Deformations of monoidal functors

TL;DR

This paper demonstrates that the deformational Hochschild complex of a monoidal functor admits a strong homotopy Gerstenhaber algebra structure, linking deformation theory of monoidal functors to algebraic deformation frameworks.

Contribution

It introduces a Gerstenhaber-Voronov type structure on the Hochschild complex of monoidal functors, extending deformation theory analogies from associative algebras.

Findings

Gerstenhaber-Voronov operations define a strong homotopy Gerstenhaber algebra

Deformation theory of monoidal functors parallels associative algebra deformations

A quasi-classical limit relates to Poisson structures

Abstract

We point out that for Yetter's deformational Hochschild complex of a monoidal functor between abelian monoidal categories the Gerstenhaber-Voronov type operations can be defined making it a strong homotopy Gerstenhaber algebra. This encodes deformation theory of monoidal functors in an analogical way as deformation theory of associative algebras is described by the strong homotopy Gerstenhaber algebra structure on the corresponding Hochschild cochains. We describe a quasi-classical limit of deformations of a symmetric monoidal functor in terms of Poisson type structure.