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Symmetric homotopy theory for operads

Malte Dehling and Bruno Vallette

Algebraic & Geometric Topology 21 (2021) 1595–1660

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 idea, in this direction, 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.

homotopical algebra, operad, Koszul duality, $E_\infty$–operad
Mathematical Subject Classification 2010
Primary: 18D50
Secondary: 18G55
Received: 1 May 2017
Revised: 22 January 2019
Accepted: 26 April 2019
Published: 18 August 2021
Malte Dehling
Mathematisches Institut
Georg-August-Universität Göttingen
Bruno Vallette
Laboratoire J A Dieudonné
Université de Nice Sophia-Antipolis
Laboratoire Analyse, Géométrie et Applications
CNRS, UMR 7539
Université Paris 13