Representations of the Homotopy Surface Category of a Simply Connected Space

At the heart of the axiomatic formulation of 1+1-dimensional topological field theory is the set of all surfaces with boundary assembled into a category. This category of surfaces has compact 1-manifolds as objects and smooth oriented cobordisms as morphisms. Taking disjoint unions gives a monoidal structure and a 1+1-dimensional topological field theory can be defined to be a monoidal functor E : S → VectC normalised so that E(cylinder) = identity. Here VectC is the category of finite dimensional complex vector spaces and linear maps, with monoidal structure the usual tensor product. It is well known that to specify a 1+1-dimensional topological field theory is the same thing as specifying a finite dimensional commutative Frobenius algebra. There is a natural generalisation of the category of surfaces, where one introduces a background space X and considers maps of surfaces to X. Once again a monoidal category emerges. In this paper we explain how for a simply connected background space, monoidal functors from this category to VectC can be interpreted in terms of Frobenius algebras with additional structure. The main result is Theorem 4.1. Although we use C throughout, any other field will give the same results. We thank Ran Levi, Paolo Salvatore and Ulrike Tillmann for helpful conversations.