Combinatorics of the Surface Category and Tqfts

The (1+1)-dimensional cobordism category of closed 1-dimensional manifolds and oriented surfaces is a most basic mathematical structure. It has played a fundamental role in string theory and conformal field theory. We give a brief account of its structure and indicate what structure it gives vector spaces and other algebraic objects on which it (or an embellished version of it) acts. We explain how such functors lead to invariants of 2and 3-dimensional manifolds and give an application to topological conformal field theory.