Are dagger-categories / categories with duality related to unoriented field theories?

Question

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.

Answer 1

The answer is yes. More specifically, dagger categories are related to unoriented one-dimensional TQFTs, a perspective I learned from Lukas Müller.

Consider the symmetric monoidal category $Vect_k$ of vector spaces over a field $k$. Anything I will say works for an arbitrary symmetric monoidal $\infty$-category, but I want to make it more concrete and connect it to the classical definition of TQFT. If we restrict to dualizable objects (finite-dimensional vector spaces), the dual makes $Vect_k$ into a 'category with duality' in the sense you mention: there is a functor $(.)^* \colon Vect_k \to Vect_k^{op}$ equipped with data telling us how $(.)^{**} \cong id_{Vect_k}$.

As Simon Henry explained in his comments, the $O(1)$-action on the $(2,1)$-category of categories trivializes on the subcategory of groupoids. Therefore, the duality induces an $O(1)$-action on $core(Vect_k)$, where $core$ takes the maximal subgroupoid. Note that $core(Vect_k)$ is the space of framed/oriented 1d TQFTs and this is the $O(1)$-action featuring in the cobordism hypothesis. The groupoid of unoriented field theories is therefore $core(Vect_k)^{O(1)}$, the groupoid of vector spaces equipped with a nondegenerate symmetric bilinear form. Explicitly, the bilinear form associated to a given 1d TQFT is the self-duality obtained by evaluating a half-circle as a bordism $pt \sqcup pt \to \emptyset$.

Simon Henry also mentioned that $(.)^*$ does not make $Vect_k$ into a dagger category because it is not the identity on objects. However, the category of vector spaces equipped with a nondegenerate symmetric bilinear form is a dagger category via the adjoint operation. Considering this bigger category also makes sense from the perspective of TQFTs with defects: natural transformations between TQFTs are automatically invertible, but defect TQFTs give a geometric interpretation to noninvertible 1-morphisms in the target category. For unoriented TQFTs, defects between them have 'no direction', which is made precise in our example by thinking of them to consist of the pair $(f,f^\dagger)$. Abstractly, we can think of unoriented 1d defect TQFTs as the flagged category $$ core(Vect_k)^{O(1)} \to Vect_k. $$ This picture is expected to generalize to $n$ dimensions using what we call $O(n)$-dagger categories in the paper https://arxiv.org/abs/2403.01651 Urs mentioned.

By the way: the only dagger $1$-categories which arise from $1$-categories with duality using the above construction are those for which every self-adjoint automorphism is of the form $f^\dagger f$ for $f$ an automorphism. In https://arxiv.org/pdf/2304.02928 we show that there is an equivalence of $2$-categories between dagger categories of this type and categories with duality such that every object admits a symmetric self-duality. It would be interesting to try and make sense of this part for $\infty$-categories.