  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 (Reshetikhin and Turaev), monoidal
categories with additional structure (Turaev and Yetter), and modular functors
(Walker and Kontsevich). These constructions are very closely related. We take
a unifying categorical approach based on a natural 2-dimensional generalization
of a topological field theory in the sense of Atiyah and Segal, and show that
the axioms defining these complex algebraic structures are a consequence of the
underlying geometry of surfaces. In particular, we show that any linear
category over a field with an action of the surface category is semi-simple and
Artinian.
