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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJun 15th 2010
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   Author: TobyBartels
   Format: MarkdownItexThe [[AnonymousCoward]] who creates blank pages in places where I ought to write stubs has been at it again, this time at [[Stone duality]].

   The AnonymousCoward who creates blank pages in places where I ought to write stubs has been at it again, this time at Stone duality.

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 15th 2010
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   Author: Harry Gindi
   Format: MarkdownItexWhy does he do that?

   Why does he do that?

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownThat's how the legends appear in history...

   That's how the legends appear in history...

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 15th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI 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 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!

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2015
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   Author: Urs
   Format: MarkdownItexI 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 18th 2015
   • (edited Jan 18th 2015)
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   Author: Todd_Trimble
   Format: MarkdownItexPursuant 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 18th 2015
   • (edited Jan 18th 2015)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexLooking 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2015
   • (edited Jan 18th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOn 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 18th 2015
   • (edited Jan 18th 2015)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexUrs, 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](http://facultypages.ecc.edu/alsani/ct99-00%288-12%29/msg00128.html) 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.

   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.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2015
    • (edited Jan 19th 2015)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI'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’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…)

11. • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 19th 2015
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    Author: Todd_Trimble
    Format: MarkdownItexI 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.

    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.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexOf 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 $V$.

    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 $V$.

13. • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 25th 2015
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexComing 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 25th 2015
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    Author: Mike Shulman
    Format: MarkdownItexVery nice, thanks!

    Very nice, thanks!

15. • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 25th 2015
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexI added a little bit to [[dual adjunction]].

    I added a little bit to dual adjunction.

16. • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 8th 2019
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexAdded a section on Stonean duality <a href="https://ncatlab.org/nlab/revision/diff/Stone+duality/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone+duality/25">v25</a>, <a href="https://ncatlab.org/nlab/show/Stone+duality">current</a>

    Added a section on Stonean duality

    diff, v25, current

17. • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 14th 2021
    • (edited Apr 14th 2021)
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexThis answer <https://mathoverflow.net/questions/390085/analytical-origins-of-the-stone-duality/390134#390134> should probably be copied to the nLab, but to what article? The scope of [[Stone duality]] is a bit too narrow, whereas the answer includes examples from [[Gelfand duality]], [[Serre-Swan duality]] etc. On the other hand, [[Isbell duality]] is concerned with a specific abstract formalism (with sheaves and cosheaves) and [[space and quantity]] is also using sheaves and cosheaves, which is not the focus of the asnwer. Do we have a page on the duality between spaces and algebras as well as bundles and modules (and, more generally, geometry and algebra) where this answer would be suitable?

    This answer https://mathoverflow.net/questions/390085/analytical-origins-of-the-stone-duality/390134#390134 should probably be copied to the nLab, but to what article?

    The scope of Stone duality is a bit too narrow, whereas the answer includes examples from Gelfand duality, Serre-Swan duality etc.

    On the other hand, Isbell duality is concerned with a specific abstract formalism (with sheaves and cosheaves) and space and quantity is also using sheaves and cosheaves, which is not the focus of the asnwer.

    Do we have a page on the duality between spaces and algebras as well as bundles and modules (and, more generally, geometry and algebra) where this answer would be suitable?

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexInteresting. You could create a new page and just cross-link with the existing pages. Maybe "generalized Stone duality"? Or whatever term you find fits best.

    Interesting. You could create a new page and just cross-link with the existing pages. Maybe “generalized Stone duality”? Or whatever term you find fits best.

19. • CommentRowNumber19.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 14th 2021
    • (edited Apr 14th 2021)
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexPerhaps one of these? duality between algebras and spaces duality between algebra and geometry Assigning it just to Stone may be too much. Gelfand, Zariski, Chevalley, Milnor also contributed. And von Neumann, as we can now find out from that MathOverflow thread.

    Perhaps one of these?

    duality between algebras and spaces

    duality between algebra and geometry

    Assigning it just to Stone may be too much. Gelfand, Zariski, Chevalley, Milnor also contributed. And von Neumann, as we can now find out from that MathOverflow thread.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2021
    • PermaLink
    Author: Urs
    Format: MarkdownItexSure! "duality between algebra and geometry" sounds good. The [[Isbell duality - table]] is subtitled "Isbell duality between algebra and geometry". Here I was thinking of Isbell duality as the general abstract picture behind the entries in the table. But it's a a stretch and would be better replaced by a pointer to an entry like you are envisioning.

    Sure! “duality between algebra and geometry” sounds good.

    The Isbell duality - table is subtitled “Isbell duality between algebra and geometry”.

    Here I was thinking of Isbell duality as the general abstract picture behind the entries in the table. But it’s a a stretch and would be better replaced by a pointer to an entry like you are envisioning.

21. • CommentRowNumber21.
    • CommentAuthorGuest
    • CommentTimeSep 7th 2022
    • PermaLink
    Author: Guest
    Format: MarkdownItexIs $Bool$ the same category as $BoolAlg$? Olin

    Is $Bool$ the same category as $BoolAlg$?

    Olin

22. • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 7th 2022
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexRenamed BoolAlg to Bool to ensure consistency of notation. <a href="https://ncatlab.org/nlab/revision/diff/Stone+duality/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone+duality/27">v27</a>, <a href="https://ncatlab.org/nlab/show/Stone+duality">current</a>

    Renamed BoolAlg to Bool to ensure consistency of notation.

    diff, v27, current

23. • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeSep 17th 2022
    • PermaLink
    Author: zskoda
    Format: MarkdownItex* G. Dimov, E. Ivanova-Dimova, [[W. Tholen]], _Categorical extension of dualities: From Stone to de Vries and beyond, I._ Appl Categor Struct 30, 287–329 (2022) [doi](https://doi.org/10.1007/s10485-021-09658-6) <a href="https://ncatlab.org/nlab/revision/diff/Stone+duality/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone+duality/28">v28</a>, <a href="https://ncatlab.org/nlab/show/Stone+duality">current</a>

    • G. Dimov, E. Ivanova-Dimova, W. Tholen, Categorical extension of dualities: From Stone to de Vries and beyond, I. Appl Categor Struct 30, 287–329 (2022) doi

    diff, v28, current

24. • CommentRowNumber24.
    • CommentAuthorCorbin
    • CommentTimeDec 18th 2023
    • PermaLink
    Author: Corbin
    Format: MarkdownItexSprout a new page specifically for the Stone gamut, rather than cluttering this page. <a href="https://ncatlab.org/nlab/revision/diff/Stone+duality/30">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone+duality/30">v30</a>, <a href="https://ncatlab.org/nlab/show/Stone+duality">current</a>

    Sprout a new page specifically for the Stone gamut, rather than cluttering this page.

    diff, v30, current

25. • CommentRowNumber25.
    • CommentAuthornLab edit announcer
    • CommentTimeDec 6th 2024
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexAdded reference * {#CCGM24} [[Felix Cherubini]], [[Thierry Coquand]], [[Freek Geerligs]], [[Hugo Moeneclaey]], *A Foundation for Synthetic Stone Duality* ([arXiv:2412.03203](https://arxiv.org/abs/2412.03203)) Maus <a href="https://ncatlab.org/nlab/revision/diff/Stone+duality/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/Stone+duality/32">v32</a>, <a href="https://ncatlab.org/nlab/show/Stone+duality">current</a>

    Added reference

    • {#CCGM24} Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)

    Maus

    diff, v32, current