Davydov-Yetter cohomology for Tensor Triangulated Categories

TL;DR

This paper develops a cohomology theory analogous to Davydov-Yetter cohomology for tensor triangulated categories, especially derived categories from algebraic geometry, to understand their deformation theory.

Contribution

It introduces perfect pseudo dg-tensor structures and defines a double complex to analyze infinitesimal deformations of tensor triangulated categories.

Findings

The 4th cohomology group captures first order deformation information.

Provides a framework for deformation theory in tensor triangulated categories.

Extends Davydov-Yetter cohomology concepts to a new categorical setting.

Abstract

One way to understand the deformation theory of a tensor category $M$ is through its Davydov-Yetter cohomology $H_{DY}^{\ast}(M)$ which in degree 3 and 4 is known to control respectively first order deformations of the associativity coherence of $M$ and their obstructions. \\ In this work we take the task of developing an analogous theory for the deformation theory of tensor triangulated categories with a focus on derived categories coming from algebraic geometry. We introduce the concept of perfect pseudo dg-tensor structure $\Gamma$ on an appropriate dg-category $\mathscr{T}$ as a truncated dg-lift of a tensor triangulated category structure on $H^{0}(\mathscr{T})$ and we define a double complex $DY^{\ast,\ast}(\Gamma)$ and we see that the 4th cohomology group $HDY^{4}(\Gamma)$ of the total complex of $DY^{\ast,\ast}(\Gamma)$ contains information about infinitesimal first order deformations of the tensor structure.