Algebraic operads up to homotopy

TL;DR

This paper develops a model category framework for the homotopy theory of differential graded operads, enabling a more accessible and enriched understanding of their homotopy properties through obstruction methods.

Contribution

It introduces a model category structure on curved conilpotent cooperads, Quillen equivalent to operads, simplifying the study of homotopy properties of differential graded operads.

Findings

Established a Quillen equivalence between categories

Provided a new approach to homotopy properties

Enhanced understanding of differential graded operads

Abstract

This paper deals with the homotopy theory of differential graded operads. We endow the Koszul dual category of curved conilpotent cooperads, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of operads. This allows us to describe the homotopy properties of differential graded operads in a simpler and richer way, using obstruction methods.