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 ${\mathcal{C}}^{\mathrm{\prime }}$.