Mathematical Institute of the Serbian Academy of Sciences and Arts
|Title:||A bicategorical approach to actions of monoidal categories||Journal:||Journal of Algebra and its Applications||First page:||2350073||Issue Date:||1-Jan-2022||Rank:||~M23||ISSN:||0219-4988||DOI:||10.1142/S0219498823500731||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.
|Keywords:||2-(co)monads | action of categories | bicategories | Monoidal categories | quasi-bialgebras | Yetter-Drinfel'd modules||Publisher:||World Scientific|
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