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- W3013209609 abstract "We introduce the notion of $n$-commutativity ($0le nleinfty$) for cosimplicial monoids in a symmetric monoidal category $mathbf{V}$, where $n=0$ corresponds to just cosimplicial monoids in $mathbf{V}$, while $n=infty$ corresponds to commutative cosimplicial monoids. When $mathbf{V}$ 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 $E_2$-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." @default.
- W3013209609 created "2020-04-03" @default.
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- W3013209609 date "2023-08-17" @default.
- W3013209609 modified "2023-10-17" @default.
- W3013209609 title "Cosimplicial monoids and deformation theory of tensor categories" @default.
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- W3013209609 doi "https://doi.org/10.4171/jncg/512" @default.
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