Journal
SELECTA MATHEMATICA-NEW SERIES
Volume 17, Issue 2, Pages 363-434Publisher
BIRKHAUSER VERLAG AG
DOI: 10.1007/s00029-010-0047-6
Keywords
E-n-operads; Gerstenhaber operads; Koszul duality; Cobar construction; Complete graph operad; kappa-operads
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Funding
- OBTH [ANR-06-JCJC-0042]
- Agence Nationale de la Recherche (ANR) [ANR-06-JCJC-0042] Funding Source: Agence Nationale de la Recherche (ANR)
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The goal of this paper is to prove a Koszul duality result for E (n) -operads in differential graded modules over a ring. The case of an E (1)-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an E (n) -operad is identified with the n-Gerstenhaber operad and forms another well-known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an E (n) -operad E-n defines a cofibrant model of E-n. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul duality. Most models of E (n) -operads in differential graded modules come in nested sequences E-1 subset of E-2 subset of ......subset of E-infinity homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings En-1 -> E-n at the level of cobar constructions.
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