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

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 $\mathcal C'$.