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Non-extendable 2D TQFTs correspond to finite dimensional Frobenius algebras [1].

How about 3D TQFTs? While the answer is clear for the extended ones (e.g. (3,2,1) TQFTs almost correspond to modular tensor categories [2]), I have not seen any discussion for (3,2) TQFTs.

More precisely, can one classify the functors

$$Cob_{3,2}^{oriented} \to (Vect_\mathbb{C})?$$

Reference

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Check out Andras Juhasz' paper: https://arxiv.org/pdf/1408.0668.pdf

Specifically, Theorem 1.10:

There is an equivalence between the symmetric monoidal category of (2+1)-dimensional TQFTs and the category of J-Algebras.

J-Algebras are somewhat cumbersome to define. They are graded vector spaces equipped with a Frobenius-like structure, together with an action of the genus $g$ mapping class group on the $g$-th graded piece.

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