About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Petersen, Dan
A closer look at Kadeishvili's theorem. (English). Higher Structures, vol. 4 (2020), issue 2, pp. 211-221
MSC: 16E45, 16S80, 18D50 | MR 4133168 | Zbl 1457.18020 | DOI: 10.21136/HS.2020.14
Keywords:
transfer theorem; Koszul duality; homological perturbation theory; operads
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.
References:
[1] Berger, Clemens, Moerdijk, Ieke: Axiomatic homotopy theory for operads. Comment. Math. Helv., 78(4):805–831 MR 2016697
[2] Berglund, Alexander: Homological perturbation theory for algebras over operads. Algebr. Geom. Topol., 14(5):2511–2548 MR 3276839
[3] Burke, Jesse: Transfer of A-infinity structures to projective resolutions. Preprint available at arXiv:1801.08933
[4] Dotsenko, Vladimir, Poncin, Norbert: A tale of three homotopies. Appl. Categ. Structures, 24(6):845–873 MR 3572456
[5] Dotsenko, Vladimir, Shadrin, Sergey, Vallette, Bruno: Pre-Lie deformation theory. Mosc. Math. J., 16(3):505–543 MR 3510210
[6] Gugenheim, Victor K. A. M., Lambe, Larry A.: Perturbation theory in differential homological algebra. I. Illinois J. Math., 33(4):566–582
[7] Gugenheim, Victor K. A. M., Lambe, Larry A., Stasheff, James D.: Perturbation theory in differential homological algebra. II. Illinois J. Math., 35(3):357–373
[8] Huebschmann, Johannes, Kadeishvili, Tornike: Small models for chain algebras. Math. Z., 207(2):245–280
[9] Huebschmann, Johannes, Stasheff, Jim: Formal solution of the master equation via HPT and deformation theory. Forum Math., 14(6):847–868 MR 1932522
[10] Kadeišvili, Tornike V.: On the theory of homology of fiber spaces. Uspekhi Mat. Nauk, 35(3(213)):183–188. International Topology Conference (Moscow State Univ., Moscow, 1979) MR 0580645
[11] Kontsevich, Maxim, Soibelman, Yan: Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203–263. World Sci. Publ., River Edge, NJ
[12] Lefèvre-Hasegawa, Kenji: Sur les A_(∞)-catégories. PhD thesis, Université Paris Diderot (Paris 7)
[13] Loday, Jean-Louis, Vallette, Bruno: Algebraic operads, volume 346 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg MR 2954392
[14] Manetti, Marco: A relative version of the ordinary perturbation lemma. Rend. Mat. Appl. (7), 30(2):221–238 MR 2760671
[15] Markl, Martin: Homotopy algebras are homotopy algebras. Forum Math., 16(1):129–160 MR 2034546
[16] Markl, Martin: Transferring A_(∞) (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2) Suppl., (79):139–151 MR 2287133
[17] Merkulov, Sergei A.: Strong homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices, (3):153–164
[18] Rogers, Christopher L.: Homotopical properties of the simplicial Maurer-Cartan functor. In 2016 MATRIX annals, volume 1 of MATRIX Book Ser., pages 3–15. Springer, Cham MR 3792513
[19] Spaltenstein, Nicolas: Resolutions of unbounded complexes. Compositio Math., 65(2):121–154
[20] Laan, Pepijn van der: Operads. Hopf algebras and coloured Koszul duality. PhD thesis, Universiteit Utrecht
Partner of
EuDML logo