A bicategorical approach to actions of monoidal categories

TL;DR

This paper develops a bicategorical framework to understand actions of monoidal categories on algebra representation categories, introducing cocycles, bimonads, and Yetter-Drinfeld modules within 2-category theory.

Contribution

It introduces a bicategorical approach to actions of monoidal categories, including new concepts like cocycles, (co)quasi-bimonads, and Yetter-Drinfeld modules, generalizing existing results.

Findings

Characterization of actions via 2-cocycles in Eilenberg-Moore categories

Monoidal structures on categories of Tambara modules over (co)quasi-bimonads

Connections between 2-cocycles and classical cohomology in bicategory context

Abstract

We characterize in terms of bicategories actions of monoidal categories to representation categories of algebras. For that purpose we introduce cocycles in any 2-category $\K$ and the category of Tambara modules over a monad $B$ in $\K$. We show that in an appropriate setting the above action of categories is given by a 2-cocycle in the Eilenberg-Moore category for the monad $B$. Furthermore, we introduce (co)quasi-bimonads in $\K$ and their respective 2-categories. We show that the categories of Tambara (co)modules over a (co)quasi-bimonad in $\K$ are monoidal, and how the 2-cocycles in the Eilenberg-Moore category corresponding to their actions are related to the Sweedler's and Hausser-Nill 2-cocycles in $\K$. We define (strong) Yetter-Drinfel`d modules in $\K$ as 1-endocells of the 2-category $\Bimnd(\K)$ of bimonads in $\K$, which we introduced in a previous paper. We prove that the monoidal category of Tamabra strong Yetter-Drinfel`d modules in $\K$ acts on the category of relative modules in $\K$. Finally, we show how the above-mentioned results on actions of categories come from pseudofunctors between appropriate bicategories. Our results are 2-categorical generalizations of several results known in the literature.