Symmetric homotopy theory for operads - Archive ouverte HAL
Article Dans Une Revue Algebraic and Geometric Topology Année : 2021

Symmetric homotopy theory for operads

Malte Dehling
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-01142687 , version 1 (15-04-2015)

Identifiants

Citer

Malte Dehling, Bruno Vallette. Symmetric homotopy theory for operads. Algebraic and Geometric Topology, 2021, 21 (4), pp.1595-1660. ⟨10.2140/agt.2021.21.1595⟩. ⟨hal-01142687⟩
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