S-Structures for k-Linear Categories and the Definition of a Modular Functor

Ideas from string theory and quantum field theory have beenthe motivation for new invariants of knots and 3-dimensionalmanifolds which have been constructed from complex algebraicstructures such as Hopf algebras [17, 22], monoidal categorieswith additional structure [24], and modular functors [14, 23].These constructions are closely related. Here we take a unifyingcategorical approach based on a natural 2-dimensional generalisationof a topological field theory in the sense of Atiyah [1], andshow that the axioms defining these complex algebraic structuresare a consequence of the underlying geometry of surfaces.