Abstract
In this chapter, we develop the Koszul duality theory for operads. We follow the same pattern as for associative algebras: quadratic data, Koszul dual (co)operad, Koszul complex and Koszul resolution. This last one provides us with the minimal model of the operad , thereby defining the notion of -algebra up to homotopy. In the last section, we extend this method to inhomogeneous quadratic operads.
Les maths, c’est comme l’amour, ça ne s’apprend pas dans les livres mais en pratiquant.
Adrien Douady (private communication)
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References
—, Koszul duality of operads and homology of partition posets, in “Homotopy theory and its applications (Evanston, 2002)”, Contemp. Math. 346 (2004), 115–215.
I. Galvez-Carrillo, A. Tonks, and B. Vallette, Homotopy Batalin-Vilkovisky algebras, Journal Noncommutative Geometry (2009), arXiv:0907.2246, 49 pp.
E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285.
—, Operads and moduli spaces of genus 0 Riemann surfaces, The moduli space of curves (Texel Island, 1994). Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 199–230.
E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, hep-th/9403055 (1994).
V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272.
E. Getzler and M. M. Kapranov, Cyclic operads and cyclic homology, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 167–201.
J. Hirsch and J. Millès, Curved Koszul duality theory, arXiv:1008.5368 (2010).
Mikhail M. Kapranov, The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Algebra 85 (1993), no. 2, 119–142.
—, Distributive laws and Koszulness, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323.
Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002.
—, A Koszul duality for props, Trans. of Amer. Math. Soc. 359 (2007), 4865–4993.
—, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math. 620 (2008), 105–164.
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Loday, JL., Vallette, B. (2012). Koszul Duality of Operads. In: Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol 346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30362-3_7
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DOI: https://doi.org/10.1007/978-3-642-30362-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30361-6
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