Subject: Garbled posting Date: Wed, 13 Aug 2003 12:46:28 -0500 (CDT) From: Peter May To: dmd1@lehigh.edu At some point along the line, the cutting and pasting failed, so that my posting is garbled by omissions. Would you mind reposting it? Peter Model structures SInce there has been some confusion in this thread and I happen to have just written a relevant paper, let me answer Gaucher's question and advertise some work of Mike Cole and of Johann Sigurdsson and myself. There is a relevant general theorem, unpublished, due to Mike Cole: Theorem (Cole) If C is a topologically bicomplete category such that mapping path fibrations commute with colimits of inclusions, then the homotopy equivalences, Hurewicz fibrations (homotopy lifting property) and implied cofibrations (left lifting property with respect to Hurewicz fibrations) specify a model structure on C. Here C is a topological category with limits and colimits and with tensors and cotensors with spaces (that is the meaning of ``topologically bicomplete''). The mapping path fibration hypothesis holds if you work with compactly generated spaces (weak Hausdorff k-spaces). It seems not to hold if you work with general spaces or with k-spaces. Strom proves the conclusion of the theorem for general spaces, and his argument works for k-spaces, but the details of his proof must of course differ from Cole's. One point in Strom's work is a proof that, in spaces, closed Hurewicz cofibrations (homotopy extension property) are exactly the implied cofibrations of the theorem. In compactly generated spaces, you can omit the word ``closed'' and have the same conclusion. This is where what James calls Strom structures enter into the picture (there is no mention of model categories in James's work). However, there is another quite subtle point in Strom's proof of the model axioms. The factorization of a map as the composite of a cofibration and an acyclic fibration makes use of the fiberwise join (or generalized Whitney sum), and a key point is an old observation (of somebody named Hall) that the fiberwise join of (Hurewicz) fibrations is a fibration. There is another, and amazing, relevant general theorem, also unpublished and also due to Mike Cole. Theorem (Cole) Let (W_h,Fib_h,Cof_h)$ and (W_q,Fib_q,Cof_q) be model structures on the same category. Suppose that W_h is contained in W_q and Fib_h is contained in Fib_q. Then there is a third ``mixed'' model structure (W_q, Fib_h, Cof_m). The ``mixed'' cofibrations Cof_m are the maps in Cof_h that factor as the composite of a map in W_h and a map in Cof_q. An object is m-cofibrant if and only it is h-cofibrant and of the h-homotopy type of a q-cofibrant object. Think of compactly generated spaces, say. The h-model structure is the one of the first theorem: homotopy equivalences and Hurewicz fibrations. The q-model structure (q for Quillen of course) uses weak homotopy equivalences and Serre fibrations; the q-cofibrations are the retracts of the cell complexes. The mixed model structure uses weak homotopy equivalences and Hurewicz fibrations. Its cofibrant objects are the spaces of the homotopy types of CW-complexes. Anybody interested in these matters is referred to my forthcoming paper: ``Model categories and ex-spaces''. It will be posted here in a week or two. It gives a careful overview of the relevant model structures and their parametrized generalizations, which turn out to behave far more subtly than we topologists are used to. One requires base change adjunctions in homotopy categories. Pullback along a map of base spaces has both a left and a right adjoint, and model category theory cannot give you that both of these adjunctions descend to homotopy categories, no matter what model structures you use. Precisely, if both adjunctions are Quillen adjunctions and either takes homotopy equivalences of base spaces to Quillen equivalences, then the homotopy categories are trivial: one object and one morphism. The spectrum level analogue of this theory is in a joint paper with Johann Sigurdsson, ``Parametrized equivariant stable homotopy theory'', which should also be posted in a few weeks. As the title of the second paper indicates, both papers are written equivariantly. Peter May