The adjoint algebra for 2-categories

Abstract

For any 0-cell B in a 2-category $\mathcal{B}$, we introduce the notion of adjoint algebra ${\mathcal{A}d}_B$. This is an algebra in the center of $\mathcal{B}$. We prove that if $\mathcal{C}$ is a finite tensor category, this notion applied to the 2-category ${}_{\mathcal{C}}\mathrm{Mod}$ of $\mathcal{C}$-module categories coincides with the one introduced by Shimizu. As a consequence of this general approach, we obtain new results on the adjoint algebra for tensor categories.