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.
Related
- Standard modules and Dold-Kan correspondences for generalized Reedy categories
- Normalization graphs and Dold-Kan equivalences for generalized Reedy categories
- The local Dold congruence for Fibonacci and Lucas sequences
- Degrees of Maps between Generalized Dold Manifolds
- Dynamical invariants of monomial correspondences
BibTeX
@misc{warg2026doldkan,
title = {{Dold--Kan Correspondences via Morita Normalization}},
author = {Florian Warg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28138},
eprint = {2610.28138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28138},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dold--Kan Correspondences via Morita Normalization
AU - Florian Warg
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28138
UR - https://arxiv.org/abs/2610.28138
AB - 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.
ER -