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*To*: dmd1@lehigh.edu, categories@mta.ca*Subject*: categories: Model category structures on Cat*From*: "Steven R. Costenoble" <Steven.R.Costenoble@Hofstra.edu>*Date*: Fri, 05 May 2000 10:48:58 -0400*Sender*: cat-dist@mta.ca

Last Fall I asked the Algebraic Topology and Category mailing lists what was known about model category structures on Cat, the category of small categories. For those who are interested, what follows is a summary of what I found. It is, I'm sure, incomplete, and I apologize to anyone whose work I have slighted or missed entirely. The interest and the problem stem from the observation that the nerve functor N induces an equivalence between Cat and the category of simplicial sets, after inverting weak equivalences. The weak equivalences in Cat are, by definition, those maps that become weak equivalences on applying N. This result can be found in the following two sources (but goes back further, to at least Gabriel & Zisman): Thomason, Homotopy colimits in Cat, with applications to algebraic K-theory and loop space theory, Dissertation, Princeton, 1977. Latch, The uniqueness of homology for the category of small categories, J. Pure Appl. Algebra 9, 221-237 (1977). Unfortunately, the left adjoint for N, which is the "categorical realization" functor c, does not give the inverse equivalence. Although the counit cN -> 1 is an isomorphism, the unit 1 -> Nc is far from being a weak equivalence. A related problem is that, if C is a small category, then NC is fibrant iff C is a groupoid. This raises the question: Is there an adjoint pair of functors (L,R): SimpSet -> Cat and a model category structure on Cat with the weak equivalences as above, such that (L,R) induces an adjoint pair of equivalences between the Quillen homotopy categories (i.e., (L,R) is a Quillen equivalence)? Thomason gave an affirmative answer in Thomason, Cat as a closed model category, Cahiers Topologie Geom. Differentielle Categoriques XXI, 305-324 (1980). This followed an attempt by Golasinski in a 1978 preprint which appeared in print as Golasinksi, Homotopies of small categories, Fund. Math. CXIV, 209-217 (1981). Golasinski's proposed structure failed to satisfy the factorization axiom, but does lead to a closed model category on the category of pro-objects in Cat, as shown in Golasinski, On closed models on the precategory of small categories and simplicial schemes, Uspekhi Mat. Nauk 39, 239--240 (1984) (Russian, translated in Russian Math. Surveys 39, 275-276 (1984)). More recently, Heggie defined a class of cofibrations in Cat, in Heggie, Homotopy cofibrations in CAT, Cahiers Topologie Geom. Differentielle Categoriques XXXIII, 291-313 (1992). I haven't had time to compare them, but at first glance these cofibrations appear closely related to Thomason's, if not identical. I also recommend two other papers by Heggie: Heggie, The left derived tensor product of CAT-valued diagrams, Cahiers Topologie Geom. Differentielle Categoriques XXXIII, 33-53 (1992). Heggie, Homotopy colimits in presheaf categories, Cahiers Topologie Geom. Differentielle Categoriques XXXIV, 13-36 (1993). The upshot of all this is that Thomason's is still the only model category structure known (or, at least, published) on Cat making it equivalent to the category of simplicial sets. Besides an aesthetic objection to Thomason's model category structure, I have a practical one: Thomason uses the second subdivision followed by categorical realization as the functor going from simplicial sets to small categories inducing the equivalence of homotopy categories. This functor does not preserve products, as does categorical realization by itself. For various reasons it would be really nice to have a functor that does preserve products and is the left adjoint in a Quillen equivalence. I throw this out as a problem I'd like to work on myself, when I get time. If anyone has any thoughts about it, I'd be happy to hear them. There were other references I came across, answering related but different questions. For example, there is a short, unpublished 1996 manuscript by Charles Rezk giving a very nice model structure on Cat in which the weak equivalences are the equivalences of categories (this was probably folklore for quite a while). There has also been quite a lot of work on model category structures for n-categories and related gadgets; however, when restricted to ordinary categories this work generally gives the homotopy equivalence between the category of simplicial 1-types and the category of small groupoids. --Steve Costenoble

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