Topological gravity in dimensions two and four

A (two-)category with manifolds as objects and cobordisms as maps [and diffeomorphisms as two-morphisms] becomes a very interesting topological category when its morphism categories are replaced by their classifying spaces: the result is a hybrid of Segal's category and Wheeler's superspace. The physicists' theories of topological gravity are just monoidal representations of such categories: Kontsevich-Witten theory is the `standard' example in two dimensions, but classical Abel-Jacobi theory also fits naturally in this framework. Floer-Donaldson theory yields a more complicated, and interesting, example.