Fibred sites and stack cohomology
J.F. Jardine
June 21, 2004
Introduction
A stack G is traditionally defined to be a pseudofunctor on a Grothendieck
site C which takes values in groupoids, and which satisfies the effective desce*
*nt
condition. The effective descent condition specifies that the objects of G sati*
*sfy
a pseudofunctorial sheaf condition. With this description in hand, one can form
the Grothendieck construction, here denoted by C=G, for the stack G and let it
inherit a topology from the ambient site C, so that C=G acquires the structure
of a Grothendieck site. Then one can speak of sheaves on this site, and stack
cohomology of G with coefficients in a sheaf F on C=G is the cohomology of the
site with coefficients in F in the standard way.
That said, the connection between this definition of stack cohomology and
the geometry of the stack G is a bit tenuous, at least apparently, and it has
historically been rather awkward to relate this invariant to other standard she*
*af
theoretic invariants.
The general approach to stacks (and higher stacks) has changed a great deal
in recent years, because we now understand that they are homotopy theoretic
objects. A stack G is now thought of, most generally, as a presheaf of groupoids
on a site C which is fibrant with respect to a nicely defined model structure on
the category of presheaves of groupoids on C
More explicitly, one says that a functor G ! H between presheaves of
groupoids is a weak equivalence (respectively fibration) if the induced map
BG ! BH is a local weak equivalence (respectively global fibration) in the
standard model structure on the category of simplicial presheaves on C. Thus,
G is a stack if and only if BG is a globally fibrant simplicial presheaf. This *
*de
scription of stacks was a major conceptual breakthrough which was initiated by
Joyal and Tierney [14] in the case of sheaves of groupoids and was completed by
Hollander [4] for presheaves of groupoids. Stack completion becomes a fibrant
model in this setup, and it is now well understood that path components (or
isomorphism classes) in the global sections of a stack G are in bijective corre
spondence with the set [*, BG] of morphisms in the homotopy category of sim
plicial presheaves. This gives a rather striking generalization of the early re*
*sult
that identified the homotopy invariants [*, BH] arising from sheaves of groups
H with isomorphism classes of Htorsors [8]. We also now understand what the
1
higher order analogues of Htorsors should be, and a homotopy theoretic (and
geometric) identification of these higher order torsors has been achieved [13].
This paper brings stack cohomology into this arena, by giving an homotopy
theoretic description of the invariant in terms of presheaves of groupoids. One*
* of
the more important consequences of this approach is that one can then show that
the new cohomology theory for presheaves of groupoids is homotopy invariant.
In fact, one generalizes the traditional description of the site C=G fibred
over a stack G even further, to that of the site C=A fibred over a presheaf of
categories A. This seems like a strange thing to do at first, but the concept
is painless to both define and manipulate. This expanded notion specializes
to fibred site constructions that are in standard use, including the usual sites
fibred over diagrams of schemes, and hence over simplicial schemes in standard
geometric settings. It is also interesting to observe that the idea has nontri*
*vial
content even in the case where A consists only of a presheaf of objects.
Simplicial presheaves X for the site C=A take the form of enriched con
travariant diagrams defined on A and taking values in simplicial sets, and as
such naturally determine homotopy colimits
holim!AopX ! BAop
over the nerve BAopof the opposite category Aop, or equivalently over BA. The
homotopy theory of simplicial presheaves on the fibred site C=A is actually a t*
*ype
of coarse equivariant theory of Aopdiagrams _ one says öc arse" because this
is an enriched version of the old BousfieldKan theory for diagrams of simplici*
*al
sets [1].
In the case when A is a presheaf of groupoids G, this assignment of homotopy
colimits determines an equivalence of homotopy categories
Ho(s Pre(C=G)) ' Ho(s Pre(C)=BGop) (1)
which generalizes the known relationship [3] between diagrams of simplicial sets
defined on a groupoid H and that of simplicial sets over BH. This identification
gives the homotopy invariance, because the homotopy category of simplicial
presheaves over BGop is insensitive to the homotopy type of the presheaf of
groupoids G up to equivalence.
It is a consequence of the equivalence (1) that any functor G ! H of
presheaves of groupoids which is a local weak equivalence induces an adjoint
equivalence of homotopy categories
Ho (s Pre(C=G)) ' Ho(s Pre(C=H)) (2)
With a little care (so that you don't spend a long time doing it), this adjoint
equivalence can be parlayed into an adjoint equivalence
Ho(s*Pre(C=G)) ' Ho(s*Pre(C=H)) (3)
for pointed simplicial presheaves, and then to adjoint equivalences of stable
homotopy categories
Ho (Spt (C=G)) ' Ho(Spt (C=H)) (4)
2
and
Ho (Spt (C=G)) ' Ho(Spt (C=H)) (5)
for presheaves of spectra and presheaves of symmetric spectra, respectively.
Note the level of generality: these results hold over arbitrary small Grothe*
*n
dieck sites C. One fully expects that this pattern can be replicated for catego*
*ries
of module spectra and for various derived categories of chain complexes, as the
need arises. The development given in this paper ends with the symmetric
spectrum result.
The overall aim of this paper is to introduce a very general new constructio*
*n,
namely the site fibred over a presheaf of categories, and to give some of its
applications for presheaves of groupoids. The statements which are listed as
theorems are Theorem 30, which establishes the equivalence (1), and Theorem
41, which gives (4). The equivalence (2) appears here as Corollary 32, and (3)
is Corollary 37. The equivalence (5) is a consequence of Theorem 41, and is
formally stated in Corollary 44.
My personal impression is that the homotopy invariance statements will turn
out to be quite important in applications, as one now has the ability to define
stack cohomology via a construction which comes directly out of a representing
presheaf of groupoids without passing to any form of either associated sheaf or
stack completion. The first example that comes to mind for which this may be
of some use is in the applications of the cohomology of the presheaf of formal
group laws on the flat site.
I would like to thank the American Institute of Mathematics for its hospital
ity and support during the week of the workshop "Theory of motives, homotopy
theory of varieties and dessins d'enfants", held at AIM April 2326, 2004. The
appearance of this paper is a direct result of my participation in that confere*
*nce.
Contents
1 Fibred sites 4
2 The fibred site for a presheaf 7
3 Constructions for presheaves of categories 13
4 Simplicial set constructions 19
5 Presheaves of groupoids 25
3
1 Fibred sites
Suppose that C is a small Grothendieck site, and that A is a presheaf of catego*
*ries
on C.
The category C=A has objects consisting of all pairs (U, x) where U is an
object of C and x is an element of the set of section Ob (A)(U) of the presheaf
Ob (A) of objects of A. We can and will alternatively think of x as a presheaf
morphisms x : U ! Ob(A). A morphism (ff, f) : (V, y) ! (U, x) in this category
is a pair consisting of a morphism ff : V ! U of C together with a morphism
f : y ! ff*(x) of A(U).
Given another morphism (fl, g) : (W, z) ! (V, y), the composite (ff, f)(fl, *
*g)
is defined by
(ff, f)(fl, g) = (fffl, fl*(f)g),
where the composite
*(f)
z g!fl*(y) fl!fl*ff*(x) = (fffl)*(x)
is defined by the usual sort of convolution _ this category is the result of
applying the Grothendieck construction to the diagram of categories represented
by A.
There is a canonical forgetful functor ß : C=A ! C which is defined by
sending the object (U, x) to the object U of C. Observe that any sieve R
hom ( , (U, x)) of C=A is mapped to a sieve ß(R) hom ( , U) under the functor
ß.
If S is a sieve for U 2 C write ß1(S) for the collection of all morphisms
(ff, f) with ff 2 S. The covering sieves of C=A are the sieves of the form ß1S*
* for
covering sieves S of C. If R contains a covering sieve ß1S, then R = ß1(S0)
for some covering sieve of S. Note that ß1S is the smallest sieve containing
all morphisms (ff, 1) with ff 2 S.
Lemma 1. The collection of covering sieves R for C=A satisfy the axioms for
a Grothendieck topology.
Proof. There is a relation (ff, f)1ß1S = ß1ff1S, so the covering sieves of
C=Y are closed under pullback.
Suppose that S is a covering sieve for U and that Sffis a choice of covering
sieve for V for each ff : V ! U in S. Then the local character axiom for the
site C implies that the collection of all maps W 0! U which factor through a
composite
W fi!V ff!U
with ff 2 S and fi 2 Sffis a covering sieve for U.
Suppose that R, R0 are sieves for (U, x) and that R is covering. Suppose
further that (ff, f)1(R0) is covering for all (ff, f) 2 R. Suppose that R = ß*
*1S.
Then (ff, 1)1(R0) is a covering sieve for each ff : V ! U in S, and so there is
a covering sieve Sfffor U such that ff1R0 contains all morphisms (fl, 1) with
4
fl 2 Sff. It follows that the collection of all morphisms of the form (i, 1) in*
* R0
defines a covering sieve of C for U, so that R0is covering.
The (trivial) sieve of all morphisms (ff, f) is ß1S, where S is the sieve o*
*f_
all morphisms V ! U in C. All trivial sieves are therefore covering. _*
*_
The site C=A will be called the fibred site for the presheaf of categories A.
Recall that every Grothendieck site D determines a model structure on its asso
ciated category s Pre(D) of simplicial presheaves, for which the cofibrations a*
*re
the monomorphisms, the weak equivalences are the local weak equivalences, and
the fibrations are the global fibrations. One of the primary goals of this paper
is to analyze the corresponding model structures on s Pre(C=A) for the fibred
sites in important special cases. Sites fibred over presheaves of groupoids will
be of fundamental interest in the later sections of this paper. This constructi*
*on
is quite general, and specializes to many other well known standard examples,
as the following preliminary list of examples is meant to demonstrate.
Example 2. Suppose that I is a small category and that Y : I ! Pre(C) is an I
diagram in the category of presheaves on C defined by i 7! Yi. Write C=Y for the
category whose objects consist of presheaf morphisms (ie. sections) x : U ! Yi,
and whose morphisms are commutative diagrams of presheaf morphisms
V __ff_//_U
y x
fflffl fflffl
Yi__`*_//Yj
where ` : i ! j is a morphism of I. Denote such a morphism by (ff, `).
There is a presheaf of categories EY which is defined by setting EY (U) to
be the translation category for the functor Y (U) : I ! Set, where Y (U)i =
Yi(U).The objects of EY (U) consist of all pairs (i, x) where x is an element of
Yi(U). Equivalently, x is a presheaf map U ! Yi. If y : V ! Yj is an object of
EY (V ) and ff : V ! U is a morphism of C, then ff*(x) is the composite
V ff!U x!Yj,
so that a morphism f : (i, y) ! (j, ff*(x)) in EY (V ) is morphism ` : i ! j su*
*ch
that the diagram
V __ff_//_U
y x
fflffl fflffl
Yi__`*_//Yj
commutes. The category C=Y , as defined above, therefore coincides with the
category C=EY .
5
Example 3. The previous example specializes to the "standard" description of
the site C=X for a simplicial presheaf X.
Suppose that X is a simplicial presheaf and that Z is a globally fibrant
simplicial presheaf on the site C=X. For each n, there is a presheaf 1Xn which *
*is
represented by the identity on Xn in the sense that the sections corresponding
to y : V ! Xk are the commutative diagrams of Cpresheaf maps
V ______//Xn (6)
y 1
fflffl fflffl
Xk __`*_//Xn
where `* is a simplicial structure map. The maps
*
Xk __`__//Xn
1 1
fflffl fflffl
Xk __`*_//Xn
gives the family 1X = {1Xn } the structure of a simplicial presheaf on C=X.
Note that the set of all maps (6) can be identified with the collection of ordi*
*nal
number maps n ! k, and it follows that there is an isomorphism of simplicial
sets
1X (y) ~= k
The canonical simplicial presheaf map 1X ! * is therefore a local weak equiva
lence.
If F is a presheaf on C=X, a presheaf map f : 1Xn ! F is completely
determined by the images under f of the sections
U ______//Xn
 
 1
fflffl fflffl
Xn __1__//Xn
so that
hom(1Xn , F ) ~= lim Fn(x) ~= *Fn,
x:U!Xn
where Fn denotes the restriction of F to the site C=Xn.
Thus, if Z is a globally fibrant simplicial presheaf on C=X, there is a weak
equivalence
*Z = hom (*, Z) '!hom (1X , Z),
where the function space hom (1X , Z) is a homotopy inverse limit of the simpli
cial sets *Zn, computed on the respective sites C=Xn. This is, effectively, an
old observation _ see [6].
6
Example 4. Suppose that J is a small category, and identify J with a constant
presheaf of categories on C. The category C=J has, for objects, all pairs (U, x)
where U is an object of C and x is an object of J, since Ob (J) is a constant
presheaf. The morphisms (ff, f) : (U, x) ! (V, y) are pairs consisting of a
morphism ff : U ! J of C and a morphism f : x ! y of J. In other words,
C=J = C x J, and it's easy to see that this identification gives C x J the prod*
*uct
topology, with J discrete. A sheaf on C x J (and hence on C=J) can therefore
be identified with a Jopdiagram of sheaves on C.
Write Xj for the simplicial presheaf X( , j) on C. A weak equivalence X ! Y
of simplicial presheaves on C x J is a map which induces a weak equivalence
weak equivalences Xj ! Yj of simplicial presheaves on C for all j 2 J. This can
be proven directly by using the observation that F is a sheaf on C x J if and
only if each Fj is a sheaf, or by using Lemma 14 below. It is also plain that
a map A ! B of simplicial presheaves on C x J is a cofibration if and only if
each map Aj ! Bj is a cofibration of simplicial presheaves on C. The model
structure for simplicial presheaves on C x J therefore coincides with one of the
standard model structures (due to BousfieldKan [1]) for Jopdiagrams in the
category s Pre(C) of simplicial presheaves on C.
If X is a globally fibrant simplicial presheaf on C x Jop, then X ! * has
the right lifting property with respect to all trivial cofibrations A ! B of
Jopdiagrams of simplicial sets, interpreted as trivial cofibrations of (C x J)
presheaves which are constant in the C direction. Then the global sections
simplicial set *X = limX can be written as an inverse limit
*X = lim *Xj
j
of the global sections of the Cpresheaves Xj. The indicated lifting property f*
*or
X means that the Jopdiagram j 7! *Xj of simplicial sets has the right lifting
property with respect to all trivial cofibrations of Jop diagrams. It follows t*
*hat
*X is the homotopy inverse limit of the Jopdiagram *Xj.
Observe that the functor Y 7! Yi preserves weak equivalences, and has a
left adjoint defined by A 7! A x hom Jop( , i). This adjoint preserves trivial
cofibrations, so that Y 7! Yi preserves global fibrations. In particular, if X *
*is
a globally fibrant simplicial presheaf on C x J, then all Xj are globally fibra*
*nt
simplicial presheaves on C.
2 The fibred site for a presheaf
Suppose that X is a presheaf on C. The corresponding category C=X has as
objects all pairs (U, x) with x 2 X(U). The morphisms (U, x) ! (V, y) of C=X
consist of morphisms ff : U ! V of C such that ff*(y) = x. Such morphisms
can be identified with commutative diagrams of presheaf morphisms
U __ff_//::V
x ÆÆ: y
X
7
Suppose that ß : Y ! X is a map of presheaves on C, and that x : U ! X
is an object of C=X. Then Y represents a presheaf ß* on C=X by setting ß*(x)
to be the set of sections
>Y>~
oe~~~~ß
~~ fflffl
U __x_//_X
of ß over x.
Conversely, if F is a presheaf on C=X, define
G
F*(U) = F (x).
x:U!X
Then any map ff : U ! V in C defines a function ff* : F*(V ) ! F*(U), which is
the unique function making the diagrams
F
F (y)______//_y:V !XF (y)
 ff
  *
fflffl F fflffl
F (y . ff)__//x:U!X F (x)
commute, where the horizontal functions are canonical. There is a canonical
function ßF : F*(U) ! X(U) which sends the summand F (x) to the section
x : U ! X in X(U), and all such functions form the components of a presheaf
map ßF : F* ! X.
The assignments ß 7! ß* and F 7! ßF are functorial, and define an equiva
lence of categories
Pre(C)=X ' Pre(C=X).
Note that this equivalence specializes to an equivalence of presheaves on C=U
with morphisms of presheaves Z ! U for each object U of C.
A presheaf morphism ff : X ! Y induces a functor ff : C=X ! C=Y by
composition with ff: the object x : U ! X maps to the composite
U x!X ff!Y.
Suppose that F is a presheaf defined on C=Y . Then composition with ff de
termines a presheaf F ff on C=X, and it is easily seen that there is a pullback
diagram
(F ff)*____//F*
ß ß
fflffl fflffl
X __ff__//_Y
Any object x : U ! X determines a functor OEx = OEU,x: C=U ! C=X. As
the notation suggests, if F is a presheaf on C=X, then the induced presheaf
8
Fx = FU,xon C=U is defined by composition with x. It follows that there is a
pullback diagram
(Fx)*_____//F* (7)
ß ß
fflffl fflffl
U ___x___//X
in the category of presheaves on C.
Note that a presheaf map ß : Y ! X represents a sheaf on C=X if and only
if all presheaves (ß*)U,xof sections are sheaves on C=U. Equivalently, the map
ß : Y ! X represents a sheaf on C=X if and only if, given a section x 2 X(U)
and a compatible family of sections
>Y>
oei"""
""" 
" fflffl
Ui__xi_//X
defined over the restrictions xi of x along some covering family Ui ! U, there
is a unique section
Y>>
oe~~~
~~ 
~~ fflffl
U __x__//X
which restricts to all oei.
Lemma 5. The collection of all presheaf maps ß : Y ! X which represent
sheaves on C=X is stable under base change.
The statement of the Lemma (which is easy to prove) means that, given a
pullback square
Z xX Y _____//Y
ß* ß
fflffl fflffl
Z _______//_X
if ß represents a sheaf on C=X, then ß* represents a sheaf on C=Z.
Lemma 6. A map ß : Y ! X represents a sheaf on C=X if and only if in all
pullback diagrams
U xX Y _____//Y
ß* ß
fflffl fflffl
U ___x___//_X
arising from sections x 2 X(U), U 2 C, the map ß* represents a sheaf on C=U.
9
Example 7. Suppose that G is a sheaf on the site C, and let q : C=U ! C
denote the canonical forgetful functor. Then the composite Gq is a sheaf on
C=U. Explicitly, if x : V ! U is an object of C=U, then Gq(x) = G(V ). It
follows that G
(Gq)*(V ) = G(V ) = U(V ) x G(V ),
x:V !U
and the canonical map (Gq)*(V ) ! U(V ) is just the projection onto U(V ).
In other words, (Gq)* ~= G x U, and the canonical map ß is the projection
G x U ! U. This object represents a sheaf on C=U in the sense described
above, as one can check directly, but the product G x U need not be a sheaf on
the site C.
We do, however, have the following:
Lemma 8. Suppose that X is a sheaf and that ß : Y ! X is a presheaf map.
Then ß represents a sheaf on C=X if and only if Y is a sheaf.
Proof. The map ß : Y ! X represents a sheaf on C=X if and only if, given a
section x 2 X(U) and a compatible family of sections
>Y>
oei"""
""" 
" fflffl
Ui__xi_//X
defined over the restrictions xi of x along some covering family Ui ! U, there
is a unique section
Y>>
oe~~~
~~ 
~~ fflffl
U __x__//X
which restricts to all oei.
If Y is a sheaf on C, then there is a unique element oe : U ! Y which restri*
*cts
to all oei. Since X is a sheaf, ß(oe) = x.
Suppose that ß represents a sheaf on C=X, and let oei : Ui ! Y be a com
patible family of elements defined on a covering Ui! U of U. Then ß(oei) = xi
and the xi uniquely determine a section x : U ! X since X is a sheaf on C.
But then a lifting oe : U ! Y of x exists and is uniquely determined since ß
represents a sheaf on C=X. Any such lifting oe extending the oei must map_to x,
since X is a sheaf. __
Say that a map
f
Z :____//_:W
:ÆÆ
X
of simplicial presheaves over X is a local weak equivalence fibred over X if it
represents a local weak equivalence of simplicial presheaves on C=X.
10
Lemma 9. Suppose that X is a presheaf on C. Suppose that
f
Z :____//_:W
:ÆÆ
X
is a commutative diagram of simplicial presheaves. Then f represents a local
weak equivalence of simplicial presheaves on C=X if and only if the map Z ! W
is a local weak equivalence of simplicial presheaves on C.
Proof. Recall that if Z ! X is a map of simplicial presheaves, then the presheaf
that it represents on C=X associates to x : V ! X, the fibre Zx over x for the
simplicial set map Z(V ) ! X(V ). Certainly, Z(V ) = tx2X(V )Xx and it's clear
that Ex1 Z(V ) = tx2X(V )Ex1 Zx. In particular, the map Z ! Ex1 Z fibres
over X, and represents a sectionwise weak equivalence of simplicial presheaves *
*on
C=X. The map Z ! Ex1 Z is also a sectionwise weak equivalence of simplicial
presheaves on C. It suffices, therefore, to assume that Z and W are presheaves
of Kan complexes on C.
In that case, the map f has the standard factorization
j
Z ______//AAT
AAA p
f* AA__Afflffl
W
where p is a sectionwise Kan fibration and j is left inverse to a sectionwise
trivial Kan fibration. Furthermore, this factorization is fibred over X and has
the same properties in each fibre. In particular j represents a sectionwise tri*
*vial
map of simplicial presheaves on C=X. It suffices, therefore, to assume that f
is a sectionwise Kan fibration between presheaves of Kan complexes, and show
that it is locally trivial for the site C if and only if it is locally trivial *
*for the site
C=X.
Suppose given a commutative diagram
@ n _____//Zx
 
 f
fflffl fflffl
n _____//_Wx
where x : V ! X is an object of C=X. Then there is a covering family OEi: Vi!
V for which the displayed liftings exist in the diagram
@ n _____//Zx_____//_Z(V_)___//_Z(Vi)33ggggg
ggg
 oeigggggg f
 gggggg 
fflfflggggggg fflffl
n _____//_Wx____//W (V_)___//W (Vi)
11
But then oei factors through the summand ZOE*i(x), since its image in W (Vi)
factors through the summand WOE*i(x).
Conversely, suppose given a diagram
@ n __ff_//_Z(V )
 f
 
fflffl fflffl
n __fi_//_W (V )
Then fi factors through a summand Wx for some x : V ! X in C since n is
connected, and it follows that ff factors through the summand Zx. Thus if the
lifting problem can be solved locally over C=X it can be solved locally over C.
It follows that if f represents a local trivial fibration on the site C=X,_then*
*_f is
a local trivial fibration on C. __
Corollary 10. Suppose that X is a presheaf on C. The model structure on the
category s Pre(C)=X which arises from the topology on C=X is induced from the
model structure on the category s Pre(C) of simplicial presheaves. In particula*
*r,
a map
f
Z :____//_:W
:ÆÆ
X
is a weak equivalence (respectively cofibration, fibration) if and only if the *
*map
f : Z ! W is a weak equivalence (respectively cofibration, global fibration) of
simplicial presheaves on C.
Proof. The statement about cofibrations amounts to the observation that cofi
brations are defined fibrewise. The statement for weak equivalences is Lemma__
9, and then the fibration statement is a formal consequence. __
Lemma 11. 1) Suppose that ff : X0 ! X is a morphism of presheaves. Then
the functor s Pre(C)=X ! s Pre(C)=X0 defined by pullback preserves weak
equivalences.
2)Suppose that X is a presheaf on C. Then a map
f
Z _____//::W
ÆÆ:
X
of simplicial presheaves over X represents a local weak equivalence on C=X
if and only if all pullbacks
f
U xX;Z _____//U xX W
;;
;;;
ÆÆ;
U
12
over sections x : U ! X, U 2 C represent local weak equivalences on C=U.
Proof. Statement 2) implies statement 1). We shall prove statement 2).
Recall that G
Z(V ) = Zx
x2X(V )
and that G
U xX Z(V ) = ZxOE.
OE:V !U
Once again, the Ex1 construction is performed fibrewise, so it suffices to assu*
*me
that Z and W are presheaves of Kan complexes. The canonical replacement of
a map by a fibration is also a fibrewise construction, so it suffices to assume*
* that
f is a Kan fibration in each section, and hence in each fibre. But then f has t*
*he
local right lifting property with respect to all inclusions @ n n if and only
if all pullbacks of f along sections x : U ! X have the same local right liftin*
*g_
property, by the argument that appears in the proof of Lemma 9. __
Note that statement 1) of Lemma 11 is not true if X0 and X are replaced by
simplicial presheaves. One can see counterexamples easily in ordinary simplicial
sets.
3 Constructions for presheaves of categories
Suppose that A is a presheaf of categories. An enriched diagram X on A consists
of setvalued functors X(U) : A(U) ! Set defined by x 7! A(U)x, one for
each U 2 C, such that each morphism OE : V ! U of C induces functions
OE* : X(U)x ! X(V )OE*(x)and all diagrams
X(U)x ___ff*__//X(U)y
OE* OE*
fflffl fflffl
X(V )OE*(x)(OE*(ff))//_X(V )OE*(y)
*
commute, where ff : x ! y is a morphism of A(U).
Let F be a presheaf on the fibred site C=A. Then F assigns a set F (U)x =
F (U, x) to each object x : U ! Ob (A). Every morphism fl : x ! y in
A(U) determines a morphism (1, fl) : (U, x) ! (U, y), and hence induces a
function (1, fl)* : F (U)y ! F (U)x. In particular, F determines a functor
F (U) : A(U)op ! Set. Any morphism ff : V ! U of C induces a morphism
(ff, 1) : (V, ff*(x)) ! (U, x) in C=A, and hence induces a function ff* : F (U)*
*x !
F (V )ff*(x). If ff : V ! U is a morphism of C and fl : x ! y is a morphism of
13
A(U) then the diagram
(1,ff*(fl)) *
(V, ff*(x))___//(V, ff (y))
(ff,1) (ff,1)
fflffl fflffl
(U, x)__(1,fl)//_(U, y)
commutes in C=A, so that the diagram
(1,fl)*
F (U)y________//F (U)x
(ff,1)* (ff,1)*
fflffl fflffl
F (V )ff*(y)(1,ff*(fl))*//_F (U)ff*(x)
commutes. In other words, F defines an enriched diagram F on the presheaf of
categories Aop.
Suppose that G is an enriched diagram on the presheaf of categories Aop.
Write G(U, x) = G(U)x for each object (U, x) of C=A. Let (ff, fl) : (V, y) ! (U*
*, x)
be a morphism of C=A. Then (ff, fl) has a factorization
(1,fl) *
(V, y)____//(V, ff (x))
KKK 
KKK (ff,1)
(ff,fl)%%KKKfflffl
(U, x)
Associate to (ff, fl) the composite
* fl*
G(U)x ff!G(V )ff*(x)!G(V )y.
If (fi, !) : (W, z) ! (V, y) is another choice of morphism of C=A, there is a
commutative diagram
(1,!) *(1,fi*(fl)) * *
(W, z)____//_(W, fi (y))_//(W, fi ff (x))
LLL
LLL (fi,1) (fi,1)
(fi,!)L%%LLLfflffl fflffl
(1,fl) *
(V, y)_______//_(V, ff (x))
OOO
OOOO (ff,1)
(ff,fl)O''OOOOfflffl
(U, x)
It follows that the assignment (U, x) 7! G(U)x defines a presheaf on the catego*
*ry
C=A.
We have proved the following:
14
Lemma 12. Suppose that A is a presheaf of categories on a small site C. Then
the category Pre(C=A) is equivalent to the category of enriched diagrams on the
presheaf of categories Aop.
Note that a presheaf F on C=A consists of a presheaf of objects F0 ! Ob(A)
on C= Ob(A), with extra structure.
There is a canonical functor _ : Ob (A) ! A, and the assignment F 7! F0
coincides with the restriction functor
_* : Pre(C=A) ! Pre(C= Ob(A))
which is defined by composition with the canonical functor _, under the equiv
alence
Pre(C= Ob(A) ' Pre(C)= Ob(A)
of the last section.
The object (U, x) of the site C=A determines a functor
OEU,x: C=U ! C=A
which sends an object OE : V ! U to the object (W, OE*x). This functor sends
the morphism
V99_ff//_V 0
OEøø9 OE0
U
to the morphism
(W, OE*(x)) (ff,1)!(W 0, (OE0)*(x)).
When F is a presheaf on C=A, write FU,x for the presheaf on C=U which is
defined by composition with OEU,xin the sense that
FU,x= F . OEU,x.
Now here are some observations:
Lemma 13. 1) A presheaf F on C=A is a sheaf if and only if all restricted
presheaves FU,xare sheaves on C=U.
2)A presheaf F on C=A is a sheaf if and only if it restricts to a sheaf F0 !
Ob (A) on C= Ob(A).
3)The restrictions F 7! FU,xand F 7! F0 commute with the associated sheaf
functor on C=U and C= Ob(A) respectively, up to natural isomorphism.
In the same way, a simplicial presheaf X on the fibred site C=A consists of a
simplicial presheaf of objects X0 ! Ob(A) over the presheaf Ob (A) with extra
structure. The restriction functor
_* : s Pre(C=A) ! s Pre(C= Ob(A))
15
can be identified up to equivalence with the object functor
s Pre(C=A) ! s Pre(C)= Ob(A)
which takes a simplicial presheaf X (aka. enriched diagram in simplicial sets)
to the simplicial presheaf of objects X0 ! Ob(A) over Ob (A).
Lemma 14. The object functor _* : s Pre(C=A) ! s Pre(C)= Ob(A) preserves
and reflects local weak equivalences.
Proof. We show that a map f : X ! Y of simplicial presheaves on C=A is
a local weak equivalence if and only if the induced map X0 ! Y0 is a local
weak equivalence of simplicial presheaves. The statement of the result is a
generalization of Lemma 9, and the proof involves the same ideas. This works
because the topology on C=A only involves the topology on C= Ob(A).
The forgetful functor preserves sectionwise weak equivalences. A map f :
X ! Y is a local weak equivalence if and only if the induced map Ex1 X !
Ex 1 Y is a local weak equivalence. The canonical map j : X ! Ex1 X is a
sectionwise weak equivalence. The Ex1 construction and the associated sec
tionwise equivalence are preserved by the forgetful functor. Thus it suffices to
assume that X and Y are presheaves of Kan complexes.
In that case the map f : X ! Y has a standard factorization
X __i__//@@Z
@@ p
f@@@__fflffl
Y
where p is a Kan fibration in each section and i is right inverse to a sectionw*
*ise
trivial Kan fibration. It therefore suffices to assume that f is a Kan fibration
in each section, and show that f is a local trivial fibration if and only if the
induced map f0 : X0 ! Y0 is a local trivial fibration. But this is now clear:_t*
*he
argument is finished as in the proof of Lemma 9. __
The object functor s Pre(C=A) ! s Pre(C)= Ob(A) also preserves and reflects
monomorphisms.
The restriction functor _* has a left adjoint
_* : s Pre(C= Ob(A)) ! s Pre(C=A).
which is defined by left Kan extension along the inclusion _ : Ob(A) ! A, and
we identify this with a left adjoint
_* : s Pre(C)= Ob(A) ! s Pre(C=A).
for the object functor. For a fixed simplicial presheaf X ! Ob(A) over Ob (A),
the map _*X0 ! Ob(A) can be identified with the composite
X xOb(A)Mor (A) ! Mor(A) t!Ob(A)
16
where t is the target map, and both the indicated pullback and the projection
are defined by the pullback diagram
X xOb(A)Mor (A) _____//Mor(A)
 s
 
fflffl fflffl
X ____________//Ob(A)
Here, s is the source map. The map s is a local fibration since Mor (A) and
Ob (A) are simplicial presheaves which are constant in the simplicial direction*
*. It
follows that the indicated pullback is a homotopy cartesian diagram of simplici*
*al
presheaves. It follows that the functor which sends the simplicial presheaf map
X ! Ob(A) to (_*X)0 = X xOb(A)Mor (A) preserves local weak equivalences.
It also preserves cofibrations. This suffices for a proof of the following:
Lemma 15. The object functor _* : s Pre(C=A) ! s Pre(C)= Ob(A) defined by
sending X to the map X0 ! Ob(A) preserves global fibrations.
In particular, a global fibration X ! Y in s Pre(C=A) consists of a global
X0 ! Y0 over Ob (A) which is Aequivariant in an enriched sense.
For a fixed simplicial presheaf (or enriched functor) X on C=A, applying the
homotopy colimit functor in each section gives a simplicial presheaf holim!A*
*opX
and a canonical simplicial presheaf map ß : holim!AX ! BAop. This assign
ment is plainly functorial in X.
Lemma 16. The homotopy colimit functor s Pre(C=A) ! s Pre(C)=BAop pre
serves weak equivalences.
Proof. Note first of all that Ob (Aop) = Ob(A).
There is a presheaf Morn(Aop) which consists of strings of arrows of length n
in the presheaf of categories Aop, and holim!AopX is the diagonal of a bisim*
*plicial
sheaf which is given by the object X xOb(A)Mor nAop in horizontal degree n.
Here, the map s0 : Morn(Aop) ! Ob(A) is defined by picking out the first object
in the string, and is a local fibration. It follows that the pullback diagram of
simplicial presheaf maps
X xOb(A)Mor n(Aop)_____//Morn(Aop)
 s0
 
fflffl fflffl
X _______________//Ob(A)
is homotopy cartesian, so that any local weak equivalence X ! Y over Ob (A)
induces a local weak equivalence
X xOb(A)Mor n(Aop) ! Y xOb(A)Mor n(Aop).
This is true in all horizontal degrees n, and so the map
holim!AopX ! holim!AopY
is a local weak equivalence. ___
17
The model structure that we have been using so far on s Pre(C=A) is the
natural "injective" structure, for which a map f : X ! Y of enriched diagrams
in simplicial presheaves is a weak equivalence (respectively cofibration) if and
only if the induced map
f0
X0 _________//FFY0
y
FF"" __yyy
Ob(A)
is a weak equivalence (respectively cofibration) of s Pre(C)= Ob(A). There is a*
*lso
a projective structure on s Pre(C=A) which has the same weak equivalences, but
for which a map f is a fibration if and only if the induced diagram as above is*
* a
fibration of s Pre(C)= Ob(A). Say that such a map is a projective fibration, and
say that a projective cofibration is a map which has the left lifting property *
*with
respect to all maps p : X ! Y which are simultaneously projective fibrations
and local weak equivalences.
Lemma 17. The category s Pre(C=A) of enriched diagrams on Aop, together
with the local weak equivalences, projective fibrations and projective cofibrat*
*ions
as defined above, satisfies the axioms for a closed model category.
Proof. A map p : X ! Y is a projective fibration (respectively trivial projecti*
*ve
fibration) if and only if it has the right lifting property with respect to all*
* maps
i* : _*A ! _*B where i : A ! B is a trivial cofibration (respectively cofibra
tion) over Ob (A). We have already seen that the functor A 7! _*A preserves
local weak equivalences. The factorization axiom is now an easy consequence
of these observations, along with the standard fact that the "injective" model
structure for the category of simplicial presheaves is cofibrantly generated._T*
*he
lifting axiom CM4 follows by a standard argument. __
Suppose that OE : A ! B is a functor of presheaves of categories. Then
precomposition with OE defines a restriction functor
OE* : s Pre(C=B) ! s Pre(C=A).
In effect, an enriched diagram X on B taking values in simplicial sets consists*
* of
contravariant simplicial setvalued functors X(U) : B(U) ! S, U 2 C which fit
together along morphisms of C, and then OE*X consists of the composite functors
A(U) OE!B(U) X!S.
The following result is a corollary of Lemma 11 and Lemma 14:
Corollary 18. The restriction functor OE* preserves local weak equivalences for
any functor OE : A ! B of presheaves of categories.
Proof. The object functor _* : s Pre(C=A) ! s Pre(C)= Ob(A) detects weak
equivalences, and there is a relation OE*_* = _*OE*. The functor OE* induced by
the objectlevel morphism OE : Ob(A) ! Ob(B) preserves weak equivalences_by
Lemma 11. __
18
Note that there is a pullback diagram of simplicial presheaves
(OE*X)0_______//X0
 
 
fflffl fflffl
Ob(A) __OE_//_Ob(B)
The functor X 7! OE*X preserves projective fibrations almost by definition, and
it follows from Corollary 18 that OE* preserves trivial projective fibrations. *
*The
functor OE* therefore determines a derived functor
ROE* : Ho(s Pre(C=B)) ! Ho(s Pre(C=A))
which is defined by ROE*(X) = OE*F X, where the trivial cofibration j : X ! F X
is a projective fibrant replacement for X.
The left adjoint
OE* : s Pre(C=A) ! s Pre(C=B),
preserves projective cofibrations and weak equivalences between projective cofi
brant objects, and therefore has an associated derived functor
LOE* : Ho(s Pre(C=A)) ! Ho(s Pre(C=B)),
which is left adjoint to the derived functor ROE*. The derived functor LOE* is
defined by LOE*(Y ) = OE*CY , where the trivial projective fibration p : CY ! Y
is a projective cofibrant replacement for Y .
4 Simplicial set constructions
Suppose that G is a groupoid and that A : G ! sSet is a Gdiagram in the
category of simplicial sets _ write sSetG for the category of all such objects.
The diagram A determines a canonical simplicial set map holim!GA ! BG,
where holim!GA is identified with the diagonal of the usual bisimplicial set.
In general, if f : X ! BG is a simplicial set map, then f can be identified
with a setvalued functor oe 7! Xoedefined on the simplex category =BG of
BG, where Xoe= f1 (oe) is the fibre over oe for the function Xn ! BGn if oe is
an nsimplex of BG. Note that a morphism
m SS`*oeS
`  SSS))
 kBG55
fflffloekkkkk
n
induces a function Xoe! X`*(oe)in the obvious way. Suppose that oe is the
string
a0 ! a1 ! . .!.an
19
of morphisms of G. Then a map
g
X 2______//_2holim!GA
22 """
f 222 "ß"
ßß~~""
BG
of simplicial sets over BG can be identified with a natural transformation g :
Xoe! (Aa0)n over the simplex category of BG; the naturality means that all
diagrams
g
Xoe________//(Aa0)n (8)

 `*
 
 fflffl
`* (Aa0)m

 
 `*
fflffl fflffl
X`*(oe)_g_//_(Aa`(0))m
commute, where `* is induced by the map a0 ! a`(0)of G.
Suppose that y is an object of G, let f : X ! BG be a simplicial set map,
and define pb(X)y by the pullback diagram
pb(X)y _____//_X (9)
 
 f
fflffl fflffl
B(G=y) ____//_BG
where B(G=y) ! BG is induced by the forgetful functor G=y ! G. An n
simplex of pb(X)y consists of a triple
(x, oe : a0 ! . .!.an, ff : an ! y),
where x 2 Xn, f(x) = oe and ff is a morphism of G. Since G is a groupoid,
all morphisms in the string oe are invertible, and we can instead identify the
nsimplex of pb(X)y displayed by the triple above, with a triple of the form
(x, oe : a0 ! . .!.an, fl : a0 ! y).
Of course, the assignment y ! pb(X)y defines a functor pb(X) : G ! sSet.
Observe that there is an inclusion
coe: Xoe! (pb(X)a0)n
which is defined by sending x to the triple (x, oe, 1 : a0 ! a0). It is not har*
*d to
show that diagrams of the form (8) commute for the list of functions {coe}, and
20
so these functions define a natural map j : X ! holim!Gpb(X) of simplicial
sets over BG.
There is a simplicial map ffly : pb(holim!GA)y ! Ay which is defined on
nsimplices by sending the triple
(x 2 Aa0, oe, fl : a0 ! y)
to the element fl*(x) 2 (Ay)n. This map is natural in y and in A, and therefore
defines a natural map of Gdiagrams ffl : pb(holim!GA) ! A. It is not diffic*
*ult
to show that the natural maps j and ffl satisfy the triangle identities, so tha*
*t we
have proved
Lemma 19. Suppose that G is a groupoid. Then the functor pb is left adjoint
to the homotopy colimit functor holim!: sSetG ! sSet=BG.
Lemma 20. The canonical map c : holim!Gpb(X) ! X is a weak equivalence,
for all objects f : X ! BG of the category sSet=BG of simplicial sets over BG.
Proof. The map c is induced by a map of bisimplicial sets which is specified in
horizontal degree n by the simplicial set map
G
pb(X)y0! X.
y0!...!yn
which will also be denoted by c. Note that BG ~= lim!y2GB(G=y), so that
X ~= lim!y2Gpb(X)y. If x 2 Xoe= f1 (oe), where oe is the ksimplex z0 !
. .!.zk of BG, then the preimage of x under c can be identified with a copy
of B(zk=G), which is contractible. It follows that the bisimplicial set map c
is a weak equivalence in each vertical degree, and therefore induces a weak_
equivalence of associated diagonals. __
Corollary 21. The map j : X ! holim!Gpb(X) is a weak equivalence.
Proof. The map holim!Gpb(X) ! X of Lemma 20 is a left inverse for j. ___
Corollary 22. The counit map ffl : pb(holim!GA) ! A is a weak equivalence
for all Gdiagrams A.
Proof. The induced map holim!Gpb(holim!GA) ! holim!GA is a weak equiv
alence, by Corollary 21 together with the fact that j and ffl satisfy the trian*
*gle
identities. At the same time, all diagrams
By _____//holim!GB
 
 ß
fflffl fflffl
0 ___y___//_BG
are homotopy cartesian since G is a groupoid, by Quillen's Theorem B. It follow*
*s_
that ffl is a weak equivalence of Gdiagrams. __
21
It is shown in [3, VI.4.2 (p.330)] that the homotopy colimit functor A 7!
holim!GA takes pointwise fibrations to fibrations over BG. We know that both
the homotopy colimit functor and the pullback functor X 7! pb(X) preserve
weak equivalences, and so we have the following:
Lemma 23. The functors
holim!G: sSetG ø sSet=BG : pb
induce an adjoint equivalence of the associated homotopy categories.
Suppose that M is a right proper closed model category. Every morphism
f : X ! Y of M induces a functor
f* : M=X ! M=Y
by composing with f. There is a functor
f* : M=Y ! M=X
which is defined by pullback along f, and f* is left adjoint to f*. The com
position functor plainly preserves cofibrations and weak equivalences, so the
pullback functor preserves fibrations and trivial fibrations. The pullback func
tor therefore preserves weak equivalences of fibrant objects _ note that a fibr*
*ant
object of M=Y is a fibration Z ! Y .
Each object ff : Z ! Y of M=Y has a fibrant model, meaning a factorization
jff
Z _____//AAZff
AA p
ffAA__AAfffflffl
Y
where jffis a trivial cofibration and pffis a fibration. Form the pullback
X xY Zff_____//Zff
pff* pff
fflffl fflffl
X ____f___//_Y
Then the assignment ff 7! pff*preserves weak equivalences by the properness
assumption for M, and defines the derived functor
Rf* : Ho(M=Y ) ! Ho(M=X)
Of course, composition with f preserves weak equivalences and induces a functor
Lf* : Ho(M=X) ! Ho(M=Y )
22
Then one shows by chasing explicit homotopy classes that Lf* is left adjoint
to Rf* . The map j : fi ! Rf*Lf*fi is the map Z ! X xY Zffiwhich is
determined by the diagram
jffi
Z _____//Zffi
fi pffi
fflffl fflffl
X __f__//_Y
The map ffl : Lf*Rf*ff ! ff is represented in the homotopy category, for an
object ff : Z ! Y , by the composite
X xY Zff! ZffjffZ.
Lemma 24. Suppose that M is a right proper closed model category, and sup
pose that f : X ! Y is a weak equivalence of M. Then the functors
Lf* : Ho(M=X) ø Ho(M=Y ) : Rf*
form an adjoint equivalence of categories.
Proof. Since pffiis a fibration and f is a weak equivalence, the map f* : X xY
Zffi! Zffiis a weak equivalence. The map jffi: Z ! Zffiis a weak equivalence
by construction, so that the map j : Z ! X xY Zffiis a weak equivalence.
Since pffis a fibration and f is a weak equivalence, the map f* : X xY Zff!
Zffis a weak equivalence. It follows that ffl is an isomorphism in the_homotopy
category. __
The following sequence of results (Corollary 25  Corollary 27) is perhaps of
interest in its own right. It is also a prototype for a series of results conce*
*rning
presheaves of groupoids which appears in the next section.
Corollary 25. Suppose that the morphism of groupoids f : G ! H induces
a weak equivalence f : BG ! BH. Then the composition with f functor f*
and the pullback functor f* together induce an adjoint equivalence of homotopy
categories
Lf* : Ho(sSet=BG) ø Ho(sSet=BH) : Rf*.
Corollary 26. Suppose that the map f : G ! H of groupoids induces a weak
equivalence f : BG ! BH. Then the functor
Rf* : Ho(sSetH ) ! Ho(sSetG )
which is defined by composition with f is an equivalence of categories.
Proof. There is a commutative diagram of functors
sSetH ____//_sSet=BH
f* f*
fflffl fflffl
sSetG _____//sSet=BG
23
where the horizontal functors are defined by homotopy colimit, and hence induce
equivalences of homotopy categories according to Lemma 23. The functor
f* : sSet=BH ! sSet=BG
is defined by pullback along the map f : BG ! BH, and hence induces an __
equivalence of homotopy categories by Corollary 25 __
The restriction functor f* : sSetH ! sSetG has a left adjoint f* defined by
left Kan extension. The functor f* preserves pointwise weak equivalences and
pointwise fibrations, so that the functor f* preserves cofibrations and trivial
cofibrations, and thus preserves pointwise weak equivalences between cofibrant
objects. It follows that if CX denotes a cofibrant replacement for a diagram X
on the groupoid G, then the assignment X ! f*CX induces a functor
Lf* : Ho(sSetG ) ! Ho(sSetH )
which is left adjoint to the functor
Rf* : Ho(sSetH ) ! Ho(sSetG )
The functor Rf* is part of an equivalence on the homotopy category level, with
inverse G, say. But every equivalence of categories is an adjoint equivalence
[17, p.93], so that Lf* is naturally isomorphic to G as a functor Ho(sSetG ) !
Ho (sSetH ). We have therefore proved the following:
Corollary 27. Suppose that f : G ! H is a morphism of groupoids such
that f : BG ! BH is a weak equivalence of simplicial sets. Then the left
Kan extension f* of the restriction functor f* : sSetH ! sSetG has a derived
functor
Lf* : Ho(sSetG ) ! Ho(sSetH )
which is an inverse up to natural isomorphism for the derived restriction funct*
*or
Rf* : Ho(sSetH ) ! Ho(sSetG ).
Here's a result that is well known [18], but stated and proved in a complete*
*ly
functorial manner. We will need the functoriality for a corresponding result on
presheaves of categories which will be used in the next section of this paper.
Lemma 28. There are canonical natural weak equivalences BCop ' dX(C) '
BC for a suitably defined natural simplicial set dX(C).
Proof. The simplicial set BCop has nsimplices given by strings of arrows
b0 b1 . . .bn
with simplicial structure maps defined in the obvious way. Consider the bisim
plicial set X(C) with (m, n)bisimplices given by all strings
bm ! . .!.b0 ! a0 ! . .!.an.
24
Assigning the msimplex
bm ! . .!.b0
to this bisimplex defines a function OE : X(C)m,n ! BCopm, and this list of
functions defines a bisimplicial set map OE : X(C) ! BCop. Assigning the
nsimplex
a0 ! . .!.an.
to the same bisimplex defines a function _ : X(C)m,n ! BCn, and the list of
functions defines a bisimplicial set map OE : X(C) ! BC.
The fibre of _ : X(C)*,n! BCn over a fixed nsimplex a0 ! . .!.an can
be identified with the simplicial set B(a0=Cop), which is contractible. It foll*
*ows
that _ induces a weak equivalence of associated diagonal simplicial sets.
The fibre of OE : X(C)m,* ! BCopmover a fixed msimplex bm ! . .!.b0
can be identified with the simplicial set B(b0=C), which is again contractible.*
* It
follows that OE induces a weak equivalence of associated diagonal simplicial se*
*ts.
We have therefore constructed natural weak equivalences
BCop OEdX(C) _!BC,
as required. Here, d denotes the diagonal functor. ___
5 Presheaves of groupoids
Suppose that G is a presheaf of groupoids, and let C=G be the corresponding site
fibred over G. Recall from Lemma 12 that a presheaf on C=G can be identified
with an enriched diagram on the presheaf of groupoids Gop. It follows that a
simplicial presheaf on C=G can be identified with an enriched diagram X on Gop
taking values in simplicial sets.
This means that X consists of functors X(U) : G(U)op! sSet, x 7! X(U)x,
one for each object U 2 C, such that each morphism OE : V ! U of C induces
simplicial set maps OE* : X(U)x ! X(V )OE*x. In addition we require the diagram
of simplicial sets
X(U)x ___ff*__//X(U)y
OE* OE*
fflffl fflffl
X(V )OE*(x)(OE*(ff))//_X(V )OE*(y)
*
to commute for each morphism ff : x ! y of G(U)op.
Bundling the simplicial sets X(U)x together over Ob (Gop(U)) for all U
defines the object map X0 ! Ob (Gop) of simplicial presheaves. Recall that
X0 ! Ob (Gop) = Ob (G) represents the simplicial presheaf _*X, where _ :
Ob (G) ! G is the canonical functor. Lemma 14 implies that a map f : X ! Y
25
is a local weak equivalence of enriched Gopdiagrams if and only if the corre
sponding map
f
X0 ___________//=Y0
==
===
=OEOE
Ob (Gop)
is a local weak equivalence of simplicial presheaves on the fibred site C= Ob(G*
*op).
A similar observation holds for monomorphisms: a map g : A ! B is a
monomorphism of enriched diagrams if and only if the objectlevel map A0 ! B0
is a monomorphism of simplicial presheaves. Note that Lemma 15 implies that
a global fibration p : X ! Y of enriched diagrams is an object level global
fibration X0 ! Y0 which is Gopequivariant.
Given an enriched diagram X, taking homotopy colimits in each section
defines an enriched homotopy colimit holim!GopX and a canonical map of sim
plicial presheaves
ß : holim!GopX ! BGop
Conversely, one can start with a map f : Y ! BGop and produce an enriched
Gopdiagram pb(Y ): one applies the construction which associates the Gop(U)
diagram pb(Y (U)) to the simplicial set map Y (U) ! BGop(U) in each section.
By working section by section, one sees that there are natural maps j : Y !
holim!Gop~Yand ffl : pb(holim!GopX) ! X, and that these two maps satisfy *
*the
triangle identities. We know from Corollary 21 that the map j is a sectionwise
weak equivalence. Corollary 22 says that all maps
fflx : pb(holim!Gop(U)X)x ! Xx
are weak equivalences of simplicial sets, for all x 2 Ob (Gop(U)) and all U 2
C. It follows that ffl is a natural weak equivalence of simplicial presheaves *
*on
C= Ob(Gop).
Lemma 16 says that the homotopy colimit functor
s Pre(C=G) ! s Pre(C)=BGop
preserves local weak equivalences.
Lemma 29. The functor s Pre(C)=BGop! s Pre(C=G) defined by X 7! pb(X)
preserves local weak equivalences.
Proof. Suppose that Y ! B is a simplicial set map, where is a groupoid.
26
Quillen's Theorem B implies that the square portion of the diagram
F
x2Ob( )pb (Y )x_____//Y
 
 
F fflffl fflffl
x2Ob( )B( =x) _____//B


fflffl
Ob ( )
is homotopy cartesian. Applying this construction in each section to a simplici*
*al
presheaf map X ! BGop gives a diagram of simplicial presheaf maps
pb(X)0 ________//X
 
 
fflffl fflffl
pb(BGop)0 _____//BGop


fflffl
Ob (Gop)
in which the square is homotopy cartesian. Thus if f : X ! Y is a local weak
equivalence of simplicial presheaves over BGop, the induced map pb(X)0 !
pb (Y )0 is a local weak equivalence of simplicial presheaves over Ob (Gop)._The
desired statement is then a consequence of Lemma 14. __
We have assembled a proof of the following:
Theorem 30. Suppose that G is a presheaf of groupoids on a site C. Then the
homotopy colimit and pullback functors determine an adjoint equivalence
holim!: Ho(s Pre(C=G)) ' Ho(s Pre(C)=BGop) : pb
We now have a list of corollaries which is analogous to the sequence Corolla*
*ry
25 _ Corollary 27.
Corollary 31. Suppose that the map f : G ! H of presheaves of groupoids
induces a local weak equivalence f : BG ! BH. Then the derived functor
Rf* : Ho(s Pre(C=H)) ! Ho(s Pre(C=G))
defined by composition with f is an equivalence of categories.
Proof. There is a commutative diagram of functors
s Pre(C=H)_____//s Pre(C)=BHop
f* f*
fflffl fflffl
s Pre(C=G)_____//s Pre(C)=BGop
27
where the horizontal functors are defined by homotopy colimit, and hence induce
equivalences of homotopy categories according to Theorem 30. The functor
f* : s Pre(C)=BHop ! s Pre(C)=BGop
is defined by pullback along the map f : BGop! BHop. This map f is a local
weak equivalence by Lemma 28, and pullback along f : BGop! BHop induces __
an equivalence of homotopy categories by Lemma 24. __
Recall (see the remarks following Lemma 17) that the derived functor
Rf* : Ho(s Pre(C=H) ! Ho(s PreC=G)
has a left adjoint
Lf* : Ho(s Pre(C=G) ! Ho(s PreC=H)
which is the homotopy left Kan extension with respect to the projective model
structure. Corollary 31 implies that the derived functors Rf* and Lf* therefore
determine an adjoint equivalence of homotopy categories, and so we have proved
the following:
Corollary 32. Suppose that f : G ! H is a morphism of presheaves of
groupoids such that f : BG ! BH is a local weak equivalence of simpli
cial presheaves. Then the left Kan extension f* of the restriction functor
f* : s Pre(C=H) ! s Pre(C=G) has a derived functor
Lf* : Ho(s Pre(C=G) ! Ho(s PreC=H)
which is an inverse up to natural isomorphism for the derived restriction funct*
*or
Rf* : Ho(s Pre(C=H) ! Ho(s PreC=G)
Corollary 32 says that the Quillen adjunction determined by the functor
f : G ! H is a Quillen equivalence if f : BG ! BH is a weak equivalence. The
following is essentially a reformulation of that statement.
Corollary 33. Suppose that f : G ! H is a morphism of presheaves of
groupoids which induces a local weak equivalence BG ! BH. Then the fol
lowing statements hold:
1)Suppose that X is a projective cofibrant enriched Gdiagram and that ff :
f*X ! F f*X is a weak equivalence of enriched Hdiagrams with F f*X
projective fibrant. Then the composite
X j!f*f*X f*ff!f*F f*X
is a weak equivalence of enriched Gdiagrams.
28
2)Suppose that Y is a projective fibrant enriched Hdiagram and that fi :
Cf*Y ! f*Y is a weak equivalence of enriched Gdiagrams with Cf*Y
projective cofibrant. Then the composite
*fi ffl
f*Cf*Y f! f*f*Y ! Y
is a weak equivalence of enriched Hdiagrams.
The following result (Lemma 35) requires an independent proof, because the
terminal object * of s Pre(C=G) is not projective cofibrant in general.
Example 34. Suppose that K is a group. Then K acts freely on the space EK
and the map EK ! * is a Kequivariant trivial fibration, while a Kequivariant
map * ! EK would pick out a fixed point. There are no such fixed points, and
it follows that the trivial projective fibration EK ! * does not have a section.
Lemma 35. Suppose that f : G ! H is a morphism of presheaves of groupoids
such that the induced map BG ! BH is a weak equivalence of simplicial
presheaves. Then the canonical map f*(*) ! * is a local weak equivalence
of simplicial presheaves on C=H.
Proof. For a fixed object U 2 C, the (simplicial) set f*(*) is defined for y 2
Hop(U) by the assignment
f*(*)(y) = lim!*
f(x)!y
where the colimit is computed over the index category f=x, and where f :
G(U)op ! H(U)op is the corresponding groupoid morphism. In other words,
there is a natural isomorphism
f*(y) ~=ß0B(f=y).
Each diagram
F op
y2Ob(H)op(U)B(f=y) _______//_BG(U)
 
 f
F fflffl fflffl
y2Ob(H)op(U)B(Hop(U)=y) _____//BH(U)op
is homotopy cartesian by Quillen's Theorem B, and so the diagram of simplicial
presheaf maps
pb(BGop)0 _____//BGop
 
 f
fflffl fflffl
pb(BHop)0 _____//BHop
29
is homotopy cartesian. The simplicial presheaf map f is a weak equivalence by
assumption, and so it follows that there are local weak equivalences
pb(BGop)0 '!pb(BHop)0 '!Ob(Hop).
In particular the presheaf map
ß0pb (BGop)0 ! Ob(Hop)
induces an isomorphism of associated sheaves. But we also know that there is
an isomorphism
ß0pb (BGop)0 ~=f*(*)0,
and the resulting map
f*(*)0 ! Ob(Hop)
is induced by the canonical morphism f*(*) ! *, and it follows that this_canon_
ical morphism is a weak equivalence. __
Write s*Pre(C=G) for the category of pointed simplicial presheaves on the
site C=G). Pointed simplicial presheaves X on C=G restrict objects X0 ! Ob(G)
with a fixed choice of section s : Ob (G) ! X0 and one can work with this
internally, but it's much easier to work directly with the restriction functor
_* : s*Pre(C=G) ! s*Pre(C= Ob(G)).
A similar remark can be made about presheaves of spectra.
The functor f* restricts to a functor
f* : s*Pre(C=H) ! s*Pre(C=G)
relating pointed simplicial presheaves for the two sites. The functor f* has a
left adjoint
f~*: s*Pre(C=G) ! s*Pre(C=H)
which is defined for a pointed simplicial presheaf X by
~f*(X) = f*(X)=f*(*)
Fibrant models are formed in pointed simplicial presheaves just as in simpli
cial presheaves, and we know from Lemma 35 that the map f*(*) ! * is a weak
equivalence if f : G ! H induces a weak equivalence BG ! BH. Suppose that
BG ! BH is a weak equivalence, and suppose that X is a projective cofibrant
pointed simplicial presheaf on C=G. Then, in the diagram
j *f*(ff) *
X _____//EEf*f_X___//f*F f X
EEE  
jEEE""fflffl fflffl
f*f~*X_f*(ff)//_f*F ~f*X
30
the map f*F f*X ! f*F ~f*X is a weak equivalence, so Corollary 33 implies
that the bottom composite
X j!f*f~*X f*(ff)!f*F ~f*X
is a weak equivalence if ff : ~f*X ! F ~f*X is a projective fibrant model for ~*
*f*X.
Suppose that Y is a projective fibrant pointed simplicial presheaf on C=H.
Form the diagram
f*Ff~*fi f*Fffl
f*F ~f*Cf*Y_____//f*FO~f*f*Y___//f*FOYOOOO
f*ff f*ff f*ff
 f*~f*fi  f*ffl 
f*f~*Cf*Y______//_f*f~*f*Y____//_f*YOOOO99s
ss
j j ssss
  sss 1
Cf*Y ____fi____//_f*Y
by making suitable choices of fibrant models ff and cofibrant models fi. Then
the composite
Cf*Y j!f*f~*Cf*Y f*ff!f*F ~f*Cf*Y
is a weak equivalence from what we have just seen, since Cf*Y is projective
cofibrant. The map f*ff : f*Y ! f*F Y is a weak equivalence by Corollary 18,
and of course the map fi : Cf*Y ! f*Y is a weak equivalence. It follows that
the top composite in the diagram is a weak equivalence. The composite map
f~*Cf*Y ~f*fi!~f*f*Y ffl!Y
is therefore a weak equivalence on account of Corollary 32, since f* must then
reflect weak equivalences between projective fibrant objects.
We have therefore proved the following:
Lemma 36. Suppose that f : G ! H is a morphism of presheaves of groupoids
which induces a local weak equivalence BG ! BH. Then the following state
ments hold:
1)Suppose that X is a projective cofibrant pointed simplicial presheaf on C=G
and that ff : f*X ! F f*X is a weak equivalence of pointed simplicial
presheaves on C=H with F f*X projective fibrant. Then the composite
X j!f*f~*X f*ff!f*F ~f*X
is a weak equivalence of pointed simplicial presheaves on C=G.
31
2)Suppose that Y is a projective fibrant pointed simplicial presheaf on C=H
and that fi : Cf~*Y ! f~*Y is a weak equivalence of pointed simplicial
presheaves on C=G with Cf~*Y projective cofibrant. Then the composite
~f*Cf*Y ~f*fi!~f*f*Y ffl!Y
is a weak equivalence of pointed simplicial presheaves on C=H.
Corollary 37. Suppose that f : G ! H is a morphism of presheaves of
groupoids such that f : BG ! BH is a local weak equivalence of simpli
cial presheaves. Then the left Kan extension f~*of the restriction functor
f* : s*Pre(C=H) ! s*Pre(C=G) has a derived functor
Lf~*: Ho(s*Pre(C=G)) ! Ho(s*Pre C=H))
which is an inverse up to natural isomorphism for the derived restriction funct*
*or
Rf* : Ho(s*Pre(C=H)) ! Ho(s*Pre C=G))
There is one final thing to know about pointed simplicial presheaves on C=G,
which will be of some use later on:
Lemma 38. Suppose that K is a pointed simplicial set. Then the following
hold:
1)If p : X ! Y is a projective fibration (respectively trivial projective fi*
*bra
tion) then the induced map of pointed function complexes
p* : hom *(K, X) ! hom *(K, Y )
is a projective fibration (respectively trivial projective fibration).
2)The functor X 7! X ^ K preserves projective cofibrations and trivial pro
jective cofibrations.
Proof. Statement 1) follows from the fact that restriction along the functor
_ : Ob(G) ! G preserves the displayed pointed function complex constructions._
Statement 2) is equivalent to Statement 1), by an adjointness argument. __
Suppose again that G is a presheaf of groupoids on the site C, and write
Spt (C=G) for the category of presheaves of spectra on the fibred site C=G.
Let _ : Ob (G) ! G denote the canonical functor, and recall that a map
f : X ! Y of pointed simplicial presheaves is a local weak equivalence (re
spectively cofibration) if and only if its restriction f* : _*X ! _*Y is a local
weak equivalence (respectively cofibration) on the site C= Ob(G). By definition,
f is a projective fibration if and only if f* is a global fibration on C= Ob(G).
We also know, from Lemma 15, that the restriction functor _* preserves global
fibrations.
Recall [9] that a map g : Z ! W of presheaves of spectra is a stable equiva
lence if the induced map QJX ! QJY is a levelwise weak equivalence, where
32
X ! JX is a natural choice of strictly fibrant model and QY for a level fibrant
object Y is the result of the usual stabilization construction. In particular, *
*QY n
is the colimit of the diagram
Y n! Y n+1! 2Y n+2! . . .
Restriction along the canonical functor _ : Ob (G) ! G preserves level fibrant
models and the stabilization construction (the latter by Lemma 35). The re
striction functor also reflects level weak equivalences. It follows that a map
g : Z ! W of presheaves of spectra on the site C=G is a stable equivalence if
and only if its restriction g* : _*Z ! _*W is a stable equivalence of presheaves
of spectra on the site C= Ob(G). It is also relatively easy to see that g is a
cofibration of presheaves of spectra on C=A if and only if g* is a cofibration *
*of
presheaves of spectra on C= Ob(G).
It can be shown that a map p : X ! Y of presheaves of spectra is a stable
fibration if and only if the following conditions hold:
1)All level maps p : Xn ! Y nare fibrations of pointed simplicial presheaves.
2)Given any commutative diagram
j
X _____//_Z
p 
fflffl fflffl
Y ___j_//W
where the maps labelled j are stable equivalences and Z and W are stably
fibrant, then all diagrams
j
Xn _____//_Zn
p 
fflffl fflffl
Y n__j__//W n
are homotopy cartesian diagrams of pointed simplicial presheaves.
In particular, if X and Y are already stably fibrant, then a stable fibration
p : X ! Y is a map such that all level maps p : Xn ! Y n are fibrations of
pointed simplicial presheaves.
Say that a map p : X ! Y of presheaves of spectra on C=G is a projective
fibration if the restriction p* : _*X ! _*Y is a stable fibration on C= Ob(G).
One can see by using the criteria 1) and 2) above that the functor _* preserves
stable fibrations, so that every stable fibration is a projective fibration. A
projective cofibration of presheaves of spectra on C=G is a map which has the
left lifting property with respect to all maps which are simultaneously stable
equivalences and projective fibrations.
33
The restriction functor _* preserves stable fibrations and trivial stable fi*
*bra
tions, so that its adjoint _* preserves cofibrations and stably trivial cofibra*
*tions.
The stable model structure of presheaves of spectra is cofibrantly generated, so
we are therefore entitled to the following analogue of Lemma 17:.
Lemma 39. The category Spt(C=A) of presheaves of spectra on the site C=G,
together with the stable weak equivalences, projective fibrations and projective
cofibrations as defined above, satisfies the axioms for a closed model category.
A presheaf of spectra X is stably fibrant if all objects Xn are fibrant and *
*all
adjoint bonding maps Xn ! Xn+1 weak equivalences. Furthermore, a map
f : X ! Y between stably fibrant presheaves of spectra is a stable equivalence
if and only if all level maps Xn ! Y n are weak equivalences of simplicial
presheaves. It follows that a presheaf of spectra Z on C=G is projective fibrant
if and only if all objects Zn are projective fibrant pointed simplicial preshea*
*ves
and all morphisms Zn ! Zn+1 are local weak equivalences. It also follows
that a map g : Z ! W of projective fibrant presheaves of spectra is a stable
weak equivalence if and only if the restriction g : _*Z ! _*W is a stable weak
equivalence of presheaves of spectra on C= Ob(G). It is a further consequence
that the restriction functor
f* : Spt(C=H) ! Spt(C=G)
preserves projective fibrant presheaves of spectra, and preserves stable weak
equivalences between projective fibrant presheaves of spectra.
This characterization gives rise to an obvious recognition principle for pro
jective fibrations of presheaves of spectra on C=G, and implies that a map
q : Z ! W between projective fibrant presheaves of spectra is a projective
fibration if and only if all level maps p : Zn ! W nare projective fibrations.
It follows in particular that for any morphism f : G ! H of presheaves of
groupoids the restriction functor
f* : Spt(C=H) ! Spt(C=G)
preserves projective fibrations and stable equivalences between projective fibr*
*ant
objects. It therefore also follows that the left adjoint
f~*: Spt(C=G) ! Spt(C=H)
preserves projective cofibrations and stable equivalences between projective co*
*fi
brant objects. We are therefore entitled to derived functors
Lf~*: Ho(Spt (C=G)) ø Ho(Spt (C=H)) : Rf*
relating the associated stable categories. Furthermore, Lf~*is left adjoint to
Rf*, with the usual description of unit and counit.
Lemma 40. Suppose that f : G ! H is a morphism of presheaves of groupoids
which induces a local weak equivalence BG ! BH. Then the following state
ments hold:
34
1)Suppose that X is a projective cofibrant presheaf of spectra on C=G and
that ff : f*X ! F f*X is a stable equivalence of presheaves of spectra on
C=H with F f*X projective fibrant. Then the composite
X j!f*f~*X f*ff!f*F ~f*X
is a stable weak equivalence.
2)Suppose that Y is a projective fibrant presheaf of spectra on C=H that
fi : Cf~*Y ! ~f*Y is a stable equivalence of presheaves of spectra on C=G
with Cf~*Y projective cofibrant. Then the composite
~f*Cf*Y ~f*fi!~f*f*Y ffl!Y
is a stable equivalence.
Proof. Suppose that X is a projective cofibrant presheaf of spectra. Then there
is a level equivalence ß : X~ ! X where the pointed simplicial presheaf X~0
is projective cofibrant and all bonding maps S1 ^ ~Xn! X~n+1 are projective
cofibrations. In particular all pointed simplicial presheaves X~n are projecti*
*ve
cofibrant. The construction of ß is the standard cofibrant replacement trick,
which takes advantage of the fact that a map p : X ! Y is a projective fibration
and a stable equivalence if and only if all level maps p : Xn ! Y n are trivial
projective fibrations of pointed simplicial presheaves. In the diagram
X~ _j__//_f*f~*~Xf*ff//_f*F ~f*~X
ß  f*Ff~*ß
fflffl fflffl
X _j__//_f*f~*Xf*ff//_f*F ~f*X
the map f*F ~f*ß is a stable equivalence, since ~f*preserves stable equivalences
between projective cofibrant objects. The top horizontal composite is a level
weak equivalence by Lemma 36, and so the bottom horizontal composite is also
a stable equivalence.
Assertion 2) has a similar proof: the cofibrant model Cf*Y can be chosen
so that it consists of projective cofibrant pointed simplicial presheaves_in all
levels. __
We have also proved the following
Theorem 41. Suppose that f : G ! H is a morphism of presheaves of
groupoids such that f : BG ! BH is a local weak equivalence of simpli
cial presheaves. Then the left Kan extension f~*of the restriction functor
f* : Spt(C=H) ! Spt(C=G) has a derived functor
Lf~*: Ho(Spt (C=G)) ! Ho(Spt (C=H))
which is an inverse up to natural isomorphism for the derived restriction funct*
*or
Rf* : Ho(Spt Pre(C=H)) ! Ho(Spt (C=G)).
35
Theorem 41 implies the corresponding result for presheaves of symmetric
spectra rather easily, subject to having appropriate projective model structures
in place.
For a fixed presheaf of groupoids G the restriction functor
_* : Spt (C=G) ! Spt (C= Ob(G))
between the respective categories of presheaves of symmetric spectra preserves
stable fibrations and trivial stable fibrations (see [10]). It follows that its*
* left
adjoint _* preserves cofibrations and trivial cofibrations. Say that a map p :
X ! Y of symmetric spectra on C=G is a projective fibration if the induced
map p* : _*X ! _*Y is a stable fibration of presheaves of symmetric spectra
on C= Ob(G). A projective cofibration is a map of Spt (C=G) which has the
left lifting property with respect to all maps which are both stable equivalenc*
*es
and projective fibrations. Note that every stable fibration of Spt (C=G) is a
projective fibration, so that every projective cofibration is a cofibration.
The category of presheaves of symmetric spectra is cofibrantly generated,
and one can prove the following:
Lemma 42. The category Spt (C=A) of presheaves of spectra on the site C=G.
together with the stable weak equivalences, projective fibrations and projective
cofibrations as defined above, satisfies the axioms for a closed model category.
Suppose that f : G ! H is a morphism of presheaves of groupoids. Then
the restriction functor
f* : Spt (C=H) ! Spt (C=G)
preserves projective fibrations and trivial projective fibrations. It follows t*
*hat
the left adjoint functor
~f*: Spt (C=G) ! Spt (C=H)
preserves projective cofibrations and trivial projective cofibrations. As in a*
*ll
other cases, one shows that the corresponding adjunction
Lf~*: Ho(Spt (C=G)) ø Ho(Spt (C=H)) : Rf*
is an adjoint equivalence.
Recall that the forgetful functor U : Spt (D) ! Spt(D) has a left adjoint
V , and that these functors form a Quillen equivalence, for any small site D. In
particular, V preserves cofibrations and trivial cofibrations while U preserves
stable fibrations and trivial stable fibrations, and the corresponding derived
functors
LV : Ho(Spt (D)) ø Ho(Spt (D)) : RU
form an adjoint equivalence of categories. In particular, there are (composite)
stable equivalences
V CUY Vfi!V UY ffl!Y
36
for all stably fibrant Y and
X j!UV X Uff!UF V X
for all cofibrant X, where fi and ff are cofibrant and fibrant models, respecti*
*vely.
Lemma 43. Suppose that the map f : G ! H of presheaves of groupoids
induces a local weak equivalence BG ! BH. Then the derived functor
Rf* : Ho(Spt (C=H)) ! Ho(Spt (C=G))
is an equivalence of categories.
Proof. The diagram of functors
Spt (C=H) _U__//_Spt(C=H)
f* f*
fflffl fflffl
Spt (C=G)__U__//Spt(C=G)
induces a commutative diagram of right derived functors
Ho(Spt (C=H)) RU'_//_Ho(Spt (C=H))
Rf* ' Rf*
fflffl ' fflffl
Ho (Spt (C=G))RU__//Ho(Spt (C=G))
by standard results about symmetric spectra and Theorem 41. ___
The left adjoint Lf~*of Rf* must coincide with the inverse of Rf* up to
natural isomorphism, and so we have the following:
Corollary 44. Suppose that f : G ! H is a morphism of presheaves of
groupoids such that f : BG ! BH is a local weak equivalence of simpli
cial presheaves. Then the left Kan extension f~*of the restriction functor
f* : Spt (C=H) ! Spt (C=G) has a derived functor
Lf~*: Ho(Spt (C=G)) ! Ho(Spt (C=H))
which is an inverse up to natural isomorphism for the derived restriction funct*
*or
Rf* : Ho(Spt Pre(C=H)) ! Ho(Spt (C=G)).
Corollary 45. Suppose that f : G ! H is a morphism of presheaves of
groupoids which induces a local weak equivalence BG ! BH. Then the fol
lowing statements hold:
37
1)Suppose that X is a projective cofibrant presheaf of symmetric spectra on
C=G and that ff : f*X ! F f*X is a stable equivalence of presheaves
of symmetric spectra on C=H with F f*X projective fibrant. Then the
composite
X j!f*f~*X f*ff!f*F ~f*X
is a stable weak equivalence.
2)Suppose that Y is a projective fibrant presheaf of symmetric spectra on C=H
that fi : Cf~*Y ! ~f*Y is a stable equivalence of presheaves of symmetric
spectra on C=G with Cf~*Y projective cofibrant. Then the composite
~f*Cf*Y ~f*fi!~f*f*Y ffl!Y
is a stable equivalence.
References
[1]A.K. Bousfield and D.M. Kan, Homotopy limits completions and localizations,*
* Springer
Lecture Notes in Math. 304 (2nd corrected printing), SpringerVerlag, Berlin*
*Heidelberg
New York (1987).
[2]J. Giraud, Cohomologie nonab'elienne, SpringerVerlag, BerlinHeidelbergN*
*ew York
(1971).
[3]P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Math.*
* 174,
Birkhäuser, BaselBostonBerlin (1999).
[4]S. Hollander, A homotopy theory for stacks, Preprint (2001).
[5]M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 1*
*3 (2000),
149208.
[6]J.F. Jardine, Simplicial objects in a Grothendieck topos, Ä pplications of *
*Algebraic K
theory to Algebraic Geometry and Number Theory I", Contemp. Math. 55,I (1986*
*),
193239.
[7]J.F. Jardine, Simplicial presheaves, J. Pure App. Algebra 47 (1987), 3587.
[8]J.F. Jardine, Universal HasseWitt classes, Ä lgebraic Ktheory and Algebra*
*ic Number
Theory", Contemp. Math. 83 (1989) 83100.
[9]J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Mathematic*
*s Vol. 146,
Birkhäuser, BaselBostonBerlin (1997).
[10]J.F. Jardine, Presheaves of symmetric spectra, J. Pure App. Algebra 150 (20*
*00), 137154.
[11]J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445552.
[12]J.F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homotop*
*y, Homology
and Applications 3(2) (2001), 361384.
[13]J.F. Jardine and Z. Luo, Higher order principal bundles, Preprint (2004).
[14]A. Joyal and M. Tierney, Strong stacks and classifying spaces, Category The*
*ory (Como,
1990), Springer Lecture Notes in Math. 1488 (1991), 213236.
[15]A. Joyal and M. Tierney, On the homotopy theory of sheaves of simplicial gr*
*oupoids,
Math. Proc. Camb. Phil. Soc. 120 (1996), 263290.
[16]G. Laumon and L. MoretBailly, Champs alg'ebriques, Ergebnisse der Mathemat*
*ik und
ihrer Grenzgebiete 3~9, Springer, BelinHeidelberg (2000).
[17]S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Ma*
*themat
ics, Vol. 5, 2nd ed., SpringerVerlag, New York, Heidelberg, Berlin (1998).
[18]D. Quillen, Higher algebraic Ktheory, I, Springer Lecture Notes in Math. 3*
*41 (1973),
85147.
38