Homological perturbation theory for algebras over operads

Abstract

We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this problem, we introduce thick maps of O-algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. As an application, we derive explicit formulas for transferring Ω(C) -algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O∞ -algebras for any Koszul operad O, in particular for A∞-, C∞-, L∞- and G∞-algebras. A key feature is that our formulas are expressed in terms of the compact description of Ω(C) -algebras as coderivation differentials on cofree C-coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy.