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We introduce a notion of $n$-commutativity ($0\le n\le \infty$) forcosimplicial 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 monoidalmodel structure we show (under some mild technical conditions) that the totalobject of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure.  Our main applications are to the deformation theory of tensor categories andtensor functors. We show that the deformation complex of a tensor functor is atotal complex of a $1$-commutative cosimplicial monoid and, hence, has an$E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex ofan associative algebra provided by Deligne's conjecture. We further demonstratethat 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 ofDelannoy paths and their noncommutative liftings. We investigate how thesestructures manifest themselves in concrete examples.

Cosimplicial monoids and deformation theory of tensor categories