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*
implicial 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 K-theory 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 space-level 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 Eilenberg-Mac 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 U-sections 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 HZ-trivial 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 E1-E7 almo*
*st
immediately below, and then sE1-sE7 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 Morel-Voevod*
*sky
A1-localization theory is an instance of the f-localization 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 E1-E7.