报告题目：Cosimplicial monoids and deformation theory of tensor categories

报 告 人：Michael Batanin

所在单位：Mathematical Institute, Czech Academy of Sciences

报告时间：2025年6月26日 14:00-14:50

报告地点：娱乐城 数学楼第5研讨室

报告摘要: We introduce a notion of n-commutativity for cosimplicial monoids in a symmetric monoidal category V, where n=0 corresponds to just cosimplicial monoids in V while n=\infty corresponds to commutative cosimplicial monoids. If V has a monoidal model structure we show (under some mild technical conditions) that the totalisation 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.

报告人简介：The speaker graduated from Novosibisk State University in 1983. He is currently a Senior Researcher at the Institute of Mathematics of Czech Republic, and a Professor of Charles University in Prague. He is specialising in Algebraic Topology, Category Theory, Operads and related topics.