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The AnonymousCoward who creates blank pages in places where I ought to write stubs has been at it again, this time at Stone duality.
Why does he do that?
That's how the legends appear in history...
I expanded Stone duality somewhat, including some examples (taken from Johnstone’s book) of algebraic theories for which profinite algebras are, and are not, equivalent to Stone topological ones. There is still a lot more to say!
I noticed that the entry Stone duality didn’t mention at all the role of $\mathbf{2}$ as a dualizing object. I found discussion of this at BoolAlg, but I seem to remember that we had more on this on the $n$Lab. If so, it seems hard to find and might need more pointers.
For the moment I have copied the paragraph titled “Stone duality” at BoolAlg to the section “Stone spaces and Boolean algebras” at Stone duality. The paragraph on profinite sets that used to be at the latter place I have moved down to the section titled “Stone spaces and profinite sets”.
More could be done here to improve the exposition, I think, but I won’t try to.
Pursuant to Urs’s remark on $\mathbf{2}$: most such dualities come under the umbrella of Chu space duality, i.e., are restrictions of the $\ast$-autonomous duality on $Chu_2$ to suitable full subcategories. I may add a remark on this and link to Pratt’s notes on this.
Looking over at the concept of “concrete duality” in duality, I don’t think this concept is explained very accurately there. Certainly the concept ought to embrace concrete dualities induced by ambimorphic (or whatever you want to call them) objects, which have to do not exactly with closed monoidal structure but with liftings of contravariant hom-functors. Maybe I’ll try my hand at some rewriting here.
On p. 121 of Lawvere-Rosebrugh, “concrete duality” is used to refer to contravariant functors that deserve to be written $V^{(-)}$. The text there is shy about stating technical details, the examples spelled out happen in $Set$, but it seemed to me that a charitable formal interpretation of p. 121 would be to read it as referring to contravariant exponentiation in closed categories. At least the point made around that p. 121 does not seem to need ambimorphicity. That would be something to add on top.
You know the established terminology better than I do. Would be great if you’d find time to expand the entry. But it seems to me that contravariant exponentiation in itself deserves to be regarded as a concept of duality, while homming particularly into ambimorphic objects is a further variant.
Urs, although I don’t have Lawvere and Rosebrugh in front of me (or even at all), I expect that their $V$ generally refers to an ambimorphic structure (so living in some concrete categories $\mathbf{C}$ and $\mathbf{D}$) so that $\mathbf{C}(-, V): C^{op} \to Set$ lifts to $C^{op} \to D$ and $\mathbf{D}(-, V): D^{op} \to Set$ lifts to $D^{op} \to C$, with these liftings forming a contravariant adjunction. “Duality” in the proper sense of the article means that the adjunction is a contravariant equivalence. For example, in this post by Lawvere, he uses the same notation V+exponentiation (NB: the (-)^V he wrote is a typo; he means V^(-)), but where in his case $V$ is ambimorphically a set and an $M$-set with $M = \hom(V, V)$).
I would regard duality in your sense via contravariant exponentiation $V^{(-)}$ as a special case of what I’m referring to. In other words, if $\mathbf{C}$ is (let’s say symmetric) monoidal closed, then the ordinary hom-functor $\hom(-, V): \mathbf{C}^{op} \to Set$ lifts through $\hom(I, -): \mathbf{C} \to Set$ to an enriched hom-functor $(-) \multimap V: \mathbf{C}^{op} \to \mathbf{C}$, so that here we are regarding $V$ as $(\mathbf{C}, \mathbf{C})$-ambimorphic. I would further note that whatever Lawvere and Rosebrugh are referring to, where you say the examples happen in $Set$, I imagine they are not talking only about sets (since there are no such dualizing objects $D$ or $V$ in $Set$, in the sense described at duality), but as sets equipped with some structures.
I’ll be happy about in whatever generality you’ll add it to the entry!
By the way, I am looking at the text via GoogleBooks. (When GoogleBooks fails me, there are other places to turn to…)
I am not able to access that page through Google books.
Nevertheless, there is no equivalence $Set \to Set$ induced by a double dualization in the sense of concrete duality described at duality. I will add to the page later.
Of course there is no such equivalence. I was wondering why you said this, now I saw that the entry had something about involutions at the beginning. I have removed that.
In that section 7.1 L-R speak about how epis are dual to monos by homming into any $V$.
Coming back to this thread, I have completed (for now) my edits at duality that I intended back in #7.
To say it all properly required a substantial revision of dualizing object, which I have also done.
Very nice, thanks!
I added a little bit to dual adjunction.
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