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 2-category. Examples include G-graded fusion 1-categories, G-crossed 1-categories. explore the properties 2-categories modules bimodules over algebra, by giving criterion for existence right left adjoints. Then, we consider separable algebras, which particularly well-behaved algebras. Specifically, given 2-category, prove that finite semisimple. Finally, define dimension connected such is if only its non-zero.