Recall that a dagger-category is a category $C$ equipped with an identity-on-objects, involutive anti-equivalence $\dagger: C^{op} \to C$. I believe that the correct $\infty$-categorical generalization is that a dagger $\infty$-category is an object of $Cat_\infty^{hC_2}$, where $C_2$ acts by taking opposite categories, which I believe has been called an "$\infty$-category with duality" in the literature.

**Question:** Is there a TQFT interpretation of dagger categories / categories with duality?

What I have in mind is that the $C_2 = O(1)$ action on $Cat_\infty$ should be viewed as the $O(1)$-action on the dualizable objects in some symmetric monoidal $\infty$-category (probably the $\infty$-category $Prof$ of categories and profunctors between them), so that to give a category the structure of a dagger category is to lift it from an oriented 1-dimensional field theory to an unoriented one. The awkward issue is that $Cat_\infty$ doesn't have duals -- so although we can embed it into $Prof$ so that every object becomes dualizable, we now have different 1-morphisms, so I'm not sure whether the notion of dagger category falls out precisely.