No Arabic abstract
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 simple objects in the braided tensor category $mathcal{C}$. When $mathcal{C}$ is a modular tensor category, the $H^2_{[rho]}(G,mathcal{A})$ and $H^3(G,U(1))$ torsorial action gives a complete generation of possible $G$-crossed extensions, and hence provides a classification. This torsorial classification can be (partially) collapsed by relabeling equivalences that appear when computing the set of $G$-crossed braided extensions of $mathcal{C}$. The torsor method presented here reduces these redundancies by systematizing relabelings by $mathcal{A}$-valued $1$-cochains.
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 braided fusion categories and their centers, which are interesting for their own sake.
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 the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.
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 generalization of the theory of $Gamma $-module extensions.
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 of the examples that we constructed are proven to be indeed new and non-easy. We interpret some of the new categories in terms of quantum group anticommutative twists.