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I have edited at HQFT, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.
Are HQFTs taking place in some slice or context from the HoTT point of view?
I think the good abstract way to think about them is given by that version of the cobordism hypothesis which I was alluding to above: $Bord^{fr}_n(X)$ is essentially the free $(\infty,n)$-category with duals not on the point, but on the $\infty$-groupoid $\Pi(X)$.
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