A bicategorical approach to actions of monoidal categories

Abstract

We characterize in bicategorical terms actions of monoidal categories on the categories of representations of algebras and of relative Hopf modules. For this purpose we introduce 2-cocycles in any 2-category . We observe that under certain conditions the structures of pseudofunctors between bicategories are in one-to-one correspondence with (twisted) 2-cocycles in the image bicategory. In particular, for certain pseudofunctors to Cat, the 2-category of categories, one gets 2-cocycles in the free completion 2-category under Eilenberg-Moore objects, constructed by Lack and Street. We introduce (co)quasi-bimonads in and a suitable bicategory of Tambara (co)modules over (co)quasi-bimonads in fitting the setting of the latter pseudofuntors. We describe explicitly the involved 2-cocycles in this context and show how they are related to Sweedler's and Hausser-Nill 2-cocycles in , which we define. This allows us to recover some results of Schauenburg, Balan, Hausser and Nill for modules over commutative rings. We fit a version of the 2-category of bimonads in , which we introduced in a previous paper, in a similar setting as above and recover a result of Laugwitz. We observe that pseudofunctors to Cat in general determine what we call pseudo-actions of hom-categories, which correspond to the whole range of a 2-cocycle, so that the described actions of categories appear as restrictions of these 2-cocycles to endo-hom categories.