Introduction to Introduction to Category Theory

by countedblessings 1mo6th Oct 20192 min read12 comments

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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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