# Are fully extended TQFTs generalized cohomology theories?

Forgive the naiveness of this question. Whatever an $n$-vector space exactly is, one expects that the basic example of fully extended $n$-dimensional tqft is a symmetric monoidal functor $Cob_n\to n$-Vect. Now, whatever an $n$-vector space exactly is, one expects $(n-1)$-Vect to be the based loop space of $n$-Vect. This suggests that the $n$-categories of $n$-vector spaces organize themselves in an hypothetical spectrum Vect and that the tqft invariants one computes are actually cohomology classes for the corresponding generalized cohomology. For instance, the fact that a fully extended tqft is completely determined by its value on a point would be in this perspective an analogue of Mayer-Vietoris. Also, the combinatorial constructions of the Dijkgraaf-Witten model would be an analogue of operations in simplicial cohomology. So it seems there is some general abstract nonsense supporting the above point of view.

Question: are there references addressing/formalizing/developing this point of view?

-
TQFTs in general are not homotopy-invariant (for example, they distinguish homotopy equivalent lens spaces), so one probably wouldn't expect them to be derived from a generalized cohomology theory. – Ian Agol Jun 18 '12 at 16:24