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Stream: theory: category theory

Topic: Vietoris Locale Research Question

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

Notification Bot (Aug 08 2025 at 10:25):

Graham Manuell has marked this topic as unresolved.

Graham Manuell (Aug 08 2025 at 10:27):

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

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.

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 $X$ can be viewed as the patch of $\mathbb{S}^X$ (for compact Hausdorff $X$). That's nice and, indeed, in the classical literature as you pointed out separately. I think I have a neat general proof:

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

$\phi:\mathbb{S}^{VA} \cong P_L(VA) \rightarrow^{P_L(i)} P_L(P_U(A)) \cong \mathbb{S}^{\mathbb{S}^A} \cong \mathbb{S}^{P_L A} \rightarrow^{\mathbb{S}^j} \mathbb{S}^{VA}$

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

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

Graham Manuell (Oct 08 2025 at 08:06):

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

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 $\phi$ formally?

$VA$ is discrete - so apply $\mathbb{S}^X \cong P_L(X)$ for any discrete $X$*. That $VA$ is discrete for discrete $A$ follows by applying (formally) the standard join-coverage result using that the finite diagonals on $A$ are open. (This was really the original question: is this step written down for the nullary diagonal $A \rightarrow 1$? 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 $Idl$ when talking about the compact Hausdorff dual to discrete $Idl$...)

• actually is the question how to get $\mathbb{S}^X = P_L(X)$ formally for discrete $X$? That's laboured in my paper on 'Aspects of Slice Stability in Locale Theory'., Prop 4.2.4; again not doing much more than just repeating Vickers original proof.

Graham Manuell (Oct 08 2025 at 12:35):

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

Christopher Townsend (Oct 09 2025 at 19:27):

MATH_VIETORIS.jpeg
i.e. just use that $\exists_X$ exists to show $\exists_{VX}$ must exist by application of (formal!) suplattice coverage.
Showing that the open diagonal is preserved is similar, using the meet semi lattice operation on $P_U(X)$.

I hope to write this up properly in due course - treat with caution!