The Dold-Kan theorem for paracyclic modules

TL;DR

This paper proves a version of the Dold-Kan theorem for paracyclic modules by analyzing the Karoubi operator and its relation to normalized chain complexes, extending previous work in cyclic and paracyclic contexts.

Contribution

It provides a direct proof of the Dold-Kan theorem for paracyclic modules, linking the Karoubi operator to the normalized subcomplex projection.

Findings

Established a direct proof of the Dold-Kan theorem for paracyclic modules.

Connected the Karoubi operator to the projection onto normalized subcomplexes.

Extended previous cyclic case results to the broader paracyclic setting.

Abstract

We study the Karoubi operator on the unnormalized chain complex of a paracyclic module; its restriction to the normalized chain complex has previously been considered by Dwyer and Kan, and in the cyclic case by Cuntz and Quillen. We obtain a direct proof of the Dold-Kan theorem for paracyclic modules of Dwyer and Kan, by directly relating the Karoubi operator to projection to the normalized subcomplex.