Koszul Duality of En-Operads

The goal of this paper is to prove a Koszul duality result for En-operads in differential graded modules over a ring. The case of an E1-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an En-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 En-operad En defines a cofibrant model of En. 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 En-operads in differential graded modules come in nested sequences E1 ⊂ E2 ⊂ · · · ⊂ E∞ 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 ↪→ En at the level of cobar constructions. Mathematics Subject Classification (2010). Primary: 55P48; Secondary: 18G55, 55S12.