Now I have `!include`

-ed the section as a Properties-subsection into the relevant entries

(e.g. at *Hausdorff space*, see the announcement here).

Except for the case of $T_1$, as that is currently redirecting just to *separation axiom*. It might be good to remove that redirect and instead make *T1-space* a page.

I have `!include`

ed (here) a Properties-section “In terms of lifting properties” as discussed in another thread: here

I see. Okay, I have edited the entry a little, to make it work well as an `!include`

-page:

added a section header and inside it Proposition-environments for various claims,

expanded out the lead-in text a little to make it more inviting,

gave each of the diagrams a brief text saying what it’s about, with a pointer to the corresponding separation axiom.

Now to include this as a Properties-section in the relevant entries, to add the following line at the desired spot

```
[[!include main separation axioms -- as lifting properties]]
```

]]>
brief `category:people`

-entry for hyperlinking references at *holographic QCD* and *neutron star*

added pointer to this review which appeared today:

- Matti Järvinen,
*Holographic modeling of nuclear matter and neutron stars*, Eur.Phys.J.C. 56 2021 (arXiv:2110.08281)

Do you think the diagrams in this page are clear to the intended reader, with the little explanation provided ?

If yes, should I add !include this page alongside the table of separation axioms , in the pages for [[normal space etc ?

Alternatively, I could add an appropriate diagram + this short explanation + link to separation axioms as lifting properties

at the pages for normal, regular, etc spaces ?

What would be most appropriate ? ]]>

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.

]]>Reworked the Serre–Swan material to say

]]>The category of measurable fields of Hilbert spaces on $(X,\Sigma,N)$ (as defined above) is equivalent to the category of countably-generated W*-modules over the commutative von Neumann algebra $\mathrm{L}^\infty(X,\Sigma,N)$.

(If we work with bundles of general, possibly nonseparable Hilbert spaces, then the W*-modules do not need to be countably generated.)

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”.

I have changed the table header from “Isbell duality” to “duality between algebra and geometry”. The former was meant to be read as a synonym for the latter, but I can see how that’s too ideosyncratic not to be confusing (as per the discussion here).

For the moment the page title (which is not displayed in the pages where this table is `!include`

-ed) remains the same. Because I am unsure if the software would correctly handle `!redirect`

s inside `!include`

s, and since I don’t have the time now to search for and change all the `!include`

-commands to this table.

Re #3: Yes, either both fibers and modules are countably generated, or not.

]]>Filled in publication details for

]]>

- {#Hyland13} Martin Hyland.
Classical lambda calculus in modern dressMathematical Structures in Computer Science, 27(5) (2017) 762-781. doi:10.1017/S0960129515000377. arxiv:1211.5762

Doi link for Scott 1976, and more details for Hyland 2017 (arxiv, doi, journal ref)

]]>Adding in doi links for papers of Dana Scott

]]>How do you reconcile the statements

The last condition restrict us to bundles of separable Hilbert spaces. One can also define bundles of nonseparable Hilbert spaces, but this cannot be done simply by dropping the last condition.

and

(If we work with bundles of separable Hilbert spaces, then W*-modules must be countably generated.)

Isn’t the definition currently given in a way such that the fibres are always separable? I would think the better way to state the Serre-Swan type duality is by default with countably-generated W*-modules, then add a comment about the generalisation to the non-separable case.

]]>Added whitespace before bullet-point list, to make formatting work

]]>Missing assumption

Anonymous

]]>Remove repetition

Anonymous

]]>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.

]]>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.

]]>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.

]]>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?

]]>Added redirect: Zariski duality. To satisfy a link at duality between geometry and algebra.

]]>Added redirect: Milnor duality. To satisfy a link at duality between geometry and algebra.

]]>