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)).
                             ~<fi  e

Hence  (abs-I-cell)    (codom (I)).




             A NON-COFIBRANTLY GENERATED MODEL CATEGORY            3


                      3. Proof of Theorem 1.1

  Let us prove first a slightly more general result.

Proposition 3.1. Let D be a small category enriched over S which admits a proper
class of orbits OD . Then the model category M on the D-shaped diagrams of spac*
 *es
generated by the orbits is not cofibrantly generated.

Proof.We argue by contradiction. Suppose the model category M generated by
the proper class of orbits OD is cofibrantly generated. Let I be the set of gen*
 *erating
cofibrations, then any cofibration is a retract of an I-cellular map. (This fol*
 *lows
from Quillen's small object argument; see [7, 2.1.15].) In particular, any orbi*
 *t of
OD  is a retract of an I-cellular complex.  Hence any orbit is a retract of some
orbit of an I-cellular space. But by 2.2 the whole collection of orbits of I-ce*
 *llular
complexes form a set, hence the contradiction.

  In particular, the model category SJ is generated by the proper class of or-
bits OJ = {X ! *}, where X runs through all the objects of S.  Therefore, by
Proposition 3.1, SJ is not cofibrantly generated, hence the main result 1.1.


                          4.More examples

  Let us conclude by giving more examples of non-cofibrantly generated model
categories. Proposition 3.1 implies that M = SD  is not cofibrantly generated i*
 *ff
OD is a proper class. We have already indicated in the introduction that if we *
 *take
D = G to be a group, then OG = {G=H|H < G} is a set, hence SG is cofibrantly
generated.  The same holds for groupoids.  However, the following proposition
provides us with a large family of examples.

Proposition 4.1. Let D be a small category which admits a fully faithful functor
i : K ! D, where K is a category with two objects k1, k2, at least one arrow
f : k1 ! k2 and no arrows in the opposite direction. Then OD consists of a prop*
 *er
class of orbits.

Proof.First define for each space X 2 S an orbit over K, i.e. a functor TX : K *
 *! S,
by TX (k1) = X, TX (k2) = *, TX (g) = idX for any g 2 mor(k1, k1) and TX on the
elements of mor(k1, k2), mor (k2, k2) has a unique definition, since * is the f*
 *inal
object of S. Obviously TX is an orbit over K.
  Next we define for each TX a D-orbit OX by extending the definition of TX to
the whole D.  More precisely, OX  = LaniTX .  We need to check that OX  is an
orbit.  It follows from the fact that colimit is itself a left Kan extension al*
 *ong a
functor into the trivial category. But any two left Kan extensions commute since
they may be represented as coends, and for the coends there is a üF bini" theor*
 *em.
See [8, X] for the details.
  The functor i is taken to be fully faithful, hence OX (i(k1)) = X; therefore *
 *we
obtain a proper class of D-orbits of the form OX .

Question 4.2. Let D be a monoid which is not a group. Is SD cofibrantly generat*
 *ed?


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4                           BORIS CHORNY


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