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#diagonalization
m---a---x · 3 months
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Welcome to the premier of One-Picture-Proof!
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This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)
Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)
The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.
Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)
Preliminaries and Definitions:
B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).
The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)
φ is just some morphism A→B^A.
Δ is the diagonal, which maps a↦(a,a).
1 is the terminal object, you can think of it as a single-point set.
We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:
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Exercises:
What is the statement of the theorem?
Work through the proof, follow the arrows in the diagram, understand how it is composed.
What is the more popular name for this technique?
What are some applications of it? Work through those corollaries in the diagram.
Can the theorem be modified for epimorphisms? Why or why not?
For the advanced: What is the precise requirement on the category, such that we can perform this proof?
For the advanced: Can you alter the proof to lessen this requirement?
Bonus question: Can you see the Sicko face? Can you unsee it now?
Expand to see the solutions:
Solutions:
This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.
Good job! Nothing else to say here.
This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.
Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.
If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.
This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.
Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.
Yes you can. No, you will never be able to look at this diagram the same way again.
We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.
If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.
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bobonbooks · 1 month
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Review: Doppelganger
Doppelganger: A Trip into the Mirror World, Naomi Klein. New York: Farrar, Straus and Giroux, 2023. Summary: Naomi Klein, a liberal activist and writer finds herself being confused with another Naomi, once a feminist now become an anti-vax advocate and darling of the extreme right. Last summer, an anonymous pretender created a fake version of a social media page I curate, stealing a picture of…
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corn6vhcbn3c · 1 year
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Darkko Deep Pussy MILFS Cherie DeVille tomando cerveja direto da perereca mijada procura no google casa dos nudes Sexy sloppy head POV SEXY BBW MILF GETTING FUCKED !!! PART 3(a) Guy removes Asian Katsumi pink panties to fuck her Marta La Croft Fuck In Public Back Area Gay Sissy shows of sexy ass, lubes and fucks tight asshole with dildo Novinho metendo o consolo no cu Straight boy first time doing mexican and fun boys hitchhiking gay Naked gay cop porn movie This was going to be a patient/ intern
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iliothermia · 3 months
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jaradraws · 10 months
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bishop takes queen
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Not! All! People! With! Your! Type! Of! Disability! Can! Do! What! You! Can! Do!
[Plaintext: Not all people with your type of disability can do what you can do! /End plaintext]
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cali · 8 months
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really a lot of the time i am onto something big. my idea for revolutionizing music:
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dinoserious · 6 months
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invertober day 2, silvery leafcutter bee!
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liimonadas · 1 month
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awaiting retribution
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obsob · 2 years
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a redraw of ‘lament for icarus’ by herbert james draper ✷
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hotvintagepoll · 18 days
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Propaganda
Jean Simmons (The Big Country, Guys and Dolls, Hamlet)— A fragile, heart-breaking, haunting beauty.
Grace Kelly (Rear Window, High Society, Dial M for Murder)—The literal princess of Hollywood (she retired at 26 to become princess of Monaco), her name said everything about why she was so hot. She carried herself with a grace and elegance you just don't see anymore. Her voice was sultry without being overbearing, and she had the ability to be sweet but suggest a deep sensuality at all times.
This is round 1 of the tournament. All other polls in this bracket can be found here. Please reblog with further support of your beloved hot sexy vintage woman.
{additional propaganda submitted under the cut]
Jean Simmons:
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Grace Kelly:
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flawlessly beautiful and a literal princess
Her facial structure? Flawless. Her eyes? Stunning. Her hair? Gorgeous. Her style? Immaculate. Every second she’s on screen, she just exudes this elegance and sophistication. It’s no wonder she ended up marrying a prince. But she’s got this mischief in her eyes that is compelling.
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She was so elegant, so beautiful and perfect I could cry for real. A fairy disguised as a woman.
Not only was she princess of Monaco she also is Stéphanie de Monaco's mother and yeah, vote for her she's soooo pretty That red dress in Dial M.... hot damn
the most beautiful of Hitchcock's "icy blondes". elegant, glamorous, she left hollywood to became an actual princess, I mean, COME ON
She's just so elegant, look at her all dressed up like a Barbie doll in the latest fashions. There's a quiet dignity about her.
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To me, she is the first and only blonde. She earned it. Paired with Edith Head's costume design she is unstoppable. I dare anyone to watch her as Lisa Carol Fremont in Rear Window and not be completely blown away by her hotness.
she's so pretty and refined and elegant! I'm pretty sure taylor swift's blonde hair red lip look is modeled partly after her
SHE IS SO PRETTY AND FASHIONABLE!! Not only that but she has an alluring aura to her in whatever film I've seen her in! Rear Window is just one of my personal favorite films she was in, especially for her costumes in that. And how many actresses can you say was a princess consort in addition to being a famous leading lady?
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b0tster · 8 months
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my thigh pouches built in snaps werent strong enough to keep it closed while moshing so i replaced them with some heavier duty belts 😈
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pangur-and-grim · 10 months
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putting this under a readmore because of the blood, but I got so fucking got last night
like dear god, I got got!
I think Pangur has a little something special about her brain. most cats will give clear outward signals if they’re about to reinforce their boundaries, but for Pangur she won’t necessarily hiss or walk away before clawing you. and the signs of her discomfort as just her subtly tensing, so you need to pay attention.
this is hard when I’m on a call to friends! because I’ll be talking loudly and unpredictably (at a volume that overstimulates pangur) with my attention elsewhere. and because this throws her off, she’ll seek to be comforted, which means she’ll actively be climbing my shirt and begging to be held. that might seem contradictory (I’m the one upsetting her, but she wants me to comfort her) but is actually fairly straightforward in Pangur logic.
unfortunately, this is a situation that leads to a boiling point, with Pangur getting more and more upset, and wanting to be closer and closer to me, until she snaps. at which point she is usually quite close to my poor vulnerable face.
this time my upper lip got clawed in two, and it took a lot of firm pressure to stop the bleeding. even though it’s not a massive cut, the whole thing was pretty gory, and a left a trail of blood splatter through the house. funnily enough, me cursing and running for the bathroom also upset Pangur, and so she was howling at my feet wanting to be comforted and asking to be picked up again.
very silly animal! 
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dragonskulls · 3 months
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i made this really quick for a friend but im posting it in case anyone else might find this useful. Basically i got this "technique" from a single image of concept art for the eagle from that "Rescuers down under" movie 😭 but imagining the head as a 3d object instead of starting with the basic flat circle REALLY helped me a lot
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vorpalfae · 7 months
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fleshdyke · 6 months
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i dont usually post my photography here bc i’m not too great at it yet but this one has gotta be my favourite pic i've ever taken tbh
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