Symmetric homotopy theory for operads

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.

Pre-Lie deformation theory

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.