Homotopy of operads and Grothendieck-Teichmüller groups
- Benoit Fresse.
- Providence, Rhode Island : American Mathematical Society, 
- Physical description
- 2 volumes : illustrations ; 27 cm.
- Mathematical surveys and monographs ; no. 217.
Science Library (Li and Ma)
Library has: pt.1-
pt.2 <v.217 in series>
|QA3 .A4 V.217:PT.1||Unknown|
|QA3 .A4 V.217:PT.2||Unknown|
- Fresse, Benoit, author.
- Includes bibliographical references and index.
- * 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.
- (source: Nielsen Book Data)9781470434816 20170621
- * 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.
- (source: Nielsen Book Data)9781470434823 20170621
- Publisher's Summary
- 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.
(source: Nielsen Book Data)9781470434816 20170621
- The ultimate goal of this 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.
(source: Nielsen Book Data)9781470434823 20170621
- Homotopy theory.
- Grothendieck groups.
- Teichmüller spaces.
- Grothendieck groups.
- Homotopy theory.
- Teichmüller spaces.
- Algebraic topology > Homotopy theory > Loop space machines, operads.
- Category theory; homological algebra > Homological algebra > Homotopical algebra.
- Algebraic topology > Homotopy theory > Homotopy equivalences.
- Algebraic topology > Homotopy theory > Rational homotopy theory.
- Manifolds and cell complexes > Homology and homotopy of topological groups and related structures > Hopf algebras.
- Group theory and generalizations > Permutation groups > Infinite automorphism groups.
- Group theory and generalizations > Special aspects of infinite or finite groups > Braid groups; Artin groups.
- Publication date
- Mathematical surveys and monographs ; volume 217
- 9781470434816 (hardcover alkaline paper ; pt. 1)
- 1470434814 (hardcover alkaline paper ; pt. 1)
- 9781470434823 (hardcover alkaline paper ; pt. 2)
- 1470434822 (hardcover alkaline paper ; pt. 2)
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