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Tag Archives: locale
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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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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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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Countably presented locales
Reinhold Heckmann showed the following in a very nice paper of 2014: every countably presented locale is spatial. What makes it even nicer is that he shows how tightly this is connected with the Baire property. This also gives a … Continue reading
The locale of random elements of a space
Alex Simpson has a lot of slides with very interesting ideas. One of them is what he calls the locale of random sequences. This is a terribly clever idea that aims at solving the question “what are random sequences?”, using … Continue reading
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Tagged frame, locale
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Isbell’s density theorem and intersection of sublocales
When I wrote my latest blog post, there were many things I thought would be useful to know about sublocales. Those eventually turned out to be useless in that context. However, I think they should be known, in a more … Continue reading
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Tagged frame, locale
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The O functor does not preserve binary products
In Exercise 8.4.23 of the book, I said: “Exercise 8.4.21 may give you the false impression that the O functor preserves binary products. This is wrong, although an explicit counterexample seems too complicated to study here: see Johnstone (1982, 2.14).” … Continue reading
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Tagged frame, locale
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Locales, sublocales III: the frame of nuclei
My goal today is to describe two elegant proofs of the fact that nuclei form a frame. There are many proofs of that. The main difficulty is that, while meets (infima) of nuclei are taken pointwise, joins (suprema) are much … Continue reading
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Tagged frame, locale
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Locales, sublocales II: sieves
Last time, I promised you we would explore another way of defining sublocales. We shall again use the naive approach that consists in imagining how we would encode subspaces of a T0 topological space X by looking at open subsets … Continue reading
Locales, sublocales I
Stone duality leads naturally to the idea of locale theory. Quickly said, the idea is that, instead of reasoning with topological spaces, we reason with frames. The two concepts are not completely interchangeable, but the O ⊣ pt adjunction shows … Continue reading