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 introduce a notion of $n$-commutativity ($0le nle infty$) for cosimplicial monoids in a symmetric monoidal category ${bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${bf V,}$ while $n=infty$ corresponds to commutative cosimplicial
monoids. If ${bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Delignes conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.
We compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $mathfrak{u}_q(mathfrak{g})$, where $q$ is a root of unity.
