Combinatorial categorical equivalences of Dold–Kan type

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let XX denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold–Puppe–Kan theorem that simplicial objects in XX are equivalent to chain complexes in XX; (b) the observation of Church, Ellenberg and Farb [9] that XX-valued species are equivalent to XX-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 XX is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourne.