Symmetric homotopy theory for operads

TL;DR

This paper develops a new homotopy theory for differential graded operads over any ring by incorporating symmetric group actions into the operad structure, introducing higher cooperads, and establishing a new bar-cobar adjunction.

Contribution

It introduces a novel approach to operad homotopy theory by considering symmetric group actions as part of the structure, along with new dual categories and adjunctions.

Findings

Higher bar-cobar construction yields cofibrant replacements for operads.

Established homotopy properties for higher homotopy operads.

Developed a conceptual framework using curved Koszul duality for colored operads.

Abstract

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric groups as part of the defining structure of an operad and not as the underlying category. We introduce a new dual category of higher cooperads, a new higher bar-cobar adjunction with the category of operads, and a new higher notion of homotopy operads, for which we establish the relevant homotopy properties. For instance, the higher bar-cobar construction provides us with a cofibrant replacement functor for operads over any ring. All these constructions are produced conceptually by applying the curved Koszul duality for colored operads. This paper is a first step toward a new Koszul duality theory for operads, where the action of the symmetric groups is properly taken into account.