Category Theory: An Introduction to Abstract Nonsense

[@arisweedler4703]

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 :)

[@cristianolopes3750]

3B1B Is it a good channel?

[@aug3842]

@@cristianolopes37503blue1brown is a great youtube channel

[@unfulfilling_entertainer]

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

[@Kveldred]

​@@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!"

[@FamilyYoutubeTV-x6d]

@@Kveldred I think there is something about finding abstract relations between things ("the black box"), and, the hope or desire or wish or thought or expectation that, with sufficient efforts, we and only we, may be able to learn what these relations are, and use them for our benefit. It comes from an interesting place of an interface between ego and abstraction and curiosity. That's how I ultimately realized where my passion for these abstract topics came from. Interestingly, once you get a basic grasp of these subjects or topics, problems in your own domain of expertise may become more tractable, interestingly without even relying on category theory, which is where the black box paradox hits back, which is somewhat mystical and amazing altogether. Did that make sense? :p

[@TepsiMorphic]

Please make more high level math videos. They are in demand! Your content is amazing!

[@gwh0]

Read the subtitle: "Abstract Nonsense".

[@TepsiMorphic]

@@gwh0 what do you mean?

[@pedrogouveia4326]

@@TepsiMorphic he means nonsense

[@TepsiMorphic]

@@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.

[@thevolatilitysmile]

​@@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."

[@nestorv7627]

I finally understand why category theory is nonsense. Thanks

[@thadtheman3751]

It's not just nonsense, it's abstract so it's abstract nonsense.

[@UdacznikDuchowy]

That is why we love category theory!

[@evan7391]

It's non-sense

[@mr.champion7304]

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

[@OnTheThirdDay]

"And here's the chalkboard. This is the ladel by which we drink from the fountain of knowledge." - Spongebob

[@feynmanschicken]

9:40 correction: Universal Property of Quotients requires ker(f) to contain ker(pi)

[@stv3qbhxjnmmqbw835]

But can you answer, "why did the chicken cross the road?" ?

[@notsojharedtroll23]

That is a nice reference referring to the domain of algebras as the "caliphate". Al Juarismi would be proud.

[@authenticallysuperficial9874]

💯

[@Ali-l7m6k]

This video changed my academic path.

[@raka9403]

this is the best introductory video on CT in youtube, it goes a little deeper still being fun to watch

[@MultiAblee]

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!

[@Quince751]

Thank you for time and energy spending to explain complicated ideas in a simple languave.

[@bissbort]

This is the most underrated channel I've seen for a while. Keep up the great work!

[@snk-js]

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

[@liquidcode1704]

a monad is a monoid in the category of endofunctors

[@kumoyuki]

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

[@snk-js]

​@@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.

[@kumoyuki]

@@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

[@TopeshMitter]

​@@snk-js are you an Indian??

[@alessandrobertulli425]

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!

[@LolSumor]

I understood next to nothing, but this was a fun and entertaining to watch.

[@madlarch]

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!

[@Hans_Magnusson]

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! 😃

[@Hans_Magnusson]

So obviously I am right now putting in the details into this generalization!

[@snk-js]

is a new way of speaking mathematics in high level, humanity needed this for sake

[@rommalb]

I absolutely loved this introduction! You explained it really well, great job!

[@universallanguageproject]

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!

[@raheem2845]

Very fundamental , great Motivations Extremely well simplified. Worth keeping the work up for sure

[@Duskull666]

I really loved the map of mathematics at the beginning

[@haseebasif100]

This is the best video I have ever seen on mathematics. Absolutely gorgeous. If i had money I wouldve comtributed

[@SimGunther]

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!

[@shoam2103]

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?

[@domenickriggio684]

Great first video, excited to see more!

[@frankg7786]

Everytime I feel somewhat self confident I like to watch videos like these to feel dumb again

[@meccamiles7816]

The caliphate of algebra.... beautiful piece of rhetoric, particularly given the historical genesis of algebra :)

[@Alan-zf2tt]

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!

[@prajitrajadhikari9548]

Love the storytelling themic explanation!

[@briannewman9285]

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.

[@laurenaeva1184]

This is such an incredible and well made video! Thank you for a wonderful introduction

[@zermelos]

Just wanting to let you know that you did an amazing job making this video! Thank you!

[@qftymasfisicateorica.2811]

Thanks, probably this is the best video in wich can undestand category rheory

[@xyzct]

Mathematics is the art of abstraction. Category theory is merely abstraction about abstraction.

[@eternaldoorman5228]

10:10 A category theorist is someone who can say "such that the diagram cmmutes" and seriously believe that this does not need any further explanation whatsoever.

[@Hans_Magnusson]

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! 👍

[@antondovydaitis2261]

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.

[@harelkariv1477]

I need more videos from this channel, this content is amazing

[@jpphoton]

wow but definitely LOVE you for the succint based no fluff synopsis. damn.

[@JayVal90]

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...

[@havenbastion]

Logic is relationships that always replicate, regardless of what they're used for, regardless if they're precise enough to do math on.

[@havenbastion]

Math is a sub-set of logic dealing exclusively with relationships of quantity, regardless of what it's applied to.

[@michaelkeys4303]

Thanks for this superb overview. Do you have the time or energy to upload any other great material? I would be very excited to watch any intricate mathematical material that you wish to explain or explore. Cheers 👍👍👍 Tony from Zurich

[@leftfootfirstpolitics]

Ah, yes. Very interesting. I understood some of those words.

[@keithpeterson4005]

Nice! I'm so happy to see this subject grow in popularity.

[@burgerxxl6115]

Fantastic video, I love it! It’s very fun to watch but still contains a lot of information. Well done!

[@tyalsvarrr]

Great video! The theme style is also very compatible with maths. I hope you'll make more videos.

[@scollyer.tuition]

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?

[@thadtheman3751]

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.

[@subhrayanbarman1654]

Well that went over my head.

[@imnimbusy2885]

Feynmann’s Chicken? Schrödinger’s cat? We need more Mathematical Animal Mascots!

[@bengobler]

Fantastic! I love this video!

[@axelperezmachado3500]

I understood nothing and I loved it. I need to learn this now. Amazing video.

[@jacobparasite]

Did you make the map! :O can we get prints of it??

[@justanormalyoutubeuser3868]

Loved the map, it's full of detail.

[@CamEron-nj5qy]

"Abstract Nonsense" is fitting

[@Hans_Magnusson]

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! 😂😂😂

[@av3stube480]

Wow. I can comprehend the words, but their meanings? That's another story.

[@stevenwilson5556]

I definitely like the top-level view but I think some videos showing examples of some of this stuff would be helpful.

[@mikeCavalle]

Dense clear and concise. thank you.

[@phenixorbitall3917]

Very interesting! Especially the last slide... I wonder... Does every mathematician has to understand category theory?

[@rosebohrer4899]

(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.

[@feynmanschicken]

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 : )

[@a-guess-at-the-riddle]

Re. #3 What on earth?

[@abj136]

@@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.

[@decaydjk8922]

@@abj136 "The claim is wrong" as asserted by someone who doesn't know either mathematics or social theory

[@aaAa-vq1bd]

@@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

[@vinvic1578]

Are you planning on making more videos ?

[@feynmanschicken]

Yes! More videos are on the way : )

[@vinvic1578]

@@feynmanschicken great ! Thank you :)

[@ericzenk4404]

Great video. The pictures/slides clearly and correctly state a lot of interesting material.

[@dagadagahonchannel]

Great sharing again sir.

[@Kveldrunari]

Love this video. Please make more! :)

[@authenticallysuperficial9874]

Love how al-jibr is a caliphate 😅

[@beatn2473]

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?

[@arsicjovan9171]

Did anybody recognise the map from Zogg from Betelgeuce?

[@orangeninjaMR]

It's not non-constructive, it's just less explicit. "Non-constructive" has a specific meaning in the foundations of math, referring to principles which provide objects whose specific values cannot be determined (such that those objects may fail to exist in settings where those principles are not valid); examples of these are proofs by contradiction or using the axiom of choice. This is different from abstracting away irrelevant details; the proof that the fundamental group of a topological group is abelian doesn't rely on any non-constructive principle.

[@kumoyuki]

I confess that I took the assertion about non-constructive proofs at face value, assuming that category theoretical proofs would mostly be non-constructive seems natural to me given the very limited structure available. Full disclosure: my PhD thesis foundered on a categorical proof that was obvious to me but non-obvious to my adviser, so I may not be the right person to come to such a conclusion

[@MrV_El_Domador_De_Capibaras]

Too much abstraction for today DX Thanks! good video

[@gregoryigbanoi8628]

Welldone to Category theory...the mathematics that is

[@WayOfTheCode]

One more great channel

[@CovidIslandDiscs]

My brain hurts.

[@theproofessayist8441]

Sees map at 0:00 - ah another MMM enjoyer I see. It is a very good map.

[@moonshine7753]

I see now why category theorists are hated so much, thank you!

[@strangeWaters]

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.

[@felooosailing957]

great video, please make more, just suscribed

[@scythe2591]

Where can we find this first illustration (math-world map)

[@viacheslavprokopev8192]

Thank you, this is fun

[@abhiramdeva1417]

nice one feynman's chicken. lets see if you can come up with a functor from feynman's diagrams to pure math.

[@stevenwilson5556]

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.

[@alextrebek5237]

Please make more videos!!!

[@tridivsharma2342]

You have to make more videos, pleasseeee

[@miketurn5]

Damn, you know you some math!

[@ranro7371]

Need to be x3 the length. What's the rush ?

[@yash1152]

how do category theory & grp theory relate to each other?

[@Monothefox]

Playing off the playful title, I now understand why Cantor went insane.

[@tomholroyd7519]

I'd love to see a cat theory description of the construction of the complex numbers from the reals using x^2 + 1 = 0

[@ravigautam8685]

Very helpful video

[@vf1941]

Excelent video!

[@tantzer6113]

“Composition makes sense for morphisms, that is ….” Is that not an automatic consequence of the definition of f o g ?

[@WWLinkMasterX]

5:19 - You meant to say "functors" instead of "morphisms," right?

[@aladdinovich]

It is a nonsense to listen to such videos at 2 am

[@diskgrinder]

So category theory is papyrus and group theory is Helvetica. Thanks

[@jimf2525]

Simon and Garfunkel came to mind: slow down you move too fast. I two minutes in there were so many new terms that I gave up. I would like to learn more about it, so my advice is to pace yourself.

[@kinesissado9636]

I hope Asaf one day sees this video 😂

[@HelloForeignWorld]

Oh no! I learned something 🙈

[@melomateus_m.r]

What a lovely world we live in, where people think youtube videos of all things can disprove shit.

[@ewwseww]

Would you let me know which program did you use for the slide, where did you get the skin? Those are my things!

[@brunaschulz_]

Hello, does anyone have a step-by-step reference to understand the group object?

[@Hans_Magnusson]

There’s lots of papers on the internet Google group theory

[@Ragnarok540]

Please make more videos. I bet you could explain what is a Monad.