Book: Homotopy of Operads & Grothendieck-Teichmüller Groups

By Benoit Fresse

Description

The first aim of this book is to give an overall reference, starting from scratch, on the applications of methods of algebraic topology to operads. To be more specific, one of the main objectives is the development of a rational homotopy theory for operads. Most definitions, notably fundamental concepts of operad and homotopy theory, are carefully reviewed in order to make this book accessible to a broad readership of graduate students and researchers interested in the applications of operads.
The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, the background required by an in-depth study of this subject is set up, and the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads is made precise. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.

Features

• Title: Homotopy of Operads & Grothendieck-Teichmüller Groups
• Author: Benoit Fresse, Université de Lille 1 (France).
• Readership: Graduate students and researchers, in algebra, topology, and in related fields
• Length: xl+534 pages (first volume), xxxi+704 pages (second volume)
• Classification: Primary: 55P48; Secondary: 18G55, 55P10, 55P62, 57T05, 20B27, 20F36.
• Keywords: Operads; Rational homotopy theory; Grothendieck-Teichmüller groups; Braid groups; E_n-operads; Homotopical algebra.
• Publication reference: volume 217 of the series Mathematical Surveys and Monographs of the American Mathematical Society. The whole book is available from the web-site http://bookstore.ams.org/surv-217/.
• Supplemental materials: List of errata.

Reading guide and overview of the content of the book

This monograph comprises three main parts which form a progression up to our ultimate mathematical goal. Part I “From operads to Grothendieck-Teichmüller groups” is mainly devoted to the algebraic foundations of our subject. In Part II “Homotopy theory and its applications to operads”, we develop our rational homotopy theory of operads after a comprehensive review of the applications of methods of homotopy theory. In Part III “The computation of homotopy automorphism spaces of operads”, we work out our problem of giving a homotopy interpretation of the Grothendieck-Teichmüller group. Three appendices A-B-C give detailed surveys on the applications of trees in operad theory, on the definition of free objects and of free resolutions in the category of operads, and on the Koszul duality of operads.
These parts are widely independent from each others. Each part of this book is also divided into subparts which, by themselves, form self-contained groupings of chapters, devoted to specific topics, and organized according to an internal progression of the level of the chapters each. There is a progression in the level of the parts of the book too, but the chapters are written so that a reader with a minimal background could tackle any of these subparts straight away in order to get a self-contained reference and an overview of the literature on each of the subjects addressed in this monograph.

Contents of the first volume, published as "Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 1: The Algebraic Theory and its Topological Background"

• Preliminaries
  • Preface
  • Mathematical objectives
  • Foundations and conventions
  • Reading guide and overview of the volume
  • Acknowledgments
• Part I. From operads to Grothendieck-Teichmüller groups
  • Part I(a). Introduction to the general theory of operads
    • Chapter 1. The basic concepts of the theory of operads
    • Chapter 2. The definition of operadic composition structures revisited
    • Chapter 3. Operads in symmetric monoidal categories
  • Part I(b). Braids and E₂-operads
    • Chapter 4. The little discs model of E_n-operads
    • Chapter 5. Braids and the recognition of E₂-operads
    • Chapter 6. The magma and parenthesized braid operad
  • Part I(c). Hopf algebras and the Malcev completion
    • Chapter 7. Hopf algebras
    • Chapter 8. The Malcev completion for groups
    • Chapter 9. The Malcev completion for groupoids and operads
  • Part I(d). The operadic definition of the Grothendieck-Teichmüller Group
    • Chapter 10. The Malcev completion of braid operads and Drinfeld's associators
    • Chapter 11. The pro-unipotent Grothendieck-Teichmüller group
    • A glimpse at the Grothendieck program
• Appendices
  • Appendix A. The construction of free operads
  • Appendix B. The cotriple resolution of operads
• Notation glossary
• Bibliography
• Subject Index

Contents of the second volume, published as "Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods"

• Preliminaries
  • Preface
  • Reminders
  • Reading guide and overview of the volume
• Part II. Homotopy theory and its applications to operads
  • Part II(a). Introduction to general methods of homotopy theory
    • Chapter 1. Model categories and homotopy
    • Chapter 2. Simplicial model categories and mapping spaces
    • Chapter 3. Simplicial structures and mapping spaces in general model categories
    • Chapter 4. Cofibrantly generated model categories
  • Part II(b). Modules, algebras, and the rational homotopy of spaces
    • Chapter 5. Differential graded modules, simplicial modules and cosimplicial modules
    • Chapter 6. Differential graded algebras, simplicial algebras and cosimplicial algebras
    • Chapter 7. Models for the rational homotopy of spaces
  • Part II(c). The (rational) homotopy of operads
    • Chapter 8. The model category of operads in simplicial sets
    • Chapter 9. The model category of (Hopf) cooperads
    • Chapter 10. Models for the rational homotopy of (non-unitary) operads
    • Chapter 11. The model category of (Hopf) Λ-cooperads
    • Chapter 12. Models for the rational homotopy of unitary operads
  • Part II(d). Applications to E_n-operads
    • Chapter 13. Complete Lie algebras and models of classifying spaces
    • Chapter 14. Rational models of E_n-operads
• Part III. The computation of homotopy automorphism spaces of operads
  • Introduction to the results of the computations for E₂-operads
  • Part III(a). The applications of homotopy spectral sequences
    • Chapter 1. Homotopy spectral sequences and mapping spaces of operads
    • Chapter 2. Applications of the cotriple cohomology of operads
    • Chapter 3. Applications of the Koszul duality of operads
  • Part III(b). The case of E_n-operads
    • Chapter 4. The applications of the Koszul duality in the E_n-operad case
    • Chapter 5. The interpretation of the computation in the E₂-operad case
  • Conclusion: a survey of further research on operadic mapping spaces and their applications
    • Chapter 6. Graph complexes and E_n-operads
    • Chapter 7. From E_n-operads to embedding spaces
• Appendices
  • Appendix C. Cofree cooperads, and the bar and Koszul duality of operads
• Notation glossary
• Bibliography
• Subject index