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

The Grothendieck-Teichmuller 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-Teichmuller 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. Cover 1 Title page 4 Contents 8 Preliminaries 12 Preface 14 Mathematical Objectives 18 Foundations and Conventions 28 Reading Guide and Overview of this Volume 40 Part I . From Operads to Grothendieck–Teichmüller Groups 48 Part I(a) . The General Theory of Operads 50 Chapter 1. The Basic Concepts of the Theory of Operads 52 1.1. The notion of an operad and of an algebra over an operad 53 1.2. Categorical constructions for operads 70 1.3. Categorical constructions for algebras over operads 84 1.4. Appendix: Filtered colimits and reflexive coequalizers 91 Chapter 2. The Definition of Operadic Composition Structures Revisited 94 2.1. The definition of operads from partial composition operations 95 2.2. The definition of unitary operads 104 2.3. Categorical constructions for unitary operads 121 2.4. The definition of connected unitary operads 127 2.5. The definition of operads shaped on finite sets 137 Chapter 3. Symmetric Monoidal Categories and Operads 146 3.0. Commutative algebras and cocommutative coalgebras in symmetric monoidal categories 147 3.1. Operads in general symmetric monoidal categories 153 3.2. The notion of a Hopf operad 159 3.3. Appendix: Functors between symmetric monoidal categories 169 Part I(b) . Braids and E2-operads 174 Chapter 4. The Little Discs Model of E_{n}-operads 176 4.1. The definition of the little discs operads 177 4.2. The homology (and the cohomology) of the little discs operads 187 4.3. Outlook: Variations on the little discs operads 197 4.4. Appendix: The symmetric monoidal category of graded modules 204 Chapter 5. Braids and the Recognition of E2-operads 206 5.0. Braid groups 207 5.1. Braided operads and E2-operads 214 5.2. The classifying spaces of the colored braid operad 224 5.3. Fundamental groupoids and operads 234 5.4. Outlook: The recognition of E_{n}-operads for n>2 241 Chapter 6. The Magma and Parenthesized Braid Operads 244 6.1. Magmas and the parenthesized permutation operad 245 6.2. The parenthesized braid operad 255 6.3. The parenthesized symmetry operad 267 Part I(c) . Hopf Algebras and the Malcev Completion 272 Chapter 7. Hopf Algebras 274 7.1. The notion of a Hopf algebra 275 7.2. Lie algebras and Hopf algebras 283 7.3. Lie algebras and Hopf algebras in complete filtered modules 305 Chapter 8. The Malcev Completion for Groups 324 8.1. The adjunction between groups and complete Hopf algebras 325 8.2. The category of Malcev complete groups 330 8.3. The Malcev completion functor on groups 340 8.4. The Malcev completion of free groups 343 8.5. The Malcev completion of semi-direct products of groups 349 Chapter 9. The Malcev Completion for Groupoids and Operads 358 9.0. The notion of a Hopf groupoid 359 9.1. The Malcev completion for groupoids 362 9.2. The Malcev completion of operads in groupoids 375 9.3. Appendix: The local connectedness of complete Hopf groupoids 381 Part I(d) . The Operadic Definition of the Grothendieck–Teichmüller Group 384 Chapter 10. The Malcev Completion of the Braid Operads and Drinfeld’s Associators 386 10.0. The Malcev completion of the pure braid groups and the Drinfeld–Kohno Lie algebras 388 10.1. The Malcev completion of the braid operads and the Drinfeld–Kohno Lie algebra operad 396 10.2. The operad of chord diagrams and Drinfeld’s associators 402 10.3. The graded Grothendieck–Teichmüller group 415 10.4. Tower decompositions, the graded Grothendieck–Teichmüller Lie algebra and the existence of rational Drinfeld’s associators 432 Chapter 11. The Grothendieck–Teichmüller Group 446 11.1. The operadic definition of the Grothendieck–Teichmüller group 447 11.2. The action on the set of Drinfeld’s associators 455 11.3. Tower decompositions 458 11.4. The graded Lie algebra of the Grothendieck–Teichmüller group 461 Chapter 12. A Glimpse at the Grothendieck Program 468 Appendices 474 Appendix A. Trees and the Construction of Free Operads 476 A.1. Trees 477 A.2. Treewise tensor products and treewise composites 489 A.3. The construction of free operads 500 A.4. The construction of connected free operads 509 A.5. The construction of coproducts with free operads 514 Appendix B. The Cotriple Resolution of Operads 524 B.0. Tree morphisms 525 B.1. The definition of the cotriple resolution of operads 532 B.2. The monadic definition of operads 545 Glossary of Notation 550 Bibliography 558 Index 568 Back Cover 581 Cover -- Title page -- Contents -- Preliminaries -- Preface -- Mathematical Objectives -- Foundations and Conventions -- Reading Guide and Overview of this Volume -- Part I . From Operads to Grothendieck-Teichmüller Groups -- Part I(a) . The General Theory of Operads -- Chapter 1. The Basic Concepts of the Theory of Operads -- 1.1. The notion of an operad and of an algebra over an operad -- 1.2. Categorical constructions for operads -- 1.3. Categorical constructions for algebras over operads -- 1.4. Appendix: Filtered colimits and reflexive coequalizers -- Chapter 2. The Definition of Operadic Composition Structures Revisited -- 2.1. The definition of operads from partial composition operations -- 2.2. The definition of unitary operads -- 2.3. Categorical constructions for unitary operads -- 2.4. The definition of connected unitary operads -- 2.5. The definition of operads shaped on finite sets -- Chapter 3. Symmetric Monoidal Categories and Operads -- 3.0. Commutative algebras and cocommutative coalgebras in symmetric monoidal categories -- 3.1. Operads in general symmetric monoidal categories -- 3.2. The notion of a Hopf operad -- 3.3. Appendix: Functors between symmetric monoidal categories -- Part I(b) . Braids and 2-operads -- Chapter 4. The Little Discs Model of _{ }-operads -- 4.1. The definition of the little discs operads -- 4.2. The homology (and the cohomology) of the little discs operads -- 4.3. Outlook: Variations on the little discs operads -- 4.4. Appendix: The symmetric monoidal category of graded modules -- Chapter 5. Braids and the Recognition of 2-operads -- 5.0. Braid groups -- 5.1. Braided operads and 2-operads -- 5.2. The classifying spaces of the colored braid operad -- 5.3. Fundamental groupoids and operads -- 5.4. Outlook: The recognition of _{ }-operads for>2 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. The ultimate goal of this book is to explain that the Grothendieck-Teichmuller 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.