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Applied Category Theory Course

math.ucr.edu

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Re: Applied Category Theory Course

#41

Post by epgui »

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?

It is used everywhere. ML/AI would be a good example.

Re: Applied Category Theory Course

#42

Post by epgui »

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…

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

Probably true. However... I wouldn't take that comment too far. IMO if more programmers took interest in the subject, they might begin to see complexity/simplicity they didn't recognize before, and make different language choices, or take a different approach to compositionality.

Re: Applied Category Theory Course

#43

Post by epgui »

stiff wrote: post #29
mjan22640 wrote: post #22

Earlier quoted context omitted.

Programming is applied algebra, the choice is not using vs not using, but understanding what you are doing vs being ignorant about it. Relevance is in the eye of the beholder. In typical business the incentive is to have a good enough code ready "yesterday", and it will be thrown away tomorrow, as the code and whatever it does are only a means to generate profit and not a goal in itself, and spending time on making i…

Programming is applied algebra just like tying your shoes is applied knot theory :)

IMO, knot theory is not as closely related to shoe-tying as programming is to algebra...

Re: Applied Category Theory Course

#44

Post by solomonb »

jiggawatts wrote: post #30
solomonb wrote: post #24

Earlier quoted context omitted.

here you go: https://arxiv.org/pdf/1009.1166.pdf

This paper is actually interesting and talks about real-world schemas. I particularly like the concept of distinguishable nulls that define a sane equality operator, unlike the traditional SQL NULL. The paper I was referring to used the category Poly, and literally just stated that Poly could be used for database migrations without elaborating further.

I don't know the paper you are talking about but the author of this one (David Spivak) does a ton of research on Polynomial Functors and he has a whole book on them: https://topos.site/poly-book.pdf

Poly is an incredibly rich category which can be used to describe a ton of stuff such as finite state machines, neural networks, game interactions, wiring diagrams and dynamical systems more generally. I wouldn't be surprised at all if Poly can describe database migrations.

Re: Applied Category Theory Course

#45

Post by stathibus »

hughesjj wrote: post #9
stathibus wrote: post #7

Why is everyone here so obsessed with category theory? It has basically no useful theorems.

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 think you're confused about what "applications" means to mathematicians

Re: Applied Category Theory Course

#47

Post by catgary »

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…

It’s a bit tough, on one hand John Baez is a great messenger because he’s such a fantastic communicator. But on the other hand, he’s a pretty horrible spokesperson for ACT because the man did theoretical physics and higher category theory for 30 years and now has decided to do “applied” work. It’s pretty evident in this course, he’s just setting up the frameworks people talk about using but isn’t going for any actual results that demonstrate the claimed application. I don’t think he’s ever actually done any applied work before he decided to revolutionize the field.

I once watched John talk down to some sort of research electrical engineer at a conference because his diagrammatic calculus didn’t line up with John’s approach, when anyone with a lick of sense would want to understand why their approach didn’t suffice and how these changes addressed its shortcomings - this person was a subject matter expert and Baez condescended to him about his own field! I think his attitude has, unfortunately, rubbed off on a lot of the people in his orbit.

Re: Applied Category Theory Course

#48

Post by bmitc »

k__ wrote: post #31
bmitc wrote: post #8

Earlier quoted context omitted.

I think it's mainly software folks who feel mathematical learning it. Mathematicians primarily use it as a linkage and general tool in some context. I feel category theory needs something to latch it on to. Most software based applications seem a little too contrived, unless I have missed something interesting.

If CT is used to connect different parts of math and software is basically math proofs, then it makes sense programmers find CT interesting. It let's you connect your software to math concepts.

> software is basically math proofs

But it's not.

Re: Applied Category Theory Course

#49

Post by JHonaker »

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…

It really depends on what your idea of applied is, but Conal Elliot's work is really interesting.

His paper on Compiling to Categories [1] pitches programming language semantics as a cartesian closed category [2] allows for really cool stuff by mapping typical evaluation semantics to alternatives like building a computational graph visualization, a pretty printer, and imbuing the original program with automatic differentiation capabilities purely through the categorical interpretation of the same program. Essentially, if you can formulate a desired output of the program as a cartesian closed category, you can do it quite easily.

[1]: http://conal.net/papers/compiling-to-categories/

[2]: A cartesian category with is a category with an operation that lets you pair things together into a new object and get out the original parts with eliminators. Think construct a tuple and have the ability to get the first and second elements out of the tuple. A cartesian closed category adds in the ability to model (partial) function application to its arguments via "apply, curry, and uncurry." See https://en.wikipedia.org/wiki/Cartesian_closed_category

Re: Applied Category Theory Course

#50

Post by throw156754228 »

Some things in Haskell seem so laboriously interwoven with Category Theory. Take for example representable functors for memoization. Pages of Category Theory to do something dead simple. In comparison memoization in javascript is just throwing things into a {}.
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