Category-valued traces for bimodule categories: a representation-theoretic realization

TL;DR

This paper provides a representation-theoretic realization of category-valued traces for bimodule categories over linear monoidal categories, generalizing known constructions like Drinfeld centers and Deligne products, using bicomodules over Hopf algebras.

Contribution

It introduces a new realization of the category-valued trace as a category of generalized Hopf bimodules, extending the understanding of bimodule categories in a representation-theoretic context.

Findings

Category-valued trace is realized as a category of generalized Hopf bimodules.

Generalizes Drinfeld centers and Deligne products within a Hopf algebra framework.

Provides a new perspective on bimodule categories through Hopf algebra bicomodules.

Abstract

The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules.