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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 14th 2021

Created.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeApr 14th 2021

Isbell duality would be worth linking to.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 14th 2021
• (edited Apr 14th 2021)

[oh, overlapped with David]

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 18th 2021
• (edited Oct 18th 2021)

I have !include-ed the Isbell duality - table. Optimally this would be merged with the new tables here, but for the moment I leave it as is.

Even before merging them, the new tables here are still lacking many of their requested hyperlinks.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 18th 2021

Re #4: The entry is now free from broken links.

There is certainly some overlap between Isbell duality and this article. But are they the same?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 18th 2021
• (edited Oct 18th 2021)

Thanks!

The table I once made seems to have the same motivation and partly the same content as your table.

Maybe just its title is misleading to readers. I adopted the habit of crediting Isbell for formalizing the general idea of duality between geometry and algebra, as I think of Isbell duality like a broad-brush template along which to look for duality between geometry and algebra. But maybe this is overly ideosyncratic and confusing terminology.

I could just as well rename that table. In any case, when you ignore its title and just look at its content, it should be clear that we were after making the same kind of table.

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 18th 2021
• (edited Oct 18th 2021)

Re #6: Yes, I agree that the two tables should be merged.

What is currently a cause for concern to me is that the Isbell duality article is currently (implicitly) asserting that all entries in the table can be obtained via the formal construction on (co)presheaves described in the first 5 sections of the article.

While this is certainly true for some entries, it is far from obvious to me that (for example) Gelfand duality for C*-algebras can be pulled out of the formalism as it is currently presented. (Which does not preclude it from being obtainable from some modified variant of Isbell duality.)

So in my mind, there are two separate articles: Isbell duality describes formal constructions on (co)presheaves and points out that some of the rows in the table at duality between geometry and algebra can be recovered in this manner, whereas duality between geometry and algebra explores the duality from a semiformal point of view, without necessarily insisting that all entries can be recovered using Isbell duality as it is currently presented.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 18th 2021

By the way, I myself wrote a substantial number of pages on what can be rightfully termed (generalized) Isbell duality here: https://dmitripavlov.org/notes/cart.pdf, but the manuscript is far from being finished.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 19th 2021

Thanks for the pointer to your note, that loooks interesting, will try to find time to have a closer look.

I have briefly scanned again over the old entry “Isbell duality” and it looks to me like it does not say anything it shouldn’t say. (But if there is something, please feel invited to edit.)

The sticking point seems to be the header line above the !include-table showing examples of the duality between algebra and geometry. So I have changed that header line now (as announced here) from saying “Isbell duality” to saying “duality between algebra and geometry”.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 19th 2021

Maybe to expand:

In speaking about duality between algebra and geometry one is typically faced with a list of plausible examples but without any general definition of what these are examples of. One relies on the informal mechanism of “we don’t know what it is, but we recognize it when we see it”.

Isbell duality was to, or at least carried the promise of, providing the missing abstract definition: Give any site, geometry is presheaves on the site which force some colimits, while algebra is copresheaves forcing some limits, and the duality between the two is just the hom-functor.

Or rather, Isbell duality says this just at the level of presheaves, without talking about (co)limits, and that’s its shortcoming which makes us have this discussison.

But somebody must have thought, or else somebody ought to think, about how to boost Isbell duality to incorporate descent.

• CommentRowNumber11.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 19th 2021

Re #10: I agree, and my note does incorporate descent. In fact, I would very much like to also see the Gelfand duality for C*-algebras expressed as Isbell duality, I just don’t know how to do it yet, which is why I think separating the two articles may be desirable for now.