Koszul duality of E_n-operads

TL;DR

This paper proves a Koszul duality theorem for E_n-operads in differential graded modules, establishing a cofibrant model that realizes the minimal model of the n-Gerstenhaber operad at the chain level.

Contribution

It introduces a new cofibrant model for E_n-operads via operadic cobar construction, connecting to the minimal model of the n-Gerstenhaber operad.

Findings

Operadic cobar construction yields a cofibrant model of E_n.

The model realizes the minimal n-Gerstenhaber operad at the chain level.

Defines a model for operad embeddings E_{n-1} --> E_n.

Abstract

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 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 of operads 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 E_n-1 --> E_n at the level of cobar constructions.