Symmetric homotopy theory for 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.

40 pages. Comments are welcome

References in corpus (6)

• On weak Lie 2-algebras
• Coloured Koszul duality and strongly homotopy operads
• Tensor products of representations up to homotopy
• Homotopy theory of homotopy algebras
• The universal Hopf operads of the bar construction
• From homotopy operads to infinity-operads

Cited by in corpus (9)

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• Deformation theory of Cohomological Field Theories
• The diagonal of the operahedra
• Moduli problems for operadic algebras
• On the Goodwillie derivatives of the identity in structured ring spectra
• Combinatorial homotopy theory for operads
• PD Operads and Explicit Partition Lie Algebras
• Coextension of scalars in operad theory
• $E_2{L}_{\infty }$-algebras, Generalized Geometry, and Tensor Hierarchies