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A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to $G$-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of $3$-categories $mathcal{C}$ equipped with a 3-functor $mathrm{B}G to mathcal{C}$ which is essentially surjective on objects and $1$-morphisms is equivalent to the $2$-category of $G$-crossed braided categories. This provides a uniform approach to various constructions of $G$-crossed braided categories.
We develop a method for generating the complete set of basic data under the torsorial actions of $H^2_{[rho]}(G,mathcal{A})$ and $H^3(G,U(1))$ on a $G$-crossed braided tensor category $mathcal{C}_G^times$, where $mathcal{A}$ is the set of invertible
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate
We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate th
If $Gamma $ is a group, then braided $Gamma $-crossed modules are classified by braided strict $Gamma $-graded categorial groups. The Schreier theory obtained for $Gamma $-module extensions of the type of an abelian $Gamma $-crossed module is a gener
We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm which we used to obtain first examples of so-called non-easy linear categories of partitions. All