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We prove a class of equivalences of additive functor categories that arerelevant to enumerative combinatorics, representation theory, and homotopytheory. Let $\mathscr{X}$ denote an additive category with finite direct sumsand split idempotents. The class includes (a) the Dold-Puppe-Kan theorem thatsimplicial 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 functorsfrom the category of finite sets and injective partial functions; (c) a resultT. Pirashvili calls of "Dold-Kan type"; and so on. When $\mathscr{X}$ issemi-abelian, we prove the adjunction that was an equivalence is now at leastmonadic, in the spirit of a theorem of D. Bourn.

Combinatorial categorical equivalences of Dold-Kan type