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Category Theory: Lecture Notes and Online Books

logicmatters.net

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Re: Category Theory: Lecture Notes and Online Books

#11

Post by meiji163 »

vkk8 wrote: post #3
elnatro wrote: post #2

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

Seven Sketches in Compositionality by Spivak and Fong ( https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf ) is supposed to be very practical. It includes many examples, but I'm not sure how useful category theory actually is in the example cases. There is the theory of monotone co-design ( https://co-design.science/index.html ) which is formulated using category theory. It seems pretty practical. Then there's…

> I'm not sure how useful category theory actually is in the example cases.

It's hard to say that category theory is "applied" to this or that problem. you'll hear many mathematicians call it "abstract nonsense" half-jokingly. More than anything it's a unified way of talking about mathematical structures that gives you a certain point of view (which is where it might be useful).

Re: Category Theory: Lecture Notes and Online Books

#12

Post by rg111 »

Seven Sketches in Compositionality is easily the best book to get started on Category Theory.

Learning something becomes highly fun and enjoyable when the authors/lecturers are deliberately trying to make it so.

Most books on anything serves to document something and looks like accumulated notes that are an overt attempt to impress the authors' peers. There is no focus on actually teaching.

This book is not like that. This book focuses entirely on teaching, tries deliberately so that you can learn.

Some other books that I have come across in my life are:

1. Intro to Electrodynamics by Griffiths

2. High School Physics book by Halliday, Resnick, Walker.

3. ToC book by Sipser.

You get the idea.

Re: Category Theory: Lecture Notes and Online Books

#14

Post by lapinot »

elnatro wrote: post #2

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

This feels to me like "is there any non rubik's cube example of group?". Category theory is a mathematical tool used by mathematicians and built for mathematicians, so you'll have trouble understanding why it's concretely relevant anyway if you're not in the business of doing maths :D The main motivation is that it's a good language for describing some kinds of objects and writing proofs about them.

I'm personaly using category in the field of logic and language semantics (describing and proving stuff about notions of equivalence between programs, results like language X can interpret language Y with such and such fidelity). At a high level, i think category theory is useful in that realm because it enables to make up new structures (algebraic structures, think some set with operations and laws) which will be specific to you (won't be something well known like ring or whatever) but thanks to category theory you'll already have results on these structures because you'll show that they have some generic constructions.

In general category theory sometimes enables mathematical statements to drastically cut down on unimportant assumptions, getting at the real core of some theorem. By doing that, you're enabling other to reuse some result in settings which are "kinda similar", where "morally stuff work in the same way", but where the classical definition would have been too rigid.

edit: concretely about these generalizations: if you know about monoids, then look at https://en.wikipedia.org/wiki/Monoid_(category_theory) which is a categorification of it.

Re: Category Theory: Lecture Notes and Online Books

#15

Post by DaiPlusPlus »

ryan93 wrote: post #9

Category theory seems like a "meme" interest among programmers.

It is, because modern programming languages today are far more expressive than previous generations of languages are, with powerful type-systems - and languages' type-systems are grounded in category-theory (and group-theory, and type-theory, and set-theory).

Modern languages are our tools, and you can't use your tools effectively without understanding them, and you can understand them better by understanding their mathematical underpinnings.

-------------

Disregarding the comparatively more esoteric languages like Haskell, consider TypeScript, Rust, Haxe, and to an extent: Kotlin and Scala: you can take a self-taught (i.e. non-CS/SE graduate) programmer and eventually they'll figure their way through, and eventually grok, that languages' type-system - and hey-presto, you've just unwittingly taught someone something an entire course of graduate-level abstract mathematics would do, and been productive in the process.

Re: Category Theory: Lecture Notes and Online Books

#16

Post by DarylZero »

lapinot wrote: post #14
elnatro wrote: post #2

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

This feels to me like "is there any non rubik's cube example of group?". Category theory is a mathematical tool used by mathematicians and built for mathematicians, so you'll have trouble understanding why it's concretely relevant anyway if you're not in the business of doing maths :D The main motivation is that it's a good language for describing some kinds of objects and writing proofs about them. I'm personaly usi…

Haskell is why it's relevant =D

Re: Category Theory: Lecture Notes and Online Books

#17

Post by lapinot »

DarylZero wrote: post #16
lapinot wrote: post #14

Earlier quoted context omitted.

This feels to me like "is there any non rubik's cube example of group?". Category theory is a mathematical tool used by mathematicians and built for mathematicians, so you'll have trouble understanding why it's concretely relevant anyway if you're not in the business of doing maths :D The main motivation is that it's a good language for describing some kinds of objects and writing proofs about them. I'm personaly usi…

Haskell is why it's relevant =D

For haskell programmers most likely! Just like there's only the obvious way in which groups are useful for speed cubers.

Re: Category Theory: Lecture Notes and Online Books

#18

Post by Jhsto »

elnatro wrote: post #2

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

This series has some C++ examples alongside Haskell: https://bartoszmilewski.com/2014/10/28/category-theory-for-p... and acquirable as a PDF at https://github.com/hmemcpy/milewski-ctfp-pdf

Re: Category Theory: Lecture Notes and Online Books

#19

Post by cashsterling »

elnatro wrote: post #2

I find this topic fascinating but hard to start with. Does anyone have any non-Haskell example of category theory? Something that shows why is useful with a use-case.

Cambridge Quantum Computing applies Category Theory to development of quantum computing algorithms. Recent paper by Bob Coecke: https://arxiv.org/abs/2110.05327. Look up more papers by Bob Coecke and co-authors on Arxiv.

Cambridge Quantum Computing and Honeywell Quantum Solutions recently merged to form Quantinuum. I work on the hardware side at Quantinuum. I don't actually use or apply Category Theory on a day to day basis but I know it is useful.

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