Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some target category, usually your favorite version of $n$-Vect. When $k=0$, a full "classification" of TQFTs with a given target category is given in:
- Lurie, Jacob. On the classification of topological field theories. Current developments in mathematics, 2008, 129--280, Int. Press, Somerville, MA, 2009. 58Jxx (57Rxx) MR2555928. arXiv:0905.0465.
Or, rather, Lurie first provides reasonable definitions for a number of things, end then proves that there is an equivalence of $n$-categories between the $(0,\dots,n)$-TQFTs with target $\mathcal V$ and the $n$-groupoid of ("fully") dualizable objects in $\mathcal V$. (The classification is not particularly effective in two ways: given a dualizable object, which is the value the TQFT assigns to a point, it can be still very hard to understand the functor on complicated manifolds; and given a category, it can be still very hard to classify its dualizable objects.) For a review, see nLab: cobordism hypothesis.
But Lurie's result does not describe all gadgets that deserve to be called "TQFT"s. For example, it is a classical folk theorem that $(1,2)$-TQFTs are the same as commutative cocommutative Frobenius algebras. I think that there are other similar results of this nature, but I don't know of any theory that puts them all into a single framework. Hence:
Question: Is there a classification, similar to Lurie's, for $(k,\dots,k+n)$-TQFTs with a give target $n$-category?