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?

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    $\begingroup$ 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. $\endgroup$ – Ian Agol Jun 18 '12 at 16:24

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").


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


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