Homotopy of Operads and Grothendieck–Teichmüller Groups, Part 1: Part 1: The Algebraic Theory and its Topological Background
About this Title
Benoit Fresse, Université de Lille 1, Villeneuve d’Ascq, France
Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 217.1
ISBNs: 978-1-4704-3481-6 (print); 978-1-4704-3755-8 (online)
MathSciNet review: MR3643404
MSC: Primary 55P48; Secondary 17B55, 18D50, 20F38, 20F40, 55P10, 55P62, 57T05
The Grothendieck–Teichmüller group was defined by Drinfeld in quantum group theory with insights coming from the Grothendieck program in Galois theory. The ultimate goal of this book is to explain that this group has a topological interpretation as a group of homotopy automorphisms associated to the operad of little 2-discs, which is an object used to model commutative homotopy structures in topology.
This volume gives a comprehensive survey on the algebraic aspects of this subject. The book explains the definition of an operad in a general context, reviews the definition of the little discs operads, and explains the definition of the Grothendieck–Teichmüller group from the viewpoint of the theory of operads. In the course of this study, the relationship between the little discs operads and the definition of universal operations associated to braided monoidal category structures is explained. Also provided is a comprehensive and self-contained survey of the applications of Hopf algebras to the definition of a rationalization process, the Malcev completion, for groups and groupoids.
Most definitions are carefully reviewed in the book; it requires minimal prerequisites to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.
Graduate students and researchers interested in algebraic topology and algebraic geometry.
Table of Contents
From operads to Grothendieck–Teichmüller groups
The general theory of operads
- The basic concepts of the theory of operads
- The definition of operadic composition structures revisited
- Symmetric monoidal categories and operads
Braids and $E_2$-operads
- The little discs model of $E_n$-operads
- Braids and the recognition of $E_2$-operads
- The magma and parenthesized braid operators
Hopf algebras and the Malcev completion
The operadic definition of the Grothendieck–Teichmüller group
- The Malcev completion of the braid operads and Drinfeld’s associators
- The Grothendieck–Teichmüller group
- A glimpse at the Grothendieck program