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categories: Model category structures on Cat



    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