Obstruction theory for algebras over an operad
Annales mathématiques Blaise Pascal, Volume 23 (2016) no. 1, pp. 75-107.

The goal of this paper is to set up an obstruction theory in the context of algebras over an operad and in the framework of differential graded modules over a field. Precisely, the problem we consider is the following: Suppose given two algebras $A$ and $B$ over an operad $\mathsf{P}$ and an algebra morphism from ${H}_{*}A$ to ${H}_{*}B$. Can we realize this morphism as a morphism of $\mathsf{P}$-algebras from $A$ to $B$ in the homotopy category? Also, if the realization exists, is it unique in the homotopy category?

We identify obstruction cocycles for this problem, and notice that they live in the first two groups of operadic $\Gamma$-cohomology.

Published online:
DOI: 10.5802/ambp.355
Classification: 55S35,  18D50,  55P48
Keywords: Obstruction theory, algebras over operads
Eric Hoffbeck 1

1 Université Paris 13, Sorbonne Paris Cité LAGA, CNRS, UMR 7539 99 avenue Jean-Baptiste Clément 93430 Villetaneuse, France
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Eric Hoffbeck. Obstruction theory for algebras over an operad. Annales mathématiques Blaise Pascal, Volume 23 (2016) no. 1, pp. 75-107. doi : 10.5802/ambp.355. https://ambp.centre-mersenne.org/articles/10.5802/ambp.355/

[1] Clemens Berger; Benoit Fresse Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc., Volume 137 (2004) no. 1, pp. 135-174 | DOI

[2] Clemens Berger; Ieke Moerdijk Axiomatic homotopy theory for operads, Comment. Math. Helv., Volume 78 (2003) no. 4, pp. 805-831 | DOI

[3] Clemens Berger; Ieke Moerdijk The Boardman-Vogt resolution of operads in monoidal model categories, Topology, Volume 45 (2006) no. 5, pp. 807-849 | DOI

[4] D. Blanc; W. G. Dwyer; P. G. Goerss The realization space of a $\Pi$-algebra: a moduli problem in algebraic topology, Topology, Volume 43 (2004) no. 4, pp. 857-892 | DOI

[5] W. G. Dwyer; J. Spaliński Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 73-126 | DOI

[6] Benoit Fresse Modules over operads and functors, Lecture Notes in Mathematics, 1967, Springer-Verlag, Berlin, 2009, x+308 pages | DOI

[7] Benoit Fresse Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, Alpine perspectives on algebraic topology (Contemp. Math.), Volume 504, Amer. Math. Soc., Providence, RI, 2009, pp. 125-188 | DOI

[8] Benoit Fresse Props in model categories and homotopy invariance of structures, Georgian Math. J., Volume 17 (2010) no. 1, pp. 79-160

[9] Ezra Getzler; J. D. S. Jones Operads, homotopy algebra and iterated integrals for double loop spaces (1994) (http://arxiv.org/abs/hep-th/9403055)

[10] Victor Ginzburg; Mikhail Kapranov Koszul duality for operads, Duke Math. J., Volume 76 (1994) no. 1, pp. 203-272 | DOI

[11] P. G. Goerss; M. J. Hopkins Moduli spaces of commutative ring spectra, Structured ring spectra (London Math. Soc. Lecture Note Ser.), Volume 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200 | DOI

[12] Stephen Halperin; James Stasheff Obstructions to homotopy equivalences, Adv. in Math., Volume 32 (1979) no. 3, pp. 233-279 | DOI

[13] Vladimir Hinich Homological algebra of homotopy algebras, Comm. Algebra, Volume 25 (1997) no. 10, pp. 3291-3323 | DOI

[14] Philip S. Hirschhorn Model categories and their localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, Providence, RI, 2003, xvi+457 pages

[15] Eric Hoffbeck $\Gamma$-homology of algebras over an operad, Algebr. Geom. Topol., Volume 10 (2010) no. 3, pp. 1781-1806 | DOI

[16] Mark Hovey Model categories, Mathematical Surveys and Monographs, 63, American Mathematical Society, Providence, RI, 1999, xii+209 pages

[17] T. V. Kadeišvili On the theory of homology of fiber spaces, Uspekhi Mat. Nauk, Volume 35 (1980) no. 3(213), pp. 183-188 International Topology Conference (Moscow State Univ., Moscow, 1979)

[18] Muriel Livernet On a plus-construction for algebras over an operad, $K$-Theory, Volume 18 (1999) no. 4, pp. 317-337 | DOI

[19] Martin Markl; Steve Shnider Associahedra, cellular $W$-construction and products of ${A}_{\infty }$-algebras, Trans. Amer. Math. Soc., Volume 358 (2006) no. 6, p. 2353-2372 (electronic) | DOI

[20] Alan Robinson Gamma homology, Lie representations and ${E}_{\infty }$ multiplications, Invent. Math., Volume 152 (2003) no. 2, pp. 331-348 | DOI

[21] Alan Robinson; Sarah Whitehouse Operads and $\Gamma$-homology of commutative rings, Math. Proc. Cambridge Philos. Soc., Volume 132 (2002) no. 2, pp. 197-234 | DOI

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