Operads, graph complexes, and Grothendieck-Teichmüller groups

The main purpose of this course is to give an introduction to the computation of the rational homotopy of mapping spaces of operads, with motivations coming from the calculus of embeddings and the study of Grothendieck-Teichmüller groups.

In short, an operad P is a structure formed by a collection of objects P(n) whose elements model operations with n inputs and one output. The structure of an operad can be used to define a category of algebras and one can observe that the usual categories of algebras, the category of commutative algebras, the category of associative algebras, the category of Lie algebras, are associated to operads in vector spaces.

For the motivating applications of the course, we consider the classes of E_n-operads, where n = 1,2,... is a dimension parameter. The class of E_n-operads consists of the operads in topological spaces that are (weakly) homotopy equivalent to a reference model, the operads of litlle n-cubes. This notion of operads was introduced for the study of iterated loop spaces. The relationship with the calculus of embeddings come from the Goodwillie-Weiss calculus, which implies that operadic mapping spaces associated to E_n-operads can be used to compute the homotopy of certain approximations embedding spaces and the homotopy of the embedding spaces themselves when the codimension of the embeddings is large enough. The relationship with the Grothendieck-Teichmüller groups comes from the observation that the rational (resp. profinite) Grothendieck-Teichmüller group represents the group of classes of homotopy equivalences of a rationalization (a profinite completion) of E₂-operads. In a recent collaboration (with Victor Turchin and Thomas Willwacher), we proved that, in the realm of rational homotopy, we can compute the homotopy of these mapping spaces and of these homotopy automorphism spaces associated to E_n-operads in terms of spaces of Maurer-Cartan forms on graph complexes.

The mini-course consists of three lectures, recorded by IHP. The first lecture includes a survey of the motiviting result and a basic introduction to operads. The second lecture is a continuation of the study of operads, and an introduction to methods of homotopy theory. In the third lecture, I explain the applications of these methods of homotopy theory to operads, and the ideas which make graph complexes appear in the case of E_n-operads.

Recordings

Lecture 1 (May 3, 2023)

Part 1: Introduction. Motivating applications to embedding calculus and Grothendieck-Teichmüller groups. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-1

Part 2: The general definition of the notion of an operad. Trees and composition of operations. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-2

Lecture 2 (May 10, 2023)

Part 1: The general definition of the notion of an operad (continued). The construction of free objects. The definition of the operads governing commutative, associative an Lie algebras. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-3

Part 2: Methods of homotopy theory. The definition of the notion of a model category. The example of topological spaces, simplicial sets and differential graded modules. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-4

Lecture 3 (May 17, 2023)

Part 1. The applications of methods of homotopy theory to operads. The model categories of operads. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-5

Part 2. The graph operad as a model for the rationalization of En-operads. https://www.carmin.tv/en/video/operads-graph-complexes-and-grothendieck-teichmuller-groups-part-6

References

1. BF, Homotopy of operads & Grothendieck-Teichmüller groups. Mathematical Surveys and Monographs 217, American Mathematical Society, 2017.
2. BF, T. Willwacher, The intrinsic formality of E_n-operads. J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 2047-2133.
3. BF, T. Willwacher, Mapping spaces for dg Hopf cooperads and homotopy automorphisms of the rationalization of E_n-operads. Preprint arXiv:2003.02939 (2020).
4. BF, V. Turchin, T. Willwacher, The rational homotopy of mapping spaces of E_n-operads. Preprint arXiv:1703.06123 (2017).