Characterizing Isbell self-dual objects

Question

It is well known that under the Isbell duality $\text{Spec}\dashv {\cal O} : {\cal V}^{A^°} \leftrightarrows \big({\cal V}^A\big)^°$ representable functors are self-dual, i.e. fixed by the unit and counit of the adjunction: $$\text{Spec}({\cal O}(\hom(-,x)))\cong \hom(-,x)\qquad\qquad {\cal O}(\text{Spec}(\hom(y,-)))\cong \hom(y,-)$$

No mention is made about representables being the unique self-dual objects. Is it true?

If not, how to characterize a bigger class of Isbell self-dual objects? My initial guess was that at least a class of colimits of representables are Isbell-self-dual. They are not, even for coproducts, if this computation is correct:

1. ${\cal O}(\hom(-,x)\amalg \hom(-,y)) = \hom(x,-)\times \hom(y,-)$;

2. $\text{Spec}$ of this evaluated in $a$ now equals $$ Nat(\hom(-,x)\times \hom(y,-), \hom(a,-)) $$ that doesn't seem related to the initial presheaf. If you assume that $A$ has coproducts, in fact, it is true that this is $\hom(a,x\amalg y)$, so that $\text{Spec}({\cal O}(h_x\amalg h_y)) \cong h_{x\amalg y}$, different from $h_y\amalg h_y$ in general.

This question has already been asked on math.SE, and it was me that pointed the OP to another MO thread discussing the topic, but I'm seeking some explicit answer now.

Answer 1

A quick answer for now, which I might add more to later. Recall that for every adjunction $F \dashv G: C \to D$ there is the notion of "fixed point" of the adjunction which has two faces: either it is an object $c$ of $C$ for which the counit $\epsilon_c: FGc \to c$ is an isomorphism, or it is an object $d$ of $D$ for which the unit $\eta_d: d \to GFd$. The adjunction $F \dashv G$ then induces an adjoint equivalence between the full subcategories $\text{Fix}_{FG}(C)$ and $\text{Fix}_{GF}(D)$, and thus we identify these categories. In the case of the Isbell conjugation adjunction for a small category $X$, the category of fixed points is often called the Isbell envelope or Isbell completion; let's denote it $I(C)$.

In the special case where $C$ is a preorder, i.e., a $\mathbf{2}$-enriched category, $I(C)$ is a preorder which goes by a more famous name: the Dedekind-MacNeille completion. You can find some discussion of this in the MO-answer here. In this case $I(C)$ is both complete and cocomplete (i.e. admits small limits and colimits).

In other cases, the Isbell completion need not be $\mathcal{V}$-complete/cocomplete, as you correctly surmise (see comments below the MO-answer I just mentioned), but it can be quite a bit larger (and certainly larger than the Cauchy completion). It's maybe best to point to some examples. For the case where $\mathcal{V}$ is the monoidal closed category $([0, \infty), \geq, +)$, i.e., where $\mathcal{V}$-categories are metric spaces in the sense of Lawvere, Simon Willerton has an interesting article which connects the Isbell completion to the tight span of metric spaces; you can find a Café discussion here where some examples are computed.

Isbell completions also figure in "comparative smootheology", but I'd need to study and think more before commenting on that. You can also find more references to the Isbell completion in the nLab; as I said, I may come back and add more.