Homotopy of Operads and Grothendieck-Teichmüller Groups : Part 2: The Applications of (Rational) Homotopy Theory Methods

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. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck-Teichmuller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory. Cover 1 Title page 4 Contents 6 Preliminaries 10 Preface 12 Reminders 16 Reading Guide and Overview of the Volume 24 Part II . Homotopy Theory and its Applications to Operads 38 Part II(a) . General Methods of Homotopy Theory 40 Chapter 1. Model Categories and Homotopy Theory 42 1.0. Introduction: The problem of defining homotopy categories 43 1.1. The notion of a model category 45 1.2. The homotopy category of a model category 51 1.3. The example of topological spaces and of simplicial sets 62 1.4. The model category of operads and of algebras over operads 74 Chapter 2. Mapping Spaces and Simplicial Model Categories 82 2.0. The definition of functors on the category of simplicial sets 83 2.1. The notion of a simplicial model category 84 2.2. Homotopy automorphism spaces 93 2.3. Simplicial structures for operads and for algebras over operads 97 Chapter 3. Simplicial Structures and Mapping Spaces in General Model Categories 102 3.1. The Reedy model structures 103 3.2. Framing constructions and mapping spaces 118 3.3. The definition of geometric realization and totalization functors 127 3.4. Appendix: Homotopy ends and coends 137 Chapter 4. Cofibrantly Generated Model Categories 144 4.1. Relative cell complexes and the small object argument 145 4.2. The notion of a cofibrantly generated model category 149 4.3. Cofibrantly generated model categories and adjunctions 153 4.4. Outlook: Combinatorial model categories 157 Part II(b) . Modules, Algebras, and the Rational Homotopy of Spaces 162 Chapter 5. Differential Graded Modules, Simplicial Modules, and Cosimplicial Modules 164 5.0. Background: dg-modules and simplicial modules 165 5.1. The model category of cochain graded dg-modules 176 5.2. Monoidal structures and the Eilenberg–Zilber equivalence 183 5.3. Hom-objects on dg-modules and simplicial modules 189 5.4. Appendix: Contracting chain-homotopies and extra-degeneracies 196 Chapter 6. Differential Graded Algebras, Simplicial Algebras, and Cosimplicial Algebras 200 6.1. The definition of unitary commutative algebras 200 6.2. The model category of unitary commutative algebras 207 6.3. The bar construction in the category of commutative algebras 215 Chapter 7. Models for the Rational Homotopy of Spaces 220 7.1. The Sullivan cochain dg-algebra associated to a simplicial set 220 7.2. The adjunction between dg-algebras and simplicial sets 232 7.3. Applications of the Sullivan model to the rational homotopy theory of spaces 235 Part II(c) . The (Rational) Homotopy of Operads 248 Chapter 8. The Model Category of Operads in Simplicial Sets 250 8.0. The category of operads in simplicial sets 252 8.1. The model category of symmetric sequences 253 8.2. The model category of non-unitary operads 263 8.3. The model category of -sequences 277 8.4. The model category of augmented non-unitary Λ-operads 292 8.5. Simplicial structures and the cotriple resolution of operads 302 Chapter 9. The Homotopy Theory of (Hopf) Cooperads 310 9.1. Cooperads 310 9.2. The model category of cochain graded dg-cooperads 321 9.3. Hopf cooperads 331 9.4. Appendix: The totalization of cosimplicial (Hopf) cochain dg-cooperads 338 Chapter 10. Models for the Rational Homotopy of (Non-unitary) Operads 356 10.0. The model category of connected operads 356 10.1. The Hopf cochain dg-cooperad model 358 10.2. Applications to the rational homotopy of operads 367 Chapter 11. The Homotopy Theory of (Hopf) Λ-cooperads 370 11.1. The notion of a coaugmented Λ-cooperad 370 11.2. The adjunction with the category of plain cooperads 382 11.3. The model category of coaugmented Λ-cooperads 387 11.4. The model category of Hopf Λ-cooperads 394 Chapter 12. Models for the Rational Homotopy of Unitary Operads 404 12.0. The model category of connected Λ-operads 404 12.1. The Hopf Λ-cooperad models 407 12.2. Applications to the rational homotopy of Λ-operads 410 Part II(d) . Applications of the Rational Homotopy to E_{n}-operads 414 Chapter 13. Complete Lie Algebras and Rational Models of Classifying Spaces 416 13.0. Background 417 13.1. The Chevalley–Eilenberg cochain complex of complete chain graded Lie algebras 421 13.2. The classifying space of Malcev complete groups 437 Chapter 14. Formality and Rational Models of E_{n}-operads 448 14.0. Preliminaries on additive operads and additive cooperads 451 14.1. The graded Drinfeld–Kohno Lie algebra operads and the applications of Chevalley–Eilenberg cochain complexes 455 14.2. The chord diagram operad and the rational model of E2-operads 476 14.3. Appendix: Reminders on the Drinfeld–Kohno Lie algebra operad 483 Part III . The Computation of Homotopy Automorphism Spaces of Operads 486 Introduction to the Results of the Computations for E2-operads 488 Part III(a) . The Applications of Homotopy Spectral Sequences 498 Chapter 1. Homotopy Spectral Sequences and Mapping Spaces of Operads 500 1.0. Conventions on bigraded structures 500 1.1. Homotopy spectral sequences 501 1.2. Applications to operads 511 Chapter 2. Applications of the Cotriple Cohomology of Operads 520 2.0. Multi-complexes 522 2.1. Modules of derivations associated to operads 526 2.2. The definition and the applications of the cotriple cohomology 535 2.3. Appendix: Hom-objects on the category of Λ-sequences 541 Chapter 3. Applications of the Koszul Duality of Operads 556 3.1. The applications of the cobar-bar and Koszul resolutions 558 3.2. The applications of the Koszul derivation complex 569 Part III(b) . The Case of E_{n}-operads 578 Chapter 4. The Applications of the Koszul Duality for E_{n}-operads 580 4.1. The Koszul dual of the Gerstenhaber operads 581 4.2. The cotriple cohomology of the Gerstenhaber operads 587 Chapter 5. The Interpretation of the Result of the Spectral Sequence in the Case of E2-operads 596 5.0. Reminders on the Grothendieck–Teichmüller group 597 5.1. The degree zero homotopy of the homotopy automorphism space 603 5.2. The action of the classifying space of the additive group and the concluding result 618 5.3. Appendix: Rationalization and homotopy spectral sequences 623 Conclusion: A Survey of Further Research on Operadic Mapping Spaces and their Applications 626 Chapter 6. Graph Complexes and E_{n}-operads 628 6.1. The operads of graphs 628 6.2. Mapping spaces of E_{n}-operads and graph complexes 632 6.3. Possible generalizations of the computations in positive characteristic and in pro-finite homotopy theory 642 Chapter 7. From E_{n}-operads to Embedding Spaces 644 Appendices 652 Appendix C. Cofree Cooperads and the Bar Duality of Operads 654 C.0. Reminders on the language of trees 655 C.1. The construction of cofree cooperads 662 C.2. The bar duality of operads 677 C.3. The Koszul duality of operads 700 Glossary of Notation 708 Bibliography 716 Index 726 Back Cover 743 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–Teichmüller 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. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.