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.
Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the
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Nakajima varieties provide a natural home for geometric representation theory of simply-laced complex simple Lie algebras.
Ultimately, Nakajima theory is a theory about the interaction of symplectic geometry and representation theory.
Everyone pretty much knows that Time can be Rewritten, and yet Fixed Points are a thing. So, how can time be both malleable and concrete at the same time?
I propose it’s more about the people. Time Lords can sense Fixed Points, there is nothing that should be done about it and even their mere presence might affect it, so they try to stay away. Even more so if they know about what the Fixed Point is. But as humans don’t have time senses and don’t sense Fixed Points, as long as they don’t know about the Fixed Point, it’s not Fixed to them. Sure, most people wouldn’t interfere with the happenings of a Fixed Point, because their non-interference is part of it, but say an accidental time traveler interfered in a Fixed Point by just, I don’t know, helping a fallen man up or bumping into someone making them late to a meeting or something equally mundane.
This was born mostly out of a plot bunny in which Rose comes back, doesn’t find the Doctor but finds little Melody Pond alone in the children’s home and taking her with her, changing everything.
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**“You ever have those moments when you can feel time pass? Like, you’re just sitting silently and watching as things change.” This came from my friend, and gave me this idea for another fanfiction, so here I go again ^^) This is pretty similar to His Last Bow round the end.
It was a sunny day in the middle of spring, and I had no other business to attend to. I decided to see how the retirement to beekeeping was treating Sherlock Holmes in his Sussex cottage.
After we spent quite some time catching up over lunch in the house, we ventured outside to walk round the area. As we were walking, I happened to catch a glimpse of a bush where a leaf held a small sack of a cocoon. “Why, Holmes,” I remarked, pointing towards the bush, “it appears your bees will have some company.”
Holmes looked in the direction I indicated, then bent down a bit for a better view. “Seems a valid deduction, my dear friend,” he said with a little grin. Then he strolled over and sat leisurely in the grass near the bush, where he had command of a decent view of the cocoon. I joined him, and for several minutes we sat in simple silence, keeping a vigil of sorts over the silky white blot on the leaf.
After some time, I began to notice that the small nest was wiggling slightly. I quietly gasped and asked, “Is that the breeze, or is it opening?”
With a light chuckle, Holmes responded, “I perceive it to be our once-was caterpillar friend attempting to break out of its case.”
As we continued gazing at the trembling cocoon, it finally gave way enough for the creature to poke through. We watched the young butterfly’s head emerge from its protection, followed by the rest of its body as it climbed delicately out to the open. It stretched its wings back and forth slowly as it crawled about on top of its leaf.
I glanced at Holmes for a moment, and noticed that he suddenly appeared thoughtful and vaguely troubled. “Holmes, are you not feeling well?” I asked, a bit startled by the sudden shift.
He started and blinked, then offered a light, reassuring smile. “Everything is in constant change. I am certainly glad that you at least have remained the fixed point, my dear Watson.”
