This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures of some $D^S$ (a product of $D$ with itself many times), so they **both** carry some topology induced from the Tychonoff topology on the product, although this topology is usually discarded on one of the sides.

I understand relatively well when in the Pontryagin case the dual of a compact abelian group sits inside a power of $D$ as a discrete subgroup, so the topology honestly disappears. But this is relatively rare, right? A discrete subgroup of a compact group must be finite, so any infinite abelian group acquires some canonical non-discrete topology in this way, what is this topology? In any case, on one hand the category of finite abelian groups becomes self-dual and on the other hand the category of locally compact abelian groups becomes self-dual. After that it is easy to name additional structure on the level of individual objects which makes them self-dual.

There is a vast labyrinth of dualities for vector spaces which I understand much worse but somehow realize that in the end of the analog of the above paragraph one arrives at something like Hilbert spaces. But I have no idea what is the analog of Hilbert spaces for some of the versions, e. g. for the locally linearly compact vector spaces in the sense of Lefschetz. Some relatively recent work on the latter by Kapranov is very interesting in this respect but I do not really understand it well.

And I am at complete loss with things like Stone, Priestley and Esakia dualities for Boolean algebras, distributive lattices and Heyting algebras. What might be self-dual objects there? What is the topology on a distributive lattice $L$ induced from the embedding $L\hookrightarrow2^{\textrm{prime filters of }L}$? Or is it more appropriate to consider the topology induced from the Vietoris space of the dual? Or are these actually the same? On the other side, -- what are those Priestley spaces $(X,\leqslant)$ which happen to be distributive lattices with respect to $\leqslant$? In short,

In what sense may one speak of self-dual objects in context of Priestley and other Stone-like dualities? Is there any self-dual category at all either including or included in both Priestley spaces and distributive lattices?

There is some kind of analog of locally compact abelian group thing in this situation - the self-dual category of complete join-semilattices, but here also I do not understand why has topology disappeared altogether. There is an "induced-Stone" topology on each complete join-semilattice, why is it not in the game here? Also, what are self-dual objects in this category?

For rings, a similar question leads to looking at affine ring schemes, Hopf ring schemes and alike. Are there any real-life examples of self-dual Hopf-ring schemes?

What about Gelfand duality in this context?

Does anybody have further examples of dualities where one may either restrict or enlarge to a self-dual category, and then self-dual objects give some known and meaningful mathematical objects?

(added slightly later) Somehow the adele construction looks like relevant. It embeds a (number or function) field $F$ into a **self-dual** locally compact ring $\mathbb A_F$ as a **discrete** subfield. Then self-duality of this object seems to play important rôle in class field theory and arithmetic algebraic geometry. However I cannot quite figure out where does this self-duality take place. On the surface of it, $\mathbb A_F$ is just self-dual as an abelian group wrt addition. The multiplicative part (including the group of ideles) is somehow separate...