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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.

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    $\begingroup$ The correct $\infty$-categorical generalization of dagger categories is a little more complicated than this - you need to include the groupoids of objects and "dagger-isomorphism" between them as part of the structure. This was considered here mathoverflow.net/questions/220032/… (see Peter Lumsdain answer) $\endgroup$
    – Simon Henry
    Apr 12, 2023 at 14:53
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    $\begingroup$ The problem with the definition you propose is that it miss the key condition "acts trivially on objects". I think the correct definition can be expressed as follows: the action of $C_2$ on $Cat_\infty$ by taking the opposite category restrict to a "trivial" action on the full subcategory $Gpd_\infty \subset Cat_\infty$. That is we have a functor $Gpd_\infty \to Cat_\infty^{C_2}$. An $\infty$-dagger category is a pair [...] $\endgroup$
    – Simon Henry
    Apr 12, 2023 at 15:00
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    $\begingroup$ [...] of an $\infty$-groupoid $X$ and an object $C$ of $Cat_\infty^{C_2}$ together with a morphism $X \to C$ in $Cat_\infty^{C_2}$ which when seen as a morphism in $Cat_\infty$ is a (non-full) subcategory inclusion which is subjective on objects. And then we impose a Segal type condition that forces $X$ to be the groupoid of dagger-isomorphisms in $C$. $\endgroup$
    – Simon Henry
    Apr 12, 2023 at 15:00
  • $\begingroup$ There is apparently a 12-author article coming up with more discussion, talk notes here: ncatlab.org/nlab/show/dagger+category#ReferencesHigherDagger $\endgroup$
    – Urs Schreiber
    Nov 9, 2023 at 8:06

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