Homotopy of Operads and Grothendieck–Teichmüller Groups, Part 2: Part 2: The Applications of (Rational) Homotopy Theory Methods

About this Title

Benoit Fresse, Université de Lille 1, Villeneuve d’Ascq, France

Publication: Mathematical Surveys and Monographs
Publication Year: 2017; Volume 217.2
ISBNs: 978-1-4704-3482-3 (print); 978-1-4704-3757-2 (online)
DOI: https://doi.org/10.1090/surv/217.2
MathSciNet review: MR3616816
MSC: Primary 55P62; Secondary 18D50, 55P48, 57T05

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Table of Contents

Front/Back Matter

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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

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