Functorial Knot Theory: Categories Of Tangles, Coherence, Categorical Deformations And Topological Invariants Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants

Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory. This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber's deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations Acknowledgements......Page 8 Contents......Page 10 1. Introduction......Page 14 I Knots and Categories......Page 18 2. Basic Concepts......Page 20 3. Monoidal Categories Functors and Natural Transformations......Page 46 4. A Digression on Algebras......Page 68 5. More About Monoidal Categories......Page 78 6. Knot Polynomials......Page 90 7. Categories of Tangles......Page 94 8. Smooth Tangles and PL Tangles......Page 104 9. Shum's Theorem......Page 124 10. A Little Enriched Category Theory......Page 138 II Deformations......Page 146 11. Introduction......Page 148 12. Definitions......Page 150 13. Deformation Complexes of Semigroupal Categories and Functors......Page 156 14. Some Useful Cochain Maps......Page 160 15. First Order Deformations......Page 162 16. Obstructions and Cup Product and Pre-Lie Structures on X'(F)......Page 166 17. Units......Page 182 18. Extrinsic Deformations of Monoidal Categories......Page 188 19. Vassiliev Invariants Framed and Unframed......Page 192 20. Vassiliev Theory in Characteristic 2......Page 208 21. Categorical Deformations as Proper Generalizations of Classical Notions......Page 216 22. Open Questions......Page 220 Bibliography......Page 226 Index......Page 232 As part of a series exploring why knot theory serves as a nexus for math, physics, biology, and philosophy, this volume probes the quantum topology links between the pivotal objects of study in low- dimensional geometric topology--including classical knots--and such exotic algebraic objects as Hopf algebras and monoidal categories. Yetter (mathematics, Kansas State U.) covers basic concepts and categories, and provides proofs of some universally assumed "folk theorems," examples of knot diagrams, and 63 references. The section on deformations assumes some familiarity with algebraic deformation theory and homological algebra. c. Book News Inc A study of functorial knot theory. It discusses the key ideas in the discovery of monoidal categories of tangles as central objects in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors. David N. Yetter. Includes Bibliographical References (p. 219-224) And Index.