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.

looking up a math definition on wikipedia you gotta slog through like five paragraphs of motivation and special cases and examples from fields of math you don't give a fuck about, when really the hover-over feature they introduced a few years back should be enough to give you the definition in the first sentence

meanwhile nlab:

Nightmare loses a battle (part 4)

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This cringe website doesn't support unicode latex smh

proof by stirring it then blowing a bubble

N Lab食宴式 | 開創彰化在地新活力！第二場11/1開放報名～

《N Lab食宴式》活動圓滿落幕！今年的N Lab食宴式真的太精彩了，不僅多了超過50家優質的彰化在地商家、店家、文創者，活動過程也是精采萬分，阿偉是參加第一場的《N Lab食宴式》，熱鬧的攤商吸引無數人的目光與駐留，別具意義的成果發表食之饗宴，更是在我舌尖上颳起了一場風暴，真的太棒啦～ Continue reading Untitled

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Gee, Thanks nLab.

you're on the doctor who wiki page for breast cancer... I'm on the nlab page for phenomenology of spirit... we are not the same.

I feel like nlab is just fucking with me.

Do you have any recommendations for texts for a first reading of category theory? :))

So this is a super common question and I think it's a little hard to give a good answer to! I'll tell you what I did.

Back in the summer of the beautiful year 2019 I had just finished my first year of university mathematics education. I don't remember where I had even heard of the subject, but the first math book I ever self-studied out of was Mac Lane's Categories for the Working Mathematician (the second edition, specifically). It's a good book, I think! But I only really appreciated it on my reread (i.e. my more-than-the-first-three-chapters read) last year. For one, I had not done a lick of topology by this point, and I really do recommend knowing basic topology and basic abstract algebra before getting into categories; it gives you a nice two-pronged approach to view every construction.

So I washed out of that one pretty quickly. I learned the basic definitions of categories, functors, and natural transformations. I could work out what it meant for a product to be the limit of a discrete diagram (super neat when you just learned what a group was five months ago!), but if you asked me to give an example of a pair of adjoint functors that wasn't in the text I would've been stumped (natural transformations between hom-functors? what?).

Over the course of the next semester I got familiar with the basics of topology (and I made this blog!). Now I could see why a lot of Mac Lane's examples were so nice. Compositions of homotopic maps are homotopic, so you can take a quotient category the same way you take a quotient group. The fundamental group is so useful exactly because it is functorial. A map into a product space is continuous precisely when its composition with the projections is continuous. And so on. I sampled some more books. I read the first few chapters of Awodey's Category Theory and of Adámek, Herrlich, and Strecker's Abstract and Concrete Categories. Each offers a slightly different perspective, but every time I would wash out after the first several chapters.

But it was exactly rereading all of these different perspectives that made me grok what it was all about. How to work with these structures. I had limits and colimits pretty much down by this point. Ah, it's not a coincidence that the least upper bound and greatest lower bound are categorical limits, they're expressing the same idea. I realized on my own that not only is the discrete space functor to topological spaces left adjoint to the forgetful functor to sets, the indiscrete (or chaotic, or trivial) space functor is right adjoint. The abelianization of a group or the completion of a metric space are left adjoint to the inclusion functors of abelian groups and complete metric spaces, because a map from a group/metric space into an abelian group/complete metric space determines a unique map (in fact it factors through this map) out of the abelianization/completion. Even though the abelianization makes the group smaller, and the completion makes the space bigger, they're expressing the same relationship.

This went on for a while. I read Bartosz Milewski's blog on Category Theory for Programmers, but I didn't know Haskell so that didn't get me very far. I read a lot of nLab pages (I still do). I read a small bit of Borceux's Handbook of Categorical Algebra, which is very nicely written if you can get your hands on it. I spent a summer in a reading group for Tai-Danae Bradley's Topology: A Categorical Approach (this one's probably a good pick for you in particular!). Like two years ago I played the Natural Number Game (which they changed recently apparently!), which got me really into Lean and proof assistants in general. Playing around with Lean gave me a great appreciation for what category theory brings to the table for logic and computational mathematics. Reading Dan Marsden's blog on monads got me really into monad theory, which is just super cool and I love it very much. I had a brief fling with bicategories, mostly because of the monads. Around this time I made a bunch of explanatory posts on the things I was learning about, and that was just terrifically helpful for my understanding. I read Mac Lane and Eilenberg's original paper introducing categories, General Theory of Natural Equivalences, and it's fascinating to see the perspective that it takes. Last year I got really into sheaf theory by reading Mac Lane and Moerdijk's Sheaves in Geometry and Logic, which finally made me understand the Yoneda Lemma.

These last few examples are what I would call pretty intermediate, as far as categories go. Somehow by reading all these different sources (and progressing in my studies in other mathematical subjects!) I had gathered up enough experience to understand the higher-level structures built on top of the basics. I still don't understand Kan extensions (though I tried reading Riehl's Category Theory in Context to understand it) or fibrations, or model categories, or co/end calculus, but I'm sure I will once the need arises.

So that's my advice! Read as many different introductions to the subject as you can, in as many different contexts as you have the experience for! The power and beauty of category lies in the bridging over the gaps between these contexts. Any of the books I mentioned I would recommend, but really the most important rule of category theory is to have fun and be yourself :-)

the least I could do this late at mornight ✌️✌️✌️✌️ (voting for Zesty in a pointless nlab poll)

nlab sweats nlab sweats nlab sweats

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