Earlier quoted context omitted.
Not addressing your question, but the Yoneda Lemma is kind of a charlatan. On first reading, it seems magical and deep, but once you grok the proof, it feels like a relatively trivial observation. The whole thing is just about arrow composition! In a way, once you're on the other side, the Yoneda Lemma feels a bit like a checkpoint during the accimatization period where your brain gets used to thinking in categories…
Yoneda just says that we may exchange an object for all of the arrows which point to (dually, from) it. This is extremely deep; it is rather surprising that objects and arrows would have such a duality or exchange!
What Is Applied Category Theory? (2018)
51–60 of 71 posts
Re: What Is Applied Category Theory? (2018)
#52Earlier quoted context omitted.
Yoneda just says that we may exchange an object for all of the arrows which point to (dually, from) it. This is extremely deep; it is rather surprising that objects and arrows would have such a duality or exchange!
+1. The lemma is trivial not because the result isn't deep but because we have the right definitions.
Re: What Is Applied Category Theory? (2018)
#53Just a nice set of ideas to be used for social signalling. In with type systems, a typeclass is all you need. The mantra is "to be an X (being substituted for an X) is to be able to perform (implement) such and such actions (or have this or that biochemical properties). It is that general, that deep, it could be even seen in molecular biology. Category theory, on the other hand, is just a few nested abstract concepts…
Re: What Is Applied Category Theory? (2018)
#54Category Theory is one of those things which I know are important and has applications in many engineering fields. I also have an intuition that its obviously because things need to flow into each other and composability is how we achieve that. But man, whenever I try to get into this I feel like someone is hitting me on the head with equations.
Maybe for a value of "has applications" that doesn't necessarily include anyone actually applying it. As a generalisation, I don't think anyone by people into category theory actually cares about or applies category theory.
However, Set Theory is essential to build mathematics.
Taking points from: https://mathoverflow.net/questions/10334/what-practical-appl...
Analysis, Number theory, most of modern mathematics rely on set theory, even computer science in the definition of Turing machines and dealing with infinities there are implicit axioms of set theory.
Likewise, perhaps category theory may not yield immediate applications, but it can be useful as a language (like a programming language, natural or mathematical) to express mathematical ideas, and to build mathematical systems and foundations.
I'm not sure how closely it relates to Category theory, but Homotopy Type Theory has yielded the fantastic Univalent foundations, a seemingly more modern and simple foundation for mathematics. This line of work also yielded a solid basis for Proof assistants.
I don't know enough about functional programming to comment...
Re: What Is Applied Category Theory? (2018)
#55I'm not a natural mathematician (emphatically not!) and pretty much every attempt I've made to wrap my head around Category Theory has foundered on an onslaught of formalism by paragraph 2. To be clear, I'm not knoocking that: for those who can and do grok things from that sort of treatment, all power to your elbow (and no small amount of admiration). But it's not how my brain works. I need an intuitive understanding…
The style is super friendly and builds up intuition slowly, while being completely rigorous. For a great example, check out the "Crema di Mascarpone"[1] chapter which introduces all the tools for writing rigorous mathematical proofs using flow diagrams.
[0]: https://graphicallinearalgebra.net/
[1]: https://graphicallinearalgebra.net/2015/05/06/crema-di-masca...
Re: What Is Applied Category Theory? (2018)
#56Earlier quoted context omitted.
I don't have any resources to share on this, but I was also interested in this type of thing at your age (still am). If it interests you, never assume something is out of your league. You have time on your side. Sustained effort compounded over time is unreasonably powerful. Keep pushing!
> Sustained effort compounded over time is unreasonably powerful. Thank you, this hits close to home and is great advice, I appreciate it.
On my cube walls, whenever I get back to my cube: “magic is just cleverly concealed patience.” The approximate η-expansion: you can generally figure out what professional magicians do, you just dismiss it, “I mean of course she coulda known what card I was going to pull if she bought 52 decks of cards and assembled an entire card made out of the Jack of Spades... but that’s expensive and difficult and who would do that.” Penn Jillette puts the same point a different way, saying that he is going to have to perform the same trick over and over again and so it cannot be 99% likely to not injure him, that is far, far too low, he has to be 100% certain his tricks are safe. If he is firing a nail gun at Teller’s neck it must surely be incapable of firing a nail at Teller, and if he waxes on about how much he has memorized the pattern of the nails in the gun then he must be cleverly concealing whatever patience went into making that nail gun and the illusion that it is loaded.
But the key is that once you see this one place you see it everywhere. A restaurant works by the same magic. You say what you want and it magically appears before you. How did that happen? Prep work for hours in advance of the meal, so that when the time comes I just need to put tab A in slot B. Like a beef wellington; pre-cooked steak wrapped into a nice little bundle that just needs to be finished in the oven for 20 minutes and sliced. Something to think about next date night—you can impress your date immensely if you make the wellingtons the night before, so that your focus is on your lover and not on your cutting board.
You don’t usually think about the prep work that went into the Mother of All Demos, but you should. Sometimes when Alan Kay shows a clip from there, he mentions something like, “you might notice that they get sub-second latency on all these operations, but the terminal was here and the computer was over there so I like to ask students how they got sub-second latency on all these operations, and only one time have I got the right answer, one student said ‘because they wanted sub-second latency?’ and that is absolutely correct, they-goddamn-wanted-subsecond-latency, these things are there for you if you really want them.”
Re: What Is Applied Category Theory? (2018)
#57If anybody interested the best book to learn the category theory: https://www.maths.ed.ac.uk/~v1ranick/papers/maclanecat.pdf
Why do you think this is the best book? And would someone's background, or lack thereof, in other fields of mathematics not have a bearing on that assessment?
Re: What Is Applied Category Theory? (2018)
#58Earlier quoted context omitted.
Not addressing your question, but the Yoneda Lemma is kind of a charlatan. On first reading, it seems magical and deep, but once you grok the proof, it feels like a relatively trivial observation. The whole thing is just about arrow composition! In a way, once you're on the other side, the Yoneda Lemma feels a bit like a checkpoint during the accimatization period where your brain gets used to thinking in categories…
> The whole thing is just about arrow composition! There's much much more to it. For example, a version of the yoneda lemma also holds for metric spaces (instead of a set of arrows between to things, you simply have a number indicating a distance between two things). Here's how I like to think about the yoneda lemma: If you have some kind of objects you want to talk about, one way to do this is by relating these obje…
That *is* a arrow. Arrow composition is adding up the distance along a path.
Re: What Is Applied Category Theory? (2018)
#59Earlier quoted context omitted.
Maybe for a value of "has applications" that doesn't necessarily include anyone actually applying it. As a generalisation, I don't think anyone by people into category theory actually cares about or applies category theory.
I'm applying it.
Anyone wanna talk about my applied category theory paper at the fall AAAI symposium? https://arxiv.org/abs/2011.11063
Re: What Is Applied Category Theory? (2018)
#60Earlier quoted context omitted.
I'm applying it.
Getting to feel a bit sour about how I get downvoted for talking about hacking on Hacker News and upvoted when I write about politics rubbish. Anyone wanna talk about my applied category theory paper at the fall AAAI symposium? https://arxiv.org/abs/2011.11063