A tale of three homotopies

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.

22 pages, final version

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• Convolution algebras and the deformation theory of infinity-morphisms
• Gauge equivalence for complete $L_{\infty }$-algebras
• Derived homotopy algebras
• Homological sections of Lie algebroids
• Homotopy probability theory on a Riemannian manifold and the Euler equation
• Stability of fixed points of Dirac structures