  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 $\mathcal C'$.
