Category Theory: An Introduction to Abstract Nonsense
ฝัง
- เผยแพร่เมื่อ 25 ก.ค. 2024
- Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
8:17 Universal Properties
11:57 Proof using Category Theory
13:27 Shortcomings of Category Theory
Please make more high level math videos. They are in demand! Your content is amazing!
Read the subtitle: "Abstract Nonsense".
@@gwh0 what do you mean?
@@TepsiMorphic he means nonsense
@@pedrogouveia4326 oh really? Thank you! I didn't know that.
I mean what does he mean with the abstract nonsense. Is he calling category theory abstract nonsense and that's why we don't see more of it? Because this is kind of ignorant.
@@TepsiMorphic no it is not nonsense, what Pedro says is nonsense. "Labeling an argument "abstract nonsense" is usually not intended to be derogatory,[1][2] and is instead used jokingly,[3] in a self-deprecating way,[4] affectionately,[5] or even as a compliment to the generality of the argument."
I finally understand why category theory is nonsense. Thanks
It's not just nonsense, it's abstract so it's abstract nonsense.
That is why we love category theory!
This is an incredible video. I am a programmer and I fell into a category theory rabbit hole a while back when trying to fully grok what a monad is. I get that now, but I got bit by the category theory bug.
It should be no surprise that, as a programmer, I just enjoy learning about a combination of novel ways to abstract and build the most powerful building blocks - so every problem becomes a familiar or common one. Category theory is just… fun to have as a mental model. I’ve really enjoyed a series by Bartosz Milewski, but it helps to take a step back every once in a while. Also it helps to have the ability to give an intro to this stuff without sharing a 10 hour playlist.
This is the most coherent overall explanation I have seen to date, and I think only a small percent of the content went over my head (my background in math mostly ends after college level calculus/discrete math/some linear algebra, and then a TON of pop-math youtube videos: 3b1b, mathologer, reducible, numberphile, etc.).
Wanted to say - thanks for creating this! I enjoyed it a lot :)
3B1B Is it a good channel?
@@cristianolopes37503blue1brown is a great youtube channel
Dude same! I’m reading (or at least trying to) a book called Category Theory for Programmers and now I got delusional and wanna do pure maths lol
@@unfulfilling_entertainer lmao what is it about this stuff? same effin' thing happened to me - never gave much of a damn about math outside of "what immediate practical goal is served by learning this?"...
...and now I'm all like "hey maybe I should go back and major in pure abstraction!"
That is a nice reference referring to the domain of algebras as the "caliphate". Al Juarismi would be proud.
💯
9:40 correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:54 There's another version of this map that I saw which reflects that very idea. It replaces the "coast of category theory" with the "coast of universal algebra", and promotes category theory to a "moon", with the intent that the moon creates the waves in the ocean of logic
"And here's the chalkboard. This is the ladel by which we drink from the fountain of knowledge."
- Spongebob
this is the best introductory video on CT in youtube, it goes a little deeper still being fun to watch
I understood next to nothing, but this was a fun and entertaining to watch.
I can clearly see you put a lot of effort in this group, and the result can be seen (tho I couldn't understand everything, given my layman's math background). Great work, thanks!
I love how you've been able to do such a comprehensive video on category theory with so much content. Such an awesome job. Fantastic!
I am currently doing my masters in pure math and I have a bunch of algebraic topology etc under my belt and I have to say. I love that this was high level and at the same time approachable. Would love a deeper dive into more Category Theory or matter of fact any other higher level concepts in the future!
Thank you for time and energy spending to explain complicated ideas in a simple languave.
Wow - from basic definitions to functors and natural transformations and on to infinity categories in less than 10 minutes! Quite remarkable. Really hope you choose to make more videos, because this was one of the best introductions to CT that I have come across!
I really loved the map of mathematics at the beginning
It's very clear instructions, if you have some higher level maths
This is the most underrated channel I've seen for a while. Keep up the great work!
Very fundamental , great Motivations
Extremely well simplified.
Worth keeping the work up for sure
Great first video, excited to see more!
Just wanting to let you know that you did an amazing job making this video! Thank you!
This is such an incredible and well made video! Thank you for a wonderful introduction
Great video! The theme style is also very compatible with maths. I hope you'll make more videos.
Fantastic video, I love it! It’s very fun to watch but still contains a lot of information. Well done!
as a developer and after watching this and other videos about category theory many times, I just end up in a conclusion that this is just consistent and composable way of programming and defining entities that could be used to create strongly-typed-well-defined systems
a monad is a monoid in the category of endofunctors
exactly so. Once you realize that programs are proofs of the theorems inherent in the types of the functions that define them, the role of category theory is almost obvious. The interesting thing is that programming is almost entirely concerned with constructive proofs, but as Feynman's Chicken mentions, category theoretical proofs tend to be non-constructive. That probably says something about which programs can be written leading usto another version of Goedel's incompleteness, I'm sure
@@kumoyuki what do you mean by new versions of Goedel's incompleteness? I understand this as to: the translation between non-constructive typing to constructive and thus applied to a Turing machine, like if it is something that went out from the deepest darkness of abstraction realm to the highest enlightened surface of practical applicability. I think Goedel's debate is trivial for this debate unless you can explain yourself, because before we even touch on the problems of mathematics itself we must think on the law of excluded middle or the axiom of choice on these translations.
@@snk-js I was hardly being rigorous, but if you will take that as a given. Goedel's incompleteness is fundamentally a problem with self-reference - it is echoed in lots of places from Quine's paradox to the halting problem, including numerous other sub-problems along the way (IIRC, Chaitin discovered a variation regarding the construction of the irrational numbers). Category theory provides a deep mechanism of self-reference for all of mathematics, and computing is sufficiently general that it has similarly self-referential power. Interestingly, type systems generally need to place some restrictions on self-reference and/or provide operators for some of the most common self-referencing operations (the Y combinator &cet)> It seems relatively obvious to me that a constructive proof (e.g. a program) of a category-theoretic version of Quine's paradox ought to be possible to construct, leading to yet another incompleteness theorem.
Of course, none of this has had any rigor applied to it, we're just having a bit of fun in the comments of a YT video. But it seems likely to me. YMMV, of course
Thanks, probably this is the best video in wich can undestand category rheory
I'm trying to awlf-study this stuff so that I can, later, try to apply it to cliodynamics, behavioral economics, quantitative finance, and economic anthropology. But, so far, I've found Category Theory HARD. Your video helped me understand it better. I hope that you keep making these sorts of videos.
is a new way of speaking mathematics in high level, humanity needed this for sake
This is the best video I have ever seen on mathematics. Absolutely gorgeous. If i had money I wouldve comtributed
Ah, yes. Very interesting. I understood some of those words.
The caliphate of algebra.... beautiful piece of rhetoric, particularly given the historical genesis of algebra :)
0:50 this is my way of approaching the world. I am a top down guy. This is why I appreciate this clip as an introduction. I get the high level stuff and could just drill into the details.
My natural inclination is to sort and sieve details to get the generalizations!
😃
So obviously I am right now putting in the details into this generalization!
Nice! I'm so happy to see this subject grow in popularity.
Love this video. Please make more! :)
Everytime I feel somewhat self confident I like to watch videos like these to feel dumb again
Fantastic! I love this video!
Dense clear and concise. thank you.
Thank you, this is fun
Well that went over my head.
Mathematics is the art of abstraction. Category theory is merely abstraction about abstraction.
I need more videos from this channel, this content is amazing
Profound! And some reflections: ...
abstract algebra by nature is -emm- abstract so easily lends itself to transformations by "changing the labels of like things"?
algebraists look for standards between mathematical things?
Mathematical things depend a lot on labels applied to them and so remain consistent when labels are changed?
analysts on the other hand seem to be drawn to things of inconsistency especially when analysis gives explanation for the inconsistencies? And at this point attract research funding because pointwise events do not readily lend themselves to generalities until those generalities are identified or reduced by a new analytical definition of some sort
I agree there is a horribly magnificent something living in math that seems to show a rainbow effect: the closer one approaches the more rapidly the rainbow disappears or re-locates
Q: Is Category Theory the way to go?
A: hell yeah! It seems a very good way to go
Excellent work!
Excelent video!
Hope this doesn't come true, but the idea that you'd "make only ONE awesome video on the channel and choose not to elaborate on it" would be sour because you have such awesome potential for 1M+ subs!
While a screwdriver is a very useful tool to carry but not useful as a sandwich maker, a pocketknife can be used as both a screwdriver and a sandwich maker. Therefore...
Thank you so much!
6:36 up to this point it actually was a very good explanation.
Quite a bit easier than reading it in formal language in any language, but especially challenging in French language!
👍
I definitely like the top-level view but I think some videos showing examples of some of this stuff would be helpful.
Too much abstraction for today DX
Thanks! good video
Great video. The pictures/slides clearly and correctly state a lot of interesting material.
Great sharing again sir.
One more great channel
Very helpful video
great video, please make more, just suscribed
Wow. I can comprehend the words, but their meanings? That's another story.
Loved the map, it's full of detail.
Knowing nothing about category theory, I found this rather more useful than other videos I have seen on the topic.
The idea of a functor makes me wonder if we can use its functorial properties to "transfer" a proof about objects in one category to objects in another e.g. maybe we can prove something about groups then use the fact that a functor exists from the Grp to Rng categories to immediately show that an analogous proof works for rings? Or am I hopelessly confused?
In part of category theory initially came from homotopy theory. In homotopy theory you construct groups on a topological space. The simplest example of a homotoy group, is the set of continuous curves that begin and end a fix point on the space. You say two curves are identical if they can be be continuously deformed into each other.
You add curves by running along one and then running along the other curve. You make the inverse of a curve by rouning around the curve in the other direction. These cirves form a group. You can show that if you have a continuous function between topological spaces you have a homomorphism between their homotopy groups. SO you can prove that a continuous function between two space cannot exist if you can show that any of their homotopy groups are not homomorphic.
Love how al-jibr is a caliphate 😅
"Abstract Nonsense" is fitting
Gracias 🙏
Did you make the map! :O can we get prints of it??
Just brilliant.
I understood nothing and I loved it. I need to learn this now. Amazing video.
Very interesting! Especially the last slide... I wonder... Does every mathematician has to understand category theory?
Wow.
I haven't looked at Category Theory for decades. I actually felt nostalgic.
Thank you.
I got into a mix of programing and teaching, and never applied category theory outside of class.
Would you let me know
which program did you use for the slide,
where did you get the skin?
Those are my things!
Logic is relationships that always replicate, regardless of what they're used for, regardless if they're precise enough to do math on.
Math is a sub-set of logic dealing exclusively with relationships of quantity, regardless of what it's applied to.
I understood a fair bit of this but there were bits where I got lost. I need to study some of the underlying math for some of these areas.
Did anybody recognise the map from Zogg from Betelgeuce?
I like category theory but I'm always annoyed that it doesn't have a built-in way to talk about multivalued or partial functions. For this reason my favorite (bi-)category is (Sets, Relations, Inclusions). It has a number of lovely properties: for instance, its monads are transitive relations, and its adjunctions are functions. Sadly I don't know of any elementary introductions to it, maybe I should write one.
(Commenting here instead of by email *for the algorithm*)
Great explanation! This would be a perfect resource to send to students who want a first primer on category theory. Some thoughts/reactions:
1) Duck good
2) I find it interesting that you describe a lot of category theory proofs as non-constructive, since in my area category-theoretic approaches are most popular among constructivists. Perhaps the difference is that if you're working at the foundational level, then high-level properties (like the functorial property you mentioned) become constructive - they have proofs, and you use these properties as lemmas
3) Though the limitation you cited isn't necessarily wrong (categories are complicated and you don't always need complicated things), my more fundamental criticism would tie back to the "universal properties" section. Category theory reflects the same formalist underpinnings as logic, which is intimately tied to the formalist/structuralist schools of philsophy (the Modernists that post-modernism rebelled against). Those schools of philosophies draw universal conclusions about humanity, which are consistently the conclusions of the most privileged class, used to erase the rest of us. Though I wouldn't jump to immediately calling category theory a colonizer, the limitations of universalism show up clearly in both cases: so much of what makes each topic interesting is its particulars, which do not fall out of the universals.
Duck good! Yeah most of the proofs I encountered (in algebra) are non-constructive but I'm definitely interested in exploring the constructive ones. And although I did see lots of discussions on philosophy & category theory, I haven't seen the criticisms on universality yet - I wonder if there is a post-modern interpretation on category theory : )
Re. #3 What on earth?
@@a-guess-at-the-riddle postmodernist social theory claims everything is a power game, including math. The claim is wrong, but it’s influencing thought greatly.
@@abj136 "The claim is wrong" as asserted by someone who doesn't know either mathematics or social theory
@@abj136 that’s.. true. but their critique of universalism isn’t wrong for those reasons per se. although so many of these tendencies: critiquing universalism, foundationalism, and the general postmodernist preoccupation with power (or “the will to power”) as an ideal category with no actual basis (I know you mathematicians like ideals and categories but here this is a pejorative) come straight from Nietzsche. A grade-A piece of shit
真棒ㄟ
3:20 ".. Isomorphism, which is exactly what you think it is.."
If I didn't know exactly what it was, I don't think I'd have any understanding of it. Is a bijection exactly what you'd think it is?
Please make more videos!!!
Sees map at 0:00 - ah another MMM enjoyer I see. It is a very good map.
You have to make more videos, pleasseeee
how do category theory & grp theory relate to each other?
Damn, you know you some math!
Fun! Clear and succinct.
Oh no! I learned something 🙈
12:20: how can GL_n(R) have a fundamental group at all as it is not even path connected ? Shouldn't it be over C?
Nice! Okay now do addition and subtraction.
Succinct and clear. Very well done!
Feynmann’s Chicken? Schrödinger’s cat? We need more Mathematical Animal Mascots!
Lovely video, It was a ton of fun!
14:44 agree on the screwdriver 😂😂😂
That also means that you should not spend too much time figuring out how to make a screwdriver…
Especially if you are making a sandwich 🥪 😂😂😂
Thank you for for that one 😂😂😂
Next time I do my screwdriver analogy!
😂😂😂
Need to be x3 the length. What's the rush ?
Hello, does anyone have a step-by-step reference to understand the group object?
There’s lots of papers on the internet Google group theory
nice one feynman's chicken. lets see if you can come up with a functor from feynman's diagrams to pure math.
I'd love to see a cat theory description of the construction of the complex numbers from the reals using x^2 + 1 = 0
Welldone to Category theory...the mathematics that is
Are you planning on making more videos ?
Yes! More videos are on the way : )
@@feynmanschicken great ! Thank you :)
“Composition makes sense for morphisms, that is ….” Is that not an automatic consequence of the definition of f o g ?
9:57 bugged my mind
I need that map please
Very interesting intro. Category theory seem to generalize the idea of elements and relationships between them.
That's an excelelnt video! Thank you!
So category theory is papyrus and group theory is Helvetica. Thanks
This is a wonderful video
My brain hurts.
I see now why category theorists are hated so much, thank you!
Any recommended books for category theory
There are quite a few good ones. If you want an easy to understand book with applications, I recommend "Category Theory for the Scienes." -by David Spivak
"No no good god don't read Category Theory for Scientists, Spivak isn't finished with it, it's full of buggy bits, and probably will do more to confuse than enlighten.
I would recommend you to start with Conceptual Mathematics: A first introduction to categories by Lawvere and Schanuel."