Category Theory: An Introduction to Abstract Nonsense 

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

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

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 :)

I finally understand why category theory is nonsense. Thanks

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

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

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

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

I really loved the map of mathematics at the beginning

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 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.

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!

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

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!

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

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

It's very clear instructions, if you have some higher level maths

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

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

Great first video, excited to see more!

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

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

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

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.

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

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

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!

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

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 :)

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!

Well that went over my head.

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

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! 😃

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...

Love this video. Please make more! :)

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

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

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! 👍

Dense clear and concise. thank you.

Thank you, this is fun

Fantastic! I love this video!

"Abstract Nonsense" is fitting

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

Too much abstraction for today DX Thanks! good video

Excelent video!

One more great channel

Loved the map, it's full of detail.

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

Thank you so much!

Love how al-jibr is a caliphate 😅

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.

great video, please make more, just suscribed

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

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

Just brilliant.

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

(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.

Damn, you know you some math!

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?

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

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

Gracias 🙏

Very helpful video

Great sharing again sir.

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

Oh no! I learned something 🙈

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?

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! 😂😂😂

Fun! Clear and succinct.

Welldone to Category theory...the mathematics that is

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

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

I hope Asaf one day sees this video 😂

Succinct and clear. Very well done!

Please make more videos!!!

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

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.

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

You have to make more videos, pleasseeee

My brain hurts.

真棒ㄟ

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

Lovely video, It was a ton of fun!

9:57 bugged my mind

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?

I am an Electrical Engineer. I do not know why I am watching this video, but I am experimenting brain damage trying to understand this nonsense. Even do, I like it.

Are you planning on making more videos ?

Did anybody recognise the map from Zogg from Betelgeuce?

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

Nice! Okay now do addition and subtraction.

Very interesting intro. Category theory seem to generalize the idea of elements and relationships between them.

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

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.

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.

That's an excelelnt video! Thank you!

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

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

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