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Applied Category Theory Course

math.ucr.edu

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Re: Applied Category Theory Course

#11

Post by j2kun »

I'm writing a book [1] about "bona fide" applications of mathematics to practical programming problems, where bona fide is my own personal definition [2]. I've had my eye on claims of applications of category theory for many years, and none of them have quite fit the bill (sorry, John, if you're reading this). When they talk about applications to databases it seems quite unrealistic or lacking in sufficient detail to evaluate. And when I talk to people who are actually on the front lines that try to use category theory to solve some problem, they tell me that it ended up being completely useless.

I'm open to having my mind changed. The one application that's come the closest is UMAP, [3] though the last time I checked, a few years ago, the jury was still out on whether the structure it produces in applied contexts like biology is spurious.

So what specific applications do you know of category theory to practical problems? For the purpose of this post "application" means used by someone in a production software setting to solve a problem not related to category theory. Someone who doesn't care about category theory should care about the problem being solved.

And, to be sure, category theory is fine an dandy in its applications to other areas of math, as a unifying language for homology and whatnot. I'm talking about the kind of claims category theory folks sometimes make like "category theory is ushering in a programming revolution and you need to learn it to stay relevant"

[1]: http://pmfpbook.org/

[2]: https://buttondown.email/j2kun/archive/whats-in-production/

[3]: https://umap-learn.readthedocs.io/en/latest/

Re: Applied Category Theory Course

#12

Post by ekidd »

behnamoh wrote: post #6

I’ve seen many “applied” category theory posts and books but it’s important to remember that pure mathematicians’ definition of “applied” is way different than and more boring than you’d think. I still have to see some insights that CT gave us about a real world application that wouldn’t have been discovered without CT. Most of CT to me seems like definitions on top of definitions.

Honestly, category theory is more useful for programming language designers than for programmers in general.

Here are some of the more interesting things it's good for:

1. The lambda calculus is the basis for most functional programming and for closures. The lambda calculus is also tightly linked to Cartesian closed categories. And a huge number of structures in math are either Cartesian closed categories, or at least topoi, which are closely related. This connection can be used to quickly sketch out designs for exotic programming languages.

2. Many simpler categorical ideas map cleanly to programming languages. Functors are parameterized types with `map` functions. Natural transformations correspond to certain kinds of tidy parameterized types. Even monads are basically just nested parameterized types that support `flatten`. And so on.

Now, most of this stuff is pretty shallow category theory. It's mostly definitions. But they're definitions that allow you to draw interesting connections between functional programming, parameterized types, and distant branches of math.

Used right, you can design some pretty unusual languages that fit together cleanly.

Re: Applied Category Theory Course

#13

Post by max_ »

j2kun wrote: post #10
hughesjj wrote: post #9

Earlier quoted context omitted.

Odd take on a post called "applications of category theory" I mean topology didn't have so many applications 100 years ago, but now that shits everywhere. Math is the ultimate r&d. It's literally the language of causality. Imho it can't not be useful.

I have looked into applications of topology a lot and "everywhere" seems like a stretch, though perhaps you mean it in a particular sense that differs from mine. Could you give some examples of what you think of as the most useful practical applications of topology?

I believe it's used in cosmology to depict the "shape" of the universe

Re: Applied Category Theory Course

#14

Post by fithisux »

ekidd wrote: post #12
behnamoh wrote: post #6

I’ve seen many “applied” category theory posts and books but it’s important to remember that pure mathematicians’ definition of “applied” is way different than and more boring than you’d think. I still have to see some insights that CT gave us about a real world application that wouldn’t have been discovered without CT. Most of CT to me seems like definitions on top of definitions.

Honestly, category theory is more useful for programming language designers than for programmers in general. Here are some of the more interesting things it's good for: 1. The lambda calculus is the basis for most functional programming and for closures. The lambda calculus is also tightly linked to Cartesian closed categories. And a huge number of structures in math are either Cartesian closed categories, or at leas…

I agree, especially for programming language semantics or abstract compilation.

Also it is very useful for theoretical computer scientists.

Re: Applied Category Theory Course

#15

Post by fithisux »

j2kun wrote: post #11

I'm writing a book [1] about "bona fide" applications of mathematics to practical programming problems, where bona fide is my own personal definition [2]. I've had my eye on claims of applications of category theory for many years, and none of them have quite fit the bill (sorry, John, if you're reading this). When they talk about applications to databases it seems quite unrealistic or lacking in sufficient detail to…

I ended up there too.

Nice for theoretical work, bad for practical work.

Functor is useful though.

Re: Applied Category Theory Course

#17

Post by jiggawatts »

j2kun wrote: post #11

I'm writing a book [1] about "bona fide" applications of mathematics to practical programming problems, where bona fide is my own personal definition [2]. I've had my eye on claims of applications of category theory for many years, and none of them have quite fit the bill (sorry, John, if you're reading this). When they talk about applications to databases it seems quite unrealistic or lacking in sufficient detail to…

> When they talk about applications to databases it seems quite unrealistic

I read a paper that claimed that the category of polynomial-like things ("Poly") was somehow useful for database migrations! Without a concrete example, that's just an absurd statement to make. If you don't explicitly connect the dots between SQL schemas and polynomials, then you're just hand-waving. It's the equivalent of: "I have a marvellous proof but alas it is too long to fit into this margin."

Re: Applied Category Theory Course

#20

Post by zmgsabst »

j2kun wrote: post #11

I'm writing a book [1] about "bona fide" applications of mathematics to practical programming problems, where bona fide is my own personal definition [2]. I've had my eye on claims of applications of category theory for many years, and none of them have quite fit the bill (sorry, John, if you're reading this). When they talk about applications to databases it seems quite unrealistic or lacking in sufficient detail to…

Developing conceptual maps/diagrams, ubiquitous language, and then implementing such a system is topos theory:

We’re starting with categorical maps and then transporting that model into a type theory, while maintaining the semantics.

Effective patterns in doing so are “applied category theory”.

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