Infinity Operads and Monoidal Categories with Group Equivariance

"This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras"-- Provided by publisher Contents Preface Operads with Group Equivariance 1. Introduction 1.1 Overview and Prospects 1.2 Categorical Conventions 1.2.1 Categories 1.2.2 Monoidal Categories 1.2.3 Adjoint Lifting Theorem 2. Planar Operads 2.1 Planar Operads as Monoids 2.2 Coherence for Planar Operads 2.3 Algebras 2.4 Examples of Planar Operads 3. Symmetric Operads 3.1 Symmetric Operads as Monoids 3.2 Coherence for Symmetric Operads 3.3 Algebras 3.4 Little Cube and Little Disc Operads 3.5 Operad in Non-Commutative Probability 3.6 Phylogenetic Operad 3.7 Planar Tangle Operad 4. Group Operads 4.1 Action Operads 4.2 Group Operads as Monoids 4.3 Coherence for Group Operads 4.4 Parenthesized Group Operads 4.5 Group Operads from Translation Categories 5. Braided Operads 5.1 Braid Groups 5.2 Braided Operads as Monoids 5.3 Examples of Braided Operads 5.4 Universal Cover of the Little 2-Cube Operad 6. Ribbon Operads 6.1 Ribbon Groups 6.2 Ribbon Operads as Monoids 6.3 Examples of Ribbon Operads 6.4 Universal Cover of the Framed Little 2-Disc Operad 7. Cactus Operads 7.1 Cactus Groups 7.2 Direct Sum Cacti 7.3 Block Cacti 7.4 Cactus Group Operad 7.5 Examples of Cactus Operads 7.6 Relationship with Braid Group Operad Constructions of Group Operads 8. Naturality 8.1 Change of Action Operads 8.2 Symmetrization and Other Left Adjoints 8.3 Change of Base Categories 8.4 Change of Algebra Categories 9. Group Operads as Algebras 9.1 Planar Trees 9.2 Group Trees 9.3 Symmetric Operad for Group Operads 10. Group Operads with Varying Colors 10.1 Category of All Group Operads 10.2 Symmetric Monoidal Structure 10.3 Closed Structure 10.4 Comparing Symmetric Monoidal Structures 10.5 Non-Strong Monoidality 10.6 Local Finite Presentability 11. Boardman-Vogt Construction for Group Operads 11.1 Substitution Category 11.2 Vertex Decoration 11.3 Internal Edge Decoration 11.4 Boardman-Vogt Construction 11.5 Augmentation 11.6 Change of Action Operads Infinity Group Operads 12. Category of Group Trees 12.1 Morphisms of Group Trees 12.2 Naturality of Group Trees 12.3 From Group Trees to Group Operads 12.4 Change of Action Operads 13. Contractibility of Group Tree Category 13.1 Closed Group Trees 13.2 Closure as Reflection 13.3 Output Extension 13.4 Contractibility 14. Generalized Reedy Structure 14.1 Coface and Codegeneracy 14.2 Dualizable Generalized Reedy Structure 14.3 Reedy-Type Model Structures 14.4 Eilenberg-Zilber Structure 15. Realization-Nerve Adjunction for Group Operads 15.1 Realization and Nerve 15.2 Group Operads as Colimits 15.3 Nerve is Fully Faithful 15.4 Symmetric Monoidal Structure on Presheaf Category 15.5 Nerve is Symmetric Monoidal 15.6 Change of Action Operads 15.7 Comparing Symmetric Monoidal Structures 16. Nerve Theorem for Group Operads 16.1 Nerve Satisfies Segal Condition 16.2 Segal Condition Implies Strict Infinity 16.3 Strict Infinity Implies Segal Condition 16.4 Characterization of the Nerve 17. Coherent Realization-Nerve and Infinity Group Operads 17.1 Coherent Realization and Coherent Nerve 17.2 Coherent Realization of the Nerve 17.3 Nerve to Coherent Nerve 17.4 Change of Action Operads 17.5 Planar BV Construction of Planar Trees 17.6 Coherent Nerves are Infinity Group Operads Coherence for Monoidal Categories with Group Equivariance 18. Monoidal Categories 18.1 Monoidal Categories and Monoidal Functors 18.2 Monoidal Category Operad 18.3 Operadic Coherence for Monoidal Categories 18.4 Strict Monoidal Functors as Algebra Morphisms 18.5 Strict Monoidal Category Operad 18.6 Monoidal Functor Operad 19. G-Monoidal Categories 19.1 G-Monoidal Category Operad 19.2 Coherence for G-Monoidal Categories, I 19.3 Strict G-Monoidal Categories 19.4 G-Monoidal Functors 19.5 G-Monoidal Functor Operad 20. Coherence for G-Monoidal Categories 20.1 Strictification of G-Monoidal Categories 20.2 Strictification of Free G-Monoidal Categories 20.3 Free Strict G-Monoidal Categories 20.4 Free G-Monoidal Categories 21. Braided and Symmetric Monoidal Categories 21.1 Braided Monoidal Categories are B-Monoidal Categories 21.2 Braided Monoidal Functors are B-Monoidal Functors 21.3 Coherence for Braided Monoidal Categories 21.4 Symmetric Monoidal Categories are S-Monoidal Categories 21.5 Symmetric Monoidal Functors are S-Monoidal Functors 21.6 Coherence for Symmetric Monoidal Categories 22. Ribbon Monoidal Categories 22.1 Ribbon Monoidal Categories are R-Monoidal Categories 22.2 Ribbon Monoidal Functors are R-Monoidal Functors 22.3 Coherence for Ribbon Monoidal Categories 23. Coboundary Monoidal Categories 23.1 Natural Actions on Coboundary Monoidal Categories 23.2 Coboundary Monoidal Categories are Cac-Monoidal Categories 23.3 Coboundary Monoidal Functors are Cac-Monoidal Functors 23.4 Coherence for Coboundary Monoidal Categories List of Notations Bibliography Index "The book gives a self-contained introduction to the theory of lambda-rings and closely related topics, including Witt vectors, integer-valued polynomials, and binomial rings. Many of the purely algebraic results about lambda-rings presented in this book have never appeared in book form before. This book concludes with a chapter on open problems related to lambda-rings."--Jacket