properties of both the projective and injective structures. In the inje*
*ctive
model structures, it is very convenient that the cofibrations are easy to descr*
*ibe and
every object is cofibrant, but the disadvantage is that the fibrations are not *
*so easy
to describe. On the other hand, the projective model structures have the advant*
*age
that the fibrations are easy to describe, but the cost is that the cofibrations*
* become
more complicated.
The flasque model structures lie somewhere between the projective and injecti*
*ve
model structures. By "between", we mean that the weak equivalences are the
same, while the class of projective cofibrations is contained in the class of f*
*lasque
cofibrations and the class of flasque cofibrations is contained in the class of*
* injective
cofibrations. Dually, this means that the class of injective fibrations is cont*
*ained in
the class of flasque fibrations and the class of flasque fibrations is containe*
*d in the
class of projective cofibrations.
One of the confusing aspects of motivic homotopy theory is that there is a wi*
*de
choice of foundational approaches [B ] [DHI ] [Hu ] [MV ] [M ], involving presh*
*eaves or
sheaves, projective or injective cofibrations, sheaves of homotopy groups or lo*
*caliza
tions with respect to hypercovers, etc. The positive aspect of this circumstanc*
*e is
that in any situation, one can choose the foundational approach that is best su*
*ited
to the problem at hand.
The flasque model structures are yet another possible foundational choice. As
an example of a result that can most easily be proven with the flasque model
structures, we show in Theorem 6.8 that motivic stable homotopy groups commute
1
2 DANIEL C. ISAKSEN
with filtered colimits (see also [DI]). In fact, this problem originally motiv*
*ated
this project. The key ingredient in the proof of this theorem is that there are
explicit (and verifiable) conditions for an object to be flasque fibrant, altho*
*ugh this
description is slightly more complicated than for projective fibrant objects. T*
*his is
entirely unlike the injective model structures, where there is no explicit desc*
*ription
of the fibrant objects.
As an another example, we consider Jardine's work on the foundations of stable
motivic homotopy theory [J1]. Here the projective model structure doesn't suffi*
*ce
for the simple reason that the maps * ! P1 and * ! A1=A1  0 are not projective
cofibrations. These are the usual models for the motivic sphere S2,1, and it's *
*much
more difficult to define spectra with noncofibrant spheres.
Consequently, the approach in [J1] is to use the injective model structure, w*
*here
the two models for S2,1are cofibrant. However, many of the proofs in [J1] begin
with the observation that injective fibrant presheaves are also flasque. This s*
*uggests
that the flasque model structure is in fact even more suitable in this context *
*than
the injective model structure. Note that the maps * ! P1 and * ! A1=A1  0 are
still flasque cofibrations.
In short, the flasque model structures should be added to the collection of f*
*rame
works for motivic homotopy theory (along with the projective and injective model
structures) because it is just the right tool in some situations.
This paper also answers some questions that arose in [J3] about model struc
tures that are intermediate between the injective and projective. Namely, we sh*
*ow
by example that reasonable intermediate model structures do exist, and they are
wellbehaved in the sense that they are cofibrantly generated (in fact, cellula*
*r).
Our approach to the local flasque model structure is to perform a left Bousfield
localization of the objectwise flasque model structure, rather than to work dir*
*ectly
with sheaves of homotopy groups. This localization approach gives the same class
of weak equivalences, but it is easier to verify abstract model theoretic prope*
*rties
of the local model structure.
1.1. Related work. We mention a few papers that were absolutely essential in
the development of this project. First, we acknowledge [J2] as the seminal paper
on the subject of model structures for simplicial presheaves.
Brown and Gersten [BG ] defined a flasque model structure for simplicial shea*
*ves
rather than presheaves. In a sense, we're doing the presheaf analogue of what
they did thirty years ago. In Theorem 4.6, we generalize part of their main re
sult on the existence of certain model structures. Our approach has a significa*
*nt
advantage over [BG ]. Namely, there is no need for a Noetherian hypothesis on
the Grothendieck topology. However, without some kind of such hypothesis, it is
not possible to obtain the elegant description of fibrant objects in terms of c*
*ertain
squares being homotopy cartesian. The work of [V ] should also be mentioned. Th*
*is
is an axiomatization of the approach of Brown and Gersten. Again, we get the
existence of model structures in greater generality but do not obtain the simple
description of fibrant objects.
L'arusson [L] has recently used a flasque model structure in the context of c*
*omplex
analysis. His work inspired ours. Actually, his situation is a slight variation*
* on the
one that we consider (see Remark 3.4).
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 3
The paper [J3] is about model structures intermediate between the projective
and injective. The flasque model structures are a perfect example of the kind of
structures studied there.
We should also mention that the definition of flasque presheaf is lifted dire*
*ctly
from [J1], which in turn is inspired by [BG ].
Finally, we thank Dan Dugger for many helpful observations.
2.Simplicial presheaves
Recall that a simplicial presheaf on a small site C is just a contravariant f*
*unc
tor from C to the category sSet of simplicial sets. We'll denote the category *
*of
presheaves by sP re(C).
Many of our results will be stated for arbitrary small sites, but we're really
interested in the Nisnevich site of smooth schemes over a fixed ground scheme S
because that's the situation that applies to motivic homotopy theory.
The category sP re(C) is enriched over simplicial sets. This means that for e*
*very
F in sP re(C) and every simplicial set K, there is a tensor F K and a cotensor
F K with certain adjointness properties. Note that F K is constructed by taki*
*ng
the objectwise product with K, and F K is constructed by taking the objectwise
simplicial mapping space out of K. The enrichment over simplicial sets also mea*
*ns
that there are simplicial mapping spaces Map (F, G) for every F and G in sP re(*
*C).
Every object X of C represents a simplicial presheaf of dimension zero. We wi*
*ll
intentionally confuse X with the presheaf that it represents.
If F is any simplicial presheaf in sP re(C) and X is any object of C, then F *
*(X)
is naturally isomorphic to Map (X, F ). We'll almost always use the latter nota*
*tion,
but it's good to keep in mind that it's just another name for F (X).
We recollect one more construction on simplicial presheaves. For any presheav*
*es
F and G, there is an internal function object Hom__(F, G), which is a simplicial
presheaf such that maps A ! Hom__(F, G) correspond bijectively to maps A x F !
G.
Definition 2.1.
(a)A map f in sP re(C) is an objectwise weak equivalence if Map (X, f) is a
weak equivalence of simplicial sets for each X in C.
(b)A map f in sP re(C) is an injective cofibration if Map (X, f) is a cofibrati*
*on
of simplicial sets for each X in C.
(c)A map f in sP re(C) is a projective fibration if Map (X, f) is a fibration of
simplicial sets for each X in C.
The projective cofibrations are defined by a left lifting property with respe*
*ct
to the maps that are both objectwise weak equivalences and projective fibrations
(i.e., objectwise acyclic fibrations). Dually, the injective fibrations are de*
*fined
by a right lifting property with respect to the maps that are both objectwise w*
*eak
equivalences and injective cofibrations.
The following is a wellknown result about homotopy theories of diagram cate
gories (see, for example, [BK ], [D ], and [Jo]).
Theorem 2.2.
(a)The projective cofibrations, objectwise weak equivalences, and projective fi*
*bra
tions form a simplicial proper cellular model structure on sP re(C).
4 DANIEL C. ISAKSEN
(b)The injective cofibrations, objectwise weak equivalences, and injective fibr*
*ations
form a simplicial proper cellular model structure on sP re(C).
(c)The identity is a left Quillen equivalence from the projective objectwise mo*
*del
structure to the injective objectwise model structure.
The proof of part (a) relies on a standard recognition principle for cofibran*
*tly
generated categories [Hi, Thm. 11.3.1]. The projective generating cofibrations *
*are
maps of the form X @ n ! X n, where n 0 and X belongs to C. The
projective generating acyclic cofibrations are maps of the form X n,k! X n,
where n k 0 and X belongs to C.
The proof of part (b) is quite a bit harder and relies on some complicated se*
*t
theoretic arguments. As a result, there is no explicit description of the inje*
*ctive
fibrations. This is the chief disadvantage of the objectwise injective model st*
*ructure.
The proof of part (c) is easy. A projective cofibration is an injective cofib*
*ration,
and the classes of weak equivalences in the two model categories are identical.
3.Objectwise flasque model structure
In this section, we work with an arbitrary small indexing category C. For now,
it is not important that C have a Grothendieck topology. In a few places (such *
*as
Theorem 3.7(c) and Corollary 3.15) we need the mild hypothesis that C contains
all finite products. In practice, this is not really a restriction of generalit*
*y.
Definition 3.1. Let X be any object of C, and let U be a (possibly empty) finite
collection of monomorphisms Ui! X in C. Define [U = [ni=1Ui to be the presheaf
that is the coequalizer of the diagram
` ____//_`
i,jUixX Uj____//_iUi
where the top arrow is projection onto the first factor and the bottom arrow is
projection onto the second factor.
Beware that [ni=1Ui is not the presheaf represented by the union (in C) of the
Ui's. There is no guarantee that this union even exists in C. In fact, in our m*
*ain
application to the category of smooth schemes, these unions generally don't exi*
*st:
the union of two smooth subschemes of a smooth scheme does not have to be
smooth.
Note that there is a canonical map [U ! X, and this map is a monomorphism
of presheaves.
When U is empty, [U is equal to the empty presheaf.
Recall that if f : F ! G is a map of presheaves and g : K ! L is a map of
simplicial sets, then the pushout product f g is the map
a
G K F L ! G L.
F K
A similar definition applies to two maps of simplicial presheaves, except that *
*the
tensors are replaced by ordinary products.
Definition 3.2.
(a)Let I be the set of maps of the form f g, where f : [U ! X is induced by a
finite collection of monomorphisms into an object X of C and g is a generati*
*ng
cofibration @ n ! n.
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 5
(b)Let J be the set of maps of the form f g, where f : [U ! X is induced by a
finite collection of monomorphisms into an object X of C and g is a generati*
*ng
acyclic cofibration n,k! n.
We will eventually show that I serves as a set of generating flasque cofibrat*
*ions,
while J serves as a set of generating acyclic flasque cofibrations.
Recall that an Iinjective is a map having the right lifting property with re*
*spect
to all elements of I, and an Icofibration is a map having the left lifting pro*
*perty
with respect to all Iinjectives (and similarly for Jinjectives and Jcofibrat*
*ions).
Definition 3.3. A map f in sP re(C) is a flasque fibration if it is a Jinjecti*
*ve.
By the usual adjointness arguments, a map f : F ! G is flasque if and only if
the map
Map (X, F ) ! Map (X, G) xMap([U,G)Map ([U, F ).
is a fibration of simplicial sets for all finite collections U of monomorphisms*
* into an
object X of C.
If f is a flasque fibration, then Map (X, F ) ! Map (X, G) is a fibration for*
* all X
in C. This follows from the existence of the empty collection of monomorphisms
into X.
A special case of Definition 3.3 tells us that a presheaf F is flasque fibran*
*t if and
only if the map Map (X, F ) ! Map ([U, F ) is a fibration for all finite collec*
*tions U
of monomorphisms into an object X of C.
Note also that if Y ! X is a monomorphism in C and f : F ! G is a flasque
fibration, then the map
Map(X, F ) ! Map (X, G) xMap(Y,G)Map(Y, F )
is a fibration. Similarly, if F is flasque fibrant, then Map (X, F ) ! Map (Y, *
*F ) is a
fibration.
Remark 3.4. There are many possible ways to vary the definition of flasque fi
brations. The results proved below about flasque model structures would hold ju*
*st
as well for these variations. We let the interested reader check that the argum*
*ents
below do carry over in the following two situations.
One possible variation is to require that a flasque fibration have the right *
*lifting
property with respect to [U ! X only for collections U of monomorphisms of size
0 and 1 (or for that matter, for collections up to size n). Another possible va*
*riation
is to only consider collections U such that each monomorphism Ui! X belongs to
some special class of monomorphisms. We have to assume that this special class *
*is
closed under base changes, though.
The situation of [L, x 12] is a combination of the two variations described a*
*bove,
where one considers collections of "Stein inclusions" (a special kind of monomo*
*r
phism) of size 0 or 1.
More abstractly, in order to make the inductive proof of Lemma 3.9 work, we
need a class C of finite collections of monomorphisms such that C is closed und*
*er
taking subcollections and is closed under base changes. This last condition mea*
*ns
that if {Ui! X} belongs to C and Y ! X is any map, then {UixX Y ! Y } also
belongs to C.
Remark 3.5. The traditional definition of a flasque presheaf F of sets (on the
Grothendieck topology of open subsets of a fixed topological space X) requires
6 DANIEL C. ISAKSEN
that each map F (U) ! F (V ) be a surjection of sets for each inclusion V ! U of
open sets of X. When F is a discrete simplicial presheaf, our definition of a f*
*lasque
fibrant presheaf does not exactly correspond to this traditional definition bec*
*ause
every map between discrete simplicial sets is a fibration. However, the philoso*
*phy is
the same because intuitively fibrations can be viewed as a certain kind of surj*
*ection,
even though this is not technically precise.
Definition 3.6. A map f in sP re(C) is a flasque cofibration if it has the left
lifting property with respect to all objectwise acyclic flasque fibrations.
Theorem 3.7.
(a)The flasque cofibrations, objectwise weak equivalences, and flasque fibratio*
*ns
form a proper cellular model structure on sP re(C). The set of maps I and
J (see Definition 3.2) serve as generating cofibrations and generating acycl*
*ic
cofibrations.
(b)The identity functor on sP re(C) is a left Quillen equivalence from the obje*
*ctwise
projective model structure to the objectwise flasque model structure and fro*
*m the
objectwise flasque model structure to the objectwise injective model structu*
*re.
(c)If C contains finite products, then the model structure of part (a) is simpl*
*icial.
Proof.As for the objectwise projective model structure of Theorem 2.2(a), the
proof is an application of a standard recognition principle for cofibrantly gen*
*erated
model categories [Hi, Thm. 11.3.1]. In order to apply this principle, we have *
*to
check a few things.
First, the category sP re(C) contains all small limits and colimits (they're *
*con
structed objectwise). Second, the objectwise weak equivalences satisfy the two
outofthree axiom and the retract axiom. The rest of the hypotheses of [Hi,
Thm. 11.3.1] are proved below in Lemmas 3.9, 3.10, and 3.11.
The simplicial structure of part (c) (of which, as usual, only axiom SM7 is n*
*on
trivial) is handled below by Corollary 3.16.
For right properness, first note that every flasque fibration is an objectwis*
*e fi
bration. Since pullbacks are constructed objectwise in sP re(C), right propern*
*ess
follows from right properness of simplicial sets.
For left properness, we can apply the same argument as in the previous paragr*
*aph
if we can show that flasque cofibrations are objectwise cofibrations. This is p*
*roved
below in Lemma 3.8.
The rest of this section is dedicated to proving the technical conditions nee*
*ded
in the proof of Theorem 3.7.
Lemma 3.8. A projective cofibration is a flasque cofibration, and a flasque cof*
*i
bration is an injective cofibration. An injective fibration is a flasque fibrat*
*ion, and
a flasque fibration is a projective fibration.
Proof.It follows from the definitions that an objectwise acyclic flasque fibrat*
*ion is
an objectwise acyclic projective fibration and that a flasque fibration is a pr*
*ojective
fibration. Now projective cofibrations are determined by the left lifting prope*
*rty
with respect to all objectwise acyclic projective fibrations (and similarly for*
* flasque
cofibrations), so a projective cofibration is a flasque cofibration.
If f : [U ! X is a monomorphism and g : n,k! n is an acyclic cofibration,
then f g is an objectwise acyclic injective cofibration. This implies that eve*
*ry
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 7
injective fibration is a flasque fibration, which implies in turn that every fl*
*asque
cofibration is an injective cofibration.
Lemma 3.9. A map f : F ! G in sP re(C) is an objectwise acyclic flasque fibrati*
*on
if and only if the map
Map (X, F ) ! Map (X, G) xMap([U,G)Map ([U, F )
is an acyclic fibration of simplicial sets for every finite collection U of mon*
*omor
phisms into an object X of C. Equivalently, a map is an objectwise acyclic flas*
*que
fibration if and only if it is an Iinjective.
Proof.The second statement follows from the first by the usual adjointness tric*
*ks.
To simplify the notation, let Map (U, f) be the map under consideration.
First suppose that Map (U, f) is an acyclic fibration for every finite collec*
*tion
U. This immediately implies that f is a flasque fibration. Now Map (X, F ) !
Map (X, G) is an acyclic fibration for every X in C (consider the empty collect*
*ion
of monomorphisms into X). Thus, f is an objectwise weak equivalence.
For the other direction, suppose that f is an objectwise acyclic flasque fibr*
*ation.
From the definitions, this immediately implies that Map (U, f) is a flasque fib*
*ration.
We use induction to show that Map (U, f) is an objectwise weak equivalence.
When n = 0, the presheaf [U is empty. In this case, Map (U, f) is just the map
Map (X, F ) ! Map (X, G), which was assumed to be a weak equivalence.
Now assume that the lemma has been proved for collections of monomorphisms
of size at most n  1. Let U0`= {U1, . .,.Un1}, and let U = {U1, . .,.Un1, V*
* }.
First note that [U equals [U0 [U00V , where U00= {UixX V } is viewed as a
collection of monomorphisms into V . Thus Map ([U, F ) equals
Map ([U0, F ) xMap([U00,F)Map(V, F )
(and similarly for Map ([U, G)). Hence we have a diagram
Map (X, F )


fflffl
Map (X, G) xMap([U,G)Map ([U, F_)______________//_Map(V, F )
 
 
fflffl fflffl
Map (X, G) xMap([U0,G)Map([U0, F_)___//Map(V, G) xMap([U00,G)Map([U00, F )
in which the square is a pullback square. The induction assumption tells us tha*
*t the
right vertical arrow is an acyclic fibration, so the lower left vertical arrow *
*is also.
The induction assumption also tells us that the composition of the left column
is an acyclic fibration, so the twooutofthree axiom allows us to conclude th*
*at
Map (U, f) is a weak equivalence.
Lemma 3.10. The domains of the maps in I or J (see Definition 3.2) are !small.
`
Proof.Let F be the presheaf X K [U K [U L, where [U ! X is induced by a
finite collection of monomorphisms into an object X of C and K ! L is an inclus*
*ion
of finite simplicial sets. It suffices to show that each of the three presheave*
*s X K,
[U K, and [U L are !small.
Now  K is left adjoint to the cotensor ()K , and ()K commutes with filtered
colimits because K is finite. Thus it suffices simply to show that X and [U are
8 DANIEL C. ISAKSEN
!small. Note that Hom (X, F ) is equal to the set of 0simplices of the simpli*
*cial
set F (X); this shows that X is !small.
For [U, the proof is by induction on the size of U. When n = 0, the empty
presheaf is certainly !small.
Now assume that the lemma has been proved for collections of monomorphisms
of size at most n  1. Let U0`= {U1, . .,.Un1}, and let U = {U1, . .,.Un1, V*
* }.
First note that [U equals [U0 [U00V , where U00= {UixX V } is viewed as a
collection of monomorphisms into V . As above, it suffices to observe that [U0,
[U00, and V are all !small, which follows from the induction assumption.
Lemma 3.11. If a map is a Jcofibration, then it is an objectwise acyclic flasq*
*ue
cofibration.
Proof.Suppose that f is a Jcofibration. It has the left lifting property with *
*respect
to all flasque fibrations by definition. Therefore, f has the left lifting pro*
*perty
with respect to all objectwise acyclic flasque fibrations, which makes it a fla*
*sque
cofibration.
Now we just have to show that a Jcofibration is an objectwise weak equivalen*
*ce.
There are two ways of proceeding. One way is to just observe that Jcofibration*
*s are
objectwise acyclic injective cofibrations because injective fibrations are Jin*
*jectives
by Lemma 3.8. The problem with this approach is that it relies on the existence
of the injective model structure, which has a complicated proof. We'll take a m*
*ore
elementary approach.
Recall that a relative Jcell complex is a transfinite composition of cobase
changes of maps in J. Standard arguments imply that every relative Jcell compl*
*ex
is a Jcofibration. We will first show that relative Jcell complexes are objec*
*twise
weak equivalences.
By direct inspection a map in J is an objectwise weak equivalence and an ob
jectwise cofibration. Next, a cobase change of a map in J is also an objectwise
weak equivalence and an objectwise cofibration. Finally, a transfinite composit*
*ion
of maps that are both objectwise weak equivalences and objectwise cofibrations *
*is
again an objectwise weak equivalence and objectwise cofibration.
Now that we know that relative Jcell complexes are objectwise weak equiva
lences, we consider an arbitrary Jcofibration f. First use the small object ar*
*gument
(with respect to J) to factor f as pi, where i is a relative Jcell complex and*
* p is
a Jinjective. Now f lifts with respect to p by assumption, which implies by the
retract argument that f is a retract of i. Since i is an objectwise weak equiva*
*lence
by the previous paragraph, so is f.
We used in an essential way in the above proof that the map [U ! X is a
monomorphism. Without that, we wouldn't be able to conclude that a cobase
change of a map in J is an objectwise weak equivalence. Thus, it's critical tha*
*t we
consider finite collections U of monomorphisms into an object X of C, not all f*
*inite
collections of maps into X.
It's not surprising that we're forced to consider monomorphisms in C. Other
wise, we would be producing classes of cofibrations that aren't monomorphisms.
Although such cofibrations are not axiomatically ruled out, in practice cofibra*
*tions
are some kind of monomorphism for most useful model categories.
3.12. Internal Function Objects. Our next goal is to show that the objectwise
flasque model structure interacts well with respect to internal function object*
*s, in
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 9
the same way that the projective and injective objectwise model structures do. *
*We
will also show that the objectwise flasque model structure is simplicial.
Lemma 3.13. If K ! L is any cofibration of simplicial sets, then
a
[U L X K ! X L
[U K
is a flasque cofibration for any finite collection U of monomorphisms into an o*
*bject
X of C, and it is objectwise acyclic if K ! L is a weak equivalence.
Proof.The map K ! L is a transfinite composition of cobase changes of maps of
the form @ n ! n, so the map under consideration`is a transfinite composition
of cobase changes of maps of the form [U n [U @ n X @ n ! X n.
Thus the map is a relative Icell complex, so it is a flasque cofibration.
If K ! L is a weak equivalence, then it is a retract of a transfinite composi*
*tion
of cobase changes of maps of the form n,k! n, so the map under consideration
is a retract`of a transfinite composition of cobase changes of maps of the form
[U n [U n,kX n,k! X n. Thus the map is a retract of a relative
Jcell complex, so it is an objectwise acyclic flasque cofibration.
Recall that we have been working so far with an arbitrary small indexing cate*
*gory
C. The next proposition requires the mild hypothesis on C that it contain fini*
*te
products. In applications to motivic homotopy theory, this will be no problem.
Proposition 3.14. Let C be an arbitrary small category that possesses finite pr*
*od
ucts. If f : F ! G and g : A ! B are flasque cofibrations, then f g is again a
flasque cofibration, and it is objectwise acyclic if either f or g is.
Proof.Recall that the flasque cofibrations are the retracts of relative Icell *
*com
plexes, and the objectwise acyclic flasque cofibrations are the retracts of rel*
*ative
Jcell complexes. If f is any map and g is a retract of a transfinite compositi*
*on of
cobase changes of maps in I, then f g is a retract of a transfinite composition*
* of
cobase changes of maps of the form f g0, where g0 belongs to I (and similarly f*
*or
J). Thus it suffices to assume that g belongs to I (for the first part) or to J*
* (for
the second part). By the symmetric argument, we can also assume that f belongs
to I.
Suppose that f is the map f0 f00, where f0 is a map [U ! X and f00is the
generating cofibration @ n ! n. Similarly, suppose that g is g0 g00, where g0 *
*is
[V ! Y and g00is @ m ! m . Then careful inspection of the definitions shows
that f g is isomorphic to h0 h00, where h0 is the map f0 g0 and h00is the map
f00 g00. Now h00is the map
a
n x @ m @ n x m ! n x m ,
@ nx@ m
which is a cofibration of simplicial sets, and h0is the map [W ! X x Y , where W
is the collection consisting of maps of the form Uix Y ! X x Y and maps of the
form X x Vi! X x Y . Thus, h0 h00is a flasque cofibration by Lemma 3.13.
The same argument works when g belongs to J, except that h00becomes
a
n x m,k @ n x m ! n x m ,
@ nx m,k
which is an acyclic cofibration of simplicial sets.
10 DANIEL C. ISAKSEN
Note that the above proof (and therefore the following two corollaries) requi*
*re us
to work with arbitrary finite collections of monomorphisms in C, not just colle*
*ctions
of size 0 and 1. This is the essential reason that it is important to work with*
* all
finite collections.
Corollary 3.15. Let C be an arbitrary small category that possesses finite prod*
*ucts.
If i : A ! B is a flasque cofibration and f : F ! G is a flasque fibration, the*
*n the
map
Hom__(B, G) ! Hom__(A, G) xHom_(A,F)Hom_(B, F )
is a flasque fibration, and it is an objectwise weak equivalence if either i or*
* f is.
Proof.This follows immediately by adjointness from Proposition 3.14.
The following corollary shows that the flasque model structure of Theorem 3.7
is simplicial.
Corollary 3.16. Let C be an arbitrary small category that possesses finite prod*
*ucts.
If i : A ! B is a flasque cofibration and f : F ! G is a flasque fibration, the*
*n the
map
Map(B, G) ! Map (A, G) xMap(A,F)Map (B, F )
is a fibration of simplicial sets, and it is a weak equivalence if either i or *
*f is an
objectwise weak equivalence.
Proof.Recall that the space of global sections of the presheaf Hom__(F, G) is e*
*qual
to the simplicial mapping space Map (F, G). The result now follows from Corolla*
*ry
3.15.
3.17. The projective, flasque, and injective model structures are distinct.
We present an elementary example showing that the classes of projective fibrati*
*ons,
flasque fibrations, and injective fibrations are mutually distinct in general.
Let C be the category with two objects 0 and 1 and two nonidentity morphisms
from 0 to 1; it is indicated by the diagram
0____//_//_1.
Now a map F ! G is a projective fibration if and only if the maps F0 ! G0
and F1 ! G1 are fibrations. It can be checked that F ! G is a flasque fibration
if and only if the maps F0 ! G0, F1 ! G1, and both maps F1 ! F0 xG0 G1 are
fibrations. Finally, F ! G is an injective fibration if and only if the maps F0*
* ! G0
and
F1 ! (F0 x F0) x(G0xG0)G1
are fibrations.
These three sets of conditions are distinct, as is easily verified by element*
*ary
examples. The reader may find it an instructive exercise to show directly that *
*the
conditions for injective fibrations implies the conditions for flasque fibratio*
*ns. This
gives a direct verification of Lemma 3.8 in this special case.
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 11
4. Local model structures
In the previous section, we worked with an index category C that did not nec
essarily have a Grothendieck topology. From here on, we must assume that C does
have a Grothendieck topology.
Recall that a hypercover is a certain kind of map of simplicial presheaves [S*
*GA4 ,
Expos'e V, 7.3] [AM ] [DHI ]. The precise definition is technical and not relev*
*ant for
us, so we will skip it.
Definition 4.1. The local projective (resp., flasque, injective) model struc
ture on sP re(C) is the left Bousfield localization [Hi, Defn. 3.3.1] of the ob*
*jectwise
projective (resp., flasque, injective) model structure at the class of all hype*
*rcovers.
This is not the way that the local projective [B ] and local injective [J2] m*
*odel
structures are usually defined. It is more usual (and probably more intuitive) *
*to
define local weak equivalences with sheaves of homotopy groups. See [DHI ] for a
proof that this Bousfield localization approach gives the same class of weak eq*
*uiv
alences for the injective and projective model structures. The analogous statem*
*ent
for the flasque model structure appears below in Theorem 4.3.
It is formal that these local model structures are left proper, cellular, and*
* sim
plicial [Hi, Thm. 4.1.1]. In fact, these model structures are right proper. One*
* way
to see this is to note that the fibrations in any of these model structures are*
* ob
jectwise fibrations. It follows from computing stalks that local weak equivalen*
*ces
are preserved by base change along objectwise fibrations.
Theorem 4.2. The identity functor is a left Quillen equivalence from the local
projective model structure to the local flasque model structure and from the lo*
*cal
flasque model structure to the local injective model structure.
Proof.In constructing the three left Bousfield localizations, one must choose m*
*odels
for the homotopy colimits of the hypercovers. It doesn't matter which models are
chosen; the localizations will be isomorphic.
Since projective cofibrations are flasque cofibrations and flasque cofibratio*
*ns are
injective cofibrations, we can choose models for the homotopy colimits in the p*
*ro
jective model structure, and these models will work for the other two structures
as well. Thus, all three local model structures are constructed by localizing a*
*t the
same set of maps.
In order to show that the identity functor gives Quillen equivalences, it suf*
*fices
because of Theorem 3.7(b) to apply [Hi, Thm. 3.3.20], which tells us that local*
*iza
tions of Quillen equivalent model categories are Quillen equivalent.
The previous theorem does not guarantee that the class of weak equivalences in
the local flasque model structure is actually equal to the class of weak equiva*
*lences
in the local injective or local projective model structure. However, this equal*
*ity is
now not hard to prove.
Theorem 4.3. The local flasque weak equivalences are detected by sheaves of ho
motopy groups.
Proof.Let f be any local flasque weak equivalence. Factor it into a local acycl*
*ic
flasque cofibration i followed by a local acyclic flasque fibration p. Theorem*
* 4.2
tells us that i is a local acyclic injective cofibration. Thus, i induces isomo*
*rphisms
12 DANIEL C. ISAKSEN
on sheaves of homotopy groups. On the other hand, p is in fact an objectwise we*
*ak
equivalence, so it also induces isomorphisms on sheaves of homotopy groups.
Now suppose that f is any map that induces isomorphisms on sheaves of ho
motopy groups. Factor it into a local acyclic projective cofibration i followed*
* by
a local acyclic projective fibration p. Theorem 4.2 tells us that i is a local *
*acyclic
flasque cofibration. On the other hand, p is an objectwise weak equivalence, so*
* it
is certainly a local flasque weak equivalence.
Proposition 4.4. If i : A ! B is a flasque cofibration and f : F ! G is a local
flasque fibration, then the map
Hom__(B, G) ! Hom__(A, G) xHom_(A,F)Hom_(B, F )
is a local flasque fibration, and it is a local weak equivalence if either i or*
* f is.
Proof.This follows by adjointness from a local version of Proposition 3.14. The
first part of that result is no problem because the cofibrations in the local f*
*lasque
model structure are identical to the cofibrations in the objectwise flasque mod*
*el
structure.
For the second part of the local version of Proposition 3.14, note that
a
B x F A x G ! B x G
AxF
is a local weak equivalence for any pair of injective cofibrations A ! B and F *
*! G,
provided at least one of them is a local weak equivalence. One way to see this *
*is
to compute stalks, noting that taking stalks commutes with finite products and
pushouts.
4.5. Simplicial sheaves. Recall that the category sP re(C) of simplicial preshe*
*aves
has a full subcategory sSh(C) of simplicial sheaves. The inclusion functor has *
*a left
adjoint a : sP re(C) ! sSh(C), which is called the sheafification functor.
Theorem 4.6. The category sSh(C) has a local flasque model structure in which
the weak equivalences are the local weak equivalences, the fibrations are the m*
*aps
that are flasque fibrations when considered as presheaves, and the cofibrations*
* are
generated by the set aI (i.e., the sheafifications of all the maps in I). Sheaf*
*ification
is a left Quillen equivalence from the local flasque model structure on preshea*
*ves to
the local flasque model structure on sheaves.
Proof.This is an application of a general result about using adjoint functors to
translate a model structure from one category to another [Hi, Thm. 11.3.2]. That
result has several hypothesis, only one of which is not immediately obvious. Na*
*mely,
we must explain why relative aJcell complexes are local weak equivalences. This
follows from the observation that the local weak equivalences are closed under
transfinite compositions.
To see that a is a Quillen equivalence, one just needs to use that the natural
map F ! aF is always a local weak equivalence for any simplicial prehseaf F [J2,
Lem. 2.6].
The previous theorem reproves the part of [BG ] that establishes a model stru*
*c
ture on the category of sheaves on a Noetherian topological space. At first gla*
*nce,
Theorem 4.6 does not appear to apply because [BG ] makes no mention of finite
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 13
collections of monomorphisms. However, in the Grothendieck topology of open sets
of a fixed topological space, the object [U is always representable.
In fact, Theorem 4.6 is much more general because there is no Noetherian hy
pothesis on the site. However, we cannot recover the simple identification of f*
*ibrant
objects in terms of certain homotopy pullback squares. This identification is *
*an
important part of [BG ] (and its generalization in [V ]).
4.7. Nisnevich local model structures. From now on, we specialize to the case
of the Nisnevich topology on the category SmS of smooth schemes over a ground
scheme S because this is the situation that arises in motivic homotopy theory. *
*In
this context, it is not necessary to localize with respect to all hypercovers a*
*s in
Section 4. Recall that there is a certain class of elementary Nisnevich squares*
* [MV ,
Defn. 3.1.3]
U xX V ____//_V
 
 
fflffl fflffl
U _______//_X.
The full definition is not so important, but we will use the fact that U ! X (a*
*nd
thus also U xX V ! V ) is a monomorphism.
The following theorem is proved in [B , Lem. 4.2].
Theorem 4.8. The local projective model structure on sP re(SmS) is the left Bou*
*s
field localization of the objectwise projective model structure at the set of c*
*onsisting
of maps P ! X, where P is the homotopy pushout of
Uoo___U xX V _____//V
for all elementary Nisnevich squares.
The homotopy pushouts in the previous theorem are a bit of an annoyance. It
would be nicer to have a more concrete description of the maps with respect to
which we are localizing. In both the flasque and injective cases, such a descri*
*ption
is possible.
Theorem 4.9. The local flasque (resp., injective) model structure on sP re(SmS)
is the left Bousfield localization of the objectwise flasque (resp., injective)*
* model
structure at the set of maps
`
U UxXV V ! X
for all elementary Nisnevich squares.
Proof.Recall that localizations of Quillen equivalent model categories are Quil*
*len
equivalent [Hi, Thm. 3.3.20]. Therefore, if we localize the flasque or injectiv*
*e ob
jectwise model structure at the set of maps P ! X (as in the previous theorem),
then Theorems 3.7(b) and 4.8 tell us that we get the desired local model struct*
*ure.
But in the flasque and injective`model structures, U xX V ! V is a cofibration,*
* so
the ordinary pushout U UxXV V is a model for the homotopy pushout.
The above theorem demonstrates one of the ways in which the flasque (or in
jective) model structure is more convenient than the projective model structure.
The localization is slightly more concrete _ there aren't any homotopy pushouts
involved.
14 DANIEL C. ISAKSEN
Theorem 4.8 formally leads to the following characterization [B , Lem. 4.1]. *
*An
object F of sP re(SmS) is local projective fibrant if and only if:
(1) F is projective fibrant;
(2)
F (X) _______//_F (U)
 
 
fflffl fflffl
F (V )___//_F (U xX V )
is homotopy cartesian for all elementary Nisnevich squares.
Because of Theorem 4.9, we can improve on this characterization for the flasq*
*ue
and injective model structures.
Corollary 4.10. An object F of sP re(SmS) is local flasque (resp., injective) f*
*ibrant
if and only if:
(1) F is flasque (resp., injective) fibrant;
(2) the map
F (X) ! F (U) xF(UxXV )F (V )
is an acyclic fibration for all elementary Nisnevich squares.
Proof.This follows from formal properties of left Bousfield localizations [Hi,
Thm. 4.1.1(2)] (see [DHI , Cor. 7.1] for a similar argument).
The previous result shows exactly why the flasque model structure is a good
compromise between the projective and injective model structures. Condition (1)
for the injective model structure is very complicated. However, condition (1) f*
*or
the flasque structure is much simpler. Condition (2) doesn't work for the proje*
*ctive
structure because a messier homotopy pullback statement is required.
5. Motivic model structures
Definition 5.1. The motivic projective (resp., flasque, injective) model struct*
*ure
on sP re(SmS) is the left Bousfield localization of the local projective (resp.*
*, flasque,
injective) model structure at the set of maps X ! X x A1.
The maps X ! X x A1 are induced by the inclusion 0 ! A1.
It is formal that these motivic model structures are left proper, cellular, a*
*nd
simplicial [Hi, Thm. 4.1.1]. In fact, they are also right proper, but this req*
*uires
some extra work [MV , 2.2.7] (beware that this reference reverses the definitio*
*n of
left proper and right proper).
Similarly to the local projective model structure, an object F is motivic pro*
*jec
tive fibrant if and only if:
(1) F is projective fibrant;
(2)
F (X) _______//_F (U)
 
 
fflffl fflffl
F (V )___//_F (U xX V )
is homotopy cartesian for all elementary Nisnevich squares;
(3) F (X x A1) ! F (X) is a weak equivalence for all X in C.
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 15
As in Corollary 4.10, we can improve on this characterization for the flasque*
* and
injective model structures.
Corollary 5.2. An object F is motivic flasque (resp., injective) fibrant if and*
* only
if:
(1) F is flasque (resp., injective) fibrant;
(2) the map
F (X) ! F (U) xF(UxXV )F (V )
is an acyclic fibration;
(3) F (X x A1) ! F (X) is an acyclic fibration for all X in C.
Proof.As in the proof of Corollary 4.10, this follows from [Hi, Thm. 4.1.1(2)].
We present one consequence of Corollary 5.2, which will be important later. N*
*ote
that the injective analogue of this statement is false.
Proposition 5.3. The class of motivic flasque fibrant objects is closed under f*
*iltered
colimits.
Proof.Condition (1) of Corollary 5.2 is preserved by filtered colimits because *
*the
generating objectwise acyclic flasque cofibrations (i.e., the set J of Definiti*
*on 3.2)
have compact domains. Conditions (2) and (3) are straightforward (see also [J1,
x 1.3]).
Proposition 5.4. If A is flasque fibrant and F is motivic flasque fibrant, then
Hom__(A, F ) is also motivic flasque fibrant.
Proof.From Proposition 4.4, we know that Hom__(A, F ) is local flasque fibrant.*
* By
[Hi, Thm. 4.1.1(2)], we only need to show that the map Hom__(A, F ) ! * has the
right lifting property with respect to every map of the form
a
X n (X x A1) n,k! (X x A1) n.
X n,k
Now this map is an objectwise acyclic flasque cofibration, so the lifting prope*
*rty is
satisfied.
In analogy to Corollary 3.15 and Proposition 4.4, it is natural to consider t*
*he
map
Hom__(B, G) ! Hom__(A, G) xHom_(A,F)Hom_(B, F )
for i : A ! B a flasque cofibration and f : F ! G a motivic flasque fibration.
However, we do not yet know how to show that this map is a motivic flasque
fibration.
6.Stable motivic model structures
6.1. Pointed model structures. In stable motivic homotopy theory, we're actu
ally interested in the pointed version of what we've been discussing so far, i.*
*e., the
category sP re(SmS)* of presheaves of pointed simplicial sets. Of course, this*
* is
just an undercategory of sP re(SmS), so we obtain formally the pointed objectwi*
*se
flasque model structure, the pointed local flasque model structure, and the poi*
*nted
motivic flasque model structure [Hi, Thm. 7.6.5].
These pointed model structures are still simplicial model categories with the
obvious pointed analogue of tensor. Namely, if F is any pointed presheaf and K *
*is
16 DANIEL C. ISAKSEN
any simplicial set, then F K is the pointed presheaf defined by (F K)(U) =
F (U) ^ K+ .
Recall that the fibrant objects in the pointed categories are simply the poin*
*ted
presheaves that are fibrant in the corresponding unpointed category. On the oth*
*er
hand, the cofibrant objects are different; instead of studying OE ! A, we now l*
*ook
at * ! A. We're going to need to know that some of the basic objects of motivic
homotopy theory are flasque cofibrant pointed presheaves.
Lemma 6.2. If X ! Y is any monomorphism in SmS, then Y=X is a flasque
cofibrant pointed presheaf.
Proof.Just consider the pushout square
X ______//Y
 
 
fflffl fflffl
* ____//_Y=X,
in which the top arrow is a flasque cofibration. Therefore, the bottom arrow is*
* also
a flasque cofibration.
The objects mentioned in the following lemma are central to the construction *
*of
stable motivic homotopy theory.
Lemma 6.3. In sP re(SmS)*, the following pointed presheaves are flasque cofi
brant:
(1) The presheaf P1, pointed at 1 (or at any other rational point).
(2) The presheaf A1  0, pointed at 1 (or at any other rational point).
(3) The presheaf A1=A1  0.
(4) The presheaf (A1  0).
Proof.The first three examples are special cases of the previous lemma. For the
last one, we just need to show that the map
a
* S1 (A1  0) x * ! (A1  0) S1
* *
is a flasque cofibration (by the definition of suspension). This is exactly wh*
*at
Lemma 3.13 says.
The category sP re(SmS)* still has internal function objects, which we denote*
* by
Hom__*(F, G). However, we can't prove the pointed analogues of Proposition 3.14*
* and
Corollary 3.15 because that would require taking smash products of representable
functors instead of products of representable functors. Nevertheless, we have t*
*he
following useful result.
Proposition 6.4. If A is a flasque cofibrant pointed presheaf and F is a flasque
(resp., local flasque, motivic flasque) fibrant pointed presheaf, then Hom__*(A*
*, F ) is
also a flasque (resp., local flasque, motivic flasque) fibrant pointed presheaf.
Proof.Consider the pullback square
Hom__*(A, F_)__//_Hom_(A, F )
 
 
fflffl fflffl
*_________//_Hom_(*, F ).
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 17
We are interested in the left vertical arrow, which is a base change of the rig*
*ht
vertical arrow. Thus, we just have to show that the right vertical arrow is a f*
*lasque
(resp., local flasque, motivic flasque) fibration. For the first two cases, thi*
*s follows
from Corollary 3.15 and Proposition 4.4.
The third (motivic) case requires only slightly more work. First note that bo*
*th
objects Hom__(A, F ) and Hom__(*, F ) are motivic flasque fibrant by Propositio*
*n 5.4.
We already know that the map between them is a flasque fibration from the pre
vious paragraph. From [Hi, Prop. 3.3.16(1)] (a general result describing fibrat*
*ions
between fibrant objects in a localization), we conclude that the map is in fact*
* a
motivic flasque fibration.
Recall that for any pointed presheaf F , the presheaf 2,1F is defined to be
Hom__*(T, F ), where T is some chosen model for the motivic sphere S2,1. For di*
*fferent
purposes, authors have chosen T to be P1, A1=A10, or (A10). In the following
corollary, it doesn't matter which of these three are used.
Corollary 6.5. If F is a motivic flasque fibrant pointed presheaf, then 2,1F is
also a motivic flasque fibrant pointed presheaf.
Proof.This is a straightforward combination of Lemma 6.3 with Proposition 6.4.
The above corollary is a great example of the advantage of the flasque model
structures over the projective model structures. Since * ! P1 and * ! A1=(A10)
are not projective cofibrations, one has to choose carefully a model for S2,1wh*
*en
working with the projective structures. With the flasque (or injective) structu*
*res,
there is no such problem.
6.6. Motivic spectra. One of the key applications of the motivic flasque model
structure is for stable motivic homotopy theory. Normally people use the inject*
*ive
structure because they need * ! P1 (or * ! A1=A1  0) to be a cofibration. Of
course, this map is also a flasque cofibration, so the general machinery for pr*
*oducing
stable model structures [Ho ] works fine.
To illustrate this point, we will describe a theorem that can be proved only *
*with
the flasque model structure (as far as we know). The proof of the theorem will
require certain properties that neither the projective nor injective model stru*
*ctures
possess.
We briefly review the construction of naive BousfieldFriedlander motivic spe*
*ctra.
All of what follows would work just as well for motivic symmetric spectra, but *
*we
use naive motivic spectra because they are easier to describe and they suffice *
*for
our specific purpose.
A naive motivic spectrum E is a sequence {En} of pointed simplicial presheaves
together with structure maps En ! 2,1En+1. The definition of maps of naive
motivic spectra is obvious.
In order to construct the stable motivic model structure, we can start with t*
*he
projective, flasque, or injective model structure on pointed simplicial preshea*
*ves.
We'll start with the flasque model structure.
We only need a few facts about the stable motivic model structure on naive
motivic spectra. First, a spectrum E is fibrant if and only if each En is flas*
*que
fibrant and the maps En ! 2,1En+1 are (unstable) motivic weak equivalences.
Second, a map E ! F of fibrant spectra is a stable motivic weak equivalence if *
*and
18 DANIEL C. ISAKSEN
only if each En ! Fn is a motivic weak equivalence of pointed simplicial preshe*
*aves.
Third, the standard models for the sphere spectra Sp,qare cofibrant spectra.
Proposition 6.7. Fibrant motivic spectra are closed under filtered colimits.
Proof.Let {Ei} be filtered system of naive motivic spectra, and let E be colimi*
*Ei.
Then En equals colimiEin, so it is a motivic flasque fibrant pointed simplicial
presheaf by Proposition 5.3.
By Corollary 6.5, 2,1Ein+1is a motivic flasque fibrant pointed simplicial
presheaf, so the structure map Ein! 2,1Ein+1is a motivic weak equivalence
between motivic flasque fibrant pointed simplicial presheaves. Therefore, it i*
*s in
fact an objectwise weak equivalence (this is a general property of localization*
*s).
It follows that colimiEin! colimi 2,1Ein+1is also an objectwise weak equivlence
and hence a motivic weak equivalence. This last map is just the structure map
En ! 2,1En+1 because 2,1() commutes with filtered colimits.
Theorem 6.8. Let {Ei} be a filtered system of motivic spectra. The natural map
colimißp,qEi ! ßp,q(hocolimiEi) is an isomorphism for all p and q in Z.
Proof.We may assume that each Ei is a fibrant motivic spectrum. By Proposition
6.7, so is colimiEi. Consider the map
colimiß0Map (Sp,q, Ei) ! ß0Map (Sp,q, colimiEi).
This is an isomorphism because Map (Sp,q, ) commutes with filtered colimits.
It only remains to show that the natural map hocolimiEi ! colimiEi is a
stable motivic weak equivalence. It suffices to show that if {Ei} and {F i} are
filtered systems of motivic fibrant spectra and each Ei ! F iis a stable motivic
weak equivalence, then colimiEi ! colimiF iis a stable motivic weak equivalence.
Since each Ei ! F iis a stable motivic weak equivalence between fibrant spect*
*ra,
each Ein! Fniis a motivic weak equivalence between motivic flasque fibrant poin*
*ted
simplicial presheaves. As observed in the previous proof, each map Ein! Fniis
in fact an objectwise weak equivalence. It follows that colimiEin! colimiFniis
still an objectwise weak equivalence, and this implies that colimEi! colimFi is*
* a
stable motivic weak equivalence.
References
[SGA4] M. Artin, A. Grothendieck, and J. L. Verdier, Th'eorie des Topos et Coho*
*mologie 'Etale
des Schemas, Lecture Notes in Math., Vol. 270, Springer, 1972.
[AM] M. Artin and B. Mazur, 'Etale Homotopy, Lecture Notes in Math., Vol. 100*
*, Springer,
1969.
[B] B. A. Blander, Local projective model structures on simplicial presheave*
*s, KTheory 24
(2001), no. 3, 283301.
[BG] K. S. Brown and S. M. Gersten, Algebraic Ktheory as generalized sheaf c*
*ohomology, in
Algebraic Ktheory, I: Higher Ktheories (Proc. Conf., Battelle Memorial*
* Inst., Seattle,
1972), 266292, Lecture Notes in Math., Vol. 341, Springer, 1973.
[BK] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localiza*
*tions, Lecture
Notes in Math., Vol. 304, Springer, 1972.
[D] D. Dugger, Universal homotopy theories, Adv. in Math. 164 (2001), 14417*
*6.
[DHI] D. Dugger, S. Hollander, and D. C. Isaksen, Hypercovers and simplicial p*
*resheaves,
Math. Proc. Cambridge Philos. Soc., to appear.
[DI] D. Dugger and D. C. Isaksen, Motivic cell structures, preprint.
[Hi] P. S. Hirschhorn, Model Categories and Their Localizations, Mathematical*
* Surveys and
Monographs, Vol. 99, Amer. Math. Soc., 2003.
FLASQUE MODEL STRUCTURES FOR PRESHEAVES 19
[Ho] M. Hovey, Spectra and symmetric spectra in general model categories, J. *
*Pure Appl.
Algebra 165 (2001), no. 1, 63127.
[Hu] P. Hu, Smodules in the category of schemes, Mem. Amer. Math. Soc. 161 (*
*2003), no.
767.
[J1] J. F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445552.
[J2] J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), n*
*o. 1, 3587.
[J3] J. F. Jardine, Intermediate model structures for simplicial presheaves, *
*preprint.
[Jo] A. Joyal, Letter to A. Grothendieck, 1984.
[L] F. L'arusson, Model structures and the Oka principle, preprint.
[M] F. Morel, Th'eorie homotopique des sch'emas, Ast'erisque, Vol. 256, 1999.
[MV] F. Morel and V. Voevodsky, A1homotopy theory, Inst. Hautes Etudes Sci. *
*Publ. Math.
90 (2001), 45143.
[V] V. Voevodsky, Homotopy theory of simplicial sheaves in completely decomp*
*osable topolo
gies, preprint.
Department of Mathematics, Wayne State University, Detroit, MI 48202
Email address: isaksen@math.wayne.edu