Are fully extended TQFTs generalized cohomology theories?

Question

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?

Answer 1

A fully extended tqft is not quite a generalized homology theory... But almost. You can find a preliminary reference here (notes of a talk by Hiro Tannaka at the mit Talbot workshop): http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology(hiro).pdf The precise statements you might be interested in are Theorems 2.16 and 2.20.

(side remark: the notes of the whole workshop are worth reading: http://math.mit.edu/conferences/talbot/2012/notes/talbot_2012_notes(claudia).pdf).

EDIT : the work announced in the above talk is now partly available on John Francis' webpage: http://www.math.northwestern.edu/~jnkf/writ/ (see "Factorization homology of topological manifolds" and "Structured singular manifolds and factorization homology").

Answer 2

If you mean 'generalised cohomology'as in 'all but dimension axiom', then no. Those animals are additive functors, while tqft's are multiplicative.