# Homotopy Gerstenhaber algebras are strongly homotopy commutative

## TL;DR

This paper demonstrates that homotopy Gerstenhaber algebras inherently possess a strongly homotopy commutative structure, connecting different notions of homotopy commutativity through additional operations.

## Contribution

It establishes a natural link between homotopy Gerstenhaber algebras and strongly homotopy commutative algebras, clarifying their structural relationship.

## Key findings

- Homotopy Gerstenhaber algebras are naturally shc algebras.
- Additional operations induce homotopy commutativity.
- Connects different frameworks of homotopy algebra structures.

## Abstract

We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a cup-1 product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.