Davydov-Yetter cohomology, comonads and Ocneanu rigidity

TL;DR

This paper links Davydov-Yetter cohomology of finite tensor categories to comonad cohomology, providing a new proof of Ocneanu rigidity and enabling computations for certain Hopf algebras.

Contribution

It establishes an equivalence between Davydov-Yetter cohomology and comonad cohomology for finite tensor categories, offering new computational tools.

Findings

Proves Davydov-Yetter cohomology is equivalent to comonad cohomology.

Provides a conceptual proof of Ocneanu rigidity.

Computes cohomology for specific non-semisimple Hopf algebras.

Abstract

Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad arising from the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity. Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra.