Homotopy of Operads and Grothendieck-Teichmuller Groups

Part 1: the Algebraic Theory and Its Topological Background

Benoit Fresse (Autor)

Buch | Hardcover

563 Seiten

2017
American Mathematical Society (Verlag)
978-1-4704-3481-6 (ISBN)

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The Grothendieck-Teichmuller 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-Teichmuller 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.

Benoit Fresse, Universite de Lille 1, Villeneuve d'Ascq, France.

From operads to Grothendieck-Teichmuller 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 $E_n$-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-Teichmuller group: The Malcev completion of the braid operads and Drinfeld's associators
The Grothendieck-Teichmuller group
A glimpse at the Grothendieck program
Appendices: Trees and the construction of free operads
The cotriple resolution of operads
Glossary of notation
Bibliography
Index

Erscheinungsdatum	08.06.2017
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	1180 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-3481-4 / 1470434814
ISBN-13	978-1-4704-3481-6 / 9781470434816
Zustand	Neuware