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What I like about your videos is you always start from somewhere simple where I can definitely understand, and you gradually ramp up without loosing me, and by the end you are summoning demons and somehow I understand it.
0:27 "Introduction to That Thing, but only for people who already know it" As someone who tried (key word: tried) to read Mac Lane's text on category theory, that hits close to home.
Mac Lane's is more a reference than learning material. I've heard much better things about Riehl's Categories in Context if you're still trying to learn it.
idk, I tried to read Category theory in context just before my first year of graduate school with absolutely no prior experience, and I thought it was too hard. It was only once I had gotten into differential geometry and started studying homological algebra that any of it started to make sense. I can't imagine the material in that book (especially the first few chapters) can be very enlightening if the reader hasn't already seen any interesting examples where they can apply what they're learning
@@cybergoth2002 "The Joy of Abstraction" by Eugenia Cheng is a pretty decent category theory intro for those with less mathematical background, I find. It's not that lightweight - it goes up to the Yoneda lemma with proofs along the way - but it is designed for a less-technical audience.
While Mac Lane can be a primary text, it can be more than little difficult (and occasionally outdated) so I don't really recommend it unless you've had a lot of other math prior to it. Riehl's Category Theory in Context is great. I also recommend Awodey's Category Theory, which is basically a "rewrite" of Mac Lane's in more elementary & modern terms. I've heard Leinster's Basic Category Theory is pretty nice too. Finally, if you're interested in the more applied side of things, Fong & Spivak's Seven Sketches in Compositionality is rather different in flavor and totally accessible to beginners.
This is the best explanation of limits I have ever seen, thank you for making this. You motivated all of the abstract nonsense so well that it doesn't feel like nonsense anymore! The bit at the end about how limits are just sequences of things with conditions was also really illuminating.
I'm absolutely delighted to say that I got many of the jokes. A few months ago I wouldn't have been able to tell where the math stopped and the abstract nonsense started. Good stuff. Thanks for sharing.
commutative diagrams are needed for every introduction to categorical limits! though I will say I’m impressed that (g+)+ explained the idea of limits without universal properties
This video was the best so far, in my opinion! Thanks to you I'm begging to appreciate for the first time how category theory works... Great videos, great explanations, thank you again!!
0/10 video did not construct the Bolzano notion of a limit by checking that a certain natural transformation is a limit in the category of filters on the real numbers (nice video, I enjoyed it).
This is a masterpiece and a pleasure to watch, especially as I'm going through Riehl's book and trying to reason more categorically in everyday life 14:39 i would love a video about \infty-categories, perhaps "What does it mean for two things to be equal" and the thumbnail is an interrogative equal sign between two similar emojis and philosphers faces in the background, or as an alternative something misleading about "higher dimensions" with 4-dimensional geometric clickbait
There are similarities (for instance, how often the word `projection` comes up), but there's a small difference: I easily understand & digest Acerola content. :')
The reason I opened this video is to check my understanding on limits after graduating from university. And the reason I finished this video is I understand none of it but enjoy the show
@@SheafificationOfG if you're getting ad revenue than most probably not. i know having a low bit rate in a video deem it to be low quality so the "holy algorithm" tends to push less the video, but I'm not sure if audio also has the same effect. Although personally speaking I feel very pleased hearing the fan noise in background. It's quite soothing tbh.
the puns are strong with this one Is there a name for the property that you can calculate limits with a single object? And does it apply to topological or metric spaces? If not, those could've made for a better example than chain complexes (and if it does, I'll be surprised and understand your choice of example a bit better).
That the limits are "calculated" using a single object is a bit disingenuous on my part. That being said, an object S is usually called a "separator" if mapping out of it can distinguish arrows of your category, which is *kinda* like being able to use a single object to identify the "underlying set" of any object of your category. The category of topological spaces does have a separator (the singleton), as does the category of metric spaces (the singleton, again), so the underlying set of the limit of any diagram in either category can be determined by probing with the singleton. However, it takes a bit more thinking to recover the rest of the structure. For example, in the category of topological spaces, knowing 1. the underlying set of the limit, and 2. that the projection maps are continuous is not sufficient to determine the topological structure of the limit, as there may be several spaces satisfying both conditions (e.g., the discrete space on the set will always satisfy these conditions). However, it's pretty much enough: the correct topology is the coarsest topology that makes the projection maps continuous. The story is similar for metric spaces (the metric in the limit is the supremum of the metrics of the components), except now it's possible that limits don't exist. [As mentioned in the video, the category of chain complexes also has a separator, but this is like choosing the set of natural numbers as the separator for Set.]
@@SheafificationOfG it being the coarsest is of course the natural result, since all the other candidates have to map into it, so that could've been an entry point to the general concept, but it also feels somewhat contrived, so not much better than what you settled on imo. OTOH, Topology and Contrivances, name a better duo. To be slightly clearer on my wording, by "calculate with a single object", I meant "fully determined by probing against a single object, making all the others redundant", just for the record.
@@decare696 I think some formalisation of your second paragraph is asking for an object S such that Hom(S, -) is conservative. In this case, I can at least say that any category C monadic over Set has a canonical choice of such an object, since the forgetful functor C -> Set is conservative, and it is corepresentable by the free object generated by the singleton set (i.e., U = Hom(S, -), where S = F(*)). This covers Vect_R (giving us R again), as well as other varieties of algebras (groups, rings, etc.).
Thanks a lot for this rather easy introduction to limits in category theory! Having no education at all in category theory but a master's degree in functional analysis, and a teaching degree at undergraduate level, I had to rewatch some bits of the video until I got the meaning though.
Hey G, thanks so much for making these videos on category theory. Have you considered making a series similar to what Bartosz Milewski did 7 or so years ago where he worked on a whiteboard to help programmers understand category theory a bit better. I really wish he would have continued this series or at least did a refresh on it because this stuff is super interesting to programmers like myself.
I've never actually looked at Milewski's work (though I've heard of it). I'll probably stick to something close to my current style, but you can at least expect more category theory flavoured videos in the future!
that point about the empty set is interesting. on the one hand, of course, it's true that all ZF sets are well-founded. on the other hand, though, you can't really express in ZFC something like "the set of all sets that contain themselves," because obstructing such comprehensions is the reason ZFC exists in the first place. I think if you let yourself talk about classes too, you can prove that the class of sets containing themselves must be a set (of course, the empty set). just to say you need more than ZFC to express that such a set really exists in ZFC, I think
These are like MLG montages for math nerds who drink loose-leaf tea instead of COD players who drink mountain dew. Trying to realize the treasure planet triangle meme at 2.5x speed is the true zoomer experience
Good category theory can only come from having a solid background in more "real" fields of maths (like in abstract algebra or related fields), so I def agree that you should start with group theory!
Replacing the singleton * with the 1-dimensional vector space over R: Why R, rather than an arbitrary field k (with characteristic other than 2, of course...) of course replacing 1 with the unit of k ? Love your spoof of mathematical astronautics (ascending to greater and greater heights of abstraction until you run out of oxygen)!
I chose the category of vector spaces over R to keep it real 😉. If vector spaces were allowed to be over arbitrary fields, then idk what reasonable maps between vector spaces of mixed characteristics would be (and then I would probably have to kiss arbitrary products goodbye). In the category Vect_k, for *any* field k, I could've replaced the singleton with k. This works even if k has characteristic 2/
Yeah but the picture is at least here to see in its completeness all the time. With words, on the other hand, you end up forgetting how it began by the time you reach the end. Or at least that's what I have, my short-term memory is abysmal. Maybe it's just that the video is so good, but I thought that I was really getting it with the help of the diagrams (they were kind of tame in comparison to more complex ones, however - but those would be exponentially more difficult to put into words as well)
You become more grateful for diagrams when you start dealing with many more than just 3-4 objects and arrows between them. For instance, a lot of homological algebra would be a huge pain to parse. Also, it's much more difficult to be comprehensive about relationships between compositions of arrows using words, but it's all there in a diagram.
@@SheafificationOfG Please never remove the fan noise, that's the funniest part of the videos(I don't understand most of the category theory jokes though)
I'm not a fan of all the meme's. Call me a boomer if you want(I'm not). It's distracting IMO. Maybe one odd one here and there but not a constant barage.
Thank you Brilliant for sponsoring!
To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/GSheaf/ . You’ll also get 20% off an annual premium subscription!
Hi! The 30 day trial period doesn't seem to work. The discount is applied as expected though
@@aleksanderadamczyk4191 wdym? you didn't get contents? or system just skips free 30 days?
this is the first time i see a math video and think “this title is actually morally bad”. and i love how diabolical this is.
If you play the video backwards the title changes from ’introduction to limits’ to ‘introduction to colimits’ & the head of every arrow becomes a tail
going into calc this year, this video has been a great help
With limits done, you're well on your way to learn how to derive funct....ors!
@@SheafificationOfG pls make a video about deriving functors
you should *totally* make a video about deriving functors, it would be a wonderful (kan) extension to this one
I seriously doubt that, lmao
hi
What I like about your videos is you always start from somewhere simple where I can definitely understand, and you gradually ramp up without loosing me, and by the end you are summoning demons and somehow I understand it.
0:27 "Introduction to That Thing, but only for people who already know it"
As someone who tried (key word: tried) to read Mac Lane's text on category theory, that hits close to home.
Mac Lane's is more a reference than learning material. I've heard much better things about Riehl's Categories in Context if you're still trying to learn it.
I had the same experience with Mac Lane's book lol. I'd also recommend Category Theory in Context, it's really good.
idk, I tried to read Category theory in context just before my first year of graduate school with absolutely no prior experience, and I thought it was too hard. It was only once I had gotten into differential geometry and started studying homological algebra that any of it started to make sense. I can't imagine the material in that book (especially the first few chapters) can be very enlightening if the reader hasn't already seen any interesting examples where they can apply what they're learning
@@cybergoth2002 "The Joy of Abstraction" by Eugenia Cheng is a pretty decent category theory intro for those with less mathematical background, I find. It's not that lightweight - it goes up to the Yoneda lemma with proofs along the way - but it is designed for a less-technical audience.
While Mac Lane can be a primary text, it can be more than little difficult (and occasionally outdated) so I don't really recommend it unless you've had a lot of other math prior to it. Riehl's Category Theory in Context is great. I also recommend Awodey's Category Theory, which is basically a "rewrite" of Mac Lane's in more elementary & modern terms. I've heard Leinster's Basic Category Theory is pretty nice too. Finally, if you're interested in the more applied side of things, Fong & Spivak's Seven Sketches in Compositionality is rather different in flavor and totally accessible to beginners.
As someone with a bachelor in math, I understood about 1/3 of this, but your charm made it very enjoyable
bro embodies chaotic good
I think this is now officially my favorite channel :D
This is the best explanation of limits I have ever seen, thank you for making this. You motivated all of the abstract nonsense so well that it doesn't feel like nonsense anymore! The bit at the end about how limits are just sequences of things with conditions was also really illuminating.
I'm absolutely delighted to say that I got many of the jokes. A few months ago I wouldn't have been able to tell where the math stopped and the abstract nonsense started.
Good stuff. Thanks for sharing.
the math never stopped :p
@@schweinmachtbree1013 So you're saying the math is unlimited?! ... ;)
abstract nonsense is maths smh
I'm not proud to say that I almost thought this was an unironic video until you started bringing out the commutative diagrams
commutative diagrams are needed for every introduction to categorical limits! though I will say I’m impressed that (g+)+ explained the idea of limits without universal properties
This video was the best so far, in my opinion! Thanks to you I'm begging to appreciate for the first time how category theory works... Great videos, great explanations, thank you again!!
Glad you liked it!
0/10 video did not construct the Bolzano notion of a limit by checking that a certain natural transformation is a limit in the category of filters on the real numbers (nice video, I enjoyed it).
This is a masterpiece and a pleasure to watch, especially as I'm going through Riehl's book and trying to reason more categorically in everyday life
14:39 i would love a video about \infty-categories, perhaps "What does it mean for two things to be equal" and the thumbnail is an interrogative equal sign between two similar emojis and philosphers faces in the background, or as an alternative something misleading about "higher dimensions" with 4-dimensional geometric clickbait
"everyday life" lol.
Whoops, accidentally starting to understand category theory thanks to your videos. (I was saving it for later)
I need to sit down and learn some category theory so that I can get all the jokes.
wow, that opening was brilliant!
This is my new favorite TH-cam channel
I am at that point of my academic career where once I see math that I have not seen before, I should be genuinely worried. I love category theory lmao
I didn't understand this at all, great video👍. /pos
I'm also at my limit!!! 💪💪💪
sheafification video when
He'll have to make it twice though
Finally, Acerolla for math.
There are similarities (for instance, how often the word `projection` comes up), but there's a small difference:
I easily understand & digest Acerola content. :')
The reason I opened this video is to check my understanding on limits after graduating from university. And the reason I finished this video is I understand none of it but enjoy the show
yours is the only channel that doesn't come to my feed still i intentionally search it up regularly just so i don't miss a video.
Damn, TH-cam out there shadowbanning me or smt?
@@SheafificationOfG if you're getting ad revenue than most probably not. i know having a low bit rate in a video deem it to be low quality so the "holy algorithm" tends to push less the video, but I'm not sure if audio also has the same effect. Although personally speaking I feel very pleased hearing the fan noise in background. It's quite soothing tbh.
the puns are strong with this one
Is there a name for the property that you can calculate limits with a single object? And does it apply to topological or metric spaces? If not, those could've made for a better example than chain complexes (and if it does, I'll be surprised and understand your choice of example a bit better).
That the limits are "calculated" using a single object is a bit disingenuous on my part.
That being said, an object S is usually called a "separator" if mapping out of it can distinguish arrows of your category, which is *kinda* like being able to use a single object to identify the "underlying set" of any object of your category.
The category of topological spaces does have a separator (the singleton), as does the category of metric spaces (the singleton, again), so the underlying set of the limit of any diagram in either category can be determined by probing with the singleton.
However, it takes a bit more thinking to recover the rest of the structure. For example, in the category of topological spaces, knowing
1. the underlying set of the limit, and
2. that the projection maps are continuous
is not sufficient to determine the topological structure of the limit, as there may be several spaces satisfying both conditions (e.g., the discrete space on the set will always satisfy these conditions). However, it's pretty much enough: the correct topology is the coarsest topology that makes the projection maps continuous.
The story is similar for metric spaces (the metric in the limit is the supremum of the metrics of the components), except now it's possible that limits don't exist.
[As mentioned in the video, the category of chain complexes also has a separator, but this is like choosing the set of natural numbers as the separator for Set.]
@@SheafificationOfG it being the coarsest is of course the natural result, since all the other candidates have to map into it, so that could've been an entry point to the general concept, but it also feels somewhat contrived, so not much better than what you settled on imo.
OTOH, Topology and Contrivances, name a better duo.
To be slightly clearer on my wording, by "calculate with a single object", I meant "fully determined by probing against a single object, making all the others redundant", just for the record.
@@decare696 I think some formalisation of your second paragraph is asking for an object S such that Hom(S, -) is conservative. In this case, I can at least say that any category C monadic over Set has a canonical choice of such an object, since the forgetful functor C -> Set is conservative, and it is corepresentable by the free object generated by the singleton set (i.e., U = Hom(S, -), where S = F(*)).
This covers Vect_R (giving us R again), as well as other varieties of algebras (groups, rings, etc.).
Thanks a lot for this rather easy introduction to limits in category theory! Having no education at all in category theory but a master's degree in functional analysis, and a teaching degree at undergraduate level, I had to rewatch some bits of the video until I got the meaning though.
I'm glad you find it so helpful!
Hey G, thanks so much for making these videos on category theory. Have you considered making a series similar to what Bartosz Milewski did 7 or so years ago where he worked on a whiteboard to help programmers understand category theory a bit better. I really wish he would have continued this series or at least did a refresh on it because this stuff is super interesting to programmers like myself.
I've never actually looked at Milewski's work (though I've heard of it). I'll probably stick to something close to my current style, but you can at least expect more category theory flavoured videos in the future!
Incredible title
This was truly our Ran Academy
Lan Academy when?
Watch this one at -1x speed in the meantime!
that point about the empty set is interesting. on the one hand, of course, it's true that all ZF sets are well-founded. on the other hand, though, you can't really express in ZFC something like "the set of all sets that contain themselves," because obstructing such comprehensions is the reason ZFC exists in the first place. I think if you let yourself talk about classes too, you can prove that the class of sets containing themselves must be a set (of course, the empty set). just to say you need more than ZFC to express that such a set really exists in ZFC, I think
this is so cool! really motivates to study it properly. but why are they called limits tho?
I might be trippin', but are the compositions at 17:28 reversed? Are you using a different convention? I could also just be sleepy, it's 4:00 AM here.
Nah you're right
The diagram went through some changes that were not reflected in the compositions
@SheafificationOfG Thanks for the clarification 👍, really good video 👌
I had to step away once I realized this wasn't going to be calculus, you brilliant prankster
These are like MLG montages for math nerds who drink loose-leaf tea instead of COD players who drink mountain dew.
Trying to realize the treasure planet triangle meme at 2.5x speed is the true zoomer experience
Wife left you? Doesn't matter, Sheafification of G just dropped a video!
more category theory or homological algebra pls!
I don't know anything about category theory. Is there a book or introduction I ought to refer to? Should I learn group theory first?
Good category theory can only come from having a solid background in more "real" fields of maths (like in abstract algebra or related fields), so I def agree that you should start with group theory!
And if you want to start learning abstract algebra then I advise you to Take a look at Herstein's "Topics in algebra".
...I was really hoping for Kan extensions
Where is the gsheaf new mic fund?
Please cover filters next
Colimit video next?
Ah yes always a good appendix for scheme theory video
Replacing the singleton * with the 1-dimensional vector space over R: Why R, rather than an arbitrary field k (with characteristic other than 2, of course...) of course replacing 1 with the unit of k ?
Love your spoof of mathematical astronautics (ascending to greater and greater heights of abstraction until you run out of oxygen)!
I chose the category of vector spaces over R to keep it real 😉. If vector spaces were allowed to be over arbitrary fields, then idk what reasonable maps between vector spaces of mixed characteristics would be (and then I would probably have to kiss arbitrary products goodbye).
In the category Vect_k, for *any* field k, I could've replaced the singleton with k. This works even if k has characteristic 2/
thanks
I've never seen a category arrow diagram that was clearer and more concise than saying what you want to convey with words 🤷🏽♂️
Yeah but the picture is at least here to see in its completeness all the time. With words, on the other hand, you end up forgetting how it began by the time you reach the end.
Or at least that's what I have, my short-term memory is abysmal. Maybe it's just that the video is so good, but I thought that I was really getting it with the help of the diagrams (they were kind of tame in comparison to more complex ones, however - but those would be exponentially more difficult to put into words as well)
You become more grateful for diagrams when you start dealing with many more than just 3-4 objects and arrows between them. For instance, a lot of homological algebra would be a huge pain to parse.
Also, it's much more difficult to be comprehensive about relationships between compositions of arrows using words, but it's all there in a diagram.
Coming next: Kan't Academy
where we discuss the Spectral theorem for Commutative C* algebras
You can escape limits; it is called non-standard analysis. Also, limits weren't scary at all; they were super easy, barely an inconvenience
I like your videos but please apply basic noise removal to them. You can get Audacity for free and remove noise that way.
You sure noise removal won't just cut out my voice?
(jk, I'll give it a go next video)
@@SheafificationOfGoh I thought that was actually a part of the gig, perhaps even the only fan of yours itself blowing in the background
@pmmeurcatpics the fan was definitely part of the gig originally, but it's because my mic kept picking it up haha
@@SheafificationOfG Please never remove the fan noise, that's the funniest part of the videos(I don't understand most of the category theory jokes though)
"will always satisfy a weak terminal object"... WILL ALWAYS BE THERE IN THE END!!
That was great. Now do it again, slower and in more detail. Assume that I am ignorant and dim, but persistent.
this seems like a really good video but for future reference you lost me at 3:48 ish
someday i will understand these videos
woah 😳
Kan academy love it!
Let's make it big together (with a discord?)
23:41 in fact you need products of zero elements too.
Zero is pretty finite :)
limitless puns i’m dying
These stuff are trivial compared to what I want to study, and I'm scared. 😢
😊
Woah there buddy, thats a bit much. I'm dead.
Ah yes, category theory, where the "concrete example" is still two-tier abstract.
Sets seem pretty concrete (they're definitionally concrete, in fact!)
I got confused between nu and v lol. I paused for so long...
v_ν
so glad i left pure math
I'm not!! :(
@@SheafificationOfG :(, maybe you'll eventually see things differently
cardinality
is this real math?
hi :3 UwU
I don’t understand this
better than wodzicki
Kan't Academy
😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂
so many puns i dont understand
Have you ever considered that you have a negative impact on mathematics youtube?
I'm not a fan of all the meme's. Call me a boomer if you want(I'm not). It's distracting IMO. Maybe one odd one here and there but not a constant barage.