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

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 $H{C}^p(\mathcal{C})\otimes H{C}^q({\mathcal{C}}^{\prime })⟶H{C}^{p+q}(\mathcal{C}\otimes {\mathcal{C}}^{\prime })$ on cyclic cohomology groups for small $k$-linear categories $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$.

Changes to the definition of "cycle", several updates in Sections 5 and 7