I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of cobordisms to a suitable $\left(\infty,n\right)$-category of vector spaces.


The original Atiyah-Witten definition was a functor for the category of $n$ dimensional cobordisms to $\mathrm{Vect}\_{\mathbb C}$. This definition was justified from the path integral in physics.


Can we similarly get an physicsy intuition of extended TQFT from a path integral-like formulation? Any references to a general construction from physics that gives rise to such a functor, starting from the path integral? 


Note: I don't want specific example relating to Chern-Simons theory or any other TQFT, but a general construction deriving the extended TQFT axioms from the path integral, or something similar.



Path integral derivation of extended TQFT
The physics motivation for extended QFTs (and not just TQFTs) comes from the locality principle (no spooky action at a distance).


The mathematical expression of locality is the descent property for extended QFTs.
For an early source, see, for instance, [Higher Algebraic Structures and Quantization](https://arxiv.org/abs/hep-th/9212115) by Daniel S. Freed
and [Triangulations, Categories and Extended Topological Field Theories](http://dx.doi.org/10.1142/9789812796387_0011) by Ruth J. Lawrence.


The descent property is proved in full generality in [arXiv:2011.01208](https://arxiv.org/abs/2011.01208).


Specifically, the assignment to X of the symmetric monoidal (∞,n)-category of extended QFTs with bordisms equipped with a map to X is a stack of symmetric
monoidal (∞,n)-categories with respect to X.


This claim fails unless we work with QFTs extended all the way down to points,
since proving the descent property requires cutting bordisms all the way
down to points.


An informal path integral construction produces an extended QFT
starting from another functorial field theory (quantum or classical) on Y
and performing pushforward along a map Y→X.
The pushforward simply integrates over spaces of n-dimensional manifolds
mapping to Y that have the same image in X.


