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Category Archives: Uncategorized
Scott’s formula
There is a wellknown formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f’ defined on the whole of X and … Continue reading
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The fundamental theorem of compact semilattices
Boundedcomplete domains, or bcdomains, are an amazingly rich kind of continuous domains. They form a Cartesianclosed category, and they are the densely injective topological spaces, among other properties. One characterization of bcdomains which I have not included in the book … Continue reading
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Tagged compact pospace, semilattice
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Exponentiable locales II: the exponentiable locales are the continuous frames
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 … Continue reading
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Tagged continuous lattice, exponentiability, locale
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The BanaschewskiLawsonErshov observation on separate vs. joint continuity
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) … Continue reading
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Tagged cspace, locally finitary compact, separate continuity
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Exponentiable locales I: every exponentiable locale is continuous
The exponentiable objects of Top are exactly the corecompact spaces. Through Stone duality, the corecompact 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 … Continue reading
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Tagged cartesian closeness, category theory, continuous lattice, exponentiability, locale
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The Seminar on Continuity in Semilattices
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 … Continue reading
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Topological Functors II: the Cartesianclosed category of Cmaps
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 … Continue reading
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Tagged cartesian closeness, category theory, exponentiability, topological functor
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Localic products and Till Plewe’s game
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 … Continue reading
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Tagged consonance, counterexample, frame, game, locale, ordinal
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On Till Plewe’s game and Matthew de Brecht’s nonconsonance arguments
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, … Continue reading
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Tagged consonance, counterexample, game, ordinal
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The space S0
S0 is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasiPolish spaces, published in 2018. S0 is very simple: it is an infinite countablybranching tree, and if you order it so that the root is at … Continue reading