  Motivated by the Moore-Segal axioms for an open-closed topological field
theory, we consider planar open string topological field theories. We
rigorously define a category 2Thick whose objects and morphisms can be thought
of as open strings and diffeomorphism classes of planar open string
worldsheets. Just as the category of 2-dimensional cobordisms can be described
as the free symmetric monoidal category on a commutative Frobenius algebra,
2Thick is shown to be the free monoidal category on a noncommutative Frobenius
algebra, hence justifying this choice of data in the Moore-Segal axioms. Our
formalism is inherently categorical allowing us to generalize this result. As a
stepping stone towards topological membrane theory we define a 2-category of
open strings, planar open string worldsheets, and isotopy classes of
3-dimensional membranes defined by diffeomorphisms of the open string
worldsheets. This 2-category is shown to be the free (weak monoidal) 2-category
on a `categorified Frobenius algebra', meaning that categorified Frobenius
algebras determine invariants of these 3-dimensional membranes.
