How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

Question

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor $$ d: \mathcal{C} \longrightarrow \mathcal{C}^{op} $$ such that $d(x)$ is dual to $x$ for all objects $x \in \mathcal{C}$. How can one construct such a functor?

Of course, the above is only a very weak formulation of what one would actually expect. In fact, it should be possible to choose $d$ such that there is a coherent homotopy $d^{op} \circ d \simeq \operatorname{Id}_{\mathcal{C}}$. By 'coherent' I mean that $\mathcal{C}$ should have homotopy fixed-point data for the involution $$ (\ )^{op}: \mathit{Cat}_\infty^{\otimes} \longrightarrow \mathit{Cat}_\infty^{\otimes}. $$ And while I'm on it, let me give the most general version of the quesion:

Is there a canonical (or maybe even essentially unique) trivialisation of the $(\ )^{op}$ involution when restricted to the full subcategory $\mathit{Cat}_\infty^{\otimes,d}\subset \mathit{Cat}_\infty^{\otimes}$ of those symmetric monoidal $(\infty,1)$-categories in which every object is dualisable?

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I am aware of a solution to this question in the $1$-categorical context; it goes as follows:

For a symmetric monoidal $1$-category $\mathcal{C}$ one constructs a category $D(\mathcal{C})$ where objects are tuples $(x,y,e,c)$ with $x,y$ two objects of $\mathcal{C}$ and $(e,c)$ is an evaluation-coevaluation pair that exhibits them as dual. The morphisms $\varphi:(x,y,e,c) \to (x',y',e',c')$ in this category are pairs $(f:x \to x',g:y' \to y)$ such that $f$ is dual to $g$ with respect to the duality data specified. This category admits canonical projections $$ \mathcal{C} \longleftarrow D(\mathcal{C}) \longrightarrow \mathcal{C}^{op}. $$ If $\mathcal{C}$ has duals, one can use essential uniqueness of duals to show that both projections are categorical equivalences. Using this construction it should be rather straight-forward to show all claims made above for the special case of $1$-categories. (Except for uniqueness of the trivialisation.)

Now the problem is that I have no idea how to generalise this proof to the $\infty$-categorical situation. For instance, for the two morphisms $f$ and $g$ to be dual would then be additional structure instead of a property.

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I would be happy to use the cobordism hypothesis in dimension $1$, if that is of any help. (It can be used to construct $d$ on the maximal subgroupid $\mathcal{C}^\sim \subset \mathcal{C}$, but it seems hard to use it to say anything about duals of non-invertible morphisms. As a side-note: Where can I find a proof or even a proof sketch of the cobordism hypothesis in dimension $1$?)

I'm interested in answers at any stage of generality.

Answer 1

One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a pairing of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're talking about a self-pairing on ${\cal C}$, which by definition is a right fibration $\mu:{\cal M} \to {\cal C} \times {\cal C}$, classified by a functor $b: {\cal C}^{\rm op} \times {\cal C}^{\rm op} \to {\cal S}$ to spaces. We say that $\mu$ is left representable if for every $x \in {\cal C}$ the functor $y \mapsto b(x,y)$ is representable in ${\cal C}$, and right representable if for every $y \in {\cal C}$ the functor $x \mapsto b(x,y)$ is represetable in ${\cal C}$. If $\mu$ is a pairing which is both left and right representable then it determines an adjunction between ${\cal C}$ and ${\cal C}^{\rm op}$. We say that $\mu$ is a perfect pairing if it is both left and right representable and the associated adjunction is an equivalence. One can then show that the data of a perfect pairing $\mu: {\cal M} \to {\cal C} \times {\cal C}$ is in fact equivalent to the data of an equivalence ${\cal C} \to {\cal C}^{\rm op}$. The advantage of this point of view is that the $\mathbb{Z}/2$-action on the space of equivalences ${\cal C} \to {\cal C}^{\rm op}$ becomes explicit: it is given by sending $\mu: {\cal M} \to {\cal C} \times {\cal C}$ to ${\rm swap} \circ \mu: {\cal M} \to {\cal C} \times {\cal C}$, where ${\rm swap}:{\cal C} \times{\cal C} \to {\cal C} \times {\cal C}$ is the equivalence that swaps the two components. In particular, to construct an equivalence ${\cal C} \stackrel{\simeq}{\to} {\cal C}^{\rm op}$ which is self-dual in a homotopy coherent manner is equivalent to constructing a perfect pairing ${\cal M} \to {\cal C} \times {\cal C}$ together with a $\mathbb{Z}/2$-action on ${\cal M}$ which lifts the swap action on ${\cal C} \times {\cal C}$. Note that this swap action can itself be encoded as a right fibration $p:{\rm Sym}({\cal C}) \to {\rm B}(\mathbb{Z}/2)$, where ${\rm B}(\mathbb{Z}/2)$ is the groupoid with one object whose endomorphism group is $\mathbb{Z}/2$, and the fiber of $p$ over this one object is ${\cal C} \times {\cal C}$. One can then encode a lift of the swap action to ${\cal M}$ by a right fibration $\widetilde{\cal M} \to {\rm Sym}({\cal C})$ whose restriction to ${\cal C} \times {\cal C}$ is ${\cal M}$. In general, we may refer to right fibrations $\widetilde{\cal M} \to {\rm Sym}({\cal C})$ as symmetric pairings, and say that a symmetric pairing is perfect when its base change to ${\cal C} \times {\cal C} \subseteq {\rm Sym}({\cal C})$ is perfect. The notion of a perfect symmetric pairing then encodes a duality on ${\cal C}$, i.e., a self-dual equivalence $d:{\cal C} \to {\cal C}^{\rm op}$ (with all the higher coherences taken into account).

Now in the case of a symmetric monoidal $\infty$-category with duals, the pairing encoding the duality is the right fibration ${\cal M} \to {\cal C} \times {\cal C}$ classified by the functor $b_{\cal C}: {\cal C} \times {\cal C} \to {\cal S}$ sending $(x,y)$ to ${\rm Map}_{\cal C}(x \otimes y,1_{\cal C})$. There is a direct way to construct this as a symmetric perfect pairing. Indeed, suppose that $\pi:{\cal C}^{\otimes} \to {\rm Fin}_*$ is the coCartesian fibration encoding the symmetric monoidal structure on ${\cal C}$, and let ${\cal C}^{\otimes}_{\rm act} \to {\rm Fin}$ be its active part, i.e., the base change to the category ${\rm Fin}$ of finite sets via the functor $I \mapsto I_+$. Then we can identify ${\rm B}(\mathbb{Z}/2)$ as the subcategory of ${\rm Fin}$ consisting of the object $\left<2\right>^{\circ} = \{1,2\} \in {\rm Fin}$ and all its automorphisms, and the base change of ${\cal C}^{\otimes}_{\rm act}$ to ${\rm B}(\mathbb{Z}/2) \subseteq {\rm Fin}$ is naturally equivalent to ${\rm Sym}({\cal C})$. If $1_{\cal C} \in ({\cal C}^{\otimes}_{\rm act})_{\left<1\right>^{\circ}}$ now denotes the unit then the right fibration $$ ({\cal C}^{\otimes}_{\rm act})_{/1_{\cal C}}\times_{{\rm Fin}} {\rm B}(\mathbb{Z}/2) \to {\cal C}^{\otimes}_{\rm act} \times_{{\rm Fin}}{\rm B}(\mathbb{Z}/2) \simeq {\rm Sym}({\cal C}) $$ is a symmetric pairing whose underlying pairing is $b_{\cal C}$ above. When ${\cal C}$ has duals this symmetric pairing is perfect, yielding the associated self-dual equivalence $d:{\cal C} \to {\cal C}^{\rm op}$.

Answer 2

I don't consider this an actual answer to the question, as it uses a hypothesis that is, to my knowledge, not yet proven. I hope that there is a simpler answer to this question that is more elementary and does not rely on such an assumption.

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I've come up with the following argument and, even though it does not answer the question, I feel like it would be helpful to share it. The argument has the following downsides:

• It is rather lengthy when spelled out in detail. For now I will only provide a sketch, but I'm happy to fill in details if requested.

• It constructs $d$ only after forgetting the symmetric monoidal structure. (This part is not actually hard to fix.)

• It relies on the following hypothesis:

Hypothesis: We have a model $B_1([n])$ of the symmetric monoidal $\infty$-category with duals freely generated by the poset $[n]$ that we understand well enough to give an isomorphism $$ \rho_n: B_1([n]) \cong B_1([n]^{op}). $$ (There also is a canonical isomorphism $B_1([n]^{op}) \cong (B_1([n]))^{op}$. After composing with this $\rho_n$ should send objects to their duals.)

Assuming the $1$d-cobodism hypothesis with sigularities such a model can be given in terms of a category of framed one-dimensional bordisms with certain labels. (See this question.) In the specific model I have in mind $\rho_n$ can be defined by simply reversing the orientation.

However, I'm not aware of a complete proof of the cobordism hypothesis with singularities, even in dimension $1$. If someone is, I'd be very glad to here about it.

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Sketch of the argument:

Let's fix some symmetric monoidal $\infty$-category with duals $\mathcal{C}^\otimes$ and write $\mathcal{C}$ for its underlying $\infty$-category. Now let $X: \Delta^{op} \to \mathit{Spaces}$ denote the simplicial space that represents $\mathcal{C}$ as a complete Segal space. By definition this is given by $$ X_{[n]} := (\mathit{Fun}_\infty( N[n], \mathcal{C}) )^\sim. $$ (Here the $(\ )^\sim$ takes the underlying $\infty$-groupoid of the functor category and interprets it as a space. The functor $N$ is the inclusion $\Delta \to \mathit{Cat} \to \mathit{Cat}_\infty$. )

By our hypothesis the symmetric monoidal $\infty$-category $B_1([n])$ is free with duals on $[n]$, so there is an equivalence of spaces $$ Y_{[n]}:= (\mathit{Fun}_\infty^\otimes( B_1([n]), \mathcal{C}^\otimes) )^\sim \ \simeq \ (\mathit{Fun}_\infty( N[n], \mathcal{C}) )^\sim =X_{[n]} $$

Hence $Y_{\bullet}: \Delta^{op} \to \mathit{Spaces}$ defines a Segal space equivalent to $X$ and therefore also models the $\infty$-category $\mathcal{C}$. By our hypothesis we have $\rho_n:B_1([n])\cong B_1([n]^{op})$. Assuming that these isomorphisms play well with the fuctoriality in $[n]$ this induces an equivalence $$ d: \mathcal{C} \sim Y_\bullet \cong (Y_\bullet)^{op} \sim \mathcal{C}^{op}. $$

One can now check that the assumptions of $\rho_n$ imply that $d$ sends objects and morphisms to their duals.

With a bit more care $d$ can be made symmetric monoidal and functorial in $\mathcal{C}$ such that it indeed gives a canonical trivialisation of $\ ^{op}$ on $\mathit{Cat}_\infty^{\otimes,d}$. I don't see how one could use this approach to show that the trivialisation is essentially unique.