Cosimplicial monoids and deformation theory of tensor categories.

We introduce the notion of n-commutativity (0 ≤ n ≤ ∞) for cosimplicial monoids in a symmetric monoidal category V, where n = 0 corresponds to just cosimplicial monoids in V, while n = ∞ corresponds to commutative cosimplicial monoids. When V has a monoidal model structure, we endow (under some mild technical conditions) the total object of an n-cosimplicial monoid with a natural and very explicit 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 E₂-algebra structure similar to the E₂-structure on Hochschild complex of an associative algebra provided by Deligne's 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 E₃-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. [ABSTRACT FROM AUTHOR]