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Hardcover ISBN:  9781470434816 
Product Code:  SURV/217.1 
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Book DetailsMathematical Surveys and MonographsVolume: 217; 2017; 532 ppMSC: Primary 55; Secondary 18; 57; 20
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 2discs, 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 selfcontained 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.
ReadershipGraduate students and researchers interested in algebraic topology and algebraic geometry.
This item is also available as part of a set: 
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

Hopf algebras

The Malcev completion for groups

The Malcev completion for groupoids and operads

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

Appendices

Trees and the construction of free operads

The cotriple resolution of operads


Additional Material

Reviews

Even if the main goal of the first volume is to provide the reader with the necessary prerequisites to understand the deep theory developed by the author in the second volume, it nevertheless provides the literature with an interesting treatise, accessible to graduate students and to researchers working in any field. It should first be mentioned that the mathematical content covered here is absolutely beautiful.In the end, the present first volume of this treatise on the homotopy of operads and GrothendieckTeichmüller groups represents a huge amount of work and is a valuable addition to the current mathematical literature.
Bruno Vallette, Mathematical Reviews 
This volume provides a clear and comprehensive introduction to the theory of operads and some of its applications, and it should indeed achieve the author's aim 'to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.'
Steffen Sagave, Zentralblatt MATH


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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 2discs, 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 selfcontained 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.

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

Hopf algebras

The Malcev completion for groups

The Malcev completion for groupoids and operads

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

Appendices

Trees and the construction of free operads

The cotriple resolution of operads

Even if the main goal of the first volume is to provide the reader with the necessary prerequisites to understand the deep theory developed by the author in the second volume, it nevertheless provides the literature with an interesting treatise, accessible to graduate students and to researchers working in any field. It should first be mentioned that the mathematical content covered here is absolutely beautiful.In the end, the present first volume of this treatise on the homotopy of operads and GrothendieckTeichmüller groups represents a huge amount of work and is a valuable addition to the current mathematical literature.
Bruno Vallette, Mathematical Reviews 
This volume provides a clear and comprehensive introduction to the theory of operads and some of its applications, and it should indeed achieve the author's aim 'to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.'
Steffen Sagave, Zentralblatt MATH