
> 
> 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?
> 
> 
> 




---


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.




---


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.**



How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?
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 $d:{\cal C} \to {\cal C}^{\rm op}$.


