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Homological perturbation theory for algebras over operads

Alexander Berglund

Algebraic & Geometric Topology 14 (2014) 2511–2548

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.

operads, strong homotopy algebras
Mathematical Subject Classification 2010
Primary: 18D50, 55P48
Received: 30 November 2011
Revised: 27 November 2013
Accepted: 11 February 2014
Published: 6 November 2014
Alexander Berglund
Department of Mathematics
Stockholm University
SE-106 91