Algebraic & Geometric Topology

Homological perturbation theory for algebras over operads

Alexander Berglund

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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

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

Received: 30 November 2011
Revised: 27 November 2013
Accepted: 11 February 2014
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48] 55P48: Loop space machines, operads [See also 18D50]

operads strong homotopy algebras


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.

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  • D W Barnes, L A Lambe, A fixed point approach to homological perturbation theory, Proc. Amer. Math. Soc. 112 (1991) 881–892
  • C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805–831
  • J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer (1973)
  • E H Brown, Jr, Twisted tensor products, I, Ann. of Math. 69 (1959) 223–246
  • R Brown, The twisted Eilenberg–Zilber theorem, from: “Simposio di Topologia”, Edizioni Oderisi, Gubbio (1965) 33–37
  • S Eilenberg, S Mac Lane, On the groups of $H(\Pi,n)$, I, Ann. of Math. 58 (1953) 55–106
  • B Fresse, Koszul duality of operads and homology of partition posets, from: “Homotopy theory: Relations with algebraic geometry, group cohomology, and algebraic $K\!$–theory”, (P Goerss, S Priddy, editors), Contemp. Math. 346, Amer. Math. Soc. (2004) 115–215
  • B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer, Berlin (2009)
  • B Fresse, Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, from: “Alpine perspectives on algebraic topology”, (C Ausoni, K Hess, J Scherer, editors), Contemp. Math. 504, Amer. Math. Soc. (2009) 125–188
  • E Getzler, J D S Jones, Operads, homotopy algebra and iterated integrals for double loop spaces
  • V K A M Gugenheim, On the chain complex of a fibration, Illinois J. Math. 16 (1972) 398–414
  • V K A M Gugenheim, L A Lambe, Perturbation theory in differential homological algebra, I, Illinois J. Math. 33 (1989) 566–582
  • V K A M Gugenheim, L A Lambe, J D Stasheff, Algebraic aspects of Chen's twisting cochain, Illinois J. Math. 34 (1990) 485–502
  • V K A M Gugenheim, L A Lambe, J D Stasheff, Perturbation theory in differential homological algebra, II, Illinois J. Math. 35 (1991) 357–373
  • S Halperin, J Stasheff, Obstructions to homotopy equivalences, Adv. in Math. 32 (1979) 233–279
  • K Hess, Perturbation and transfer of generic algebraic structure, from: “Higher homotopy structures in topology and mathematical physics”, (J McCleary, editor), Contemp. Math. 227, Amer. Math. Soc. (1999) 103–143
  • J Huebschmann, On the construction of $A\sb \infty$–structures, Georgian Math. J. 17 (2010) 161–202
  • J Huebschmann, The Lie algebra perturbation lemma, from: “Higher structures in geometry and physics”, (A S Cattaneo, A Giaquinto, P Xu, editors), Progr. Math. 287, Birkhäuser/Springer (2011) 159–179
  • J Huebschmann, The sh-Lie algebra perturbation lemma, Forum Math. 23 (2011) 669–691
  • J Huebschmann, T Kadeishvili, Small models for chain algebras, Math. Z. 207 (1991) 245–280
  • J Huebschmann, J Stasheff, Formal solution of the master equation via HPT and deformation theory, Forum Math. 14 (2002) 847–868
  • L Johansson, L Lambe, E Sk öldberg, On constructing resolutions over the polynomial algebra, Homology Homotopy Appl. 4 (2002) 315–336
  • M W Johnson, D Yau, On homotopy invariance for algebras over colored PROPs, J. Homotopy Relat. Struct. 4 (2009) 275–315
  • B Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. 27 (1994) 63–102
  • M Kontsevich, Y Soibelman, Homological mirror symmetry and torus fibrations, from: “Symplectic geometry and mirror symmetry”, (K Fukaya, Y-G Oh, K Ono, G Tian, editors), World Sci. Publ., River Edge, NJ (2001) 203–263
  • L Lambe, J Stasheff, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987) 363–376
  • K Lefèvre-Hasegawa, Sur les $A$–infini catégories
  • J-L Loday, B Vallette, Algebraic operads, Grundl. Math. Wissen. 346, Springer, Heidelberg (2012)
  • S Mac Lane, Categories for the working mathematician, 2nd edition, Graduate Texts in Mathematics 5, Springer (1998)
  • M Markl, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004) 129–160
  • S A Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices (1999) 153–164
  • M Schlessinger, J Stasheff, Deformation theory and rational homotopy type