Rigid and Separable Algebras in Fusion 2-Categories

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include $G$-graded fusion 1-categories, and $G$-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.