# Cosimplicial monoids and deformation theory of tensor categories

### Michael Batanin

Institute of Mathematics of the Czech Academy of Sciences, Prague, Czechia### Alexei Davydov

Ohio University, Athens, USA

## Abstract

We introduce the notion of $n$-commutativity ($0≤n≤∞$) for cosimplicial monoids in a symmetric monoidal category $V$, where $n=0$ corresponds to just cosimplicial monoids in $V$, while $n=∞$ 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 $E_{n+1}$-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.

## Cite this article

Michael Batanin, Alexei Davydov, Cosimplicial monoids and deformation theory of tensor categories. J. Noncommut. Geom. 17 (2023), no. 4, pp. 1167–1229

DOI 10.4171/JNCG/512