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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeJul 9th 2013

Did we really not have continuous poset?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 9th 2013

No, we really did not – but you’ve now ungrayed some links; thank you!

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 9th 2013
• (edited Jul 9th 2013)

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 21st 2013

Added to continuous poset the statement that continuous lattices are monadic over sets, via the filter monad.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 21st 2013

Incidentally, I see a gray link to locally compact locale. It might make sense to absorb this proposed article into locally compact space, because it seems to be a theorem that locally compact locales are in fact spatial. See for example Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory (ed. M.C. Pedicchio, W. Tholen), chapter II (Locales, written by Jorge Picado, Aleš Pultr, and Anna Tozzi), section 7 (p.97). Might also be in Johnstone’s Stone Spaces; haven’t checked.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 21st 2013

It sounds like the sort of theorem that might be true classically but not constructively. I don’t have SS in front of me right now to check what’s in there.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJul 21st 2013

Good call, Mike. I think you’re right.

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeJul 21st 2013

I would be inclined to fold locally compact locale into locally compact space anyway, since ‘space’ to me does not necessarily mean a topological space. In particular, the text ‘In classical mathematics, every locally compact locale is spatial, hence a locally compact space.’ now at the former page seems a non sequitur to me; classically or not, locally compact or not, every locale is a space. (This is also a problem with ‘spatial’ as applied to locales, but at least that is a technical term with an established meaning, unlike ‘space’.)

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJul 21st 2013

In that case, the page locally compact space would have to be significantly rewritten to be about more general kinds of “space”. If you object to “space” meaning “topological space” then I would be more inclined to rename the page locally compact space to locally compact topological space, have a separate page locally compact locale, and make “locally compact space” a redirect-with-hatnote to locally compact topological space.

• CommentRowNumber10.
• CommentAuthorTobyBartels
• CommentTimeJul 21st 2013

Yes, rewriting locally compact space is one thing that I've wanted to do for a while but haven't got around to, for various reasons.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJul 21st 2013

Section C4.1 of the Elephant remarks “it can be shown using the axiom of choice that if a continuous lattice $A$ is distributive, then the locale $X$ defined by $\mathcal{O}(X)=A$ is spatial, and its space of points is locally compact.” For a proof it cites “The spectral theory of distributive continuous lattices” by Hoffman and Lawson.

• CommentRowNumber12.
• CommentAuthorTodd_Trimble
• CommentTimeJun 22nd 2014

I added a bit more to continuous poset. One point which hadn’t been made before is that different people mean different things by the category of continuous lattices. People who think continuous lattices are monadic over sets think that the morphisms preserve directed joins and arbitrary meets. People who think the category of continuous lattices is cartesian closed think the morphisms are the maps that preserve directed joins.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeJun 23rd 2014

I put in an explicit discussion of possible morphisms. Are there common names for the various categories of continuous lattices?

1. Added example that the lattice of open subsets of a topological space is a continuous lattice if and only if the sobrification of the topological space is locally compact (i.e. the topology has a basis of compact neighborhoods).

I am going to add this information at a few other places (where it is relevant)

2. Also made clear in the case of frames, that already by definition a frame is locally compact if it continuous.