Finite Tensor 2-Categories

Abstract

We propose a definition of a finite linear 2-category, as a 2-categorical analogue of the notion of a finite abelian category of Etingof-Ostrik. We go on to prove that finite linear 2-categories with adjunctible projectives are precisely the Morita 2-categories of (all) algebras in a finite multitensor category. Thus, finite linear 2-categories with adjunctible projectives can be viewed as `non-semisimple' generalizations of the finite semisimple 2-categories of Douglas-Reutter. Secondly, we study finite multitensor 2-categories, that is, finite linear 2-categories equipped with a monoidal structure for which every objects has duals. We extend several results about fusion 2-categories to our present setting. In particular, we investigate the properties of the 2-category of bimodules over a rigid algebra $\mathcal{A}$ in a finite tensor 2-category $\mathfrak{C}$. We show that, if $\mathfrak{C}$ has adjunctible projectives, then the 2-category of $\mathcal{A}$-$\mathcal{A}$-bimodules in $\mathfrak{C}$ is a finite multitensor 2-category. This construction provides many examples of finite tensor 2-categories. For instance, the Drinfeld center of a finite tensor 2-category with adjunctible projectives is a finite tensor 2-category.