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A closer look at Kadeishvili’s theorem. (English) Zbl 1457.18020

Summary: We give a proof of the Homotopy Transfer Theorem following Kadeishvili’s original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an \(A_infty\)-structure on homology, we will see that a small modification of his argument proves the general case of transferring any kind of \(\infty\) -algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.

MSC:

18M50 Bimonoidal, skew-monoidal, duoidal categories
16E45 Differential graded algebras and applications (associative algebraic aspects)
16S80 Deformations of associative rings
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[1] Clemens Berger and Ieke Moerdijk. Axiomatic homotopy theory for operads.Comment. Math. Helv., 78(4):805-831, 2003. · Zbl 1041.18011
[2] Alexander Berglund. Homological perturbation theory for algebras over operads.Algebr. Geom. Topol., 14(5):2511-2548, 2014. · Zbl 1305.18030
[3] Jesse Burke. Transfer of A-infinity structures to projective resolutions. Preprint available at arXiv:1801.08933.
[4] Vladimir Dotsenko and Norbert Poncin. A tale of three homotopies.Appl. Categ. Structures, 24(6):845-873, 2016. · Zbl 1375.18076
[5] Vladimir Dotsenko, Sergey Shadrin, and Bruno Vallette. Pre-Lie deformation theory.Mosc. Math. J., 16(3):505-543, 2016. · Zbl 1386.18054
[6] Victor K. A. M. Gugenheim and Larry A. Lambe. Perturbation theory in differential homological algebra. I.Illinois J. Math., 33(4):566-582, 1989. · Zbl 0661.55018
[7] Victor K. A. M. Gugenheim, Larry A. Lambe, and James D. Stasheff. Perturbation theory in differential homological algebra. II.Illinois J. Math., 35(3):357-373, 1991. · Zbl 0727.55012
[8] Johannes Huebschmann and Tornike Kadeishvili. Small models for chain algebras.Math. Z., 207(2):245-280, 1991. · Zbl 0723.57030
[9] Johannes Huebschmann and Jim Stasheff. Formal solution of the master equation via HPT and deformation theory.Forum Math., 14(6):847-868, 2002. · Zbl 1036.17016
[10] Tornike V. Kadeišvili.On the theory of homology of fiber spaces.Uspekhi Mat. Nauk, 35(3(213)):183-188, 1980. International Topology Conference (Moscow State Univ., Moscow, 1979).
[11] Maxim Kontsevich and Yan Soibelman. Homological mirror symmetry and torus fibrations. InSymplectic geometry and mirror symmetry (Seoul, 2000), pages 203-263. World Sci. Publ., River Edge, NJ, 2001. · Zbl 1072.14046
[12] Kenji Lefèvre-Hasegawa.Sur lesA∞-catégories. PhD thesis, Université Paris Diderot (Paris 7), 2003.
[13] Jean-Louis Loday and Bruno Vallette.Algebraic operads, volume 346 ofGrundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 2012. · Zbl 1260.18001
[14] Marco Manetti. A relative version of the ordinary perturbation lemma.Rend. Mat. Appl. (7), 30(2):221-238, 2010. · Zbl 1238.16011
[15] Martin Markl. Homotopy algebras are homotopy algebras.Forum Math., 16(1):129-160, 2004. · Zbl 1067.55011
[16] Martin Markl. TransferringA∞(strongly homotopy associative) structures.Rend. Circ. Mat. Palermo (2) Suppl., (79):139-151, 2006. · Zbl 1112.18007
[17] Sergei A. Merkulov. Strong homotopy algebras of a Kähler manifold.Internat. Math. Res. Notices, (3):153-164, 1999. · Zbl 0995.32013
[18] Christopher L. Rogers. Homotopical properties of the simplicial Maurer-Cartan functor. In 2016 MATRIX annals, volume 1 ofMATRIX Book Ser., pages 3-15. Springer, Cham, 2018. · Zbl 1448.18035
[19] Nicolas Spaltenstein. Resolutions of unbounded complexes.Compositio Math., 65(2):121- 154, 1988. · Zbl 0636.18006
[20] Pepijn van der Laan.
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