On localization of inverse limits
Alexander Nofech
Abstract. For the categories of pointed spaces, pointed simplicial sets
and simplicial groups and for some fixed cofibrant object A closed model
category structures are constructed that specialize to Quillen's struc-
tures when A = S0 for spaces, or A = F S0 for simplicial groups. Closed
model category structures are also defined for diagrams in such a way
that for diagrams over a contractible category under certain conditions
the factorization of a map of diagrams into a cofibration followed by a
trivial fibration commutes with holim-up to an equivalence. This is used
to prove a weakened version of a conjecture of Farjoun characterizing
spaces of the homotopy type of a CW complex.
Introduction
For a given cofibrant object A of a closed simplicial model category
C we define a new closed model category structure in which cofibrant
objects are "A-cellular". In particular if C is the category of pointed
spaces or pointed simplicial sets, then the cofibrations are relative A-
cellular complexes and their retracts, and weak equivalences are maps
that induce isomorphisms on A-homotopy. When A = S0 this specializes
to Quillen's closed model category structure [Q1].
Factoring a map X ! * into a cofibration followed by a trivial fibra-
tion gives an A-localization of X, and an A-colocalization is obtained by
similar factoring of * ! X. (Note that this is different from localizing in
the closed model category structure of [B2] where the factoring is into
a trivial cofibration followed by a fibration).
Closed model category structures are also constructed for diagrams
in C in such a way that if the classifying space of the index category is
contractible, with certain additional conditions the holim- of a factor-
ization of a map of diagrams is "almost" homotopy equivalent to the
factorization of the induced map of holim- -s, see Th. 5.7, Th. 5.4.
Applying this to pullbacks we answer a question posed by Farjoun
in [F]. One of the results of [F] is Theorem 0.1: If CWA X; PA X are
respectively the colocalization and localization of a pointed space X with
respect to a pointed space A, then CWA X is weakly equivalent to the
homotopy fiber of the localizing map e : X ! PA X, with an additional
condition that [A; X] = {*}.
Farjoun conjectured that this is a homotopy equivalence, and this
implies for A = S0 the following characterization of spaces having the
Typeset by AM S-TEX
homotopy type of a CW complex: A pointed connected space X is
homotopy equivalent to its CW approximation CWS0 X if and only if
the space PS1 X obtained from X by glueing cells so as to kill all of its
homotopy is contractible (Conjecture 0.4, [F]).
We prove in Theorem 6.4 a slightly weaker statement, namely that
there is a right homotopy. In other words one of the compositions is
homotopic to the identity of CWS0 X, and the other is right homotopic
[Q1] to the identity of X. This is done using the closed model category
structures of x1, x3 and the theorems on interchange of factorization and
holim- of x5.
In the category of pointed spaces localizations and colocalizations with
respect to a cofibrant space A include the Quillen plus construction,
Anderson localization, Postnikov pieces and n-connected covers, but not
localization with respect to homology (see [F, C]).
Notation
Throughout CWA X and PA X stand respectively for colocalization
and localization of a pointed space X with respect to a pointed space A.
A lower bar indicates diagrams or morphisms of diagrams and a lower
case "+" stands for a space or a diagram with a disjoint basepoint. As
usual LLP and RLP stand for "left lifting property" and "right lifting
property" and we also use the shortened "i % '" for "i has the LLP
with respect to '", or equivalently "' has the RLP with respect to i",
applying this to classes of maps as well.
We often specify the kind of a weak equivalence (cofibration, fibration)
such that for example X__~A-!Y__means a weak A-equivalence of diagrams
A
and X Cof-! Y means an A-cofibration of spaces, with respect to a fixed
space A.
If C is a category and A 2 |C| is an object, CA will denote the closed
model category structure on C associated to a localization with respect
to A, see x2.
Acknowledgements
I wish to thank Prof. Farjoun for his advice and help, Prof. J. Smith
for many useful conversations and for teaching me the use of closed
model categories, and Prof. Bousfield for carefully reading the preprint.
2
x1. Localization with respect to a space.
Let A be a pointed space. One can say that a pointed space X is A-
local if M ap*(A; X) is contractible, and A-colocal if it can be expressed
as a pointed hocolim-! of copies of A and its suspensions in the same way
as a reduced CW complex is a pointed hocolim-! of spheres, see [F]. This
is expressed by the notation CWA X for the colocalization and PA X for
the localization (PSn+1 X is the n-th Postnikov piece of X).
We will construct closed model category structures in which the equiv-
alences are maps that induce ordinary equivalences on Hom*(A; _) (at
least for fibrant spaces), fibrations are the usual ones and the cofibra-
tions are defined by LLP. In such a closed model category structure the
cofibrant objects are colocal and the trivially fibrant objects are local.
In [B2] the local spaces are fibrant, and the colocal spaces are trivially
cofibrant.
One can call this a localization with respect to an object, which needs
to be cofibrant in the initial closed model category structure.
Such closed model category structures are constructed here for six
categories: pointed spaces, pointed simplicial sets, simplicial groups and
diagrams of such, so that the functor holim- preserves the factorization
of maps into a cofibration followed by a trivial fibration up to homo-
topy in the usual sense if the classifying space of the small category
is contractible, and if a pushout of a certain weak equivalence along a
cofibration is again a weak equivalence, see x5.
Definition 1.0. Let C be a pointed closed simplicial model category
and a A a cofibrant object of C. We call CA a closed model category
structure associated with localization of C with respect to A if the cofi-
brations, fibrations and weak equivalences in CA are defined as follows:
WCA = {'| Hom(A; 'f ) 2 WC }
F ibCA = F ibC
CofCA = {j| j % (WCA \ F ibCA }
Theorem 1.1. There exist closed simplicial model category structures
associated to localizations:
(1) In the category T* of pointed spaces with respect to a space A such
that A is cofibrant in the ordinary sense [Q1];
(2) In the category S* of pointed simplicial sets with respect to A*
*, a
pointed simplicial set;
(3) In the category sGr of simplicial groups with respect to a cofibrant
simplicial group;
3
(4) In the category (T*)I of I-diagrams in T* with respect to A ^ (I=_)+ .
(5) In the category (S*)I of diagrams of pointed simplicial sets over I, a
small category, with respect to the diagram A ^ (I=_)+ where (I=_)+
is the overcategory with a disjoint basepoint, see [B-K].
(6) In the category sGr I of I-diagrams of simplicial groups, with respect
to A (I=_), where A is a cofibrant simplicial group.
For convenience we reproduce the axioms that need to be checked
[Q2]:
CM1 The category is closed under finite limits and finite colimits;
CM2 If in a commutative triangle fl = ff . fi two of the maps are weak
equivalences, then so is the third;
CM3 The three classes of fibrations, cofibrations and weak equivalences are
closed under retractions;
CM4I Cofibrations have LLP with respect to trivial fibrations;
CM4II Trivial cofibrations have LLP with respect to fibrations;
CM5I Any map can be factored as a cofibration followed by a trivial fibra-
tion;
CM5II Any map can be factored as a trivial cofibration followed by a fibra-
tion.
x2. Localization with respect to a cofibrant object.
We suppose the existence in C of a set of "generators of trivial cofib*
*ra-
tions", i.e. trivial cofibrations {tj} such that a morphism ' is a fibrat*
*ion
if and only if {tj} % ', with s-definite domains and codomains. This
implies functorial CM5II factorizations.
Theorem 2.1. Let C be a pointed closed simplicial model category with
a set {tj} of generators of trivial cofibrations, with arbitrary colimits*
* and
let A be a cofibrant, s-definite object.* Then there exists a closed model
category structure associated with localization with respect to A, as in
Definition 1.0, which admits functorial factorizations.
Proof: CM1, CM3 and CM4I are immediate. CM2 follows from func-
torial factorizations (actually using only CM5II factorizations) and the
definition of equivalences.
_______________________________
*see [B1], 4.2.
4
CM4II: We need to show the existence of a lifting in a diagram:
.--- - ! .
? ?
(1) ~A ?yCofA ?yF ibA
.--- - ! .
Since F ibA = F ib, it would be enough to show that a trivial A-
cofibration is a trivial cofibration. To do this consider the factorization
of the left vertical map as in CM5II in C:
~Cof
.--- - ! .
? ?
(2) ~A ?yCofA ~A ?yF ibA
Id
.--- - ! .
The vertical map on the right of (2) is an A-equivalence since the
upper horizontal map is an equivalence, hence an A-equivalence. The
left vertical map is an A-cofibration, hence a lifting exists. Taking the
image of the resulting diagram in Ho - C we easily see that the image
of the left vertical map has an inverse, hence the left vertical map is an
equivalence in C, and the lifting in (1) exists.
CM5I: Let i[u] : _[n] ! [n] be the usual inclusions, and tj the gener-
ators of trivial cofibrations in C. For an arbitrary map f : X ! Y take
a factorization X - i! Z - '! Y obtained by using the set of morphisms
{A i[n]; tj} at every successor step of the small object argument. For
' to have RLP with respect to this set the small object argument needs
to converge, see [B1] for details. The convergence is guaranteed by the
fact that A is s-definite and this property is preserved under colimits.
This follows from Theorem 4.1. and Remark 4.6. of [B1].
The first map i is a cofibration since one can represent A i[n] as
iA i[n] where iA : * ! A and iA i[n] is a cofibration by [B1], Lemma
6.4.
The second map ' is a fibration since it has the RLP with respect to
tj - s. To see that ' is an A-equivalence note that from A i[n] % ',
given by the small object argument, it follows that i[n] % Hom C (A; '),
hence this map is a trivial fibration and so Hom C (A; 'f ) is a weak equiv-
alence.
CM5II - follows from CM5II in C, since the fibrations and the trivial
cofibrations are the same.
5
The functorial factorizations in CA follow from the fact that the gen-
erators of both cofibrations and trivial cofibrations form a set and one
can use a small object argument construction of a uniform length.
Lemma 2.2. An equivalence of cofibrant, fibrant objects in CA is an
equivalence of cofibrant, fibrant objects in the underlying category C.
Proof: A-cofibrant ) cofibrant is a case of CofCA CofC which
follows from
WC \ F ibC WCA \ F ibCA
The fibrant objects in C and CA are the same.
To see that the morphism is in WC one follows the construction of a
homotopy inverse noting that trivial A-cofibrations are trivial cofibra-
tions and hence induce isomorphisms in Ho - C. Combining, one gets
an isomorphism in Ho - C.
Theorem 2.3. Let I be a small category, and let A be a cofibrant object
of a closed simplicial model category C as in Theorem 2.1, with arbitrary
limits and with one additional condition: there exists an s-definite object
S0 such that
HomC (S0; f ) 2 WS ) f 2 WC
Then there exists a closed model category structure on the category of
I-diagrams in C in which
W(CA )I = {f_| holim- f_) 2 WCA }
F ib(CA )I= {'_| '_i2 F ibC }
Cof(CA )I = {j_| j_% (W(CA )I\ F ib(CA )I}
Remark. The functorial factorizations imply by [D-K2] the simplicial
structure which allows the holim- to exist and be functorial.
Proof: The first step is to construct an underlying closed simplicial
model category structure in which the weak equivalences and fibrations
are defined objectwise. This is given by Theorem 2.2 of [D-K1] with
orbits being S0 F__i, where F__iis the free discrete simplicial diagram
corresponding to the object i 2 I.
The second step is to apply Th. 2.1 to this underlying closed simplicial
model category structure choosing A(I=_) as the localizing object. We
need to check that A (I=_) is s-definite and cofibrant in CI. The first
follows from Lemma 3.6, and the second from the adjunction
HomCI (A (I=_); f_) ~= holim-HomC (A; f_)
6
The generators of trivial cofibrations are S0 F__i i[n; k].
Example 2.4. In the categories T*, S* the S0 is just S0 , in the category
sGr it is`the free simplicial group F S0 , in the category of I-diagrams it
is S0 F__i.
i
x3. The Proof of Theorem 1.1.
The proof consists of checking the conditions of Theorem 2.1 for five
cases: pointed spaces, pointed simplicial sets, simplicial groups, dia-
grams of pointed spaces and of pointed simplicial sets.
3.1. The category T* of pointed topological spaces.
In this section we will use the term "s-definiteness" in a specific re-
stricted sense, since compact spaces are not small. Namely, a pointed
space K will be called s-definite if there exists a cardinal fi such that
~=
(1) lim-!HomT* (K; Yff)- ! HomT* (K; lim-!Yff):
ff