Dualizable object in the category of locally presentable categories

Question

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...

Answer 1

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 $V$.

Answer 2

We studied this question (or rather, the version linear over a field) in some detail in Brandenburg, Chirvasitu, and JF. Reflexivity and dualizability in categorified linear algebra. As you point out, it is easy to show that categories of presheaves (equivalently, categories with enough compact projectives) are dualizable. We conjecture in that paper that that's it. This conjecture seems hard.

In support of the conjecture, we were able to show that the conjecture holds for categories equivalent to the category of comodules of a coalgebra. Noah Snyder later pointed out to me that there is a simple intrinsic description of which categories are equivalent to the comodules of a coalgebra: they are (the ind-cocompletions of) the "length categories" aka "abelian categories with length". They arise naturally in the categorical linear algebra because this is the largest (in the sociological sense) class of categories closed under the Kelly–Deligne tensor product. (I have never heard the name "Bird–Lurie" for this tensor product; everyone I know calls it by a nonempty subset of {Kelly, Deligne}.)

Since I think the general conjecture turns out to be hard, I think it best to attack special cases. The most interesting, to me, next case to handle would be categories of quasicoherent sheaves on a (nice) scheme. Recall that $\mathrm{QCoh}(X)$ has enough compact projectives iff $X$ is affine (IIRC, a theorem of Deligne's); so the conjecture states that $\mathrm{QCoh}(X)$ is dualizable in $\mathrm{Pres}^L$ iff $X$ is affine. "Most" $X$ which are not affine contain a (positive dimension) closed projective subvariety, and witnessing such a subvariety is certainly the most straightforward way to show a scheme is not affine. For those $X$, we were able to verify the conjecture. This basically just leaves the quasiaffines (and in unpublished work Chirvasitu has explained how to conclude the quasiprojective version of the conjecture from the quasiaffine version). The standard example of a non-affine quasiaffine is $\mathbb{A}^2 \smallsetminus \{0\}$, and in unpublished work we did check that its category of quasicoherent sheaves was not dualizable.

---

Two other comments. First, we end our paper by showing that there are plenty of presentable categories which are reflexive (equal to their double dual) but not dualizable. So $\mathrm{Pres}^L$ feels a lot like the category of abelian groups, where it is a fun exercise to show that $\mathbb{Z}^{\oplus\infty}$ is reflexive.

Second, many (linear) 2-categories are surveyed in the appendix to Bartlett, Douglas, Schommer-Pries, Vicary. Modular categories as representations of the 3-dimensional bordism 2-category, and for those surveyed, it is shown that, although their subcategories of 1-dualizable objects depend sensitively on the ambient 2-category (e.g. how cocontinuous are the functors?), the subcategories of 2-dualizable objects are all the same. Given that appendix, Scheimbauer has proposed the name "Bestiary Hypothesis" for the proposal that all "reasonable" $n$-categories of "$n$-vector spaces" should have equivalent subcategories of $n$-dualizable objects, even though their subcategories of $(<n)$-dualizables typially differ. Sadly, $\mathrm{Pres}^L$ is not known the satisfy the Bestiary Hypothesis: just like we don't know what all the $1$-dualizables are, we also don't know what all the $2$-dualizables are.

Answer 3

The following is a completement to the answer by Yonatan. I managed to work it out after I read what he had done.

It is basically about proving the converse to the implication he showed in his answer: If I'm correct one can prove that for a locally presentable category $A$, the following are equivalent:

$(1)$ $A$ is dualizable in $Pres^L$

$(2)$ $A$ is a retract of a presheaf category in $Pres^L$

$(3)$ $A$ can be constructed from an ideal $C_0 \subset C$ as described in Yonatan answer.

This being said, I'm not excluding that a better result can be obtained (for example a more canonical presentation of dualizable objects), though the example given by Yonatan seem to suggest that this is already quite close to be optimal.

$(3) \Rightarrow (2) \Rightarrow (1) $ is discussed in Yonatan answer, I'll prove $(1) \Rightarrow (2)$ and $(2) \Rightarrow (3)$ separately.

---

For $(1) \Rightarrow (2)$. Let $A$ be a dualizable object in $Pres^L$, its dual $A^*$ is isomorphic to the category of left adjoint functors $[A,Set]$. The fact that $A$ is dualizable means that there is a coevaluation map $Set \rightarrow A \otimes A^*$, which is fully described by the image of the singleton, giving a specific object $c \in A \otimes A^*$.

As any object of $A \otimes A^*$, $c$ can be written as a colimits of pure tensor:

$$ c = colim_{i \in I} a_i \otimes \chi_i $$

where $a_i \in A$ and $\chi_i \in A^* = [A,Set]$. The evaluation-coevaluation relation translate into:

$$ colim_{i \in I} \chi_i (u) \times a_i \simeq u $$

functorially in $ u \in A$ (where $\times$ above denote the cotensoring objects by sets, i.e. the coproduct of several copies of the same object).

It means that the $\chi_i$ together form a cocontinuous functor $A \rightarrow Set^I$, And $F \mapsto colim F(i) \times a_i$ form a cocontinuous functor from $Set^I$ to $A$, whose composite $A \rightarrow Set^I \rightarrow A$ is naturally isomorphic to the identity, hence $A$ is a retract of $Set^I$.

---

I now give a sketch of proof for $(2) \Rightarrow (3)$:

Assume that one has two morphisms in $Pres^L$, $f:A \rightarrow \widehat{C}$ and $g:\widehat{C} \rightarrow A$ with $g \circ f \simeq_{\Theta} Id$.

Then $P = f \circ g : \widehat{C} \rightarrow \widehat{C}$, is idempotent $P^2 \simeq_{f \theta g} P $, and moreover the isomorphism is an associative multiplication $P^2 \rightarrow P$ (I'm goging to say that $P$ is a non-unital monad), and the category $A$ identifies with the algebras for this multiplication (in the non unital sense), whose structure maps is are isomorphisms $PX \overset{\sim}{\rightarrow} X$.

Using that $P$ commutes to coproduct, this makes $P \coprod Id$ into a cocontinuous (unital) monad on $\widehat{C}$, whose algebras are the $P$-algebra (as a non-unital monad), and hence one has a category $C_P$, whose objects are these of $C$ and morphism from $c$ to $c'$ are morphisms from $c$ to $(Id \coprod P) c'$, i.e. either morphism from $c$ to $c'$, or morphism from $c$ to $P c'$ (composition of the morphism of the second type involving the use of the multiplication maps $P^2 \rightarrow P$

The map of the form $c \rightarrow P c'$ form an ideal in $C_P$, and because of the relation $P^2 \simeq P$ they satisfies the additional condition in Yonatan answer.

Now presheaves over $C_P$ are the same as presheaves over $C$ with an algebra structure for the monad $Id \coprod P$, i.e. algebra for the non-unital monade $P$. One can then check that the additional condition in Yonatan answer (with respect to the ideal mentioned above) single out the $P$-algebra for which the structure map is an isomorphism, and hence it corresponds exactly to the category $A$ we started from.