  We introduce a notion of $n$-commutativity ($0\le n\le \infty$) 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 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.
