The Dold-Kan theorem for paracyclic modules
Ezra Getzler
TL;DR
The paper develops a direct, operator-theoretic proof of the Dold–Kan correspondence for paracyclic (duplicial) modules by relating the Karoubi operator $\kappa_n$ to the projection onto the normalized chains $N_n(M)$. It shows that the duplicial structure is equivalent to the invertibility of $\kappa_n$ on $N_n(M)$ (paracyclic) and to $\pi_n|_{N_n(M)}=\mathrm{id}$ (cyclic), with the Dwyer–Kan operator $\pi_n$ expressible as $\pi_n = p_n \mathsf{T}_n = \mathsf{T}_n p_n$ and ultimately tied to the normalization projection $p_n$ via the duchain differential framework. The results unify and extend prior work of Dwyer–Kan and Cuntz–Quillen by treating $\kappa_n$ and $\pi_n$ on the full module $M_n$ and providing explicit homotopies (Connes-type $B$ and Grove–Ginzburg–Schedler type $D$) that witness a homotopy equivalence to the normalized complex. The paper thus establishes a robust Dold–Kan equivalence for paracyclic structures, clarifying how normalization, duchain complexes, and the Karoubi/Dwyer–Kan operators interlock to characterize paracyclic versus cyclic behavior.
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.