The aim of this paper is to describe Quillen model category structures on the category CatC of internal categories and functors in a given finitely complete category C. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C.
Under mild conditions on C, the regular epimorphism topology determines a model structure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Pare). For a Grothendieck topos C we get a structure that, though different from Joyal and Tierney's, has an equivalent homotopy category. In case C is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and the model structure on CatC induces a notion of homotopy of internal crossed modules. In case C is the category Gp of groups and homomorphisms, it reduces to the case of crossed modules of groups.
The trivial topology on a category C determines a model structure on CatC where we is the class of strong equivalences (homotopy equivalences), fib the class of internal functors with the homotopy lifting property, and cof the class of functors with the homotopy extension property. As a special case, the ``folk'' Quillen model category structure on the category Cat = CatSet of small categories is recovered.
Keywords: internal category, Quillen model category, homotopy, homology
2000 MSC: Primary 18G55 18G50 18D35; Secondary 20J05 18G25 18G30
Theory and Applications of Categories,
Vol. 15, CT2004,
No. 3, pp 66-94.