This degenerate case turns out to exhibit a nontrivial amount of interesting information, in particular if regarded in the context of super QFT.
The category of 0-dimensional cobordisms is the symmetric monoidal category having the -dimensional manifold as the only object and isomorphism classes of compact -dimensional manifolds as morphisms. Clearly is equivalent to
By definition of monoidal functor, one has and so is completely (and freely) determined by the assignment . In other words, the space of 0-dimensional TQFTs is .
One can consider TQFTs with a target manifold : all bordisms are required to have a map to .
The picture becomes more interesting if one goes from topological field theory to extended topological quantum field theory. Indeed, from this point of view, to the -dimensional vacuum is assigned the symmetric monoidal 0-category , and consequently, the infinity-version of the space of all -dimensional TQFTs is the Eilenberg-Mac Lane spectrum. It follows that the space of extended -dimensional TQFTs with target (taking values in -modules) is the graded integral cohomology ring .
From the differential geometry point of view, a relation between de Rham cohomology of a smooth manifold and -dimensional functorial field theories arises if one moves from topological field theory to -supersymmetric field theory, see Axiomatic field theories and their motivation from topology.
It would be interesting to describe a direct connection between the extended and the susy theory; it should parallel the usual Cech-de Rham argument