Hardcover ISBN:  9781470434809 
Product Code:  SURV/217 
List Price:  $250.00 
MAA Member Price:  $225.00 
AMS Member Price:  $200.00 
Hardcover ISBN:  9781470434809 
Product Code:  SURV/217 
List Price:  $250.00 
MAA Member Price:  $225.00 
AMS Member Price:  $200.00 

Book DetailsMathematical Surveys and MonographsVolume: 217; 2017; 1236 ppMSC: Primary 55; 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 set 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.
The first part of this twopart set 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.
The ultimate goal of the second part of the book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.
The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2disc operad.
This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
ReadershipGraduate students and researchers interested in algebraic topology and algebraic geometry.
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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 mathematial literature.
Bruno Vallette, Mathematical Reviews 
The scope of this book is vast, with a highly interesting result as its culmination. Yet, the book will be valuable for researchers beyond simply those wanting to understand the main theorem. It is a good reference for many topics in homotopy theory and operad theory.
Julia Bergner, Mathematical Reviews


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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 set 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.
The first part of this twopart set 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.
The ultimate goal of the second part of the book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.
The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2disc operad.
This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
Graduate students and researchers interested in algebraic topology and algebraic geometry.

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 mathematial literature.
Bruno Vallette, Mathematical Reviews 
The scope of this book is vast, with a highly interesting result as its culmination. Yet, the book will be valuable for researchers beyond simply those wanting to understand the main theorem. It is a good reference for many topics in homotopy theory and operad theory.
Julia Bergner, Mathematical Reviews