函子纽结理论
2001-12
Penguin
Yetter, David N.
230
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 1. IntroductionI.Knots and Categories 2. Basic Concepts 2.1 Knots 2.2 Categories 3. Monoidal Categories, Functors and Natural Transformations 4. A Digression on Algebras 5. More About Monoidal Categories 6. Knot Polynomials 7. Categories of Tangles 8. Smooth Tangles and PL Tangles 9. Shum's Theorem 10. A Little Enriched Category TheoryII.Deformations 11. Introduction 12. Definitions 13. Deformation Complexes of Semigroupal Categories and Functors 14. Some Useful Cochain Maps 15. First Order Deformations 16. Obstructions and Cup Product and Pre-Lie Structures 17. Units 18. Extrinsic Deformations of Monoidal Categories 19. Vassiliev Invariants, Framed and Unframed 20. Vassiliev Theory in Characteristic 2 21. Categorical Deformations as Proper Generalizations of Classical Notions 22. Open Questions 22.1 Functorial Knot Theory 22.2 Deformation TheoryBibliographyIndex
