# 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

### Table of Contents

**Front/Back Matter**

**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 **

**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 **

**Appendices **