| Denote by Gamma the skeletal category of based finite sets, and by E the skeletal category of finite sets and epimorphisms. In the paper Dold-Kan type theorem for Gamma-groups, Pirashvili proves that the category of functors from Gammaop to abelian groups is equivalent to the category of functors from Eop to abelian groups. More is true, in that the same result holds if the functors take values in an arbitrary abelian category. We show that one can replace the abelian category by any cofibrantly generated stable model category C and obtain a Quillen equivalence of the functor categories.;We abstract the relevant structural properties of Gamma and E , giving the notion of a regular pair of small categories. We then prove that for every regular pair ( A,B ), the category of functors from Aop to C is Quillen equivalent to the category of functors from Bop to C . We conclude by showing that all of this machinery works in the context of abelian categories as well, thereby extending Pirashvili's theorem to other indexing categories.;Regarding functors Aop→C as right A -modules, these results take on the characteristics of classical Morita theory. Throughout, we make use of this analogy as a source of inspiration for our methods as well as our terminology. |
