#fixed point

[m---a---x]

Welcome to the premier of One-Picture-Proof!

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:

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.

[swan2swan]

Whoever conceived and animated this moment, I hope they're doing well and thriving. This is S-rank romance stuff here.

[kizziahblog]

The Banach Anchor

🃏 Triadic Metaphor Tarot Card 001 Triadic Metaphor Tarot Card 001: The Banach Anchor — “Convergence is not found—it is structured.” ✨ Aphorism (Signal) “Convergence is not found—it is structured.” 📖 Interpretation (Key) In recursive systems, stability is not discovered—it is engineered. The Banach Fixed-Point Theorem proves that a contractive function will always return to a single, stable…

[lgbtlunaverse]

To me the most fun part about fix-its is placing dominoes.

Tragedies often consist of escalating series of actions and circumstances which, in isolation, were not clearly leading to the tragic end but form a chain of cause-and-effect directly towards it in hindsight. In equal but opposite fashion, I love starting with small inoccuous changes to canon that in themselves do not obviously fix everything but start a new chain that leads to a better ending.

It's kind of impossible for fix-its to feel fully natural– the reader by definition knows what the original ending was and that this ending will be happier because the writer wants it to be– but it is possible for them to not feel contrived. A big deus-ex-machina, or a character breaking with their pre-established tragic flaws to suddenly make all the "correct" decisions almost always feels unsatisfying to me.

But a few carefully placed small domino pieces slowly knocking over bigger and bigger tiles until the entire story has radically changed? That's a lot more fun.

It recquires the author to both correctly identify the original chain of cause-and-effect and understand the characters well enough to know how they'd react to different circumstances. Because if the story feels like it's fixing the wrong problem or the characters don't act like themselves the magic is lost. But when it works? When it clicks and the reader sees the domino chain laid out in front of them? It's beautiful.

[ginnyw-potter-archive]

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yes, I'm self-aware thank you

[biggest-gaudiest-patronuses]

forgot to mention, last year i taught myself to crochet and i made a stuffed whale

[when-sanpape-arts]

some dunmeshi restaurant au doodles

[crabhousecrafting]

hermitcraft “fun facts” are so hilarious because they’re always like

etho created the hopper clock! ⏰

xisuma created bedwars! 🛏️

cubfan is a published astrophysicist.

[crowberri]

Swooned & Knighted

[kaitcake1289]

in paris you gotta sleep with one eye open

[mesopelagos]

a ponytail elastic probably won't cut it, so they have to find alternate solutions

[savanir]

DP x DC prompt [9]

Danny doesn't remember much of what happened after his fight with Pariah. he knows the suit nearly killed him. 

He knew he passed out after and had to be carried back.

But considering the fact that the sky is blue and he's in his bedroom it was pretty safe to say that it was a classic case of a job well done and everything was back to normal.

The next day however, more and more oddities started happening. 

No longer did Amity Parkers get assaulted by GIW warnings when they accessed the internet. Instead they just got… nothing, nada, zilch.

Did the GIW go all in and just disconnect them from the rest of the world completely?

But then it became clear that that was the case with everything. stores weren't getting any shipments. 

phone calls would automatically say that numbers weren't in use. 

packages and mail weren't being picked up. 

Very worryingly, credit cards also stopped working and any attempt to contact the bank went utterly nowhere. 

people gradually are starting to get more and more worried.

Amity was very independent and self sufficient but this was a bit much.

At the very least now the city was more open to the doctor's Fenton energy solution of simply using Ecto to power everything.

The guys in white didn't show up in the city anymore either. 

The same went for the other out of town ghost hunters.

and after a quick check from Danny himself (as Phantom) he confirmed that the little not so very hidden base the guys in white had set up outside of the city borders was now simply gone.

Not only that but the roads going out of Amity also just suddenly stop.

At this point Team Phantom is starting to have a certain suspicion, and Sam asks Danny to find the nearest gas station and get them some newspapers.

Back home and now with a bunch of newspapers spread out over the floor with articles about Alien invasions in a place called Metropolis or the top floors of a skyscraper being blown up in a city called Gotham, they have enough to confirm their worries.

“Guys I think we got put back wrong”

[lenin-it-to-win-it]

[courfee]

17 December 1975 / 15 May 1976

[hawberries]

alright. hear me out

[pigeonwithatinyhat]

um um hi hi, pearly pop with magical sparkles for your troubles? this took me 6 hours, which isn't too bad. :D