A tale of three homotopies

TL;DR

This paper unifies various existing notions of homotopy between homotopy morphisms of homotopy us algebras for Koszul operads, clarifying their relationships within a common framework.

Contribution

It establishes the connections between different approaches to homotopies of homotopy us algebras, using the homotopy transfer theorem for homotopy cooperads.

Findings

Unified framework for homotopies of homotopy us algebras

Clarified relationships between existing notions

Applied homotopy transfer theorem to establish links

Abstract

For a Koszul operad $\mathcal{P}$, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy $\mathcal{P}$-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.