Mathematical Surveys and Monographs Volume 217 Homotopy of Operads and Grothendieck– Teichmüller Groups Part 1: The Algebraic Theory and its Topological Background Benoit Fresse American Mathematical Society 10.1090/surv/217.1 Homotopy of Operads and Grothendieck– Teichmüller Groups Part 1: The Algebraic Theory and its Topological Background Mathematical Surveys and Monographs Volume 217 Homotopy of Operads and Grothendieck– Teichmüller Groups Part 1: The Algebraic Theory and its Topological Background Benoit Fresse American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 55P48; Secondary 18G55, 55P10, 55P62, 57T05, 20B27, 20F36. For additional information and updates on this book, visit www.ams.org/bookpages/surv-217 Library of Congress Cataloging-in-Publication Data Names: Fresse, Benoit. Title: Homotopy of operads and Grothendieck-Teichm¨uller groups / Benoit Fresse. Description: Providence, Rhode Island : American Mathematical Society, [2017]- | Series: Mathe- matical surveys and monographs ; volume 217 | Includes bibliographical references and index. Contents: The algebraic theory and its topological background – The applications of (rational) homotopy theory methods Identifiers: LCCN 2016032055| ISBN 9781470434816 (alk. paper) | ISBN 9781470434823 (alk. paper) Subjects: LCSH: Homotopy theory. | Operads. | Grothendieck groups. | Teichm¨uller spaces. | AMS: Algebraic topology – Homotopy theory – Loop space machines, operads. msc | Category theory; homological algebra – Homological algebra – Homotopical algebra. msc | Algebraic topology – Homotopy theory – Homotopy equivalences. msc | Algebraic topology – Homotopy theory – Rational homotopy theory. msc | Manifolds and cell complexes – Homology and homotopy of topological groups and related structures – Hopf algebras. msc | Group theory and generalizations – Permutation groups – Infinite automorphism groups. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Braid groups; Artin groups. msc Classification: LCC QA612.7 .F74 2017 | DDC 514/.24–dc23 LC record available at https://lccn. loc.gov/2016032055 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Un souffle ouvre des br`eches op´eradiques dans les cloisons, – brouille le pivotement des toits rong´es, – disperse les limites des foyers, – ´eclipse les crois´ees. Arthur Rimbaud Nocturne vulgaire. Les Illuminations (1875) Contents This work is divided in two volumes, referred to as “Part 1: The Algebraic The- ory and its Topological Background” and “Part 2: The Applications of (Rational) Homotopy Theory Methods”, as indicated on the cover of the volumes. But our narrative is actually organized into three parts, numbered I-II-III, and three appen- dices, numbered A-B-C, which define the internal divisions of this book. This first volume comprises Part I, “From Operads to Grothendieck–Teichm¨uller Groups”; Appendix A, “Trees and the Construction of Free Operads”; and Appendix B, “The Cotriple Resolution of Operads”. The second volume comprises Part II, “Homotopy Theory and its Applications to Operads”; Part III, “The Computation of Homo- topy Automorphism Spaces of Operads”; and Appendix C, “Cofree Cooperads and the Bar Duality of Operads”. Each volume includes a preface, a notation glossary, a bibliography, and a subject index. References to chapters, sections, paragraphs, and statements of the book are given by §x.y.z when these cross references are done within a part (I, II, and III) and by §P.x.y.z where P =I, II, III otherwise. The cross references to the sections, paragraphs, and statements of the appendices are given by §P.x.y throughout the book, where §P = §A, §B, §C. The preliminary part of the first volume of this book also includes a Foundations and Conventions section, whose paragraphs, numbered §§0.1-0.16, give a summary of the main conventions used in this work. Preliminaries xi Preface xiii Mathematical Objectives xvii Foundations and Conventions xxvii Reading Guide and Overview of this Volume xxxix Part I. From Operads to Grothendieck–Teichm¨uller Groups 1 Part I(a). The General Theory of Operads 3 Chapter 1. The Basic Concepts of the Theory of Operads 5 1.1. The notion of an operad and of an algebra over an operad 6 1.2. Categorical constructions for operads 23 1.3. Categorical constructions for algebras over operads 37 1.4. Appendix: Filtered colimits and reflexive coequalizers 44 vii viii CONTENTS Chapter 2. The Definition of Operadic Composition Structures Revisited 47 2.1. The definition of operads from partial composition operations 48 2.2. The definition of unitary operads 57 2.3. Categorical constructions for unitary operads 74 2.4. The definition of connected unitary operads 80 2.5. The definition of operads shaped on finite sets 90 Chapter 3. Symmetric Monoidal Categories and Operads 99 3.0. Commutative algebras and cocommutative coalgebras in symmetric monoidal categories 100 3.1. Operads in general symmetric monoidal categories 106 3.2. The notion of a Hopf operad 112 3.3. Appendix: Functors between symmetric monoidal categories 122 Part I(b). Braids and E2-operads 127 Chapter 4. The Little Discs Model of En-operads 129 4.1. The definition of the little discs operads 130 4.2. The homology (and the cohomology) of the little discs operads 140 4.3. Outlook: Variations on the little discs operads 150 4.4. Appendix: The symmetric monoidal category of graded modules 157 Chapter 5. Braids and the Recognition of E2-operads 159 5.0. Braid groups 160 5.1. Braided operads and E2-operads 167 5.2. The classifying spaces of the colored braid operad 177 5.3. Fundamental groupoids and operads 187 5.4. Outlook: The recognition of En-operads for n>2 194 Chapter 6. The Magma and Parenthesized Braid Operads 197 6.1. Magmas and the parenthesized permutation operad 198 6.2. The parenthesized braid operad 208 6.3. The parenthesized symmetry operad 220 Part I(c). Hopf Algebras and the Malcev Completion 225 Chapter 7. Hopf Algebras 227 7.1. The notion of a Hopf algebra 228 7.2. Lie algebras and Hopf algebras 236 7.3. Lie algebras and Hopf algebras in complete filtered modules 258 Chapter 8. The Malcev Completion for Groups 277 8.1. The adjunction between groups and complete Hopf algebras 278 8.2. The category of Malcev complete groups 283 8.3. The Malcev completion functor on groups 293 8.4. The Malcev completion of free groups 296 8.5. The Malcev completion of semi-direct products of groups 302 Chapter 9. The Malcev Completion for Groupoids and Operads 311 9.0. The notion of a Hopf groupoid 312 9.1. The Malcev completion for groupoids 315 CONTENTS ix 9.2. The Malcev completion of operads in groupoids 328 9.3. Appendix: The local connectedness of complete Hopf groupoids 334 Part I(d). The Operadic Definition of the Grothendieck– Teichm¨uller Group 337 Chapter 10. The Malcev Completion of the Braid Operads and Drinfeld’s Associators 339 10.0. The Malcev completion of the pure braid groups and the Drinfeld–Kohno Lie algebras 341 10.1. The Malcev completion of the braid operads and the Drinfeld– Kohno Lie algebra operad 349 10.2. The operad of chord diagrams and Drinfeld’s associators 355 10.3. The graded Grothendieck–Teichm¨uller group 368 10.4. Tower decompositions, the graded Grothendieck–Teichm¨uller Lie algebra and the existence of rational Drinfeld’s associators 385 Chapter 11. The Grothendieck–Teichm¨uller Group 399 11.1. The operadic definition of the Grothendieck–Teichm¨uller group 400 11.2. The action on the set of Drinfeld’s associators 408 11.3. Tower decompositions 411 11.4. The graded Lie algebra of the Grothendieck–Teichm¨uller group 414 Chapter 12. A Glimpse at the Grothendieck Program 421 Appendices 427 Appendix A. Trees and the Construction of Free Operads 429 A.1. Trees 430 A.2. Treewise tensor products and treewise composites 442 A.3. The construction of free operads 453 A.4. The construction of connected free operads 462 A.5. The construction of coproducts with free operads 467 Appendix B. The Cotriple Resolution of Operads 477 B.0. Tree morphisms 478 B.1. The definition of the cotriple resolution of operads 485 B.2. The monadic definition of operads 498 Glossary of Notation 503 Bibliography 511 Index 521 Preliminaries Preface The first purpose of this work is to give an overall reference, starting from scratch, on applications of methods of algebraic topology to the study of oper- ads in topological spaces. Most definitions, notably fundamental concepts of the theory of operads and of homotopy theory, are reviewed in this book in order to make our account accessible to graduate students and to researchers coming from the various fields of mathematics related to our subject.