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Stream: learning: questions

Topic: Dedekind completions of linear preorders

Madeleine Birchfield (Feb 08 2024 at 22:41):

Is it possible to take the Dedekind completion of a linear preorder - i.e. a linear order $(S, <)$ which doesn't satisfy the condition that for all elements $x$ and $y$ in $S$ $\neg (x < y)$ and $\neg (y < x)$ if and only if $x = y$?

John Baez (Feb 09 2024 at 00:41):

Are you familiar with the Dedekind-MacNeille completion of a poset? It's a generalization of the Dedekind completion that works for any poset. I would hope it also works for preorders, since secretly it relies on a very category-theoretic concept, namely the [[Isbell envelope]]. However, even Avery and Leinster restrict to posets in Examples 6.10 and 6.11 of their paper on this stuff. I don't know why.

Madeleine Birchfield (Feb 09 2024 at 01:04):

Actually, for my case my preorders are also pseudometric spaces, so one can simply construct the localic completion of the pseudometric space and then take the space of points of the resulting locale, which I'd imagine should be equivalent to the Dedekind completion of the preorder.

Madeleine Birchfield (Feb 09 2024 at 01:11):

But it would be great if the Dedekind completion works directly for preorders.

John Baez (Feb 09 2024 at 07:02):

I predict the Dedekind-MacNeille completion of any preorder can be defined very elegantly as an Isbell envelope, copying how Avery and Leinster (and others) do it for posets.

John Baez (Feb 09 2024 at 07:28):

The reason I predict this is that the Isbell envelope is a construction that makes sense for enriched categories quite generally. Preorders are the same as categories enriched in the category of truth values. So we can apply the Isbell envelope construction to preorders. Avery and Leinster show that for the special sort of preorders called "posets" this gives the Dedekind-MacNeille completion. But a similar fact may hold for preorders in general.

Madeleine Birchfield (Feb 09 2024 at 13:20):

Oh, now I understand where the MacNeille real numbers come from in constructive mathematics.

The Dedekind-MacNeille completion of the rational numbers with respect to the poset structure on the rational numbers yields the extended MacNeille real numbers, which in classical mathematics coincides with the extended Dedekind real numbers. Then removing the infinity yields the Dedekind real numbers in classical mathematics.

But the Dedekind completion and the Dedekind-MacNeille completions don't coincide in constructive mathematics; the strict order of the extended MacNeille real numbers is not cotranstive, while the strict order of the extended Dedekind real numbers is cotransitive.

Madeleine Birchfield (Feb 09 2024 at 13:22):

This also explains why Matteo di Miglio was talking about partially ordered fields in his article on avoiding Soler's theorem in the category of Hilbert spaces.

Madeleine Birchfield (Feb 09 2024 at 13:28):

So in constructive mathematics it looks like we can construct the Dedekind-MacNeille completion of linear preorders and then the subset of bounded and located elements in the Dedekind-MacNeille completion would be the Dedekind completion of linear preorders.

Madeleine Birchfield (Feb 09 2024 at 13:42):

Is there an analogue of the Isbell envelope in $\infty$-category theory?

Graham Manuell (Feb 25 2024 at 13:24):

I think the correct categorical analogue of the Dedekind completion, as opposed to the Dedekind-MacNeille completion, is given by this paper: https://arxiv.org/abs/2204.09285.