Dold--Kan Correspondences via Morita Normalization

Abstract

The classical Dold--Kan correspondence identifies simplicial modules with connective chain complexes. Using Morita theory, we construct a Dold--Kan-type correspondence for every linear category carrying a suitable action of the maximum-preserving simplicial operators. The construction functorially assigns a Morita-equivalent normalized category to every such linear category. In particular, we obtain Dold--Kan-type correspondences for all crossed simplicial groups. We prove a rank formula for the normalized hom-modules and show that a crossed simplicial group is self-dual precisely when its underlying simplicial set of automorphism groups has dimension one. Furthermore, we construct normalization-compatible self-dualities and recover the Dwyer--Kan correspondence for duplicial, paracyclic and N-cyclic modules.