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Stream: theory: category theory

Topic: Vietoris Locale Research Question


view this post on Zulip Christopher Townsend (Aug 05 2025 at 06:24):

[Cross post from Math overflow & categories mailing list.]

Let X be a locale. Then there is the Vietoris locale construction, V, introduced by Johnstone (it's in Stone Spaces, but see also the 1985 paper, 'Vietoris Locales and Localic Semilattices'). The functor V:Loc→Loc is something like taking the locale of all 'finite' sublocales; e.g., if X is discrete then the points of V(X) are the finite subsets of X.  In his 1985 paper Johnstone alludes to the fact that if X is compact then so is V(X). But the proof, which is not explicit in the paper, uses a transfinite induction. I believe there is a much simpler proof, but have not found any references to this in the literature. Does anyone know about this specific result being published in the last 40[!] years?

Thanks,
Christopher

view this post on Zulip Notification Bot (Aug 08 2025 at 10:25):

Graham Manuell has marked this topic as unresolved.

view this post on Zulip Graham Manuell (Aug 08 2025 at 10:27):

I don't know if this has been proved in the literature, but if XX is compact Hausdorff, isn't the Vietoris locale of XX the patch topology of SX\mathbb{S}^X? This would imply that it is compact Hausdorff.

view this post on Zulip Christopher Townsend (Aug 11 2025 at 18:11):

Thanks Graham. I'd have to think about that specific point. It rings true as some of my thinking has been towards seeing the opens of V(X) as a splitting in the same way that the patch can be constructed as a splitting (interestingly, using a dcpo idempotent for general locales X, for which I think I was going to look at your paper that expresses the localic surjection of the splitting in such a circumstances, and moreover describes the splitting locale's presentation, I think?).

Of course, we know preservation by V:Loc->Loc of compact Hausdorffness from Stone Spaces - it was the preservation of compactness absent anything else that I was after. It's not really a 'big' result - it's a bit like how binary Tychonoff becomes straight forward once you use the preframe presentation of the (opens of) locale product. There's a similar result in Vicker's paper on connected Vietoris about the preservation of openness. I just wanted to see if anyone else had actually published anything on this before I wrote anything up.

view this post on Zulip Christopher Townsend (Oct 02 2025 at 16:19):

@Graham Manuell - and anyone else who is interested - so I was not aware that the Vietoris on XX can be viewed as the patch of SX\mathbb{S}^X (for compact Hausdorff XX). That's nice and, indeed, in the classical literature as you pointed out separately. I think I have a neat general proof:

Let AA be a set (=discrete locale). Then PL(A)SAP_L(A) \cong \mathbb{S}^A (e.g. Vickers). Recalling that V(A)V(A) embeds (by construction) in PL(A)×PU(A)P_L(A) \times P_U(A), say via (j,i)(j,i) then we can construct:

ϕ:SVAPL(VA)PL(i)PL(PU(A))SSASPLASjSVA\phi:\mathbb{S}^{VA} \cong P_L(VA) \rightarrow^{P_L(i)} P_L(P_U(A)) \cong \mathbb{S}^{\mathbb{S}^A} \cong \mathbb{S}^{P_L A} \rightarrow^{\mathbb{S}^j} \mathbb{S}^{VA}

recalling that VAVA is discrete for the first bit (so that PL(VA)SVAP_L(VA) \cong \mathbb{S}^{VA}). But you can unwind to see (I hope!) that ϕ\phi sends any open of VAVA to its upper closure (using the order on VAVA). The split must be the opens of Idl(VA)Idl(VA) and of (the opens of ) PL(A)P_L(A) , so we have Idl(VA)PLAIdl(VA) \cong P_LA for discrete AA.

Now, of course, this is an incredibly long winded way of proving the incredibly well know and basic fact: idl(FA)=idl(FA)= the power set of AA, where FAFA is the set of finite subsets of AA. But the point of knowing a purely localic proof of PLAIdl(VA)P_L A \cong Idl(VA) for discrete AA is the we can then appeal to compact Hausdorff/discrete duality: PU(X)Idl(VX)P_U(X) \cong Idl(VX) for compact Hausdorff XX. And since we know SXPU(X)\mathbb{S}^X \cong P_U(X) we get that Patch(SX)V(X)Patch(\mathbb{S}^X) \cong V(X) as patch reverses ideal completion!

view this post on Zulip Graham Manuell (Oct 08 2025 at 08:06):

@Christopher Townsend Nice. How do you get the first isomorphism in the definition of ϕ\phi formally?

view this post on Zulip Christopher Townsend (Oct 08 2025 at 09:16):

Graham Manuell said:

Christopher Townsend Nice. How do you get the first isomorphism in the definition of ϕ\phi formally?

VAVA is discrete - so apply SXPL(X)\mathbb{S}^X \cong P_L(X) for any discrete XX*. That VAVA is discrete for discrete AA follows by applying (formally) the standard join-coverage result using that the finite diagonals on AA are open. (This was really the original question: is this step written down for the nullary diagonal A1A \rightarrow 1? I now think it sort of is in Vickers' paper on the connected Vietoris, though not 'formal' there, easily adapted to become formal. ).

(Note: I probably should not have used the notation IdlIdl when talking about the compact Hausdorff dual to discrete IdlIdl...)

view this post on Zulip Graham Manuell (Oct 08 2025 at 12:35):

I meant how do you get VAV A is discrete in a way that dualises to the compact Hausdorff case, but yes, as you say, I guess that was the original question.