LOCALIZATION WITH RESPECT TO A CLASS OF MAPS II -
EQUIVARIANT CELLULARIZATION AND ITS APPLICATION
BORIS CHORNY
Abstract.We present an example of a homotopical localization functor whi*
*ch
is not a localization with respect to any set of maps. Our example arise*
*s from
equivariant homotopy theory. The technique of equivariant cellularizatio*
*n is
developed and applied to the proof of the main result.
Introduction
Coaugmented idempotent functors, or localizations, occur all over mathematics
under different names (e.g., idempotent monads [11], orthogonal reflections [2,*
* 1.36]
or coreflections on the full subcategory [15, IV.3]). These are functors L: C !*
* C
equipped with a natural transformation j :Id! L, such that the natural maps
LjX, jLX :LX ' LLX are equal isomorphisms. For example, any multiplicative
system S in a commutative ring R gives rise to the localization functor L: R-mo*
*d !
R-mod and a natural coaugmentation map jM :M ! L(M) = M R R[S-1],
which justify the name localization. Another example is given by the abelianiza*
*tion
functor in the category of groups.
The purpose of such construction L is to öf rget" certain information about t*
*he
mathematical structure one studies. In the case when the information we want
to discard may be described in terms of a set of generators, the existence of t*
*he
localization is usually implied by the standard results from the category theory
[2, 1.37]. But if the description of the undesired information is available on*
*ly in
terms of a proper class of generating maps, no general results are available in*
* the
standard set-theoretical framework. However, if the underlying category is loca*
*lly
presentable [2, Def. 1.17], then it is known that the general question of exist*
*ence of
orthogonal reflection is equivalent to the weak Vop~enka's principle [2, 6.22, *
*6.23].
Moreover, if one is ready to assume a more powerful Vop~enka's principle, then *
*by
[2, 6.24] any class of maps (which is closed under limits in the category of ma*
*ps)
may be generated by just a set of maps (i.e., it is a small-orthogonality class*
* [2,
Def. 1.32]).
Vop~enka's principle and weak Vop~enka's principle are set theoretical statem*
*ents
which are known to be unprovable in the standard set theory, but not known to be
independent of the rest of the axioms. See Chapter 6 of J. Ad'amek and J. Rosic*
*k'y's
book [2] for more information.
In this paper we give a counterexample to the similar question about homotopi*
*cal
localization. We do not assume here any of the non-standard axioms.
____________
Date: December 9, 2003.
1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G*
*55.
Key words and phrases. model category, localization, equivariant homotopy.
1
2 BORIS CHORNY
Homotopical localizations played an important role in algebraic topology and
algebraic geometry over past thirty years. Most of the previously known homo-
topy idempotent functors (except few rare cases like [12], [14]) are known to be
equivalent to the localization with respect to a set of maps, though sometimes *
*it
is non-trivial to find an appropriate set (see [9, 1.E] for examples and discus*
*sion).
This lead E. Dror Farjoun to ask in [8]: whether any homotopy idempotent functor
on the category of spaces is an S-localization for some set of maps S? Like in *
*the
non-homotopical version, it turned out that this question cannot be settled usi*
*ng
only the standard axioms of set theory, but the answer is affirmative if one as*
*sumes,
additionally, Vop~enka's principle [3]. Recently these results were extended to*
* any
simplicial combinatorial(= cofibrantly generated & locally presentable) model c*
*at-
egory [6].
In this work we show that it is impossible to remove the assumption on the mo*
*del
category to be cofibrantly generated. In more detail, we consider the category *
*of
diagrams of spaces with the equivariant model structure [7], which is known to *
*be
non-cofibrantly generated [5], but still reasonable enough to admit the standard
localization theory with respect to a set of maps and even with respect to a cl*
*ass
of maps satisfying some restrictive conditions [4]. The main result of the pres*
*ent
paper is that the functor which associates the constant diagram of points to any
diagram Xeis not a localization functor with respect to any set of maps of diag*
*rams.
This is really a localization with respect to a class of maps.
We would like to stress that our counterexample is not based on some anomaly
of the underlying category (in fact, it can be chosen to be locally presentable*
*). Our
argument uses the observation that the considered model category is not cofibra*
*ntly
generated, but it does not mean that our example may be generalized to any non-
cofibrantly generated model category: there exists an example of a model catego*
*ry
which is Quillen equivalent to the trivial model category, but it is not cofibr*
*antly
generated [1], hence any localization functor in this model category is a local*
*ization
with respect to an empty set of maps - a fibrant replacement.
Together with the works [3, 6] our example provides an answer to Dror Farjoun*
*'s
question.
Organization of the paper. We continue here the discussion started in [4] and
use freely results and notions introduced there. Let us recall only the definit*
*ion of
the equivariant model structure on the category of D-shaped diagrams of spaces
[7]: a diagram Teis called an orbit if colimDTe= *; a map f :Xe! Yeis a weak
equivalence or fibration if the induced map hom(Te, f): hom (Te, Xe) ! hom(Te, *
*Ye) is
a weak equivalence or fibration of simplicial sets respectively. Here and furth*
*er in
the paper hom ( . , . ) denotes the simplicial function complex.
In the fist section we develop the theory of equivariant colocalizations or c*
*el-
lularizations, which is complementary to the theory of equivariant localizations
introduced in [4]. We do not treat here the question of the existence of the fi*
*xed-
pointwise cellularization, since we do not have applications for this notion. *
*The
category of spaces may be taken to be the category of simplicial sets or the co*
*m-
pactly generated topological spaces with the standard simplicial model structur*
*e.
In the second section we apply the results of the first section in order to p*
*rove
that the functor L which entirely discards the homotopical information of diagr*
*ams
of spaces: L(Xe) = *eis not a localization with respect to any set of arrows be*
*tween
diagrams. Our proof works only for the case of diagrams of topological spaces, *
*so
EQUIVARIANT CELLULARIZATION 3
that every diagram is fibrant in the equivariant model category. In order to ob*
*tain
an example of a category which is still locally presentable we may use the cate*
*gory
of I-generated topological spaces introduced recently by J. Smith. A useful acc*
*ount
of J. Smith's ideas on the I-generated spaces is given by D. Dugger [10].
Acknowledgements. I would like to thank Emmanuel Dror Farjoun for his support
and many helpful ideas. I am grateful to Jeff Smith for catching a mistake in an
early version of this work.
1.Construction of the equivariant colocalization functor
In this section we explain the complementary approach to the localization.
Namely, we construct for any set of cofibrant diagrams A = {Ae} the augmented
functor CWA :SD ! SD such that for each Xe2 SD , CWA (Xe) is A-colocal and
the natural map pXf:CWA (Xe) ! Xeis an A-colocal equivalence (see below). We
prove also that the natural map pXf:CWA (Xe) ! Xeis terminal (up to homotopy)
among all the maps of the A-colocal spaces into Xe, thus CWA Xeis characterized
up to a weak equivalence.
The crucial difference between the equivariant framework and the ordinary one
is that it is not pointless to consider the cellularization of non-pointed diag*
*rams
(cf. [9, p. 40], [13, 3.1.10]). Already in the case of a group G acting on topo*
*logical
spaces there are non-trivial augmented homotopy idempotent functors on SG , e.g*
*.,
for any subgroup H of G there exists the augmented functor which assigns to any
G-space Xeits subspace fixed by H with the natural inclusion pXf:(Xe)H ! Xe. The
homotopy idempotence is clear.
1.1. Preliminaries on colocal diagrams and colocal equivalences.
Definition 1.1. Let A be a set of cofibrant diagrams.
o A map f :Xe! Yeis an A-colocal equivalence (or just an A-equivalence) if
for any fibrant replacement ^fof f the induced map
hom(Ae, ^f): hom (Ae, ^Xe) ! hom(Ae, ^Ye)
is a weak equivalence of simplicial sets for every Ae2 A.
o A cofibrant diagram Beis A-colocal if for any A-colocal equivalence g :X*
*e!
Yeand any fibrant replacement ^gof g the induced map
hom(Be, ^g): hom (Be, ^Xe) ! hom(Be, ^Ye)
is a weak equivalence of simplicial sets.
Remark 1.2. The above notions are well-defined, i.e., they do not depend on the
choice of the fibrant replacement. It follows from [13, 9.7.2]. We shall use al*
*so an
A-colocal version of the Whitehead theorem (see [13, 3.2.13] for the proof).
Proposition 1.3 (Ae-colocal Whitehead theorem). A map g :Q1 ! Q2 is a weak
equivalence of A-colocal diagrams if and only if g is an A-colocal equivalence.
Proposition 1.4. A map g :Xe! Yeof diagrams is an A-colocal equivalence if and
only if there exists a fibrant replacement ^gof g which has the right lifting p*
*roperty
with respect to the following families of maps:
o generating trivial cofibrations J;
o Hor(A) = {@ n Ae,! n Ae| n 0, Ae2 A}.
4 BORIS CHORNY
Proof.If g is an A-colocal equivalence, then there exists a fibrant replacement
^g:^Xe! ^Yesuch that hom (Ae, ^g) is a weak equivalence for every Ae2 A. Consid*
*er
the factorization of ^g= ^g0i into the trivial cofibration i followed by the fi*
*bration ^g0,
which is also an A-equivalence by Remark 1.2. Hence, ^g0is a fibrant approximat*
*ion
of g, which has the right lifting property with respect to the elements of J, a*
*s a
fibration of diagrams, and with respect to the elements of Hor(A) by adjunction.
Conversely, if ^gis a fibrant replacement of g with the right lifting property
with respect to the elements of J and Hor(A), then, by adjunction, hom (Ae, ^g)*
* is
a trivial fibration of simplicial sets for every Ae2 A, therefore g is an A-col*
*ocal
equivalence.
Proposition 1.5. The class of maps K = J [ Hor(A) may be equipped with an
instrumentation.
The proof of this proposition is similar to the proof of Proposition [4, 6.6]*
* and
is left to the reader.
1.2. Construction of CWA . The naive approach which is dual to the construction
of the localization does not work: if for any diagram Xewe factor the map ; ! Xe
into a K-cellular map followed by a K-injective map, then for non-fibrant Xe we
will not be able to show that the K-injective map is an A-equivalence. See [13,
5.2.7] for a counterexample in the category of pointed simplicial sets.
The right properness of the category of diagrams is essential for the followi*
*ng
construction (compare [13, 5.3.5]). For any diagram Xechoose a functorial cofib*
*rant
fibrant approximation j :Xe,~!^Xe. Apply the generalized small object argument,
with respect to the instrumented (by Proposition 1.5) class K, to factorize the*
* map
; ! ^Xeinto a K-cellular map r followed by a K-injective map s:
; -r!W^ -s! X.
f e
Next, take W = X x ^W^; then the natural map t: W ! ^W is a weak equivalence
as a pullbafck oef Xffa weak equivalence j along ftheffibration s in the right *
*proper
model category of diagrams. (s is a fibration, since s 2 K-inj.) The natural map
v :W ! X is in K-inj as a pullback of the K-injective map s. The functorial
fibfrant ecofibrant approximation ; ,! CWA X ~iW supplies us with an augmented
e u f
functor CWA Xe, where the augmentation pXfis given by the composition
CWA Xeu-!W -v!X, pX = vu.
f e f
Summarizing, we have the commutative diagram
~
;__a_______OE__________________________________________________*
*______________________________________________________________@
______________________r_______________________________________*
*__________________________________________________
______________________________________________________________*
*__________________________________
_____________________________________________________________*
*__________________________________
_______________________________$$___________________________*
*________________
g ___--______________________________________________________*
*____CWA~X////_W~//_^W
EE u f t f *
* e
EE | |
pXfEEEE|v |s
E""""fflfflfflffl|fflfflfflffl|Ø~"
Xe__j__//^X___////_*.
e
EQUIVARIANT CELLULARIZATION 5
The map pXf:CWA Xe! Xeis an A-colocal equivalence by Proposition 1.4, since
its fibrant approximation s: ^W ! X^ is K-injective, i.e., it has the right lif*
*ting
property with respect to the fsets eJ and Hor(A).
Remark 1.6. We note, for future reference, that pXf2 K-inj. pXf= vu is a com-
position of two fibrations, hence a fibration, i.e., it has the right lifting p*
*roperty
with respect to any element of J. For any element Ce,! De of Hor(A) and any
commutative square
Ce_____//"C`WA Xe
| |p
| | Xf
fflffl| fflfflfflffl|
De_______//Xe
we construct first a lift ^h:De! W^, which exists since s 2 K-inj. Let h: D !
W be the natural map into the pufllback W . Finally, the required lift l e:D!
fCWA X exists, since the map u is a trivifal fibration and any element of Hoer(*
*A) is
a cofeibration.
It remains to show that CWA Xeis A-colocal for any diagram Xe. But CWA Xeis
cofibrant and weakly equivalent to the K-cellular complex ^W, hence it will suf*
*fice to
show that ^W is A-colocal. But any K-cellular diagram is Af-colocal. The follow*
*ing
propositiofn completes the proof.
Proposition 1.7. Any K-cellular complex Beis an A-colocal diagram.
Proof.We will prove this by the transfinite induction on the indexing ordinal of
the ~-sequence ; = Be0,! Be1,! . .,.! Bei,! . .,.whose colimit is Be.
The diagram ; is obviously A-colocal for any set of diagrams A, hence the base
of the induction.
For each k the map ik: Bek,! Bek+1is a pushout of a coproduct of the elements
of K. Each map in K has the homotopy left lifting property with respect to
the class of fibrations which are A-equivalences. For the elements of Hor(A) th*
*is
follows from [13, 9.4.8(1)] (substitute i by ; ! Ae, Ae2 A and (K, L) = (@ n, *
*n)).
Maps of J are trivial cofibrations, therefore they have the homotopy left lifti*
*ng
property with respect to all fibrations [13, 9.4.4]. Next, a coproduct of a se*
*t of
maps with homotopy left lifting property with respect to any class of maps again
has the homotopy left lifting property with respect to the same class: first no*
*te that
coproducts commute with pushouts and with the left adjoint functors . K for a*
*ny
simplicial set K, then apply [13, 9.4.7(2)]. But the homotopy left lifting prop*
*erty
is preserved under pushouts, hence the map ik: Bek,! Bek+1has the homotopy left
lifting property with respect to any fibration which is an A-equivalence.
First we prove the inductive step for successor ordinals. Suppose Bekis Ae-co*
*local.
For any Ae-equivalence g :Xe! Yetake ^g:^Xe! ^Yeto be its fibrant approximation.
6 BORIS CHORNY
In the commutative diagram
hom (Bek+1, ^Xe)ho~m(B___,^g)//hom(B k+1, ^Y)
| KKK ek+1 s99s| e e
| KK%eK 9ysss |
| m KKK sssn |
| K%%Ksss |
hom(ik,X^f)|| ssPe |hom(ik,^Ye)|
| sssss |
fflfflfflffl||yyyyssss fflfflfflffl||
hom (Bek, ^Xe)hom~(B_,^g)//_hom(B k, ^Y)
ek e e
the map hom(Bek, ^g) is a weak equivalence of simplicial sets by inductive assu*
*mption.
The map hom (ik, ^Ye) is a fibration, since ik is a cofibration and ^Yeis a fib*
*rant
diagram. Let
Pe= hom(Bek, ^Xe) xhom(B ,^Y)hom(B k+1, ^Y);
ek e e e
therefore the map n is a weak equivalence as a pullback of the weak equivalence
hom(Bek, ^g) along the fibration hom (ik, ^Ye). The map m is a weak equivalenc*
*e,
since ik has the homotopy left lifting property with respect to the map ^g, whi*
*ch is a
fibration and an A-equivalence. Finally, hom(Bek+1, ^g) = nOm is a weak equival*
*ence
as a composition of weak equivalences m and n. This proves the inductive step f*
*or
successor ordinals.
If ~ ~ is a limit ordinal, then Be~ = colimk<~Bek. For any A-equivalence
g :Xe! Ye, let ^g:^Xe! ^Yebe its fibrant approximation. If Bekis A-colocal for *
*each
k < ~, then the induced map of simplicial sets hom (Bek, ^g) is a weak equivale*
*nce.
This implies that hom (Be~, ^g) = hom (colimk<~Bek, ^g) = limk<~hom (Bek, ^g) i*
*s a
weak equivalence of simplicial sets, since limk<~hom(Bek, ^Xe) and limk<~hom(Be*
*k, ^Ye)
are homotopy inverse limits of towers of fibrations.
Hence, the step of the induction.
1.3. Universality of CWA .
Proposition 1.8 (CWA is terminal). For any map u: Ue! Xefrom an A-colocal
diagram Uethere exists a factorization Ue! CWA (Xe) ! Xewhich is unique up to
simplicial homotopy.
Proof.By Remark 1.6 the natural map pXf:CWA Xe! Xe is in K-inj. The map
; ! Ueis in K-cof, since Ueis A-colocal (one adopts the proof of [13, 3.4.1] in*
* our
case). Then the following commutative square
;"____//_`CWA