Intro to Category Theory

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Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we only briefly cover the Yoneda lemma in this presentation. Actually, we only briefly cover most of the content in here - I was intending this to be a 15 minute talk, so a lot of material is glossed over.
I'd link to my essay here, but this channel is too new and can't use external links. (also, it's not quite finished yet :c). I'll update the description here when I can.
I am aware I speak quickly - I kept it in mind when recording, but I'll try harder next time. The pacing of transitions is also a bit quick in certain places upon rewatching - I'll make sure to pause more. In the meantime, I have included subtitles which might be helpful if you prefer reading.
Graphics inspired by Oliver Lugg's 27 Unhelpful Facts about Category Theory: • 27 Unhelpful Facts Abo... .
Main reference during video creation was Basic Category Theory by Leinster and Category Theory in Context by Riehl. Examples of representable functors sourced from notes by Emanuele Dotto, PhD.
Timeline:
00:00 - Introduction
01:08 - Objects
01:40 - Morphisms
02:44 - Compositions
03:01 - Identity
03:22 - Associativity
03:30 - Examples of Categories
06:18 - Product and Dual Categories
07:12 - Duality
07:44 - Commutative Diagrams
08:17 - Isomorphism
09:02 - Functors
10:40 - Covariance and Contravariance
11:15 - Examples of Functors
13:25 - Natural Transformations
15:31 - Vertical Composition
16:53 - Functor Categories
17:18 - Natural Isomorphism
18:22 - Hom Functors
22:19 - Representables
22:40 - Examples of Representables
25:30 - Classifying Spaces
28:19 - The Yoneda Lemma
Errata:
11;13 - "...that a functor is [contravariant], than to...", not "covariant".
18;12 - "...corresponding [objects] are isomorphic...", not "morphisms".
21;57 - the upper string of mappings should be g mapsto hom(h,X)(g) = g o h mapsto hom(B,f)(g o h) = f o (g o h). That is, B and X are the wrong way around in the hom morphisms.
26;59 - "...between the [functions] 1 to R and...", not "functors". (Though, if we treat 1 as the trivial category, and R as a category under ordering, then this does hold for functors in the category of categories. But I really do just mean functions here.)
28;38 - cut audio, see transcript.
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31 июл 2024

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Комментарии : 92

@rafaelfreire3792 9 месяцев назад

It's the first time I watch a video in slower speed

@skillick Год назад

Thanks for making this! You can relax on the speed though, would be happy to see this at 50 minute length.

@brianhu6277 Год назад

I put the video at ×0.75 and it sounded perfect LOL

@Electronics4Guitar Год назад

I’ll say!

@harriehausenman8623 Год назад

@@brianhu6277 weird definition of perfect 😆

@pecfexfextus4437 Год назад

good video but you could speak with spaces

@alvargd6771 Год назад

HAVE SOME MERCY HUMAN you dont have to go so faaaaast

@tomholroyd7519 Год назад

I liked this video, after I set the playback speed to .75

@Revoker1221 Год назад

Thanks for the upload, I'd definitely be interested in any other introductory presentations or followup presentations you create, should you decide to. In the meantime, I'll be reading up on the reference(s) you left in the description.

@TreeLuvBurdpu 10 месяцев назад

One cannot talk faster than one can actually talk.

@lebong6664 Год назад

Nice video, but maybe try talking slower because some of us need a few seconds. Keep up the good work

@slowsatsuma3214 Год назад

It’s good at 0.75x speed

@derderrr7220 Год назад

this is utterly brilliant good work

@Nihil2407 3 месяца назад

I actually kind of like the fact that it's very quick paced! I mean, it's obviously hard to quickly understand what you're saying, but it would be too if you would talk slowly (I have some prior knowledge on categories, so it's fine for me). It also forces me to pay attention which... I would not do as much if it weren't presented that way, tbh I guess I'll rewatch this some time when it isn't 3am (great sleeping habits over here too)

@snk-js Год назад

I appreciate your knowledge transfer, but your speech is not so clean. Diction is something you may think to improve in your vids, despite of that, everything is good.

@brianhu6277 Год назад

Yooo awesome vid! Do a vid on basic and constructive logic? Never really understood them and their difference.

@TranquilSeaOfMath Год назад

Are you a fan of Prof. Michael Penn? Good presentation. Best wishes on your academic endeavors. Cheerful Calculations. 👨‍🏫

@jeffreyhowarth7850 7 месяцев назад

Great summary of cat theory.

@alejandromelo8245 10 месяцев назад

I dont know why the amounts of hate. Great video pal! Keep up the great work

@SM321_ Год назад

Very good video! Thank you! Are you planning to upload other videos? Perhaps something more advanced in category theory or homological algebra would be cool :))

@Desync.TheBigRee Год назад

I'd like to do more - maybe a quick follow up on the proof of naturality for Yoneda, then going into the Yoneda embedding and applications. Unfortunately, I'm taking a lot of credits this term, and unlike this main presentation, I'd find it hard to justify spending too much time on something that isn't coursework. Perhaps over summer, I can think of some ideas. As for more advanced topics, I think I need to learn more first! (I am just a student, and would hate to put up something incorrect.)

@outonado Месяц назад

Upload more videos, great channel.

@Vannishn Год назад

Great video, thank you so much ! Can’t wait for more ! I feel like at 22:00 there is a little typo in the upper diagram chasing, is that right ?

@Desync.TheBigRee Год назад

Ah, yes, the X and B in the hom morphisms are the wrong way around. Thanks!

@Anonymous-cw4yd Год назад

9:42 I think you didn't mean to write for all X ahead of F preserve composition.

@user-lu8vz9du3q 10 месяцев назад

Does anyone know which LaTeX package/which editing software was used to make this video? Would be helpful to me. Thanks in advance.

@acetylcholineuser4251 Год назад

A treasure was found

@oxy8821 Год назад

❤

@censoredamerican3331 Год назад

The sound is too quiet...your voice washes out.

@devnull5475 6 месяцев назад

This might have been pretty good. I have no idea. Could barely understand a word of fast mumbling.

@redhat4569 Год назад

how did you create this video. We need tutorials on this

@Damn_damn7 Год назад

Came to wrong place. I am leaving☠️

@brandonallen2301 7 месяцев назад

When you click fast forward on an enderpearl farm tutorial

@alileo1578 Год назад

I have a question about the last corollary: Is the equivalence on the right is in Set or some sort of "corresponding category" of the left equivalence?

@Desync.TheBigRee Год назад

hom(X,-) and hom(Y,-) are functors C -> Set, so they live in the functor category [C, Set], and the isomorphism is specifically a natural isomorphism. The dual corollary also holds with hom(-,X) and hom(-,Y) which are in [C^op, Set]. The left isomorphism is just an isomorphism of objects, so that's in C.

@TheoremsAndDreams 9 месяцев назад

I like it, but you sure are one fast-talkin’ city slicker.

@angelmendez-rivera351 11 месяцев назад

Also, another issue with your presentation is, you spoke of morphisms on singleton categories as being groups, but this is not necessarily the case. You can only conclude they are monoids. Later in the video, starting with 13:11, you made an analogous mistake, where you spoke of the group action on a set, even though you only have the monoid action on the set, and this monoid is not actually guaranteed to be a group. This is only the case when the categories you are working with are so-called "groupoids."

@davidmaaschdeyck 3 месяца назад

When you put the lecture at 2x speed

@angelmendez-rivera351 11 месяцев назад

It would have been helpful to explain more carefully the usage of "classes" in category theory. While proper classes do not exist in ZFC set theory, they are rigorously dealt with in NBG set theory, MK set theory, ARC set theory, and several other set theories.

@TheOneMaddin 10 месяцев назад

I think this would be a distraction from the topic. You don't need set theory to do category theory and you run into constant doubts only because you think too much about sets.

@angelmendez-rivera351 10 месяцев назад

@@TheOneMaddin *I think this would be a distraction from the topic.* This is an entirely baseless assertion. Mentioning the word "class" in an _introduction_ without explaining its contextual relevance to the topic being introduced is certainly far more of a distraction than merely explaining its relevance. *You don't need set theory to do category theory,...* Do not misrepresent my argument, please. I never claimed you need set theory to do category theory. Here is what I did claim: in light of the word "class" being mentioned in the video (and furthermore, in light of the distinction made from sets in the ZFC sense made in the video), the person writing the introduction has a responsibility to sufficiently explain this mention in the introduction. After all, as you yourself admit, mentioning classes is entirely unnecessary, and can lead to confusion. Therefore, one ought to actually explain their mention, if they are to be mentioned at all. Again, let me remind you: this is an _introduction._ No one who does not already know about the topic would actually know the relevance of classes to the topic, hence why an obligation exists to explain their relevance. *...and you run into constant doubts only because you think too much about sets.* This would literally be alleviated by the solution I proposed, since it would clarify the distinct between a set and a class.

@TheOneMaddin 10 месяцев назад

Actually, I think we agree for the most part. Sorry for misrepresenting you. I think the video should not have mentioned ZFC (and the difficulty with sets/classes) in the first place. In a way it struck the most unfortunate combination of distracting (by mentioning it) and confusion (by not explaining it). I do however think it would be safe to use the word "class" for ob(C) without the need to further elaborate. Since the target audience are laypeople they might not suspect a formal meaning to that term. He could have said "collection" instead of "class", but again, these terms may just be interchangeable for the audience. They are as likely to suspect a formal meaning of "collection" as for "class".

@Gabcikovo 10 месяцев назад

You speak so fast that I finally dont have to put a video on 1.75 speed

@JustNow42 7 месяцев назад

What language is it he is speaking? Serbocroatic?

@markusm2538 Год назад

Interesting topic. However, the pronunciation is inarticulate. Spent a lot of time rewinding.

@mikestrongine6111 Год назад

not everyone is from the US

@sharonlima8913 11 месяцев назад

Had to turn on subtitles for the first time.

@sharonlima8913 11 месяцев назад

@@mikestrongine6111no one's claimed so :)

@robertomunoz2155 10 месяцев назад

Yeah RIP non native English speakers

@derickd6150 9 месяцев назад

I could understand it fine

@matron9936 9 месяцев назад

Your definition of category at around 0:30 has composition backwards. In particular as written it doesn’t make sense.

@alejandroggzz8833 Год назад

How can i use this new matemátics in the study of conciusness?

@bullpup1337 9 месяцев назад

its not very new

@user-or5hk3dh9c Год назад

wat

@oumasstabdallah3424 4 месяца назад

Do you have any book SIR ?

@afavel 4 месяца назад

Keep.

@rainerausdemspring3584 Год назад

Erm - what is a class of objects?

@angelmendez-rivera351 11 месяцев назад

A class of objects is just, well, a class, of objects. Classes are the objects we study in set theory, although in ZFC set theory, we just call them sets, because all classes are sets in ZFC set theory. However, the conservative extension NBG set theory, which is finitely axiomatizable in first-order logic, has classes which are not sets. A set is a class which is a member of another class. For a foundational approach to set theory, which involves creating a set theory which can adequately found category theory, and thus, its classes, you should look at ARC set theory.

@rainerausdemspring3584 11 месяцев назад

@@angelmendez-rivera351 There are no classes in ZFC.

@angelmendez-rivera351 11 месяцев назад

@@rainerausdemspring3584 There are no *proper* classes in ZFC, but every set is a class, and there are indeed sets in ZFC, since there are no empty models of ZFC. As such, yes, there are classes in ZFC. Again, just not *proper* classes. There is a distinction.

@bullpup1337 9 месяцев назад

@@angelmendez-rivera351that is putting on your CT glasses when looking at ZFC. In the axioms of ZFC, there is no concept of classes

@angelmendez-rivera351 9 месяцев назад

@@bullpup1337 All sets are classes, by definition, so you are incorrect. In ZFC, there is no notion of *proper* classes.

@minimalisbirblog 8 месяцев назад

Hello, I would like to translate your video into Turkish and share it with your name. Would you allow this? We cannot find Turkish resources about the category and not everyone speaks English.

@Desync.TheBigRee 8 месяцев назад

Please go ahead! If you send me the translation, I'm also happy to add them as subtitles.

@minimalisbirblog 8 месяцев назад

@@Desync.TheBigRee Of course, I will translate it into Turkish and send it to you as soon as possible. Thank you for your contribution to science.

@LaureanoLuna 26 дней назад

Still wondering if diction is awful or language is not English.

@TheOneMaddin 10 месяцев назад

It is called "abstract nonsense" for a reason.

@bullpup1337 9 месяцев назад

yeah but not the one you think

@TheOneMaddin 9 месяцев назад

@@bullpup1337 Honest question. Why is it called this way?

@bullpup1337 9 месяцев назад

@@TheOneMaddin Norman Steenrod coined the term 'abstract nonsense' for some category theory concepts. It's a playful term; he didn't actually think category theory was nonsense. It's all about appreciating the beauty of abstraction!

@IsomerSoma Год назад

You should really try speaking more clearly and slower.

@feliperooartola9485 Год назад

mathematical essence is embedded to the bottom of nature

@sumdumbmick 8 месяцев назад

why are you working from anything built on ZF(C), though? at 90 seconds into the vid you invoke classes, rather than sets, because ZF(C), but ZF(C) doesn't recognize classes. and ultimately the distinction is stupid, since the classical meaning of a class is a set of sets, and they exist in order to avoid paradoxes regarding sets... that's like trying to make a glass of sea water less salty by adding salt to it, and then just having a cup of coffee instead, while proclaiming that it worked. what the hell are you doing? LoGiC aNd RiGoR!

@sumdumbmick 8 месяцев назад

if you're trying to follow the results of people like Bertrand Russell, you might note that he co-authored a publication claiming to contain a proof that 1+1=2 because he forgot that algebra exists: 1 dog +1 dog = 2 dogs; this is the only thing Russell thought was possible 1 dog +1 quail = 2 wings; right numerical result, but for the wrong reason 1 dog +1 quail = 6 legs; oops 1 half +1 third = 5 sixths; but you're still gonna try to explain how it's me that doesn't understand, right? 1 frog +1 pond = 1 pond; you're definitely losing this argument, man 1 C water +1 C dirt = some mud; this one doesn't even have a defined norm... c'mon, how'd you miss this? in truth, you can't add numbers at all. you can only add vectors. and the man who came up with the paradox for which 'class' is a solution, did not know this because he was a moron.

@jawad9757 7 месяцев назад

@@sumdumbmick did you forget to take your meds again

@harriehausenman8623 Год назад

Extremely slurry speech. at slower speeds it gets clear that lots of whole syllables are completely missing 😆

@ClumpypooCP 8 месяцев назад

Bro, slow down

@CardiganBear Год назад

Unintelligible diction.

@wdobni Год назад

i'm surprised that anybody takes this kind of stuff seriously .... as time marches on and more research happens in mathematics no doubt future systems of theories will be even far more dense, abstruse, and recondite .... is this really the sum and substance of education and higher learning? or just an advanced species of secret decoder ring meant to keep mischievous boys out of trouble

@Buddharta Год назад

This comment has just won the "Tell me you are very stupid with the most amount of words" award. To receive the 10 billion dollar award, please provide the 16 numbers of your credit card plus the three on the back and the expiration date.

@redpepper74 10 месяцев назад

Lol you’re very quick to judge based on how confusing something seems at first

@garethma7734 10 месяцев назад

@@Buddharta not understanding category theory doesn't make you "very stupid", but your comment does.

@bullpup1337 9 месяцев назад

this is the new foundation if maths

@epicmarschmallow5049 6 месяцев назад

@@garethma7734Whilst not understanding elementary category theory doesn't make you stupid by itself, dismissing all of modern mathematics out of hand because you don't understand it is enough for others to conclude that you're probably quite dim