Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cosimplicial monoids and deformation theory of tensor categories   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.

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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th