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>other than boxes being connected by arrows Well, that's because that's basically it, lol. But I get your sentiment and, sure, there's a lot of things where t

>other than boxes being connected by arrows Well, that's because that's basically it, lol. But I get your sentiment and, sure, there's a lot of things where this could be applied and that hasn't happened already. It's definitely a field with a prolific future, though. I've personally applied many ideas I got from there into my smallish ventures with good results.

whatshisface 4y ago
But that's not "basically it," there are lots of high-flying theorems that make it an entire area of mathematics.
- moralestapia 4y ago
  Well, then I have no idea why you asked your previous question. ¯\_(ツ)_/¯
  - whatshisface 4y ago
    Because applied category theory stops at translating everything into boxes and arrows. There's no "applied Yoneda lemma."
    - lgdw 4y ago
      It's the actual process of translating everything into boxes and arrows that's the core of ACT.
    - larve 4y ago
      I'm not sure that's true. This is a surface / introductory article, but if you go deeper into say, categorical databases, modeling of attacks (https://arxiv.org/pdf/2103.00044.pdf https://arxiv.org/pdf/2103.00044.pdf), there sure is some wilder CT at work. I'm leaving out programming as applied category theory because there is no shortage of refined concepts there (type theory, all the haskell libraries, many things I don't even comprehend). But to me, the value in ACT is actually learning how "simple" some of the other fields could be, because it allows me to communicate with non-software engineers, and actually have them contribute to software. I did a lot of work with state machines as composable abstractions for concurrency and flow control, and mechanical engineers were able to find some really subtle say, race conditions, just by pointing out "hey, isn't there an arrow from this to this, if the limit switch catches early?"
    - vladf 4y ago
      lol, you picked the one thing category theory actually has examples of not being abstract nonsense for https://mathoverflow.net/questions/12511/what-is-yonedas-lemma-a-generalization-of https://mathoverflow.net/questions/12511/what-is-yonedas-lem...
    - hackandthink 4y ago
      In computing you can apply Yoneda for optimizations. "Each of the steps is fairly compelling, except perhaps the second one, which rests on the Yoneda Lemma" see "Kan Extensions for Program Optimisation" https://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf https://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf
dqpb 4y ago
So, if I already model systems via boxes and arrows, and use things like petri nets, then am I good, or do I still need to learn category theory?
- larve 4y ago
  If that works for you, that’s of course perfect. I find that the study of CT pretty has very much clarified that a lot of diagrams I see in other domains (business processes, supply chain modelling) directly relate to my understand of diagrams. I think it is a property of human cognition that if we draw an arrow from a to b, and one from b to c, that there is some kind of relation between a to c as well. What this means concretely is that if I can draw my problem, in my domain, as a diagram with boxes and arrows, there is a reasonable chance that you can actually understand it, and actually contribute to it. Similarly, seeing diagrams from your problem domain is often sometimes I can relate to my design patterns and software architecture foundations, and I might be able to ask quite pointed questions when trying to say, build a mathematical model or build some supply chain software. Questions like “ok, but, is there an arrow from here to here?” and the answer “oh yes, that happens when we actually replenish the inventory by doing X, I forgot about that part”. Another thing I found people do with CT is that they use it as a toolbox of abstractions that they can try to fit to a problem. Say, “oh, is there maybe a functor from petri nets to databases? ok, there is, but it’s kind of useless. Does this formulation actually work as a monad?” This is useful for finding elegant APIs to problems, say things like optics (lenses, etc…).
- agumonkey 4y ago
  I never ran into petri net in jobs, I always thought they were still stuck at educational use. Do you get good results using them ?
YetAnotherNick 4y ago
Well, not exactly. Things till functions and functor are part of almost every big enough theory like set theory or even just natural numbers and used everywhere, and if arrows are only used for describing that, I wouldn't say it is application of category theory. Only point of category theory starts when you go higher than that and deal with more abstract and generalizable things e.g monads(which again is used in other theories), bigger infinities, ordering of infinities etc.

whatshisface 4y ago

But that's not "basically it," there are lots of high-flying theorems that make it an entire area of mathematics.

moralestapia 4y ago
Well, then I have no idea why you asked your previous question. ¯\_(ツ)_/¯
- whatshisface 4y ago
  Because applied category theory stops at translating everything into boxes and arrows. There's no "applied Yoneda lemma."
  - lgdw 4y ago
    It's the actual process of translating everything into boxes and arrows that's the core of ACT.
  - larve 4y ago
    I'm not sure that's true. This is a surface / introductory article, but if you go deeper into say, categorical databases, modeling of attacks (https://arxiv.org/pdf/2103.00044.pdf https://arxiv.org/pdf/2103.00044.pdf), there sure is some wilder CT at work. I'm leaving out programming as applied category theory because there is no shortage of refined concepts there (type theory, all the haskell libraries, many things I don't even comprehend). But to me, the value in ACT is actually learning how "simple" some of the other fields could be, because it allows me to communicate with non-software engineers, and actually have them contribute to software. I did a lot of work with state machines as composable abstractions for concurrency and flow control, and mechanical engineers were able to find some really subtle say, race conditions, just by pointing out "hey, isn't there an arrow from this to this, if the limit switch catches early?"
  - vladf 4y ago
    lol, you picked the one thing category theory actually has examples of not being abstract nonsense for https://mathoverflow.net/questions/12511/what-is-yonedas-lemma-a-generalization-of https://mathoverflow.net/questions/12511/what-is-yonedas-lem...
  - hackandthink 4y ago
    In computing you can apply Yoneda for optimizations. "Each of the steps is fairly compelling, except perhaps the second one, which rests on the Yoneda Lemma" see "Kan Extensions for Program Optimisation" https://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf https://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf

dqpb 4y ago

So, if I already model systems via boxes and arrows, and use things like petri nets, then am I good, or do I still need to learn category theory?

larve 4y ago
If that works for you, that’s of course perfect. I find that the study of CT pretty has very much clarified that a lot of diagrams I see in other domains (business processes, supply chain modelling) directly relate to my understand of diagrams. I think it is a property of human cognition that if we draw an arrow from a to b, and one from b to c, that there is some kind of relation between a to c as well. What this means concretely is that if I can draw my problem, in my domain, as a diagram with boxes and arrows, there is a reasonable chance that you can actually understand it, and actually contribute to it. Similarly, seeing diagrams from your problem domain is often sometimes I can relate to my design patterns and software architecture foundations, and I might be able to ask quite pointed questions when trying to say, build a mathematical model or build some supply chain software. Questions like “ok, but, is there an arrow from here to here?” and the answer “oh yes, that happens when we actually replenish the inventory by doing X, I forgot about that part”. Another thing I found people do with CT is that they use it as a toolbox of abstractions that they can try to fit to a problem. Say, “oh, is there maybe a functor from petri nets to databases? ok, there is, but it’s kind of useless. Does this formulation actually work as a monad?” This is useful for finding elegant APIs to problems, say things like optics (lenses, etc…).
agumonkey 4y ago
I never ran into petri net in jobs, I always thought they were still stuck at educational use. Do you get good results using them ?

YetAnotherNick 4y ago

Well, not exactly. Things till functions and functor are part of almost every big enough theory like set theory or even just natural numbers and used everywhere, and if arrows are only used for describing that, I wouldn't say it is application of category theory. Only point of category theory starts when you go higher than that and deal with more abstract and generalizable things e.g monads(which again is used in other theories), bigger infinities, ordering of infinities etc.