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

Description

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 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 𝐸₂-operads Chapter 4. The Little Discs Model of 𝐸_{𝑛}-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 𝐸₂-operads 5.0. Braid groups 5.1. Braided operads and 𝐸₂-operads 5.2. The classifying spaces of the colored braid operad 5.3. Fundamental groupoids and operads 5.4. Outlook: The recognition of 𝐸_{𝑛}-operads for 𝑛>2 Chapter 6. The Magma and Parenthesized Braid Operads 6.1. Magmas and the parenthesized permutation operad 6.2. The parenthesized braid operad 6.3. The parenthesized symmetry operad Part I(c) . Hopf Algebras and the Malcev Completion Chapter 7. Hopf Algebras 7.1. The notion of a Hopf algebra 7.2. Lie algebras and Hopf algebras 7.3. Lie algebras and Hopf algebras in complete filtered modules Chapter 8. The Malcev Completion for Groups 8.1. The adjunction between groups and complete Hopf algebras 8.2. The category of Malcev complete groups 8.3. The Malcev completion functor on groups 8.4. The Malcev completion of free groups 8.5. The Malcev completion of semi-direct products of groups Chapter 9. The Malcev Completion for Groupoids and Operads 9.0. The notion of a Hopf groupoid 9.1. The Malcev completion for groupoids 9.2. The Malcev completion of operads in groupoids 9.3. Appendix: The local connectedness of complete Hopf groupoids Part I(d) . The Operadic Definition of the Grothendieck–Teichmüller Group Chapter 10. The Malcev Completion of the Braid Operads and Drinfeld’s Associators 10.0. The Malcev completion of the pure braid groups and the Drinfeld–Kohno Lie algebras 10.1. The Malcev completion of the braid operads and the Drinfeld–Kohno Lie algebra operad 10.2. The operad of chord diagrams and Drinfeld’s associators 10.3. The graded Grothendieck–Teichmüller group 10.4. Tower decompositions, the graded Grothendieck–Teichmüller Lie algebra and the existence of rational Drinfeld’s associators Chapter 11. The Grothendieck–Teichmüller Group 11.1. The operadic definition of the Grothendieck–Teichmüller group 11.2. The action on the set of Drinfeld’s associators 11.3. Tower decompositions 11.4. The graded Lie algebra of the Grothendieck–Teichmüller group Chapter 12. A Glimpse at the Grothendieck Program Appendices Appendix A. Trees and the Construction of Free Operads A.1. Trees A.2. Treewise tensor products and treewise composites A.3. The construction of free operads A.4. The construction of connected free operads A.5. The construction of coproducts with free operads Appendix B. The Cotriple Resolution of Operads B.0. Tree morphisms B.1. The definition of the cotriple resolution of operads B.2. The monadic definition of operads Glossary of Notation Bibliography Index Back Cover