Title: Cosimplicial Monoids and Deformation Theory of Tensor Categories
Speaker: Michael Batanin
Affiliation: Mathematical Institute, Czech Academy of Sciences
Time: June 26, 2025, 14:00-14:50
Location: Seminar Room 5, Mathematics Building, Jilin University
Abstract:
We introduce the concept of n-commutativity for cosimplicial monoids in a symmetric monoidal category V, where n=0 corresponds to ordinary cosimplicial monoids in V and n=∞ corresponds to commutative cosimplicial monoids. When V possesses a monoidal model structure (under certain mild technical conditions), we demonstrate that the totalization of an n-cosimplicial monoid naturally carries an E_{n+1}-algebra structure.
Our primary applications focus on the deformation theory of tensor categories and tensor functors. We prove that:
The deformation complex of a tensor functor arises as the total complex of a 1-commutative cosimplicial monoid, consequently inheriting an E_2-algebra structure - analogous to the E_2-structure on the Hochschild complex of an associative algebra established by Deligne's conjecture.
The deformation complex of a tensor category constitutes the total complex of a 2-commutative cosimplicial monoid, thus naturally admitting an E_3-algebra structure.
These algebraic structures are made explicit through the framework of Delannoy paths and their noncommutative liftings.
Speaker's Biography:
Michael Batanin graduated from Novosibirsk State University in 1983. Currently serving as a Senior Researcher at the Institute of Mathematics of the Czech Academy of Sciences and Professor at Charles University in Prague, his research specializes in algebraic topology, category theory, operads, and related areas. His work bridges deep theoretical foundations with applications in modern mathematical structures.
