Combinatorial categorical equivalences of Dold-Kan type

Abstract

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathscr{X}$ denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathscr{X}$ are equivalent to chain complexes in $\mathscr{X}$; (b) the observation of Church, Ellenberg and Farb that $\mathscr{X}$-valued species are equivalent to $\mathscr{X}$-valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of "Dold-Kan type"; and so on. When $\mathscr{X}$ is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourn.