Localization theories for simplicial presheaves
P.G Goerss1 and J.F. Jardine2
This work was motivated in part by the following question of Soule: given a s*
*im
plicial presheaf X on a site C, how does one produce a map of simplicial preshe*
*aves
X ! LHZ X in such a way that each of the maps in sections X(U) ! LHZ X(U),
U 2 C, is an integral homology localization map in the sense of Bousfield? Seco*
*ndly,
if Y is a simplicial presheaf which is integrally homology local in a suitable *
*sense,
is it the case that the map X ! LHZ X induces an isomorphism
[LHZ X; Y ] ~=[X; Y ]
relating sets of morphisms in the homotopy category of simplicial presheaves on*
* C?
These questions are related to the definition of the Ktheory of simplicial she*
*aves
that appears in [8].
The first of these questions is easily answered by observing that associated *
*fi
brations in the closed model category describing Bousfield's homology localizat*
*ions
are created with small object constructions and are therefore natural; in parti*
*cular
there is a functorial method of picking out a fibrant model Y ! LHZ Y for arbit*
*rary
simplicial sets Y , which restricts in particular to a natural simplicial presh*
*eaf map
X(U) ! LHZ X(U), U 2 C.
The second question involves homotopy coherence, and is therefore much more
subtle: the analogous spacelevel problem can be solved by Bousfield's original*
* tech
niques, but this does not imply the functorial global solution that Soule requi*
*res.
The problem is solved by using methods introduced in this paper, and in partic
ular by applying Theorem 3.9 below. In the case corresponding to the identity
functor on the site C, the chaotic topology on C and the constant presheaf of s*
*pec
tra associated to the EilenbergMac Lane spectrum HZ, Theorem 3.9 implies that
there is a closed simplicial model structure in the sense of Quillen on the cat*
*egory
SP re(C) of simplicial presheaves on C such that the cofibrations are the point*
*wise
monomorphisms and the weak equivalences are the pointwise integral homology
isomorphisms. The map i : X ! LHZ X is then just a choice of fibrant model (ie.
trivial cofibration, taking values in fibrant object) for this closed model str*
*ucture,
and the induced maps in sections i : X(U) ! LHZ X(U) are fibrant models for
the corresponding theory on simplicial sets (ie. integral homology localization*
*s in
___________________________1
Partially supported by NSF.
2Partially supported by NSERC.
This research was also supported by a NATO Collaborative Research Grant.
1
Bousfield's original sense), because the Usections functor has a left adjoint *
*which
preserves cofibrations and takes integral homology isomorphisms to pointwise in
tegral homology isomorphisms. Furthermore, if we say that a simplicial presheaf
Y is integral homology local if it's fibrant with respect to this new closed m*
*odel
structure on SP re(C), then Y is globally fibrant in the traditional sense, an*
*d the
closed simplicial model structure gives an isomorphism
ss(LHZ X; Y ) ~=ss(X; Y )
in naive homotopy classes of maps which is induced by the HZtrivial cofibratio*
*n i
(here ss(X; Y ) = ss0hom (X; Y ), for example).
This application is based on a very special case of the results that appear h*
*ere,
which hold in striking generality. The overall point is that a very wide class*
* of
results, which includes objects as apparently diverse as Theorem 3.9, the closed
model structure for simplicial presheaves [15] (see also Remark 2.9), a general*
* f
localization theory for simplicial presheaves (Theorem 4.6), the closed model s*
*truc
tures of various stable categories (Theorem 3.7) and a homology localization te*
*ch
nique for presheaves of spectra (Theorem 3.10), all arise from a simple collect*
*ion of
axioms for classes of cofibrations and weak equivalences (see axioms E1E7 almo*
*st
immediately below, and then sE1sE7 for spectra in Section 3). The proofs, in
all cases, involve relatively simple cardinality counts which are modelled simu*
*lta
neously on Bousfield's original work on homology localization and the derivation
of the closed model structures for simplicial presheaves. Some new theories ha*
*ve
been discovered along the way, including a notion of localization along a geome*
*tric
topos morphism (Theorem 2.7, Theorem 3.10) and a resulting method of localizing
a space or a spectrum at a generalized homology theory arising from a presheaf *
*of
spectra on an arbitrary site.
There is a further application for these techniques, in that the MorelVoevod*
*sky
A1localization theory is an instance of the flocalization results of Section *
*4 (see
Remark 4.10, but note that we do not discuss properness). The collection of kno*
*wn
applications is, however, still quite small. It's rather difficult, in particul*
*ar, to know
what localization along an arbitrary geometric morphism of toposes should mean
in the context of traditional homotopy theory. The fibrant objects in all of t*
*hese
theories continue to be really quite mysterious.
The second author would like to thank Vladimir Voevodsky for a series of con
versations which helped to determine the final form of the axiom list E1E7.
2
1. Fundamental results.
Suppose that C is a small Grothendieck site, and that ff is an infinite cardi*
*nal.
A simplicial presheaf X on C is said to be ffbounded if ff is an upper bound o*
*n the
cardinality of all sets of sections of X in the sense that Xn(U) ff for all *
*n 0
and all objects U of C.
Suppose that E is a class of morphisms of SP re(C), and say that a monomorphi*
*sm
of simplicial presheaves is a cofibration. Say that a simplicial presheaf cofib*
*ration
which is also a member of E is an Etrivial cofibration.
In many examples, we shall see that the class E and the class of cofibrations
together satisfy the following axioms:
E1: The class of morphisms E is closed under retracts.
E2: Given a composable pair of morphisms
f g
X! Y! Z;
if any two of f, g and gf are in the class E, then so is the third.
E3: Every pointwise weak equivalence is in E.
E4: The class of Etrivial cofibrations is closed under pushout.
E5: Suppose that fl is a limit ordinal, and there is a functor X : fl ! SP r*
*e(C)
such that for each morphism i j of fl, the induced map X(i) ! X(j) is
an Etrivial cofibration. Then the canonical maps
oi
X(i) ! lim!X(j)
j2fl
are Etrivial cofibrations.
E6: Suppose that the morphisms fi : Xi ! Yi are Etrivial cofibrations for i*
* 2 I.
Then the morphism
G G G
fi : Xi ! Yi
i2I i2I i2I
is an Etrivial cofibration.
E7: There is an infinite cardinal ff which is an upper bound for the cardina*
*lity
of the set of morphisms of C, such that for every simplicial presheaf di*
*agram
Xy

i
u
A y________wY
3
with i an Etrivial cofibration and A ffbounded, there is a subobject B*
* Y
such that A B, the object B is ffbounded, and the inclusion B \ X ,! B
is an Etrivial cofibration.
We shall refer to condition E7 as the bounded cofibration condition. It is th*
*e only
axiom of the list that is not a standard part of a closed model structure, and *
*is
almost always the most difficult to verify.
Say that a morphism p : X ! Y is an Efibration if it has the right lifting
property with respect to all Etrivial cofibrations. An Eweak equivalence is a
member of the class E.
If K is a simplicial set and X is a simplicial sheaf, then the tensor object *
*X x K
is defined in sections for U 2 C by
(X x K)(U) = X(U) x K:
For the simplicial presheaves X and Y , the function complex hom (X; Y ) is the
simplicial set whose set of nsimplices is defined by
hom (X; Y )n = hom (X x n; Y )
where the morphism set on the right is in the category of simplicial presheaves
on C. The ordinary exponential law for simplicial sets bootstraps immediately to
a simplicial category structure on the simplicial presheaf category SP re(C). *
*The
exponential object XK associated to a simplicial presheaf X and a simplicial s*
*et
K is the simplicial presheaf which is defined in sections by the function spaces
XK (U) = hom (K; X(U))
for U 2 C.
Theorem 1.1.
(1) Under the conditions E1  E7 listed above, there is a closed model struc*
*ture
on SP re(C) such that the cofibrations are the monomorphisms, the weak
equivalences are the Eweak equivalences, and fibrations (ie. Efibrati*
*ons)
are defined by a right lifting property.
(2) Suppose further that, given an inclusion i : K ! L of finite simplicial *
*sets
and a cofibration j : X ,! Y , then the induced monomorphism
X x L [XxK Y x K ,! Y x L
is an Etrivial cofibration if either i is a weak equivalence of simplic*
*ial sets
or j is an Eweak equivalence of simplicial presheaves. Then SP re(C) h*
*as
the structure of a closed simplicial model category.
4
The proof of this result is a distillation of ideas which are common to Bous
field's work on homology localizations [2], and the homotopy theory of simplici*
*al
presheaves [12].
Proof: We only need to prove the first statement. Suppose that ff is an infinite
cardinal which is an upper bound for the cardinality of the set of morphisms of*
* a
site C. Say that a cofibration A ,! B of SP re(C) is ffbounded if the object B*
* is
ffbounded. We begin by showing that a morphism of SP re(C) is an Efibration if
and only it has the right lifting property with respect to all ffbounded cofib*
*rations
which are Eweak equivalences.
Suppose that K is a simplicial set and U is an objectFof C, then the simplici*
*al
presheaf LU K is defined for V 2 C by LU K(V ) = OE:V !UK. Observe that
morphisms of simplicial presheaves LU K ! X are in one to one correspondence
with simplicial set maps K ! X(U). If the simplicial set K is ffbounded in the
sense that Kn ff for n 0, then the simplicial presheaf LU K is ffbounded.
Suppose given a diagram
Ay________wX
 iij p
i i 
ui u
B ________wY
where i is a cofibration and an Eweak equivalence, and p has the right lifting
property with respect to all ffbounded Etrivial cofibrations. We shall show t*
*hat
the indicated dotted arrow exists, making the diagram commute. Assume that the
map i is not an isomorphism, for otherwise the problem is solved trivially.
The object B is a filtered colimit of its ffbounded subobjects, since all ge*
*nerating
simplicial presheaves LU n are ffbounded. The map i is not an isomorphism, so
there is an ffbounded subobject D of B such D is not a subobject of A. But then
the bounded cofibration condition E7 says that there is an ffbounded E such th*
*at
D E B, and such that the inclusion E \ A ,! E is an Eweak equivalence.
Form the diagram
E \ A __________wA __________wX
  hhj 
 i* h 
u u h 
E __________wE [ A p

 
 
u u
B __________wY
5
where the indicated partial lift exists since p is assumed to have the right li*
*fting
property with respect to all ffbounded cofibrations which are Eweak equivalen*
*ces.
Observe further that the map i* is an Eweak equivalence since the class of Et*
*rivial
cofibrations is closed under pushout, by E4. It follows that the category of al*
*l such
partial lifts is nonempty. This category has maximal elements, by a Zorn's lem*
*ma
argument and E5, and any such maximal element must be a solution to the lifting
problem.
Every simplicial presheaf map f : X ! Y has a factorization
Xh___________wfY
hhj ')
i ' 'p
Z
where p is an Efibration and i is a Etrivial cofibration. In effect, take a c*
*ardinal
fi > 2ff. We define a functor F : fi ! SP re(C) # Y by first setting F (0) = f *
*: X !
Y . We let
X(i) = lim!X(fl)
fl 2ff, and form the factorization
Xh________________wfY
hhj ')
ifi ''F (fi)
X(fi);
in the category of simplicial sheaves on C, by analogy with the transfinite sma*
*ll
object argument appearing in the proof of Theorem 1.1. In particular, X(fi) is *
*the
colimit in the simplicial sheaf category of a functor X : fi ! SShv(C) having
X(i) = lim!X(fl)
fl 2ff, and EsX is defined by transfinite
induction. E0X is a functorial choice of fibrant model for X _ it is most conve*
*nient
here to write E0X = Ex1 X. If s is a limit ordinal Es = lim!t > 2ff> ff:
The fact that Es+1(X) is bounded ultimately relies on the fact that
Dix hom (Ci; EsX) ff . ff= ff . (2 )ff= ff . 2.ff = :
Note that if Y is bounded, then Ex 1 Y is bounded, since is infinite.
Note that by L2 we may assume we have L(Y ) \ L(Z) L(Y \ Z) L(X).
One shows that Es(Y ) \ Es(Z) = Es(Y \ Z). The limit ordinal case follows from
the successor ordinal case because filtered colimits commute with pullbacks. T*
*he
successor ordinal case follows from fact that, degreewise, Es+1Xn has the form
G
Es+1Xn = ( (Di Ci) x hom (Ci; EsX)n) t EsXn;
C
and the image of the inclusion EsYn ,! EsXn associated to any subcomplex Y X
has the form G
( (Di Ci) x hom (Ci; EsY )n) t EsYn:
C
The Ex 1 functor preserves pullbacks, giving L6.
The statement L7 is equivalent to asserting that for all X; K 2 S , there is a
natural map L(X) x K ! L(X x K) so that the following triangle commutes
X x KA_____wjXLx1(X) x K
jXxK AAC 
u
L(X x K)
40
subject to the requirements that L(X) x 0 ~=L(X x 0), and the following two
maps agree:
L(X) x (K x L) ! L(X x (K x L)) ~=L((X x K) x L))
and
L(X) x (K x L) ! (L(X) x K) x L) ! L(X x K) x L ! L((X x K) x L)):
Again, one shows the result holds for all Es. The limit ordinal case follows *
*from
the successor ordinal case, which in turn follows by induction and the fact tha*
*t, for
sets, the pushout of
ixK jxK
B x K  A x K ! C x K
is isomorphic to (B [A C) x K. Note as well that the Ex 1 functor is continuous*
* in
the same sense.
Suppose now that f : A ! B is a cofibration of simplicial presheaves on a sma*
*ll
Grothendieck site C, and say that a simplicial presheaf Z on C is flocal if Z *
*is
globally fibrant and if the induced fibration
f* : hom (B; Z) ! hom (A; Z)
is a weak equivalence.
We construct a functor L = Lf : SP re(C) ! SP re(C) by starting with an infin*
*ite
cardinal ff which is an upper bound for the cardinality of the set of morphisms*
* of
C and that of all sets of sections of B. The method of constructing the functor*
* L is
analogous to the simplicial set construction, starting with the set C of cofibr*
*ations
having elements
A x LU n [AxY B x Y ,! B x LU n;
indexed over the set of all cofibrations Y LU n.
Observe, however, that the construction of L for simplicial sets depends on t*
*he
existence of a continuous functorial fibrant model construction. For simplicial
presheaves, we require a continuous functorial globally fibrant model jX : X ! *
*GX
in order to carry out the analogous argument. The Ex 1 functor does not produce
globally fibrant models for simplicial presheaves, so we have to do something m*
*ore
interesting:
Lemma 4.1. Suppose that the infinite cardinal ff is an upper bound for the car
dinality of the set of morphisms of a small Grothendieck site C. Then there is*
* a
functorial natural map jX : X ! GX such that the map jX is a trivial cofibrati*
*on,
GX is globally fibrant, and the following properties hold:
G1: G preserves weak equivalences.
41
G2: G preserves cofibrations.
G3: Let fi be any cardinal with fi ff. Let {Xj} be the filtered system of
subobjects of X which are fibounded. Then the map
lim!GXj ! GX
j
is an isomorphism.
G4: Let fl be an ordinal number of cardinality strictly greater than 2ff. L*
*et
X : fl ! SP re(C) be a diagram of cofibrations so that for all limit ord*
*inals
s < fl the induced map
lim!X(t) ! X(s)
t
2ff. We begin by writing Ci ! Di for the set of trivial cofibrations Y LU n,
where U varies through the set of objects of C. Note that there are at most 2ff
trivial cofibrations Y LU n, and that each such Y is ffbounded. Then GX =
lim!s 2ff, and all EsX are globally fibrant by construction. A similar
argument shows that LX is flocal. In the presence of Lemma 4.1, arguments for
the simplicial presheaf analogues of statements L1  L7 go through just as befo*
*re.
In particular, we have proved
Theorem 4.2. Let f : A ! B be a cofibration in SP re(C), and suppose that ff
is an infinite cardinal which is an upper bound for the cardinalities of both B*
* and
the set of morphisms of C. Then there is a functor L = Lf : SP re(C) ! SP re(C)
and a natural map jX : X ! L(X) so that L(X) is flocal and jX is a cofibration
which induces weak equivalences
j*X: hom (L(X); Z) ! hom (X; Z)
for all flocal simplicial presheaves Z.
43
As above, this yields a localization on the homotopy category.
We now use Theorem 1.1 to produce the flocal category structure on SP re(C).
The following result allows us to identify the class E of flocal equivalences.
Lemma 4.3. Let g : X ! Y be a morphism of simplicial presheaves in SP re(C).
The following two statements are equivalent.
1) Lg : LX ! LY is a weak equivalence
2) g* : hom (Y; Z) ! hom (X; Z) is a weak equivalence for all flocal obj*
*ects
Z in SP re(C).
3) [Y; Z] ! [X; Z] is an isomorphism for all flocal Z in SP re(C).
Proof: Note that if C ! D is any fcofibration and Z is flocal, then
hom (D; Z) ! hom (C; Z)
is a weak equivalence. Now examine the following diagram.
*
hom (LY; Z) ______wjhom(Y; Z)
Lg*  g*
u * u
hom (LX; Z) _____wjhom(X; Z):
Since j is an flocal equivalence, the horizontal maps are weak equivalences. T*
*hus
if Lg : LX ! LY is a weak equivalence, so is g*. Thus (1) implies (2). We have
that (2) implies (3) because any flocal object is globally fibrant. For (3) im*
*plies
(1) note that for all X and all flocal Z,
[X; Z] ' [LX; Z]
since L induces the localization functor Lf on the homotopy category. Thus (3)
says [LY; Z] ! [LX; Z] is an isomorphism for all flocal Z. Since LX and LY
are flocal, this implies that Lg : LX ! LY is an isomorphism in the homotopy
category; Lg is therefore a weak equivalence (see Lemma II.4.1 of [9]).
We now define E , the class of flocal equivalences, by the three equivalent *
*con
ditions of Lemma 4.3. We next want to show E satisfies the seven axioms required
by Theorem 1.1. Axioms E1E3 are obvious, while Lemmas 4.4 and 4.5 handle
axioms E4E6 and E7, respectively.
Lemma 4.4. The class of flocal trivial cofibrations is closed under cobase cha*
*nge,
colimits over ordinal numbers, and coproducts.
44
Proof: Use (2) of Lemma 4.3, and the fact that trivial fibrations in simplicial*
* sets
are closed under base change, limits over ordinal numbers, and products.
Lemma 4.5. Let = 2 , where is the defining cardinal for the functor L (and >
2ff). Then the class of flocal trivial cofibrations satisfies the bounded cofi*
*bration
condition for the cardinal .
Proof: Let X ! Y be an flocal equivalence and a cofibration, and let A Y be
a bounded subobject. We inductively define a chain of bounded subobjects
A = A0 A1 A2 . . .Y over , and a chain of subobjects
L(A) = L(A0) X1 L(A1) X2 L(A2) . .L.(Y );
also over , with the property that
L(X) \ Xs ! Xs
is a weak equivalence. Then we set B = lim!s k.
The collection of all projections
prX : X x A1 ! X
in (Schk)Nis forms a set, and represents a collection of presheaf maps and hen*
*ce
maps of (constant) simplicial presheaves. Each of the maps prX can be replaced
47
by a cofibration fX of simplicial presheaves up to weak equivalence, since the*
*re is
a factorization
RX
fX hhj ss
h  X
h u
X x A1 ________wprXX;
of prX with ssX a global fibration and a local weak equivalence and fX a cofibr*
*ation.
Let f be the disjoint union of the maps fX :
G G G
f = fX : X x A1 ! RX:
X X X
The MorelVoevodsky A1local theory [17] is the closed model structure on the
category of simplicial presheaves on (Schk)Nis which arises from Theorem 4.6 by
localizing at the cofibration f = tfX .
Note that we could equally well have started with a collection of sections X !
X x A1 of the corresponding projections. These have the advantage of being cofi
brations already.
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Mathematics Department, University of Washington, Seattle, WA 98195, USA.
Email: pgoerss@math.washington.edu
Mathematics Department, University of Western Ontario, London, Ont. N6A 5B7,
Canada. Email: jardine@uwo.ca
49