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.
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)
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.
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 :))
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.)
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.
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.
@@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.
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".
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."
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.
The video roughly follows chapter 47 of my notes (desyncthethird.github.io/Reference.pdf ), and the sections from 48.2 onwards are also relevant but more terse. I can also highly recommend reading *Category Theory in Context* by Riehl, *Basic Category Theory* by Leinster, and *Category Theory* by Awodey as excellent references. (Awodey in particular covers a lot more material before getting to Yoneda, which may be helpful; while Leinster is probably the most gentle introduction out of the three; and Riehl is a classic textbook, with lots of great exercises.)
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?
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.
@@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!
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!
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.
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
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.
@@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
It's the first time I watch a video in slower speed
Thanks for making this! You can relax on the speed though, would be happy to see this at 50 minute length.
I put the video at ×0.75 and it sounded perfect LOL
I’ll say!
@@brianhu6277 weird definition of perfect 😆
I liked this video, after I set the playback speed to .75
HAVE SOME MERCY HUMAN
you dont have to go so faaaaast
good video but you could speak with spaces
One cannot talk faster than one can actually talk.
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.
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)
great video. people complaining about his talking speed, you ever think about complaining about your brain speed
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.
Nice video, but maybe try talking slower because some of us need a few seconds. Keep up the good work
It’s good at 0.75x speed
this is utterly brilliant good work
Great summary of cat theory.
Upload more videos, great channel.
Are you a fan of Prof. Michael Penn? Good presentation. Best wishes on your academic endeavors. Cheerful Calculations. 👨🏫
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 :))
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.)
I dont know why the amounts of hate. Great video pal! Keep up the great work
The sound is too quiet...your voice washes out.
A treasure was found
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 ?
Ah, yes, the X and B in the hom morphisms are the wrong way around. Thanks!
9:42 I think you didn't mean to write for all X ahead of F preserve composition.
Your definition of category at around 0:30 has composition backwards. In particular as written it doesn’t make sense.
Interesting topic. However, the pronunciation is inarticulate. Spent a lot of time rewinding.
not everyone is from the US
Had to turn on subtitles for the first time.
@@mikestrongine6111no one's claimed so :)
Yeah RIP non native English speakers
I could understand it fine
Yooo awesome vid! Do a vid on basic and constructive logic? Never really understood them and their difference.
I like it, but you sure are one fast-talkin’ city slicker.
Came to wrong place.
I am leaving☠️
This might have been pretty good. I have no idea. Could barely understand a word of fast mumbling.
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.
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.
@@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.
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".
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."
how did you create this video. We need tutorials on this
You speak so fast that I finally dont have to put a video on 1.75 speed
Does anyone know which LaTeX package/which editing software was used to make this video?
Would be helpful to me. Thanks in advance.
Do you have any book SIR ?
When you click fast forward on an enderpearl farm tutorial
What language is it he is speaking? Serbocroatic?
When you put the lecture at 2x speed
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.
Please go ahead!
If you send me the translation, I'm also happy to add them as subtitles.
@@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.
do you have notes, or source code for the video that we can refer to as well?
The video roughly follows chapter 47 of my notes (desyncthethird.github.io/Reference.pdf ), and the sections from 48.2 onwards are also relevant but more terse. I can also highly recommend reading *Category Theory in Context* by Riehl, *Basic Category Theory* by Leinster, and *Category Theory* by Awodey as excellent references. (Awodey in particular covers a lot more material before getting to Yoneda, which may be helpful; while Leinster is probably the most gentle introduction out of the three; and Riehl is a classic textbook, with lots of great exercises.)
@@Desync.TheBigRee thank you. Really appreciate all the resources you shared as well :)
❤
Still wondering if diction is awful or language is not English.
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?
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.
wat
You should really try speaking more clearly and slower.
Keep.
It is called "abstract nonsense" for a reason.
yeah but not the one you think
@@bullpup1337 Honest question. Why is it called this way?
@@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!
How can i use this new matemátics in the study of conciusness?
its not very new
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!
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.
@@sumdumbmick did you forget to take your meds again
Extremely slurry speech. at slower speeds it gets clear that lots of whole syllables are completely missing 😆
mathematical essence is embedded to the bottom of nature
Bro, slow down
Unintelligible diction.
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
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.
Lol you’re very quick to judge based on how confusing something seems at first
@@Buddharta not understanding category theory doesn't make you "very stupid", but your comment does.
this is the new foundation if maths
@@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