G++, I have had a really, really rocky relationship with category theory. But now that I'm decaying through scheme theory, I need to get much more comfortable with it, and I cannot overstate how useful these videos have been. Even your other videos have revolutionized how I think, and just given me a lot of education. In other words, thank you so much, and please don't stop.
impressive very nice... now lets see the homotopy colimits.... look at that subtle geometric realisation... the tasteful topology of it... my god... it even has an adjoint
Going to give myself a pat on the back for seeing the matrix generators and changing my guess to infinite elements because, as I thought, in formal terms, "that 1 in the anti-diagonal looks like it is going places."
in my experience, one trouble about learning/teaching category theory is that it's a joint generalization of the theory of monoids and the theory of posets and lattices, both of which people often don't know too much about. (co)limits are much less scary if you already know about meets and joins, (co)presheaves are much less scary if you already know about left and right modules/actions of a monoid, etc etc.
3:15 I feel like colimits are much easier than limits at least in the special case of direct limits. Just in general monomorphisms seem easier to me than epimorphisms and profinite groups are much more intuitive than homology stuff. direct limit >> projective/inverse limit
In a way! The contravariant powerset functor is essentially its own adjoint, so maps X -> P(Y) naturally correspond to maps Y -> P(X). Taking Y to be the singleton, we see that maps X -> P(*) correspond to elements of P(X). (There may be some circularity in this argument.)
Is the fact that Omega = P(*) related to the table displayed at 10:08? Applying P to * -> X gives P(X) -> Omega instead of X -> Omega, so it is not immediately obvious to me whether or not Omega = P(*) is a coincidence. Does anyone have any useful insights on this?
Not sure if this would tickle your fancy, but here's one connection. The functor P : Set^op -> Set is basically its own adjoint, which means that functions X -> P(Y) correspond to functions Y -> P(X) in a natural way. In particular, subsets of X are elements of P(X), and therefore functions * -> P(X), which then naturally correspond to functions X -> P(*) = Omega.
Hijacking this comment section to ask you to use dark mode for code snippets for my poor eyes, languages like scala rust and gleam would be better fit than c++ imo also
Utterly incomprehensible, thank you
you mean mprehensible
Commenting after 16 minutes so it’s plausible I finished the video even though in reality I just started it
wait a second, has the youtube algorithm duped me into sitting through an entire semester of category theory?
G++, I have had a really, really rocky relationship with category theory. But now that I'm decaying through scheme theory, I need to get much more comfortable with it, and I cannot overstate how useful these videos have been. Even your other videos have revolutionized how I think, and just given me a lot of education. In other words, thank you so much, and please don't stop.
Limits are scissors, whereas Colimits are glue (unless you are in the opposite category).
I am rubber, you are a coequalizer!
Colimit : for any ε < 0…
This time on... Reversing the Arrows!
Today's episode: The Colimit
impressive very nice... now lets see the homotopy colimits.... look at that subtle geometric realisation... the tasteful topology of it... my god... it even has an adjoint
Australian Psycho
Oh, so in the intuitionistic type theory, the equalizers would be subtypes, and coequalizers would be quotient types! Thank you (g+)+
These videos are incredible. I’ve recommended this one, in particular, to all my friends.
Shef as a jr cs student i live for these videos they’re so funny and also presented in a great way
Going to give myself a pat on the back for seeing the matrix generators and changing my guess to infinite elements because, as I thought, in formal terms, "that 1 in the anti-diagonal looks like it is going places."
I thought the 'opposite' of finite 4 things is infinitely many things.
gaslighting myself that I understand all of that. I even faked laughing to the jokes
What a great way to wake up! A Sheafification of G video! This is just what I needed!
why tf do i watch these videos bru, another banger as always
It is the best math content on youtube i have ever seen
Just wanna let you know that your videos fucking rock! I forever grateful to UA-cam for recommending your video about monads to me.
another banger by big g
2:56 rm: cannot remove '/': Operation not permitted
Bots got off with a warning this time 🫥
-no--preserve--root
@@catmacopter8545 rude
regnab rehtona gnikam rof sknaht ,ebutuoy no tnemmoCoC a siht llaC
Your vids always kill me, top notch jokes and memes.
My new favourite youtube channel, yay
Merci pour cette vidéo ! Le rythme est soutenu du coup a revoir haha mais waouw c'est super bien expliqué et très détaillé !
in my experience, one trouble about learning/teaching category theory is that it's a joint generalization of the theory of monoids and the theory of posets and lattices, both of which people often don't know too much about. (co)limits are much less scary if you already know about meets and joins, (co)presheaves are much less scary if you already know about left and right modules/actions of a monoid, etc etc.
3:15 I feel like colimits are much easier than limits at least in the special case of direct limits. Just in general monomorphisms seem easier to me than epimorphisms and profinite groups are much more intuitive than homology stuff. direct limit >> projective/inverse limit
When do we get a video on the meaning of the sheafification of G?
Yo, we getting a video on topoi next??????
Keep up the category puns. The initial object in your category of collection of fans is certainly a zero object
Dear G, I love your videos!
Thank you so much!
I was expecting the limits video in reverse ngl
Welcome back, another great video 😊
every thing is a Kan extension, hope it is the next video in this series 🤓
The series is called "Kan academy", so it's possible. But given that he has never made a video about sheaf, so who knows.
10:15 : is this “because” {true,false} is (up to isomorphism) the powerset of the singleton set \ast ?
In a way! The contravariant powerset functor is essentially its own adjoint, so maps X -> P(Y) naturally correspond to maps Y -> P(X). Taking Y to be the singleton, we see that maps X -> P(*) correspond to elements of P(X).
(There may be some circularity in this argument.)
@ thanks!
10:18 eyo G why you changed the law of the excluded middle bit from what it was in past videos.
I'm the one assuming it this time 🙃
@@SheafificationOfG you did that in your last video too though ;)
Having cointellect, I'm still in the category of upsets 😞
Is the fact that Omega = P(*) related to the table displayed at 10:08? Applying P to * -> X gives P(X) -> Omega instead of X -> Omega, so it is not immediately obvious to me whether or not Omega = P(*) is a coincidence. Does anyone have any useful insights on this?
Not sure if this would tickle your fancy, but here's one connection.
The functor P : Set^op -> Set is basically its own adjoint, which means that functions X -> P(Y) correspond to functions Y -> P(X) in a natural way.
In particular, subsets of X are elements of P(X), and therefore functions * -> P(X), which then naturally correspond to functions X -> P(*) = Omega.
@@SheafificationOfG This certainly tickles my fancy. Thanks! I now think that Omega = P(*) is a incidence.
this video was a coenjoyer of me :3
Excellent as always!
could you do a specific video on category theory and generic co-objects ?
The Coextension of Kan
6:40 - I know your tricks by now, so I answered correctly 😝
commenting now while the video is only 8 minutes old so he knows I haven't finished it yet lol
Can you make a video on abelian categories?
Just make sure not to unclick yourself.
Your explanation of colimits gluing things together feels vaguely reminiscent of unfolds, and thus comonads? Is this a false intuition?
Category Theory is life.
15:12 *"now that I have an understanding..."
7:53 My guess is the next video is gonna be about coelements
Edit: nvm, he explained it 10 seconds later
Wait limit was not that easy thing in analysis why he's speaking in monad language
will those jokes co-ntinue?
I find this a bit hard to follow, I just wish there were some arrows or something to direct my attention.
this video makes my ear torturously itchy
clang is a c compiler, you should be compared clang++ to g++ rather than clang to g++
tomato tomato++
This was truly our Lan Academy.
Ran Academy when? Wait...
Kan academy gets me every time lol
keep up the good work ❤💯
honestly not that hard to understand, we should teach third graders this and see what happens
Bourbaki moment
Damn this is great thank you
But by assuming Ω(true;false) did you not assume the law of the excluded middle??
Edit: you did 😠😠
another (g+)+ banger
What's your background in math?
Misspent my youth doing a PhD in abstract homotopy theory 😭
@@SheafificationOfG *cospent
Wait you never finished the monoid example
geeafification of sheaf when
I would still like to know what the fuck a comonoid is
It's a coset equipped with a coassociative binary cooperation and a coidentity coelement.
ooh, nier automata reference. I see you
Why tf the end message is in French?
Is this a cochannel
cocochannel
@@rcobbable cocomelon 🤯
I coliked the video
Hijacking this comment section to ask you to use dark mode for code snippets for my poor eyes, languages like scala rust and gleam would be better fit than c++ imo also
Love it
aaaaaAAAAAAAAHAHHHHHHHHHHHHhhhhhhahaha haa. ha. goddammit. He got me again.
I love fake endings... 😅
Cokan academy
kino
Skib
stop