Gerstenhaber type structures on Davydov-Yetter cohomology with coefficients

TL;DR

This paper develops Gerstenhaber algebra structures on Davydov-Yetter cohomology with coefficients, revealing new algebraic operations and relations that extend the understanding of monoidal functors and their cohomological properties.

Contribution

It introduces Gerstenhaber structures on Davydov-Yetter cohomology with coefficients, utilizing a novel analogy with entwining of coalgebras and algebras, and constructs a subcomplex with Gerstenhaber algebra cohomology.

Findings

Davydov-Yetter complex with coefficients has a weak comp algebra structure.

Two distinct cup products $ and $ are defined and related.

A subcomplex with Gerstenhaber algebra cohomology is constructed.

Abstract

We obtain Gerstenhaber type structures on Davydov-Yetter cohomology with coefficients in half-braidings for a monoidal functor. Our approach uses a formal analogy between half-braidings of a monoidal functor and the entwining of a coalgebra with an algebra. We show that the Davydov-Yetter complex with coefficients carries the structure of a weak comp algebra. In particular, it is equipped with two distinct cup product structures $\cup$ and $\sqcup$ which are related in a manner that replaces graded commutativity. We also introduce a subcomplex of the Davydov-Yetter complex with coefficients whose cohomology forms a Gerstenhaber algebra in the usual sense.