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We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $cal B$ (consisting of invertible central $cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.
We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of
Let $mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majids automorphism braided group of $mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $mathcal{C}$. We show that the center of $mathcal{C}$ is isomorphic to the category o
For a braided fusion category $mathcal{V}$, a $mathcal{V}$-fusion category is a fusion category $mathcal{C}$ equipped with a braided monoidal functor $mathcal{F}:mathcal{V} to Z(mathcal{C})$. Given a fixed $mathcal{V}$-fusion category $(mathcal{C}, m
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modula
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