Cosimplicial monoids and deformation theory of tensor categories

We introduce the notion of n-commutativity (0 <= n <= infinity) for cosimplicial monoids in a symmetric monoidal category V, where n = 0 corresponds to just cosimplicial monoids in V, while n = infinity 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 En+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 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 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.