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.
Part 1 Knots and categories: monoidal categories, functors and natural transformations; a digression on algebras; knot polynomials; smooth tangles and PL tangles; a little enriched category theory. Part 2 Deformations: deformation complexes of semigroupal categories and functors; first order deformations; units; extrinsic deformations of monoidal categories; categorical deformations as proper generalizations of classical notions. (Part contents).