Introduction to Introduction to Category Theory

by countedblessings2 min read6th Oct 201920 comments

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Category TheoryWorld Modeling
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Category theory is so general in its application that it really feels like everyone, even non-mathematicians, ought to at least conceptually grok that it exists, like how everyone ought to understand the idea of the laws of physics even if they don't know what those laws are.

We expect educated people to know that the Earth is round and the Sun is big, even though those facts don't have any direct relevance to the lives of most people. I think people should know about Yoneda and adjunction in at least the same broad way people are aware of the existence and use of calculus.

But no one outside of mathematics and maybe programming/data science has heard of category theory, and I think a big part of that is because all of the examples in textbooks assume you already know what Sierpinski spaces and abelian groups are.

That is to say: all expositions of category theory assume you know math.

Which makes sense. Category theory is the mathematics of math. Trying to learn category theory without having most of an undergraduate education in math already under your belt is like trying to study Newton's laws without having ever seen an apple fall from a tree. You can...you're just going to have absolutely no intuition to rely on.

Category theory generalizes the things you do in the various fields of mathematics, just like how Newton's laws generalize the things you do when you toss a rock or push yourself off the ground. Except really, category theory generalizes what you do when you generalize with Newton's laws. Category theory generalizes generalizing.

Therefore, without knowing about any specific generalizations, like algebra or topology, it's hard to understand general generalities—which are categories.

As a result, there are no category theory texts (that I know of) that teach category theory to the educated and intelligent but mathematically ignorant person.

Which is a shame, because you totally can.

Sure, if you've never learned topology, plenty of standard examples will fly over your head. But every educated person has encountered the idea of generalization, and they've seen generalizations of generalizations. In fact, category theory is very intuitive, and I don't think it necessarily benefits from relating it all as quickly as possible to more familiar fields of mathematics. Instead, you should grasp the flow of category theory itself, as its own field.

So this is (tentatively, hopefully, unless I get busy, bored, or it just doesn't work out) a series on the basics of category theory without assuming you know any math. I'm thinking specifically of high school seniors.

There is no schedule for the posts. They'll just be up whenever I make them.

Why category theory? And why lesswrong?

Well, category theory is a super-general theory of everything. Rationality is also a super-general theory of everything. In fact, we'll see how category theory tells us a lot about what rationality really is, in a certain rigorous sense.

Basically...rationality comes from noticing certain general laws that seem to emerge every time you try to do something the "right" way. After a while, instead of focusing so much on the specifics, it starts being worth it to take a step back and study the general rules that seem to be emerging. And you start to notice that doing things the "right" way gets a lot easier when you start with the general rules and simply fill in the specifics, like how the quadratic formula makes quadratic equations a cinch to solve.

Category theory gives us all the general rules for doing things the "right" way.

(Don't actually hold me to demonstrating this claim.)

Why should you be interested in category theory?

One is because category theory is going to rise in importance in the future. It offers powerful new ways of doing math and science. So get started!

Two is that category theory makes it much easier to learn the rest of math. Well, maybe—this is an experiment, and a big motivation for doing this. How fast and well do people learn regular math if they can just say, "Oh, it's an adjunction" every time they learn a new concept?

Three is that referencing homotopy type theory in conversation will make you sound cool and mysterious.

Please let me know if there's any interest in this.


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I'm still buying the CT hype, so very interested to see more of this. However, I've been buying the hype for some 10+ years now and trying to learn CT on and off, and still can't point to a single instance of being able to use it either to approach a problem or understand something better, so I'm pretty skeptical about this being teachable to a mathematically naive audience in a way that they can internalize much anything about it that's both correct and usable in some practice that isn't advanced math study.

When we get to adjunction, I'm hoping the utility of the series will start to become clear.

This sounds like a very exciting project and a solution to an open exposition problem.

I look forward to reading the posts!

I also wonder if you know what kinds of things would motivate you to write them?

Interesting discussion? Getting data about your ability to tutor concepts? Money (maybe we could organize a Patreon with interested LW users if so)?

I'm motivated by the essays themselves. I believe this material should exist. It's also good for my own understanding to write them.

I still have no idea how forcing really works, but man do I like the cut of Chow's jib.

While I share your enthusiasm toward categories, I find suspicious the claim that CT is the correct framework from which to understand rationality. Around here, it's mainly equated with Bayesian Probability, and the categorial grasp of probability or even measure is less than impressive. The most interesting fact I've been able to dig up is that the Giry monad is the codensity monad of the inclusion of convex spaces into measure spaces, hardly an illuminating fact (basically a convoluted way of saying that probabilities are the most general ways of forming convex combinations out of measures).

I've searched and searched for categorial answers or hints about the problem of extending probabilities to other kinds of logic (or even simply extending it to classical predicate logic), but so far I've had no luck.

It seems odd to equate rationality with probabilistic reasoning. Philosophers have always distinguished between demonstrative (i.e., mathematical) reasoning and probabilistic (i.e., empirical) reasoning. To say that rationality is constituted only by the latter form reasoning is very odd, especially considering that it is only though demonstrative knowledge that we can even formulate such things as Bayesian mathematics.

Category theory is a meta-theory of demonstrative knowledge. It helps us understand how concepts relate to each other in a rigorous way. This helps with the theory side of science rather than the observation side of science (although applied category theories are working to build unified formalisms for experiments-as-events and theories).

I think it is accurate to say that, outside of computer science, applied category theory is a very young field (maybe 10-20 years old). It is not surprising that there haven't been major breakthroughs yet. Historically fruitful applications of discoveries in pure math often take decades or even centuries to develop. The wave equation was discovered in the 1750s in a pure math context, but it wasn't until the 1860s that Maxwell used it to develop a theory of electromagnetism. Of course, this is not in itself an argument that CT will produce applied breakthroughs. However, we can draw a kind of meta-historical generalization that mathematical theories which are central/profound to pure mathematicians often turn out to be useful in describing the world (Ian Stewart sketches this argument in his Concepts of Modern Mathematics pp 6-7).

CT is one of the key ideas in 20th century algebra/topology/logic which has allowed huge innovation in modern mathematics. What I find interesting in particular about CT is how it allows problems to be translated between universes of discourse. I think a lot of its promise in science may be in a similar vein. Imagine if scientists across different scientific disciplines had a way to use the theoretical insights of other disciplines to attack their problems. We already see this when say economists borrow equations from physics, but CT could enable a more systematic sharing of theoretical apparatus across scientific domains.

Under the paradigm of probability as extended logic, it is wrong to distinguish between empirical and demonstrative reasoning, since classical logic is just the limit of Bayesian probability with probabilities 0 and 1.

Besides that, category theory was born more than 70 years ago! Sure, very young compared to other disciplines, but not *so* young. Also, the work of Lawvere (the first to connect categories and logic) began in the 70's, so it dates at least forty years back.

That said, I'm not saying that category theory cannot in principle be used to reason about reasoning (the effective topos is a wonderful piece of machinery), it just cannot say that much right now about Bayesian reasoning

Interesting. This might be somewhat off topic, but I'm curious how would such an Bayesian analysis of mathematical knowledge explain the fact that it is provable that any number of randomly selected real numbers are non-computable with a probability 1, yet this is not equivalent to a proof that all real numbers are non-computable. The real numbers 1, 1.4, square root 2, pi, etc are all computable numbers, although the probability of such numbers occurring in an empirical sample of the domain is zero.

So far, Bayesian probability has been extended to infinite sets only as a limit of continuous transfinite functions. So I'm not quite sure of the official answer to that question.

On the other hand, what I know is that even common measure theory cannot talk about the probability of a singleton if the support is continuous: no sigma-algebra on supports the atomic elements.

And if you're willing to bite the bullet, and define such an algebra through the use of a measurable cardinal, you end up with an ultrafilter that allows you to define infinitesimal quantities

I don't know enough math to understand your response. However, from the bits I can understand, it seems leave open the epistemic issue of needing an account of demostrative knowledge that is not dependent on Bayesian probability.

I am strongly interested. I've been trying to get some real math under my belt, but to be frank it isn't my talent so I am perpetually on the lookout for exposition and exposition accessories.

I've tried to learn the basics of the category theory some years ago, already having some background in algebraic topology, mathematical physics and programming. And, presumably, in rationality. I got the glimpses of how interesting it is, how it could be useful, but was never quite able to make use of it. Very curious if your series of posts can change that for me. Keep going!

I apologize for spamming with two posts on the same day about the same content, but I actually wrote this post first. It was waiting to be moderated for several days, so I wrote the other one and submitted it, and to my surprise it was immediately published. I guessed this one must have gotten stuck in a queue or something, so I rewrote it and published it. My bad, we'll definitely go one at a time from now on.

But in a way it's fitting, as having two introductory posts should definitely indicate just how slow the pace of this series is going to be.

I am looking forward to this. I did math olympiads at high school, then switched to computer science, and then spent a few decades writing apps that read values from database, display them in HTML, and store the edited values again in the database (always in some new framework, so I have to keep running and yet remain at the same place). Sometimes I wonder where the alternative paths could have lead. It seems that my brain is still capable of understanding math; I can read about new topics and develop some intuition about them (and verify it with people who actually understand the stuff). Understanding the category theory would... well, feel good; as if I am getting a glimpse into the alternative universe where I continued doing math.

Sometimes things are explained in an unnecessarily difficult way. I understand that if you are a university professor teaching X, and you know that all your students learned Y during the previous year, it makes sense to write a textbook of X assuming deep and fresh knowledge of Y. But because of job and kids, I don't have the time to walk the exact path of a university student, so I would appreciate shortcuts.

I would recommend Lawvere's book Sets for mathematics to anybody who is interested in learning some mathematics. It's a beginner book but it covers interesting category theoretical stuff you wont find in any other beginner book.

I disagree with the idea that one doesn't have intuitions about generalization if one hasn't studied mathematics. One things that I find so interesting about CT is that it is so general it applies as much to everyday common sense concepts as it does to mathematical ones. David Spivak's ontology logs are a great illustration of this.

I do agree that there isn't a really good beginners book that covers category theory in a general way. But there are some amazing YouTube lectures. I got started on CT with this series, Category Theory for Beginners. The videos are quite long, but the lecturer does an amazing job explaining all the difficult concepts with lots of great visual diagrams. What is great about this series is that despite the "beginners" in the title he actually covers many more advanced topics such as adjunction, Yoneda's lemma, and topos theory in a way that doesn't presuppose prior mathematical knowledge.

In terms of books, Conceptual Mathematics really helped me with the basics of sets and functions, although it doesn't get into the more abstract stuff very much. Finally, Category Theory for Programmers is quite accessible if you have any background in computer programming.

yes, please! ... Eugenia Cheng's book "Cakes, Custard and Category Theory" nibbles around the edge ... perhaps you can prepare the main course? (^_^) ...

My masters degree involved a good bit of category theory. Personally, I don't see how it has any use outside of mathematics. (Note 'maths' includes 'mathematical logic' - so it's still a broad field of applicability).

I am highly motivated to be persuaded otherwise, and hence will be watching this series of posts with keen interest.

---
<disclaimer>
I am not a working mathematician, and have not published any papers. My masters thesis involved a lot of category theory - but only relatively simple category-theoretic concepts (it was an application of category theory to a subfield of mathematical logic).

Limits, free objects, adjunctions, natural transformations etc. but not higher-order categories, topoi/toposes or anything fancy like that.
</disclaimer>

<handwavy discussion of technical math>
As I understand it, the usual application of category theory is mostly to things involving natural transformations (it is said the need for a way to formalize natural transformations is what led to the invention of category theory) - and even then, it seems mostly to be applied to nice algebraic objects with (category theoretic) limits, and slight generalizations of these. So, to groups and rings and modules, and then to some categories made of stuff kinda like those things.

There's also the connection to topology and logic, via simplicial complexes, homotopies, toposes, type theory etc. which seems very interesting to me. It seems useful if you want to think about constructive mathematics (i.e., no law of excluded middle) - which is promising for maths involving some notion of 'computation' (for a given abstraction of computation) which has obvious applications to computing (especially automated theorem provers / checkers).

In these senses, I can certainly see its use as another mathematical field, and a good way of reasoning IN MATHS or ABOUT MATHS. But I don't quite understand its tremendous reputation as this amazing mathematical device, and less so its applications outside of what I've mentioned above.
</handwavy discussion of technical math>
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In particular, from my admittedly limited knowledge, category theory only seems useful:
a) If you already have a bunch of different fields and you want to find the connections.
b) If you want to start up a new field, and you need a grounding (after which the useful stuff will be specifically in the field being developed, and not a general category theory result).
c) For good notation / diagrams / concepts for a few things.

EDIT: Interested to hear the opinion of someone who actually works with category theory on a regular basis.