Homotopy of Operads and Grothendieck-Teichmuller Groups

Parts 1 and 2

Benoit Fresse (Autor)

Buch | Hardcover

1278 Seiten

2017
American Mathematical Society (Verlag)
978-1-4704-3480-9 (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 set 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. The first part of this two-part 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-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. The ultimate goal of the second part of the book is to explain that the Grothendieck-Teichmuller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc 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-Teichmuller group in the case of the little 2-disc 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.

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

Contents for Part 1:
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
Contents for Part 2:
Homotopy theory and its applications to operads. General methods of homotopy theory: Model categories and homotopy theory
Mapping spaces and simplicial model categories
Simplicial structures and mapping spaces in general model categories
Cofibrantly generated model categories
Modules, algebras, and the rational homotopy of spaces: Differential graded modules, simplicial modules, and cosimplicial modules
Differential graded algebras, simplicial algebras, and cosimplicial algebras
Models for the rational homotopy of spaces
The (rational) homotopy of operads: The model category of operads in simplicial sets
The homotopy theory of (Hopf) cooperads
Models for the rational homotopy of (non-unitary) operads
The homotopy theory of (Hopf) $/Lambda$-cooperads
Models for the rational homotopy of unitary operads
Applications of the rational homotopy to $E_n$-operads: Complete Lie algebras and rational models of classifying spaces
Formality and rational models of $E_n$-operads
The computation of homotopy automorphism spaces of operads: Introduction to the results of the computations for the $E_n$-operads
The applications of homotopy spectral sequences: Homotopy spsectral sequences and mapping spaces of operads
Applications of the cotriple cohomology of operads
Applications of the Koszul duality of operads
The case of $E_n$-operads: The applications of the Koszul duality for $E_n$-operads
The interpretation of the result of the spectral sequence in the case of $E_2$-operads
Conclusion: A survey of further research on operadic mapping spaces and their applications: Graph complexes and $E_n$-operads
From $E_n$-operads to embedding spaces
Appendices: Cofree cooperads and the bar duality of operads
Glossary of notation
Bibliography
Index

Erscheinungsdatum	05.07.2017
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	2620 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-3480-6 / 1470434806
ISBN-13	978-1-4704-3480-9 / 9781470434809
Zustand	Neuware