Homological perturbation theory for algebras over operads - Mathematics > Algebraic TopologyReport as inadecuate




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Abstract: We extend homological perturbation theory to encompass algebraic structuresgoverned by operads and cooperads. The main difficulty is to find a suitablenotion of algebra homotopy that generalizes to algebras over operads O. Tosolve this problem, we introduce what we call thick maps of O-algebras andspecial thick maps that we call pseudo-derivations, which serve as appropriategeneralizations of algebra homotopies for the purposes of homologicalperturbation theory.As an application, we derive explicit formulas for transferringCobarC-algebra structures along contractions, where C is any connectedcooperad in chain complexes. This specializes to transfer formulas forO-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 ourformulas are expressed in terms of the compact description of CobarC-algebrasas coderivation differentials on cofree C-coalgebras. Moreover, we get formulasnot only for the transferred structure and a structure on the inclusion, butalso for structures on the projection and the homotopy.



Author: Alexander Berglund

Source: https://arxiv.org/







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