Cycles over DGH-semicategories and pairings in categorical Hopf-cyclic cohomology

TL;DR

This paper develops a framework for understanding Hopf cyclic cohomology of Hopf-module semicategories, introducing categorified cycles and pairings that extend classical cyclic cohomology concepts to non-unital categories.

Contribution

It introduces a novel approach to Hopf cyclic cohomology using semicategories and categorified cycles, expanding the scope of cyclic cohomology in categorical settings.

Findings

Defined cocycles and coboundaries for Hopf cyclic cohomology of semicategories

Constructed a pairing on cyclic cohomology groups for small $k$-linear categories

Extended cyclic cohomology to categories without identity maps

Abstract

Let $H$ be a Hopf algebra and let $\mathcal D_H$ be a Hopf-module category. We describe the cocycles and coboundaries for the Hopf cyclic cohomology of $\mathcal D_H$, which correspond respectively to categorified cycles and vanishing cycles over $\mathcal D_H$. An important role in our work is played by semicategories, which are categories that may not contain identity maps. In particular, a cycle over $\mathcal D_H$ consists of a differential graded $H$-module semicategory equipped with a trace on endomorphism groups satisfying some conditions. Using a pairing on cycles, we obtain a pairing $HC^p(\mathcal{C}) \otimes HC^q(\mathcal{C}') \longrightarrow HC^{p+q}(\mathcal{C} \otimes \mathcal{C}')$ on cyclic cohomology groups for small $k$-linear categories $\mathcal C$ and $\mathcal C'$.