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Homotopy of Operads and Grothendieck-Teichmuller Groups<Part II> (Mathematical Surveys and Monographs Vol. 217.2) H 715 p. 17

Fresse, Benoit 　著

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発行年月	2017年06月
出版社／提供元	American Mathematical Society
出版国	アメリカ合衆国
言語	英語
媒体	冊子
装丁	hardcover
ページ数／巻数	715 p.
ジャンル	洋書／理工学／数学／幾何学
ISBN	9781470434823
商品コード	1024109457
本の性格	学術書
新刊案内掲載月	2017年04月
商品URL	https://kw.maruzen.co.jp/ims/itemDetail.html?itmCd=1024109457

内容

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.

* 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

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