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 $\mathcal{O}$. To solve this problem, we introduce thick maps of $\mathcal{O}$–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 $\Omega \left(\mathcal{C}\right)$–algebra structures along contractions, where $\mathcal{C}$ is any connected cooperad in chain complexes. This specializes to transfer formulas for ${\mathcal{O}}_{\infty }$–algebras for any Koszul operad $\mathcal{O}$, in particular for $A_{\infty }$–, $C_{\infty }$–, $L_{\infty }$– and $G_{\infty }$–algebras. A key feature is that our formulas are expressed in terms of the compact description of $\Omega \left(\mathcal{C}\right)$–algebras as coderivation differentials on cofree $\mathcal{C}$–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.