Homological perturbation theory for algebras over operads

TL;DR

This paper extends homological perturbation theory to algebras over operads by introducing thick maps and pseudo-derivations, enabling explicit transfer formulas for complex algebraic structures like A-infinity and L-infinity algebras.

Contribution

It introduces the concept of thick maps and pseudo-derivations to generalize algebra homotopies for operad-based algebras, facilitating explicit transfer formulas.

Findings

Derived explicit transfer formulas for Cobar(C)-algebra structures

Applicable to O-infinity algebras for any Koszul operad

Formulas expressed via coderivation differentials on cofree C-coalgebras

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 what we call thick maps of O-algebras and special thick maps that we call pseudo-derivations, which serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. As an application, we derive explicit formulas for transferring Cobar(C)-algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O-infinity algebras for any Koszul operad O, in particular for A-infinity, C-infinity, L-infinity and G-infinity algebras. A key feature is that our formulas are expressed in terms of the compact description of Cobar(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.