S-structures for K-linear Categories and the Definition of a Modular Functor

Motivation and background. Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras [17] [22], monoidal categories with additional structure [24], and modular functors [14] [23]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah [1], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.