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We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X; (b) the observation of Church, Ellenberg and Farb  that X-valued species are equivalent to 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 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. Bourne.
Bibliographical noteA corrigendum exists for this article and can be found in Journal of Pure and Applied Algebra, 224(3), p.1364-1366, doi: 10.1016/j.jpaa.2019.07.002
30/06/16 → 17/06/19
1/01/13 → 31/12/16