The bicategory of locally presentable categoires, and left adjoint functor between them, is monoidal closed for the Kelly tensor product. My question is what are the dualizable objects for this monoidal structure ?
Just to be clear, I'm talking about the monoidal stucture for which the internal hom $[A,B]$ is given by the category of left adjoint functor $A \rightarrow B$; or equivalently for which morphismw $A \otimes B \rightarrow C$ are the functors $A \times B \rightarrow C$ that commutes to colimits in each variables.
A typical example of such a dualizable object are the presheaves categories. If $C$ is small then the presheaf category $\widehat{C}$ is dualizable with dual $\widehat{C^{op}}$. The evaluation
$$\widehat{C} \otimes \widehat{C^{op}} \simeq \widehat{C \times C^{op}} \rightarrow Sets $$
Is given by the coend, and the counit:
$$ Sets \rightarrow \widehat{C} \otimes \widehat{C^{op}} \simeq \widehat{C \times C^{op}} $$
Is the unique left adjoint functor sending the singleton to $Hom_C( \_ ,\_ )$. The usual unit/counit relation being essentially the coend formulation of the Yoneda lemma.
I suspect there are some dualizable object not of this form, but this is not totally clear to me, and I wonder if it is possible to give a nice description of the dualizable objects in general…
Best Answer
I'm not sure about the linear case described in Theo's answer, but in the setting of locally presentable categories there are dualizable objects which are not presheaf categories. Instead they are non-trivial retracts of presheaf categories (note that since $\mathcal{Pr}^L$ is idempotent complete any retract of a dualizable object is dualizable). You can think of these guys as "projective" locally presentable categories which are not free (it's possible that in the linear setting this distinction disappears). I came across them when answering another question on a related topic: https://mathoverflow.net/questions/252698/does-bf-prof-admit-all-pseudolimits/252886#252886.
One way these guys come up is when you have a colimit-preserving comonad $F:\widehat{C} \to \widehat{C}$ on a presheaf category $\widehat{C}$ which is idempotent, that is, the comultiplication $F \Rightarrow F \circ F$ is an isomorphism. In this case the category ${\rm coAlg}_F(\widehat{C})$ of $F$-coalgebras identifies with the full subcategory of $\widehat{C}$ spanned by the $F$-colocal objects, that is, the presheaves $P \in \widehat{C}$ for which the counit $F(P) \Rightarrow P$ is an isomorphism. The functor $F$ then breaks as a composition $$ \widehat{C} \stackrel{\overline{F}}{\longrightarrow} {\rm coAlg}_F(\widehat{C}) \stackrel{\iota}{\longrightarrow} \widehat{C} $$ where $\iota$ is the inclusion of colocal objects. Then $\iota$ preserves and detects colimits and since $F$ preserves colimits we get that $\overline{F}$ preserves colimits, so that both $\overline{F}$ and $\iota$ are morphisms in ${\rm Pr}^L$. Since the counit map $F \Rightarrow {\rm Id}$ is an isomorphism on $F$-colocal objets we now get a retract diagram $$ {\rm coAlg}_F(\widehat{C}) \stackrel{\iota}{\longrightarrow} \widehat{C} \stackrel{\overline{F}}{\longrightarrow} {\rm coAlg}_F(\widehat{C}) $$ in ${\rm Pr}^L$, and so ${\rm coAlg}_F(\widehat{C})$ is dualizable. Its dual can be identified with a similar type of category of coalgebras on the dual $\widehat{C^{\rm op}}$, with respect to the idempotent comonad $F^*: \widehat{C^{\rm op}} \rightarrow \widehat{C^{\rm op}}$ induced from $F$ by identifying $\widehat{C^{\rm op}}$ with ${\rm Fun}(\widehat{C},{\rm Set})$.
One somewhat striking feature of this setup is that the category ${\rm coAlg}_F(\widehat{C})$ turns out to be equivalent in this situation to the category of algebras ${\rm Alg}_G(\widehat{C})$ over the monad $G: \widehat{C} \to \widehat{C}$ which is right adjoint to $F$. This monad will also be idempotent and this category of algebras is just the full subcategory of $\widehat{C}$ spanned by the $G$-local objects. This subcategory is not the same as the subcategory of $F$-colocal objects, but the functors $F$ and $G$ restrict to an equivalence between these two full subcategories. Similarly, the dual of ${\rm coAlg}_F(\widehat{C})$ is simultaneously a category of algebras and a category of coalgebras. Note also that ${\rm Alg}_G(\widehat{C})$ is a localization of $\widehat{C}$ with a localization functor which preserves all limits, and hence ${\rm Alg}_G(\widehat{C})$ is in particular a topos (and so is its dual). I have no idea if all dualizable objects in ${\rm Pr}^L$ arise in this way.
Finally, let us describe a way to construct explicit examples of idemponent colimit-preserving comonads on $\widehat{C}$. Suppose that we have a non-unital subcategory $C_0 \subseteq C$. By this I just mean that we have a designated collection of morphisms in $C$ which is closed under composition, but is not assumed to contain the identities. Suppose also that $C_0$ is an "ideal" in $C$, that is, any composition of a map in $C_0$ with a map in $C$ (from the right or left) is again in $C_0$. Define $F: \widehat{C} \to \widehat{C}$ by the formula $$ F(P)(X) = {\rm colim}_{[X \to Y] \in (C_0)_{X/}} f(Y) $$ for $P: C^{\rm op} \to {\rm Set}$ a presheaf. Here by $(C_0)_{X/}$ I mean the full subcategory of $C_{X/}$ consisting of those arrows $X \to Y$ which belong to $C_0$. Note that $F$ comes with a natural transformation $\eta: F \Rightarrow {\rm Id}$. Now assume in addition that for every arrow $f: X \to Y$ in $C_0$, the category of factorizations of $f$ in $C_0$ (that is, the full subcategory of $(C_{X/})_{/f}$ spanned by the factorizations $X \to Z \to Y$ of $f$ in which both components are in $C_0$) is weakly contractible. One can then show that under this condition the maps $$ \eta_{F(P)},F\eta_P: F(F(P)) \to F(P) $$ are both isomorphisms, which are actually the same isomorphism. In this case $F$ can in fact be endowed with a canonical idempotent comonoid structure by inverting these isomorphisms. The colocal objects are then the presheaves $P: C^{\rm op} \to {\rm Set}$ for which the natural maps ${\rm colim}_{[X \to Y] \in (C_0)_{X/}}P(Y) \to P(X)$ are isomorphisms. Alternatively, if we switch from the coalgebra description to the algebra description then one can consider instead the category of those presheaves $P$ for which the maps $P(X) \to {\rm lim}_{[Y \to X] \in (C_0)_{/X}}P(Y)$ are isomorphisms (a description which resembles more the notion of a sheaf. Indeed, this yields a topos).
Finally, as shown in the linked answer, under certain conditions on $C_0$ one can show that ${\rm coAlg}(\widehat{C})$ is not equivalent to any presheaf category. These conditions hold, for example, if one takes $C$ to be the poset of opens in a non-trivial compact Hausdorff topological space, and lets $C_0$ be the collection of those inclusions $V \subset U$ such that $U$ contains the closure of $V$.