We will use the reference http://arxiv.org/abs/math/0412149v7.
Let M be Segal's category of Riemann surfaces. Let C∗ be a symmetric monoidal functor from Top to the category of complexes of K-vector spaces, with K a base field of characteristic 0. Let Chn denote the category of n-dimensional chain complexes. A CQFT is defined as a symmetric monoidal functor C∗(M)→Ch. Note that C∗(M)=Ch3, so the CQFT is a functor Ch3→Ch. The CQFT maps Ch3 to some subset Ch3⊂Ch3. Using Lurie's classification allows us to define a CTQFT (a TQFT with the structure of a CQFT) as a functor C:3Cob→Ch3. This is a chain-complex valued TQFT, hence allowing us to conclude that
Idea: A CTQFT is a chain-complex valued TQFT. I.e., let ~3Cob denote the (∞,1) category with objects as two-dimensional manifolds, morphisms as bordisms, 2-morphisms as diffeomorphisms, and 3-morphisms as isotopies. Then, the functor C:~3Cob→Ch3⋂nHilb is a CTQFT.
My question is, can this idea be extended to the more general class of CQFTs? I.e., can we get Idea 2?
Proposed Idea 2: A CQFT is a chain-complex valued (T)QFT, in the sense mentioned above.