Dold–Kan Correspondences via Morita Normalization

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.