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

Here is how you can support an author and libraries: Go to your library and ask for my latest book. If they don't have it look perplexed. "What? You don't have Teri's latest book?" See if they'll order it.

If you check out a few of my books that helps keep them on the shelves.

(I am always embarrassed when books come out because I think they are priced too high at first.)

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.

But if a man forces his wife to take abortion pills, it's 180 days jail.

Tell me it's not about controlling women.

https://www.nbcnews.com/news/us-news/texas-attorney-poisoned-pregnant-wife-abortion-medication-sentenced-18-rcna138065

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.
I know you're trying your best, and that's all any of us can do.
Carry on.

122) Some have posited that #Election2024 will be the first #AI #Election

Looks like a certain #Texas based corporation by the name of Life Corporation and its owner Walter Monk can expect to experience some deep law enforecement scrutiny for deceptively impersonating Pres #JoeBiden's voice

Hopefuly this will pursued agressively to dissaude other dirty trick types

#USPolitics #Politics

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

Fishy and Ruby have been looking especially judgmental lately. I am simply asking you to support my Patreon so my cats don't think I'm a total failure. I think if I get a few more supporters they might finally begin to see me as only a partial failure.
https://patreon.com/lowqualityfacts

Would you be interested in you and your cat participating in a citizen science research project?

If so, please comment below or send an email to meow@cowboycatranch.org.

Sharing is much appreciated.

#science #cats #catsofMastodon

bagpipe levels are holding at 26%

(26%) ■■□□□□□□□□

Here is the appellate court's denial of Trump's claim of immunity:
https://storage.courtlistener.com/recap/gov.uscourts.cadc.40415/gov.uscourts.cadc.40415.1208593677.0.pdf

The decision opens by summarizing the allegations in the indictment, being careful to add that guilt hasn't yet been determined.

The court then dismisses the idea that they don't have jurisdiction over the issue and concludes that it does.

Then, on to immunity. The court rejects Trump's "separation of powers" argument, saying this:

That's strong language.

1/

5/ Via Scott MacFarlane:

Court: “Former President Trump has become citizen Trump, with all of the defenses of any other criminal defendant. But any executive immunity that may have protected him while he served as President no longer protects him against this prosecution.”

👉🏼Unanimous, emphatic, unequivocal ruling from appeals court👈🏼

[Me: 👆🏼THAT IS WHAT WE NEEDED TO HEAR]