Coloured Koszul duality and strongly homotopy operads

TL;DR

This paper establishes Koszul duality for coloured operads and introduces strongly homotopy operads, demonstrating their relevance through the example of configuration spaces and the little disks operad.

Contribution

It extends Koszul duality to coloured operads and defines strongly homotopy operads as a new homotopy invariant framework.

Findings

Rational chains on configuration spaces form a strongly homotopy operad

Strongly homotopy operads are quasi-isomorphic to chains on the little disks operad

Provides a new perspective on operad homotopy theory

Abstract

This paper proves Koszul duality for coloured operads and uses it to introduce strongly homotopy operads as a suitable homotopy invariant version of operads. It shows that rational chains on configuration spaces of points in the plane form a strongly homotopy operad quasi isomorphic to the chains on the little disks operad.