A bicategorical approach to actions of monoidal categories

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-Drinfeld 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-Drinfeld 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.