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
Does anybod know a way to learn and compare a bunch of different cohomologies? I only ever worked with Čech cohomology and am simply wondering what the world outside my cave looks like
Contemplating a future where you can buy some tools and create your own custom insects from scratch in a specialised bug workshop. The first rule would be always make the bugs look more scary. The second rule would be always attach more knees. Spikes too if you got em in stock. Aside from that though it would be pretty free form artistic expression, and maybe there would be a television show where people show off their most imaginative new species of insects. I know what I would invent if bug workshop were a real hobby. Do you?
I think genemodding bugs is comparably easy to diy
To get more knees you need more legs or more knees per leg. Both can be done by controlling the gene sequence responsible for the general bodyplan (the hox system). I forgot the reference, but i saw someone explain it by demonstrating how to add extra segments to a centipede (or millipede?). Your creation might not be livable anymore though and working with (pre-)embrionic insect stemmcells is probably a bit of a pain. All of that should be reasonably well understood though, there is a lot of literature about it
Its definitely common in research to mod all kinds of small animals up to mice, but that's of course different when you are doing it just for the fun of it. You don't want legally or marally to commit animal abuse. I have no idea what the status is in regards to insects, neither legally nor morally as i don't know the amount of interiority their strickladder nervous systems posess (which also varies wildely depending on the insect). There are definitely insects whose nervous system is primitive enough to be simulated on a computer, so if experimenting with them is morally wrong then we also have huge problems regarding how we treat complex software systems.
If you want to be certain that you don't get i to moral or legal quarrels you can also use plants or bacteria instead, that is also much easier to work with as far as i know.
Contemplating a future where you can buy some tools and create your own custom insects from scratch in a specialised bug workshop. The first rule would be always make the bugs look more scary. The second rule would be always attach more knees. Spikes too if you got em in stock. Aside from that though it would be pretty free form artistic expression, and maybe there would be a television show where people show off their most imaginative new species of insects. I know what I would invent if bug workshop were a real hobby. Do you?
I think genemodding bugs is comparably easy to diy
To get more knees you need more legs or more knees per leg. Both can be done by controlling the gene sequence responsible for the general bodyplan (the hox system). I forgot the reference, but i saw someone explain it by demonstrating how to add extra segments to a centipede (or millipede?). Your creation might not be livable anymore though and working with (pre-)embrionic insect stemmcells is probably a bit of a pain. All of that should be reasonably well understood though, there is a lot of literature about it
Contemplating a future where you can buy some tools and create your own custom insects from scratch in a specialised bug workshop. The first rule would be always make the bugs look more scary. The second rule would be always attach more knees. Spikes too if you got em in stock. Aside from that though it would be pretty free form artistic expression, and maybe there would be a television show where people show off their most imaginative new species of insects. I know what I would invent if bug workshop were a real hobby. Do you?
I think genemodding bugs is comparably easy to diy
To get more knees you need more legs or more knees per leg. Both can be done by controlling the gene sequence responsible for the general bodyplan (the hox system). I forgot the reference, but i saw someone explain it by demonstrating how to add extra segments to a centipede (or millipede?). Your creation might not be livable anymore though and working with (pre-)embrionic insect stemmcells is probably a bit of a pain. All of that should be reasonably well understood though, there is a lot of literature about it
I've a (possibly unfortunate) soft spot in my heart for informal reasoning in maths. Like don't get me wrong being correct and being demonstrably so is absolutely necessary. I'd never tell an undergrad to only show the vibes portion of a proof. But the so-called post-rigour mindset (don't like that word, also not a Terry Tao fan) has such a wonderful appeal.
Being one with the currents of reason as they take shape not merely within formulae and proofs but within groups of people, among hundreds and hundreds of proofs, within bodies of work. The stuff that takes you a year to learn to do in 5 mins. Dunking your head in the clear stream in which god fishes for theorems, oxygen be damned.
This is one of the things that bothers me so much about the use of LLMs for proofs (even Lean checked proofs). They output five hundred pages of stuff that no one has the time to read, is often subtly wrong (and Lean isn't the be-all end-all of proof correctness either but that's a separate can of worms), and takes from us (in the literal sense of prevents the author from doing themselves) the part of maths that I love most: the musing! The pontificating. The imagination! The imagination that comes from working through shit yourself and letting the waves take you somewhere you never saw coming! The informal reasoning that can only come out of a deep understanding of technicalities.
To me it's what animates a good talk. That the speaker has drowned themselves in that stream and come back with a parable. The best works of mathematics are emotional. They make an argument for what *should* be and *then* make an argument for why it *is*. It's a profoundly unscientific way to think, and it intersects an unfortunate amount with most LLMs' core capabilities. But it's what I love about maths, and it connects me to other people who love it.
It is one of the main things that i try to convey to undergrads in set thoery or logic classes, that pure symbol manipulation does not yield good results beyond the most basic problems. While every solvable problem may be sloved this way eventually, even for quite simple questions it takes pages upon pages of deductions with an exponential amount of brute force time.
Rather the (imo) best way to approach mathematics is to start very informal and then refine the informal reasoning down unil you have a formal proof. In some way that is also what many theorems are, a previous informal piece of reasoning that was made formal.
I need like, a casual hookup, except instead of sex it's working on math problems together and other such nerdiness, and instead of casual we keep doing it again and again and get married.
the thing is I just don't think intellectual labour is a special class of labour. I just don't think someone who created an intellectual product like a manuscript deserves the rights to extracting rent forever from physical copies of it more than the person who maintains the printers. I will just never be convinced of this.
This is such a bizarre take to me. I hate copyright but it seems for very different reasons. But also I just don't get what it's complaining about - the publishers always get their cut. In fact everyone typically complains their cut it too big. And their cut pays the person who maintains the printers. And keeps paying it as long as the book keeps getting printed and purchased.
Information wants to be free. Someone who writes a manuscript deserves to be paid for their efforts, if people read it. I fundamentally do not understand the difference between internet piracy and libraries. I value and respect the work of authors. An author contributes more towards the existence of a book than someone who maintains a printer. And especially more for a book that only exists online than someone who maintains a server. Everyone involved deserves to get paid for their work and realistically the one most at risk of not being paid for their contributions to the finished product in the real world right now is the author. If each physical book costs $20 retail, I would say that the author should end up getting more of that $20 than the printer-maintainer. Each of them should get a cut, and does.
why, though? i do porn. some material i got paid for, and i'm done with it. whoever has it can transmit the bits. the rest of the material is mine, and i use it for ads and content sales. way i do those, even if someone uploads them to a porntube, i'm not losing anything anyway ¯\_(ツ)_/¯
copyright died the day of the invention of the packet switching network, so we need to find other ways to monetize our work. figuring out who's buying what you sell is part of any job.
also, distribution is work. ongoing work. a printed book that gets a 10yr continuous run, that's work going on for that decade. (ebooks are ~zero marginal cost (inb4 infra: ipfs freenet bittorrent ao3-like-motherfuckingwebsite) but dead trees are made of atoms that must move.) during which decade the author may be done with that book, in which case it's ok if they don't get shit, or they may keep working to sell it, in which case they absolutely should get paid for that work. or maintaining a wordpress and a patreon, that's also work (which, hopefully, eventually gets some pay).
look, when i build a flogger and sell it to a pro domme, i don't get a residual every time she uses it on a client. so why should writers get a cut of the sale of every copy of a book they wrote that one time? i mean, sure, that's a thinkable pricing structure. just like as if i had to send money to bad dragon every time i fuck my tentacle on cam. it's just impractical, and enforcing it on any non-trivial level implies a dystopian cyberpunk nightmare of surveillance and punishment. so, uh, fuck that
All IP law that has any right to exist should be handeled as misinformation law. (No convincing impersonation, cheap copies of a product without labeling them as such, etc.)
Yeah in principle I'm on board with all of this, especially the idea of misinformation law replacing copyright. No one can sell a book they didn't write as their own, but handing it out for free with appropriate attribution is fine. Philosophically this is great.
@sophia-epistemia I think the difference between your examples and a book is simply the number of hours of work. To commission a book ahead of time would be quite expensive, just judging by number of hours invested. An author might easily spend a year writing a novel, and if that's how long it takes, that's how long it takes - as a reader I don't want to rush them, I like the work that's produced right now. But ahead of time who knows if a book will be good, so rather than raising money on promises it makes some amount of sense to raise money on product delivery. (And part of what a publisher does is act as a sorta combo insurer/loan giver to help smooth the process over for writers.) But this model just hasn't kept up with the times, I totally agree that the model is flawed. I don't actually have any idea of how to get authors a living wage, and I don't see how anything you said helps with that problem. And as a reader, I'm quite interested in authors having living wages. But also I agree about copyright and dystopian surveillance.
I don't have a good solution. Patreon can support some web authors, but I think that it can't handle the traditional novel, and I don't want to lose (large amounts of) that art form just for economic reasons. We have a pretty good thing going as readers.
in law school, my mom took me to this stress retreat because my family was vaguely aware i was a suicide risk. and they didnt allow phones, so i brought this huge bag of books. and one of the options for "destressing" was this fake cave grotto thing, where they'd decorated a room to make it look exactly like an underground cave and the air was like -10 degrees, but there was a like 4ft deep pool in the middle that was kept super super hot, so you would just switch between the hot and cold. and they would bring you an endless supply of this weird syrupy drink thing that was like super caffeinated and tasted like sugar and mint. and so i spent multiple days sitting half submerged in this fake grotto drinking mystery liquid and reading. and i have to be honest i really did feel less stressed
the thing is I just don't think intellectual labour is a special class of labour. I just don't think someone who created an intellectual product like a manuscript deserves the rights to extracting rent forever from physical copies of it more than the person who maintains the printers. I will just never be convinced of this.
This is such a bizarre take to me. I hate copyright but it seems for very different reasons. But also I just don't get what it's complaining about - the publishers always get their cut. In fact everyone typically complains their cut it too big. And their cut pays the person who maintains the printers. And keeps paying it as long as the book keeps getting printed and purchased.
Information wants to be free. Someone who writes a manuscript deserves to be paid for their efforts, if people read it. I fundamentally do not understand the difference between internet piracy and libraries. I value and respect the work of authors. An author contributes more towards the existence of a book than someone who maintains a printer. And especially more for a book that only exists online than someone who maintains a server. Everyone involved deserves to get paid for their work and realistically the one most at risk of not being paid for their contributions to the finished product in the real world right now is the author. If each physical book costs $20 retail, I would say that the author should end up getting more of that $20 than the printer-maintainer. Each of them should get a cut, and does.
why, though? i do porn. some material i got paid for, and i'm done with it. whoever has it can transmit the bits. the rest of the material is mine, and i use it for ads and content sales. way i do those, even if someone uploads them to a porntube, i'm not losing anything anyway ¯\_(ツ)_/¯
copyright died the day of the invention of the packet switching network, so we need to find other ways to monetize our work. figuring out who's buying what you sell is part of any job.
also, distribution is work. ongoing work. a printed book that gets a 10yr continuous run, that's work going on for that decade. (ebooks are ~zero marginal cost (inb4 infra: ipfs freenet bittorrent ao3-like-motherfuckingwebsite) but dead trees are made of atoms that must move.) during which decade the author may be done with that book, in which case it's ok if they don't get shit, or they may keep working to sell it, in which case they absolutely should get paid for that work. or maintaining a wordpress and a patreon, that's also work (which, hopefully, eventually gets some pay).
look, when i build a flogger and sell it to a pro domme, i don't get a residual every time she uses it on a client. so why should writers get a cut of the sale of every copy of a book they wrote that one time? i mean, sure, that's a thinkable pricing structure. just like as if i had to send money to bad dragon every time i fuck my tentacle on cam. it's just impractical, and enforcing it on any non-trivial level implies a dystopian cyberpunk nightmare of surveillance and punishment. so, uh, fuck that
All IP law that has any right to exist should be handeled as misinformation law. (No convincing impersonation, cheap copies of a product without labeling them as such, etc.)
