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)). ~