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 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.

Revised version, to appear in Algebr. Geom. Topol., 29 pages

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• Koszul duality of operads and homology of partition posets
• Sur les A-infini catégories
• Transferring homotopy commutative algebraic structures
• On homotopy invariance for algebras over colored PROPs
• On the construction of A-infinity structures
• Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

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