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Disquisitions on Monoidal Categories and Operads
Disquisitions on Monoidal Categories and Operads
On deformation theory of associators and braidings
On deformation theory of associators and braidings
Organizers
Speaker
Azat Gainutdinov
Time
Monday, June 8, 2026 4:00 PM - 5:00 PM
Venue
A3-1-301
Online
Zoom 559 700 6085
(BIMSA)
Abstract
I will talk about deformations of structures in tensor category theory: associativity constraints, monoidal structures of tensor functors, braiding isomorphisms, etc. As it is often in Algebra, infinitesimal deformations are controlled by Hochschild type complexes, called in this context Davydov-Yetter complex. We have reformulated the corresponding deformation cohomologies in terms of relative Ext groups of the standard adjunction between the tensor category C and its Drinfeld center. In particular, the deformation cohomology of a tensor category C is the 3rd self-extension group of the tensor unit of the Drinfeld center Z(C) relative to C. For any finite C-module category M, we prove that the deformation cohomology of mixed associators is the 2nd relative Ext group between the tensor unit of Z(C) and the full center object of M. Using relative homological algebra techniques, we furthermore describe the space of infinitesimal braidings of a braided tensor category C as an explicit (co)end involving only low-degree relative Ext’s. This allows us to calculate the Zariski tangent space to the affine variety of R-matrices for bosonization of exterior algebras in just a few lines. This is a joint work with M. Faitg and Ch. Schweigert.