The fundamental theorem of compact semilattices

Posted on August 20, 2023 by jgl

Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains. They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties. One characterization of bc-domains which I have not included in the book is that they are related in a very precise sense to Lawson semilattices, namely to compact semilattices with small semilattices. This is the fundamental theorem of compact semilattices, which I will (re)prove by relying a lot on the theory of compact pospaces and stably compact spaces: see the full post.

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Exponentiable locales II: the exponentiable locales are the continuous frames

Posted on July 20, 2023 by jgl

Two months ago, we have seen that every exponentiable locale had to be continuous, as a frame. We will see that the converse holds: the continuous frames are exactly the exponentiable locales. The result is due to Martin Hyland in 1979. Read the full post.

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The Banaschewski-Lawson-Ershov observation on separate vs. joint continuity

Posted on June 20, 2023 by jgl

Joint continuity is a stronger property than separate continuity. In what cases are those properties equivalent? The question was solved, partially, by Yuri Ershov in 1997, and completely by Bernhard Banaschewski in 1977 (apparently with a gap in the proof) and by Jimmie Lawson in 1985. The answer has to do with locally finitary compact spaces: read the full post.

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Exponentiable locales I: every exponentiable locale is continuous

Posted on May 21, 2023 by jgl

The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed true, as was proved by Martin Hyland in 1979 (published in 1981). I will concentrate on one half of the this result for this time, and I will explain why every exponentiable locale must be a continuous dcpo. The proof is very close to the similar result in Top, but, as usual, locales are so much more abstract that similar arguments tend to be harder to understand in Loc; I will do my best. Read the full post.

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The Seminar on Continuity in Semilattices

Posted on May 15, 2023 by jgl

Recently, Achim Jung sent me a message from Jimmie Lawson, and suggested that I might be interested in posting the information on this blog. The red book [1] is a precious source of information on domain theory, and if you are interested in knowing how the material there came to be discovered, Continue reading →

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Topological Functors II: the Cartesian-closed category of C-maps

Posted on April 20, 2023 by jgl

Some time ago, I gave an introduction to topological functors. They form a pretty brilliant categorical generalization of topological spaces. The point of today’s post is to give one particular example of the fact that you can somehow generalize some results on topological spaces to topological functors. I will concentrate on showing that a (pretty amazing) construction of certain Cartesian-closed full subcategories of Top, due to Martín Escardó, Jimmie Lawson, and Alex Simpson, generalizes pretty smoothly to a pretty large class of topological functors—the so-called well-fibered topological constructs. More precisely, I will concentrate on the first part of this construction, which builds a Cartesian-closed category MapC out of a so-called strongly productive class C of objects of a category C that forms the domain of a topological construct with discrete terminal objects. Read the full post.

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Localic products and Till Plewe’s game

Posted on March 20, 2023 by jgl

Products in the category of locales resemble, but do not coincide with products in the category of topological spaces. Till Plewe has a nice explanation to this, as I will explain in this month’s post: the localic product of two topological spaces coincides with their topological products if and only if player II has a winning strategy in a certain game, which I have already described last month. As a consequence, we will obtain that the localic product of S0 with itself is not its topological product (a result due to Matthew de Brecht), we will retrieve that the localic product of Q and of R–Q differs from their topological product (and more generally, a result of John Isbell’s), and finally that the localic product of Q with itself differs from the topological product, and that Q is not consonant… with a much, much simpler proof than those I have ever mentioned here. The idea of that argument is due to Matthew de Brecht. Read the full post.

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On Till Plewe’s game and Matthew de Brecht’s non-consonance arguments

Posted on February 20, 2023 by jgl

Last time I mentioned that S0 is not consonant. I will give Matthew de Brecht’s proof of that. Perhaps the most interesting part of this proof is a criterion that he proves and uses: if a space X is consonant, then player II has a winning strategy in a certain game invented by Till Plewe in order to characterize the spatiality of locale products. Read the full post.

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The space S0

Posted on January 21, 2023 by jgl

S0 is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018. S0 is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at the top, S0 comes with the upper topology of the resulting ordering. S0 is one of the four canonical examples of a non-quasi-Polish space (in a precise sense). I will describe it, and I will show how closed sets and compact saturated sets in S0 can be described through certain kinds of subtrees. With that done, we will see that S0 is sober, Choquet-complete, and completely Baire, but not locally compact, not convergence Choquet-complete, not compactly Choquet-complete, not LCS-complete, and, finally, not quasi-Polish. Read the full post.

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Aliaume Lopez’ master theorem of Noetherian spaces

Posted on December 19, 2022 by jgl

There are quite a few constructions that we can use to build new Noetherian spaces from old ones: spaces of finite words, of finite trees (as in Section 9.7 of the book), and a few others. Instead of writing a new proof each time, is there some form of master theorem that would have all those results as corollaries? This is exactly what Aliaume Lopez found this year. Read the full post.

Oh, and Season’s greetings, too!

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