The rational homotopy theory of operads (mini-course)
Seminar Room 2, Newton Institute Gatehouse
AbstractThe first purpose of this lecture is to explain the definition of an analogue of the Sullivan model for the rational homotopy of topological operads, and the definition of a rationalization functor on operads.
The Sullivan model of an operad involves both a commutative dg-algebra structure, which encodes the rational homotopy type of the spaces underlying the operad, and a cooperad structure, which models the composition structure attached to our topological operad. If we neglect the commutative algebra part of the structure, then we get a model for the stable rational homotopy of operads, and an operadic version of the Sullivan miminal model can also be defined in this setting. But the construction of minimal models fails when we deal with the combination of commutative algebra and operad structures involved in our model for the rational homotopy of operads in topological spaces. So does the definition of the Quillen model, as well as the classical approach to integrate deformation complexes into rational homotopy automorphism groups.
I will explain methods to bypass these difficulties, and which can be used to establish the main result of this lecture series in the little 2-discs case.General reference:B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"