## Algebraic & Geometric Topology

### Homological perturbation theory for algebras over operads

Alexander Berglund

#### 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 $O∞$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2511-2548.

Dates
Revised: 27 November 2013
Accepted: 11 February 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715994

Digital Object Identifier
doi:10.2140/agt.2014.14.2511

Mathematical Reviews number (MathSciNet)
MR3276839

Zentralblatt MATH identifier
1305.18030

Keywords

#### Citation

Berglund, Alexander. Homological perturbation theory for algebras over operads. Algebr. Geom. Topol. 14 (2014), no. 5, 2511--2548. doi:10.2140/agt.2014.14.2511. https://projecteuclid.org/euclid.agt/1513715994

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