Homological perturbation theory for algebras over operads

Berglund, Alexander

doi:10.2140/agt.2014.14.2511

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Homological perturbation theory for algebras over operads

Alexander Berglund

Algebraic & Geometric Topology 14 (2014) 2511–2548

DOI: 10.2140/agt.2014.14.2511

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 $O_{\infty}$ –algebras for any Koszul operad $O$ , 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 $Ω (C)$ –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.

Keywords

operads, strong homotopy algebras

Mathematical Subject Classification 2010

Primary: 18D50, 55P48

References

Publication

Received: 30 November 2011

Revised: 27 November 2013

Accepted: 11 February 2014

Published: 6 November 2014

Authors

Alexander Berglund
Department of Mathematics Stockholm University SE-106 91 Stockholm Sweden
http://people.su.se/~aberg/

Volume 14, issue 5 (2014)