Eilenberg-Watts Theorem for 2-categories and quasi-monoidal structures for module categories over bialgebroid categories

Abstract: We prove Eilenberg-Watts Theorem for 2-categories of the representation categories $\C\x\Mod$ of finite tensor categories $\C$. For a consequence we obtain that any autoequivalence of $\C\x\Mod$ is given by tensoring with a representative of some class in the Brauer-Picard group $\BrPic(\C)$. We introduce bialgebroid categories over $\C$ and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor-Zelinsky sequence exists in this setting. In this context, for a symmetric bialgebroid category $\A$, we interpret the middle cohomology group appearing in the third level of the latter sequence. We obtain a group of quasi-monoidal structures on the representation category $\A\x\Mod$.