2020/03/29 by Batanin, Michael, Davydov, Alexei
#Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2003.13039
We introduce a notion of n-commutativity (0≤ n≤ ∞) for cosimplicial monoids in a symmetric monoidal category \bf V, where n=0 corresponds to just cosimplicial monoids in \bf V, while n=∞ 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 En+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 E2-algebra structure similar to the E2-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 E3-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.