Monoidal Categories, 2-Traces, and Cyclic Cohomology

TL;DR

This paper develops a categorical framework linking monoidal categories, 2-traces, and cyclic (co)homology, unifying various theories and providing new conceptual insights into algebraic structures.

Contribution

It introduces a general categorical construction that associates cyclic modules to algebra objects in monoidal categories with 2-traces, unifying and extending existing cyclic (co)homology theories.

Findings

Recover all Hopf cyclic type (co)homologies

Provide a conceptual, formula-free definition of cyclic cohomology

Apply machinery beyond Hopf algebra modules and comodules

Abstract

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $\mathcal{C}$ endowed with a symmetric $2$-trace, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the (co)algebra "with coefficients in $F$". We observe that if $\mathcal{M}$ is a $\mathcal{C}$-bimodule category equipped with a stable central pair then $\mathcal{C}$ acquires a symmetric 2-trace. The dual notions of symmetric $2$-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories, obtain a conceptual understanding of anti-Yetter-Drinfeld modules, and give a formula-free definition of cyclic cohomology. The machinery can also be applied in settings more general than Hopf algebra modules and comodules.