函子紐結(jié)理論

內(nèi)容概要

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.

書籍目錄

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

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