The n-Category Café

Homotopy of Operads and Grothendieck-Teichmüller Groups

Posted by John Baez

Benoit Fresse has finished a big two-volume book on operads, which you can now see on his website:

• Benoit Fresse, Homotopy of Operads and Grothendieck-Teichmüller Groups.

He writes:

The first aim of this book project is to give an overall reference, starting from scratch, on the application of methods of algebraic topology to operads. To be more specific, one of our main objectives is the development of a rational homotopy theory for operads. Most definitions, notably fundamental concepts of operad and homotopy theory, are carefully reviewed in order to make our account accessible to a broad readership, which should include graduate students, as well as researchers coming from the various fields of mathematics related to our main topics.

The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, we set up the background required by an in-depth study of this subject, and we make precise the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.

The first volume is over 500 pages:

Contents: Introduction to the general theory of operads. Introduction to $E_n$-operads. Relationship between $E_2$-operads and (braided) monoidal categories. Applications of Hopf algebras to the Malcev completion of groups, of groupoids, and of operads in groupoids. Operadic definition of the Grothendieck-Teichmüller groups and of the set of Drinfeld’s associators. Appendices on free operads, trees and the cotriple resolution of operads.

The second volume is over 700 pages:

Contents: Introduction to general methods of the theory of model categories. The homotopy theory of modules, algebras, and the rational homotopy of spaces. The (rational) homotopy of operads. Applications of the rational homotopy theory to $E_n$-operads. Homotopy spectral sequences and the computation of homotopy automorphism spaces of operads. Applications to $E_2$-operads and the homotopy interpretation of the Grothendieck-Teichmüller group. Appendix on cofree cooperads and the Koszul duality of operads.