ceoln @ceoln@qoto.org

Quick remark on the patch topology for constructive point-free topologists.

There is a universal way of transforming a spectral locale into a Stone locale, known as the patch construction, which exhibits the category of Stone locales with continuous maps as a coreflective subcategory of that of spectral locales with spectral maps.

Here a continuous map of spectral locales is called spectral if its defining frame homomorphism maps compact opens to compact opens.

In other words, for any spectral locale A there is a Stone locale Patch A and a continuous map ε : Patch A → A such that any continuous map X → A from a Stone locale X factors uniquely through ε.

Now specialize X to the terminal locale 1, whose frame of opens is the object Ω of truth values.

Then the above universal property says that the "spectral points of A", namely spectral maps 1 → A, are in canonical bijection with the points of Patch A, namely the continuous maps 1 → Patch A.

This is interesting because, classically, A and Patch A have the same points.

To see how badly this fails constructively, consider A = 𝕊 = Sierpinski locale.

The defining frame of the Sierpinski locale can be defined in the following two equivalent ways, and, as is well known, it classifies open sublocales:

(1) The free frame on one generator.

(2) The Scott topology of Ω.

Now we have that Patch 𝕊 ≃ 𝟚 := 1+1.

We know that the frame of the locale 𝟚 is simply the powerset of the set 1+1 (which type theorists may write as Fin 2).

The easiest way to see this is to check that the inclusion ε : 𝟚 → Ω satisfies the above co-universal property.

1/

Several years ago we, as a village, bought 50 acres of land in our village and turned it into a nature reserve. We've planted over 5,000 trees, created three natural ponds, and laid 1.5km of paths for everyone to enjoy (level and wide for wheeled carriages of all type). Beyond this more sculpted 8 acres are a remaining 40 laid to wild. It's a joy to wander around these more rugged areas

#landscapes #scotland #countryside #LandscapePhotograpy

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So if a doctor in Texas performs an abortion, they are fined $100K, lose their medical license, and get a life sentence. If a woman in Texas gets an abortion:, she's fined $10K and sentenced to life in prison.

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The Trump-appointed Florida judge in the documents case is part of his defense team. This has been clear from the start, and almost every move she makes is plainly designed to help him. An ongoing travesty. https://joycevance.substack.com/p/mar-a-lago-update

M16 Close Up
Credits: IgnacioDiaz Bobillo
#nature #space #astrophotography

It's ok if you haven't finished your novel yet, or done the laundry, or discovered the mossy forgotten tombs of the old forest gods.
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Carry on.

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https://www.nytimes.com/2024/02/06/us/politics/biden-robocall-new-hampshire.html

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