Generalized Yetter-Drinfeld modules, the center of bi-actegories and groupoid-crossed braided bicategories

Abstract

We study the notion of the $E$-center ${\mathcal{Z}}_E(\mathcal{M})$ of a $(\mathcal{C},\mathcal{D})$-biactegory (or bimodule category) $\mathcal{M}$, relative to an op-monoidal functor $E:\mathcal{C}\to \mathcal{D}$. Specializing this notion to the case $\mathcal{M}={}_A\mathrm{Mod}$, $\mathcal{C}={}_H\mathrm{Mod}$, $\mathcal{D}={}_K\mathrm{Mod}$, and $E\simeq C{\otimes }_H-:{}_H\mathrm{Mod}\to {}_K\mathrm{Mod}$, where $H$ and $K$ are bialgebras, $A$ is an $(H,K)$-bicomodule algebra and $C$ is a $(K,H)$-bimodule coalgebra, we show that this $E$-center is equivalent to the category of generalized Yetter-Drinfeld modules as introduced by Caenepeel, Militaru, and Zhu. We introduce the notion of a double groupoid-crossed braided bicategory, generalizing Turaev's group-crossed braided monoidal categories, and show that generalized Yetter-Drinfeld modules can be organized in a double groupoid-crossed braided bicategory over the groupoids of Galois objects and co-objects.

30 pages