HOMOTOPY OF OPERADS AND GROTHENDIECK–TEICHMÜLLER GROUPS: PARTS 1 AND 2

Contents for Part 1
• From operads to Grothendieck–Teichmüller 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 $E_2$-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–Teichmüller group
The Malcev completion of the braid operads and Drinfeld's associators
The Grothendieck–Teichmüller 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
•1The 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

The Grothendieck–Teichmüller 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–Teichmüller 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–Teichmüller 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–Teichmüller 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.

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