**Mathematical Surveys and Monographs**

Volume: 217;
2017;
532 pp;
Hardcover

MSC: Primary 55;
Secondary 18; 57; 20

**Print ISBN: 978-1-4704-3481-6
Product Code: SURV/217.1**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

**Electronic ISBN: 978-1-4704-3755-8
Product Code: SURV/217.1.E**

List Price: $135.00

AMS Member Price: $108.00

MAA Member Price: $121.50

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#### Supplemental Materials

# Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 1: The Algebraic Theory and its Topological Background

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*Benoit Fresse*

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 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.

This volume 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.

#### Readership

Graduate students and researchers interested in algebraic topology and algebraic geometry.

#### Reviews & Endorsements

Even if the main goal of the first volume is to provide the reader with the necessary prerequisites to understand the deep theory developed by the author in the second volume, it nevertheless provides the literature with an interesting treatise, accessible to graduate students and to researchers working in any field. It should first be mentioned that the mathematical content covered here is absolutely beautiful.In the end, the present first volume of this treatise on the homotopy of operads and Grothendieck-Teichmüller groups represents a huge amount of work and is a valuable addition to the current mathematical literature.

-- Bruno Vallette, Mathematical Reviews

This volume provides a clear and comprehensive introduction to the theory of operads and some of its applications, and it should indeed achieve the author's aim 'to be accessible to a broad readership of graduate students and researchers interested in the applications of operads.'

-- Steffen Sagave, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Homotopy of Operads and Grothendieck-Teichmuller Groups: Part 1: The Algebraic Theory and its Topological Background

- Cover Cover11
- Title page iii4
- Contents vii8
- Preliminaries xi12
- Part I . From Operads to Grothendieck–Teichmüller Groups 148
- Part I(a) . The General Theory of Operads 350
- Part I(b) . Braids and 𝐸₂-operads 127174
- Part I(c) . Hopf Algebras and the Malcev Completion 225272
- Part I(d) . The Operadic Definition of the Grothendieck–Teichmüller Group 337384
- Chapter 10. The Malcev Completion of the Braid Operads and Drinfeld’s Associators 339386
- 10.0. The Malcev completion of the pure braid groups and the Drinfeld–Kohno Lie algebras 341388
- 10.1. The Malcev completion of the braid operads and the Drinfeld–Kohno Lie algebra operad 349396
- 10.2. The operad of chord diagrams and Drinfeld’s associators 355402
- 10.3. The graded Grothendieck–Teichmüller group 368415
- 10.4. Tower decompositions, the graded Grothendieck–Teichmüller Lie algebra and the existence of rational Drinfeld’s associators 385432

- Chapter 11. The Grothendieck–Teichmüller Group 399446
- Chapter 12. A Glimpse at the Grothendieck Program 421468

- Appendices 427474
- Back Cover Back Cover1581