Crossed-module crossed braided categories

Abstract

For a crossed module $\chi :G\to H$, we introduce the notion of $\chi$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $\chi =id:G\to G$ with the conjugation action recovers $G$-crossed braided categories; taking $\chi :G\to \{\ast \}$ for abelian $G$ yields $G$-graded braided categories; taking $\chi :\{\ast \}\to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $\chi$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $\chi$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $\chi$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $\chi$-crossed ribbon or admit an orthogonal $G$-decomposition.

60 pages