Modulated categories and their representations via higher categories

Abstract

We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting representation-theoretic information. Let $\mathcal{C}$ be a small 1-category and $\mathfrak{B}im_k$ be the 2-category of bimodules over $k$-algebras, where $k$ is a commutative ring with identity. We call a covariant (resp. contravariant) pseudofunctor from $\mathcal{C}$ into $\mathfrak{B}im_k$ a modulation (resp. comodulation) on $\mathcal{C}$, define and study its representations. This framework provides a unified approach to investigate 2-representations of finite groups, modulated quivers and their representations, as well as presheaves of $k$-algebras and their modules. Moreover, several key constructions are natural ingredients in $2\mathfrak{C}at$, and thus it exhibits an interesting application of higher category theory to representation theory.