AN EXAMPLE OF A NON-COFIBRANTLY GENERATED
MODEL CATEGORY
BORIS CHORNY
Abstract.We show that the model category of diagrams of spaces generated
by a proper class of orbits is not cofibrantly generated. In particular*
* the
category of maps between spaces may be given a non-cofibrantly generated
model structure.
1.Introduction and formulation of results
Several examples of non-cofibrantly generated model categories have appeared
recently (see [1], [2], [6]) in response to a question stated by Mark Hovey on *
*his
home page. In this note we introduce another family of such examples.
By the category of spaces, denoted by S, we mean the category of simplicial s*
*ets
(or compactly generated topological spaces). There are plenty of model structur*
*es
on categories of diagrams of spaces, with different notions of weak equivalence*
*s.
Some of them are cofibrantly generated, e.g. for weak equivalences and fibratio*
*ns
being objectwise and cofibrations obtained by the left lifting property with re*
*spect
to trivial fibrations, the corresponding model category is cofibrantly generate*
*d.
Let us remind (from [3], [4], [5]) that a diagram O of spaces is called an or*
*bit if
colimO = *. The weak equivalences which we would like to consider arise natural*
*ly
from the relation of equivariant homotopy. By the generalized Bredon theorem [5]
a map f : Xe! Yeis an equivariant homotopy equivalence between diagrams which
are both cofibrant and fibrant iff map (O, f) : map (O, Xe) ! map (O, Ye) is a *
*weak
equivalence of spaces for any orbit O. A model category, generated by the colle*
*ction
of orbits, on diagrams of spaces was constructed in [4] with a map f being a we*
*ak
equivalence (reps. fibration) iff map (O, f) is a weak equivalence (resp. fibra*
*tion)
for any orbit O. In the sequel we consider only this model category on diagrams
of spaces. The simplest example of a non-cofibrantly generated model category is
given by the following
Theorem 1.1. If J = (o ! o) is the category with two objects and only one non-
identity morphism, then the functor category M = SJ of maps of spaces with the
model structure as above is not cofibrantly generated.
However, not every small category gives rise to a non-cofibrantly generated m*
*odel
category of diagrams. For example, if we take G to be a group, then the above
model structure on SG is cofibrantly generated. We conclude the paper by using
this example to produce many other examples of the same nature.
____________
Date: December 12, 2001.
1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G*
*55.
Key words and phrases. model category, equivariant homotopy, non-cofibrantly*
* generated.
The author is a fellow of the Marie Curie Training Site hosted by the Centre*
* de Recerca
Matem`atica (Barcelona), grant nr. HPMT-CT-2000-00075 of the European Commissio*
*n.
1
2 BORIS CHORNY
Acknowledgments. I would like to thank C. Casacuberta and E. Dror Farjoun for
helpful conversations about the subject matter of this paper.
2.Preliminaries
By an orbit over a point in the colimit of a diagram Xewe mean the pull back
of the canonical map f : Xe! colimXeover g : * ! colimXe. Let D be any small
category enriched over S. We denote by O the collection of all orbits of D. By
collection we mean a set or a proper class with respect to some fixed universe *
*U.
The operator codom (.) applied to a collection of maps returns the collection of
ranges. Given a set I of maps in M = SD , we denote by I-cell the collection of
relative I-cellular complexes and by abs-I-cell the collection of (absolute) I-*
*cellular
complexes. See [7, 2.1.9] for precise definitions.
Definition 2.1. Let X = {Xeff}ff2Abe a collection of D-shaped diagrams of space*
*s.
The collection of orbits of X , denoted by (X ) O, consists of all orbits O *
*2 O
such that there exists ff 2 A and a point x 2 colimXeffwith O being the orbit o*
*ver
x.
Lemma 2.2. Let I be a set of cofibrations in the model category M of D-shaped
diagrams of spaces. Then (abs-I-cell) (codom (I)).
Proof.Let Xe2 M be any I-cellular complex. We proceed by transfinite induction
on the I-cellular filtration of Xe. Xe-1= ;, hence Xe02 codom(I) and in particu*
*lar
(Xe0) (codom (I)).
Suppose Xefisatisfies (Xefi) (codom (I)). We need to show that Xefi+1, wh*
*ich
is obtained from Xefiby attaching a map I 3 f : Ae,! Be, satisfies (Xefi+1)
(codom (I)).
Ae--'--! Xefi
? ?
f?ypush-out ?yf0
Be----! Xefi+1
Let Os be an orbit over a point s 2 colimXefi+1= colimXefiqcolimAecolimBe.
Considering two cases, s 2 colimXefi colimXefi+1and s =2colimXefi, we find out
that in the first case Os equals the corresponding orbit of Xefiand in the seco*
*nd case
Os is some orbit of Be. This follows immediately from the fact that the diagrams
0 ~=
Xefi --f--! Xefi+1 B=A ----! Xfi+1=X fi
?? ? e??e e ?? e
y ?y y y
~=
colimXefi----! colimXefi+1 colim(Be=Ae)----! colim(Xefi+1=Xefi)
are pull-backs. The first square is a pull-back by [4, 2.1] and the second by t*
*he ob-
servation that horizontal maps are isomorphisms. Hence (Xefi+1) (codom (I)).
Obviously, if fi is a limit ordinal, then
[
(Xefi) = (X ~) (codom (I)).
~