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Category Theory Illustrated – Natural Transformations
- larodi 1y agothis has surfaced HN top at least 5 times https://hn.algolia.com/?dateRange=all&page=0&prefix=true&query=abuseofnotation%20illustrated&sort=byPopularity&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
- phoronixrly 1y agoIt has been updated as per the author's Mastodon https://mathstodon.xyz/@abuseofnotation/115298450513159834 https://mathstodon.xyz/@abuseofnotation/115298450513159834
- deleted 1y ago[deleted]
- auggierose 1y agoThere are too many pictures in this for my taste. I am currently reading this one, and I like it better so far: https://doi.org/10.1142/13670 https://doi.org/10.1142/13670
- IdontKnowRust 1y agoIt's author's intention, since the title explicitly says "Illustrated" =)
- auggierose 1y agoFair enough.
- hamburgererror 1y agoWhat's the thing with category theory? I see this topic discussed quite frequently here but I don't get it why people are so into it
- jiggunjer 1y agoLets software designers use fancy words and ask for a raise.
- monarchwadia 1y agoFor me.. It's a very useful mental model for thinking about architecture & logic
- hamburgererror 1y agoWhy not simply use UML?
- ctenb 1y agoExamples? I haven't really seen many applications of CT, even though I looked for them since I find the idea of CT interesting
- moralestapia 1y agoYou have a function that does A() and another function that does B(). Upon careful inspection or after just writing/using them 10,000s of times[1] you realize they are both special cases of one general function f()[2]. Congrats, you're likely doing CT now, but barely scratching the surface, though. Let's say you find a way to do a function factory that generates explicit instances of f() -> A() and f() -> B() at runtime for your different use cases as they are needed. You do this 100 times, 1,000 times[1] with many different functions, in many different contexts. You eventually realize that if all your functions and their signatures had the same structure[3] it would be quite easy to mix some (or all?) of them with each other, allowing you to handle a perhaps infinite amount of complexity in a way that's very clean to conceptualize and visualize. Isn't this just FP? Yes, they're very intimately related. By this point you're 99.9999% doing CT now, but remember to shower regularly, touch grass etc. CT formalized these structures with mathematical language, and it turns out that this line of thinking is very useful in many fields like ours (CS), Math, Physics, etc. 1. Which is what happened to me. 2. Which sometimes is a way more elegant and simple solution. 3. This term is fundamental and has way more meaning than what I could write here and what one would think on a first approach to it.
- sesm 1y ago> In the course of this book, we learned that programming/computer science is the study of the category of types in programming languages. This is a golden quote.
- ctenb 1y agoIt's also wrong, since computer science is traditionally mostly about computation, which has nothing to do with CT
- Twey 1y agoInsofar as ‘computation’ is about mapping one state or value to another state or value, it has a lot to do with CT! The question of whether CT is _useful_ for studying computation is different, and there are certainly other lenses you can see computation through that some people would argue are better. But it's hard to deny that they're _related_.
- ctenb 1y agoI mean, technically almost all of math can be related to other math one way or another. To say the CT has a lot to do with computation is definitely a stretch. CT is not a recognized Computer Science subject. It's mostly used in the functional programming community to name certain concepts and theorems, but then applied to a specific type system (so it's not actually doing CT, since your restricting yourself to a single category, whereas CT is really about connecting different categories by generalizing over them).
- gnarlouse 1y agoWhen I read this title I thought it was going to be rings and groups in bikinis. I’m so dumb.
- michaelcampbell 1y agoHad to reduce the page to 67% to get out of "Fisher Price" font size, but otherwise quite interesting.
- YetAnotherNick 1y agoIt makes it so much more complicated than what is needed to understand natural transformation. Natural transformation is just mapping between two functors. You can discover the laws yourself just from this.
- gcr 1y agoAnyone who likes this might also like Stefan Miller’s paper, “a simple category theoretical understanding of category theory diagrams“, appearing in SIGBOVIK 2014. See https://sigbovik.org/2014/proceedings.pdf https://sigbovik.org/2014/proceedings.pdf (starts on PDF page 65, or page 57 if you go by margin page numbers)
- anal_reactor 1y agoI hate this particular mix of prose and formalism. Too complicated to be pop-sci, too informal to be, well, formal. I got to this part: > We know that two orders are isomorphic if there are two functors, such that going from one to the other and back again leads you to the same object. And I have no clue what is a functor, nor order. "Functor" wasn't defined, and "order" is defined as "thin category", which in turn remains undefined. Seems to me like in order to understand this text you already need to understand category theory. If that's the case, then why would you be reading it?
- gs17 1y agoIt's the newest chapter of a book, the previous one defines functors.
- mjh2539 1y agoI agree. There was (and still is) a trend in technical writing that began in the 2010s to be overly pedestrian and informal (and in many cases explicitly vulgar). The same impulse or geist resulted in many people naming their library or product a cute or contrived or irrelevant word. Get off my lawn! And start giving things long descriptive names that are aliased to acronyms again! db2, netcat, socat, emacs (editing macros), wget...etc.
- mrkeen 1y ago> And I have no clue what is a functor, nor order. If you press the Prev button at the top of the page it takes you back to Functors. Twice more and it will take you back to Orders.
- anal_reactor 1y agoThen why link to the middle of the whole thing, instead of the beginning?
- emorning4 1y ago>>Whatever is is, and what is not cannot be<< Well, that's not true. Particles, and thus facts, pop into and out of existence all the time.
- rck 1y agoThis is fun. But the bit at the beginning about philosophy is not correct. Parmenides did not believe in what we would call essences, but really did believe that nothing ever changes (along with his fellow Eliatic philosopher Zeno, of paradox fame). The idea that change is an illusion is pretty silly, and so Plato and especially Aristotle worked out what's wrong with that and proposed the idea of _forms_ in part to account for the nature of change. Aristotle extended Plato's idea and grounded it in material reality which we observe via the senses, and that's where the concept of essence really comes from - "essence" comes from the Latin "essentia" which was coined to deal with the tricky Greek οὐσία (ousia - "being") that Aristotle uses in his discussions of change.
- griffzhowl 1y agoOne way I've seen it presented is that the early Greek philosophers were grappling with how to reconcile two basic facts: somethings stay the same (constancy or regularity), and some things change. Heraclitus was before Parmenides and said that everything changes. Parmenides said that nothing changes, and then the atomists, most prominently Democritus, synthesised these two points of view by saying that there are atoms which don't change, but all apparent change is explained by the relative motions of the different basic atoms. Plato was influenced by all of these. But I would say the theory of forms accounts more for constancy or regularity more than change, no? Btw, the central concept of Parmenides' philosophy is always translated as "Being", but I couldn't find the original Greek word. It isn't "ousia"?
- rck 1y agoI'm not sure what motivated Parmenides because he was more of a poet than anything - it just happened that his poetry was what we would now recognize as incredibly philosophical. He didn't really argue, he just wrote down what the "goddess" told him. But I think the basic problem is that everyone back then agreed that you can't get "something from nothing," and it sure seems like change requires being to come from non-being. The statue is there now, but before it was cast there wasn't a statue, just a chunk of bronze. If being can't come from non-being, how do you account for the "coming-to-be" of the statue? The Eliatic position as I understand it is that the change is just an illusion. Plato and Aristotle both react against this position and argue that it's silly (I'm very inclined to agree). They then give alternative accounts of what change really is. I'm not sure about Plato, but the Aristotelian analysis is something like this: every thing that exists has the potential to exist in certain ways and not others, and it's said that the thing is "in potency" to exist in those potential ways. When something could exist in a certain way but right now doesn't, that's called a "privation." And the ways that the thing currently does exist are the "form" of the thing. So a substance changes when it goes from being in potency to being actual, and it does that by losing a privation. Aquinas follows Aristotle in giving the example: "For example, when a statue is made from bronze, the bronze which is in potency to the form of the statue is the matter; the shapeless or undisposed something is the privation; and the shape because of which it is called a statue is the form." Incidentally, Aquinas's short On the Principles of Nature (https://aquinas.cc/la/en/~DePrinNat https://aquinas.cc/la/en/~DePrinNat) is a good overview of this theory, which is spread all over Aristotle (in the Categories, the Physics, and the Metaphysics). As far as οὐσία is concerned, I think this is the complete Greek for Parmenides's poem: http://philoctetes.free.fr/parmenidesunicode.htm http://philoctetes.free.fr/parmenidesunicode.htm. In the places where that translation uses "being" you get slightly different words like γενέσθαι (to come into a new state of being) or εἶναι (just the infinitive "to be"). And looking at the definition of οὐσία (https://lsj.gr/wiki/%CE%BF%E1%BD%90%CF%83%CE%AF%CE%B1 https://lsj.gr/wiki/%CE%BF%E1%BD%90%CF%83%CE%AF%CE%B1) it looks like most of the uses of that term specifically come well after Parmenides.
- w10-1 1y agoI like it when teachers (e.g., Grant Sanderson) are careful to explain when they are trying to convey an intuition to motivate and guide some complex math, because it orients you without tangling you in all the misunderstanding that would come from extending analogies or cross-cultural/discipline comparisons too far. But when authors start slinging around Plato and Aristotle and especially Parmenides willy-nilly alongside modern principles, they're waving a red flag... Don't get me started!
- mallowdram 1y agoIsomorphism invariance applies to neural assemblies or syntax, not to mere symbols. The problem in math is it models. Brains do not model. Heraclitus was right if math never enters the picture to add its arbitrariness. "A man in the night kindles a light for himself when his sight is extinguished; living he is in contact with the dead when asleep, when awake he is in touch with the sleeper."
- intalentive 1y agoInteresting aside about the Vienna circle and isomorphism. I suspect that’s where Hayek got his idea that mind and representation are isomorphic, echoing Aristotle’s assertion in “On the Soul” / De Anima that the mind becomes the object of perception.
- measurablefunc 1y agoThe natural transformation α : F ⇒ G is not specified properly b/c when expressed in compositional form you also have to specify the subscript for the natural transformation & it is an equality instead of an isomorphism, i.e. if f : a → e then αₑ ∘ Ff = Gf ∘ αₐ. There are highter categories where what he has written down can make sense but in the context of the current exposition it is not correct as written.
- ibobev 1y agoThe author is Jencel P.? I saved this book sometime ago under the author name Boris Marinov? Is this the same person now writing under a different pen name?
- larodi 1y agoIs the same. He seems to obscure himself déliberately.
- VirusNewbie 1y agoThe author uses adjoint functors to explain equivalence and naturality but doesn't actually call it that?
- lostbean 1y agoI’m quite a visual learner, so I appreciate you sharing this.