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Sheafification of G
Приєднався 16 кві 2024
My university doesn't let me teach anymore, so the rest of the world is my victim now.
Let g be the functor mapping source to *.o objects. If we sheafify g, we get a functor g++ that now satisfies gluing: a system of compatible objects can be uniquely linked!
Let g be the functor mapping source to *.o objects. If we sheafify g, we get a functor g++ that now satisfies gluing: a system of compatible objects can be uniquely linked!
Algebra - It's not what you think it is!
When you hear that someone is "studying algebra". What comes to mind?
Are they drilling through thousands of factorisation problems?
Are they an undergraduate student of mathematics, pursuing studies you can't think of any real-world applications for?
Well, you're all wrong (or maybe you're not).
Suggested prerequisite:
ua-cam.com/video/roP_HC7tiXw/v-deo.html (though it's not really necessary).
__________
Timestamps:
00:00 - Introduction
00:51 - Crash course on monads (again)
05:36 - A variety of algebras
08:04 - The main claim is two claims
08:35 - The "easy" direction
12:20 - The "hard" direction
17:38 - Thx 4 watching (except 4 finitarians)
17:54 - Finitary theories
Are they drilling through thousands of factorisation problems?
Are they an undergraduate student of mathematics, pursuing studies you can't think of any real-world applications for?
Well, you're all wrong (or maybe you're not).
Suggested prerequisite:
ua-cam.com/video/roP_HC7tiXw/v-deo.html (though it's not really necessary).
__________
Timestamps:
00:00 - Introduction
00:51 - Crash course on monads (again)
05:36 - A variety of algebras
08:04 - The main claim is two claims
08:35 - The "easy" direction
12:20 - The "hard" direction
17:38 - Thx 4 watching (except 4 finitarians)
17:54 - Finitary theories
Переглядів: 823
Відео
Kan Academy: Intro to Colimits
Переглядів 8 тис.14 днів тому
A colimit is just a limit in the opposite category, what's the problem? Prerequisites: ua-cam.com/video/5uI0uJpsEhI/v-deo.html ua-cam.com/video/SrltwGJAiCM/v-deo.html Timestamps: 00:00 - Intro 00:56 - Bad ending 01:31 - Unpacking the definition 02:50 - Asymmetry in duality 03:38 - Colimit example 06:51 - Elementary perspective 09:05 - Coelementary perspective 10:28 - Computing colimits of sets ...
What is the opposite of a set?
Переглядів 78 тис.Місяць тому
The "opposite" of being finite is having a finite complement. The "opposite" of a vector is a linear functional. If a set is a small collection of elements... then what is the "opposite" of a set? Errata: 2:53 - Some of the equations are incorrect; thanks @sstadnicki! 6:18 - The preimage function g^{-1} is pointing the wrong way; it should be mapping P(X) to P(Y)... like every other instance of...
Solving one of the logic puzzles of all time!
Переглядів 26 тис.2 місяці тому
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. Boolos' Brewery: github.com/SheafificationOfG/BoolosBrewery To learn how to devise your own strategies for the various puzzles presented in the video, navigate to github.com/SheafificationOfG/BoolosBrewery/blob/main/docs/test_strategy.md Addenda...
Kan Academy: Introduction to Limits
Переглядів 16 тис.3 місяці тому
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. This video gives a (gentle?) example-driven introduction to limits. Pacing might be fast, but that's because you have a pause button ;) Timestamps: 00:00 - Introduction 00:47 - Getting started 01:22 - Sets 02:07 - Meaning through functions 03:39...
One second to compute the largest Fibonacci number I can
Переглядів 390 тис.4 місяці тому
Most of us are familiar with the Fibonacci sequence. What's the largest Fibonacci number you can compute in 1 second? I'm not setting any world records, here; I don't own a supercomputer. You can criticise my code here: github.com/SheafificationOfG/Fibsonicci Addenda: At 7:59, the e_{01}s in the bottom row are incorrect... [in my defense, the Fibonacci transition matrix is symmetric]. Thanks @a...
2Fast2Finite: Breaking the natural speed limit of finite numbers
Переглядів 24 тис.4 місяці тому
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. In a previous video, we used infinite ordinals to prove that certain finite number sequences called Goodstein sequences were necessarily finite. Now, let's take this one step further and derive a formula for computing the precise length of these...
Solving a finite number problem using infinities
Переглядів 17 тис.5 місяців тому
Here's a problem about sequences of finite (natural) numbers, subject to a simple finite rule, and all of these sequences have finite length... but we can't use the elementary language of finite numbers to prove this! Recommended prerequisite: ua-cam.com/video/dFsa4VeZ0cU/v-deo.html Addenda: (4:07) This argument might seem like a complete proof, but it's incomplete! Pure base expansions of a nu...
Infinite numbers have only finitely many (nonzero) digits
Переглядів 24 тис.5 місяців тому
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. Finite numbers can be represented with finite strings of (decimal) digits, but what happens when we try to imitate this representation in the world of infinite numbers? In the world of ordinals, this can be done, and it turns out that there is a...
What is a Comonad? - Comath and Mputer Science
Переглядів 10 тис.6 місяців тому
Despite being formally dual to monads, they don't seem to be "all the rave" like monads are. Just because you get 'em by just reversing some arrows doesn't mean that comonads aren't independently interesting! Suggested prerequisite: ua-cam.com/video/roP_HC7tiXw/v-deo.html Errata: At 2:14 (definitely not a mistake, I swear), the volume is 2-dimensional thanks @drdca8263 At 5:21, the blue "extend...
(Provably) Unprovable and Undisprovable... How??
Переглядів 19 тис.6 місяців тому
No matter how hard we try to axiomatise mathematics, there will always be strong, independent propositions that don't need no proofs... but how do we *show* that a proposition can't be proven nor disproven? Timestamps: 00:00 - Motivation(al) 01:14 - What is logical independence? 02:47 - An axiomatic foundation of "integers" 04:45 - A provable proposition 05:36 - An unprovable proposition 06:29 ...
Can programmers do math? What is a real number, really?
Переглядів 13 тис.6 місяців тому
Sure, integer arithmetic on a computer may lead to overflows, but this behaviour is manageable and easy to predict... why do programmers settle for the wacky artefacts of floating-point arithmetic? Timestamps: 00:00 - Introduction 00:36 - What is a real number? 02:02 - Why bother with infinite precision? 02:51 - Implementing real numbers 03:37 - Technical interview 04:04 - Naïve implementation ...
Can Mathematicians Code? The Intermediate Value Theorem
Переглядів 14 тис.6 місяців тому
The IVT is introduced in every first-year differential calculus course, and gives a way of proving the existence of solutions to various equations... but does it say anything about how we can algorithmically find such a solution? Mathematicians don't program, they prove... but can we extract algorithms from proofs? Timestamps: 00:00 - I want to apologise 00:43 - What is the IVT? 01:52 - Element...
A shallow grip on neural networks (What is the "universal approximation theorem"?)
Переглядів 8 тис.6 місяців тому
A shallow grip on neural networks (What is the "universal approximation theorem"?)
What is a Monad? - Math vs Computer Science
Переглядів 30 тис.7 місяців тому
What is a Monad? - Math vs Computer Science
Hehe a monad is just a lax 2-functor from 1 to Cat... what's the problem?? :^)
Ah yes, polyads with one object (well played, ya got me there).
I am starting the believe that these videos are just an outlet for frustration in an attempt to justify the many cold, lonely nights spent studying Category Theory. He doesn't need to try and tell you that he is better than us. He can mathematically prove it.
It feels like you didn't really define "monadic category". Do you just mean the Eilenberg-Moore category?
I definitely didn't define a "monadic category", my bad! As you say, a category is monadic over Set if it's equivalent to an Eilenberg-Moore category / category of T-algebras for a monad T.
*What is* an algebra? *vsauce music intensifies*
Every time I return to one of your videos, it's like trying again to reread Carl Linderholm's _Mathematics Made Difficult_. Sigh.
How else is mathematics made? (jk, but I hope you at least have fun!)
Algebra is just co-analysis duhhh
I've been deep diving into Algebraic Effects and Handlers, and I wonder how this all connects with it! I think it's clear that effects form such a free T-algebra, and the handler is a model of that algebra.
A model here corresponds to an algebra. The theory it is a model of is the monad. If I understood correctly.
wake up babe, sheag just dropped
what the fuck
I'm a huge Universal Algebra fan and seeing this video in my feed warmed my heart so much <3
Hey G, very specific question, but on 3:39 the left diagram, what is T \eta_X? And why can you apply \eta_X to TX, when its domain is X? Also the output of \eta_X is an element of TX, but T can only be applied to sets. Basically none of the input/outputs of T\eta_X applied to TX make sense to me. What am I missing?
Although T can be applied to sets (X), T can also be applied to functions. This is because it's a functor. If f : A → B, then T f : T A → T B (or the reverse if it's contravariant). Since η_X : X → T X, it must be T η_X : T X → T T X. On the left we have η_(T X), which is also T X → T T X.
These videos are in a way "nostalgic" for me - years ago (when I had learnt much less) there was lots of maths content online or in books that was well beyond my level, yet I could _feel_ was well-explained. I would watch / read the stuff anyway, just because the sensation of "skimming the surface of a deep ocean of truth" was quite exciting. Nowadays almost all math content is around my level or below it (not counting actual research docs or textbooks). Still wonderful - I learn a lot! - but it's nice to experience that feeling of "woaah... I can sense the beauty, even if I can't see it yet!" again.
was waiting for that finite thing :)
1:06 what is HA?
It's the bible, obvs :^) (It's Lurie's "Higher Algebra" book! 😀)
@ thank you very much! It looks like an interesting read (perhaps equipped with a mandatory religious conversion but we will have to see I suppose).
Lovely to see some (categorified) universal algebra here ❤
Im just a lowly chemist who wanted to understand the character tables we use in molecular orbital theory, fuxk me right? Because the group theory course i took (while rad) didnt get anywhere near that
Time to categorify and take a course on representation theory! Character tables should be a walk in the park after that ;)
Love your video 😊!!!
Category Theory dominating every math subjects.
I'm 7 minutes into the video, with practically no knowledge on higher math, and all I can say is, an algebra is an algebra is an algebra.
This moves so fast I'm going to have to watch this on 1x speed aren't I.
👎👎👎Watch via UA-cam the course ABSTRACT ALGEBRA with Socratica and enjoy the competence of Socratica. She knows how to teach so that students generate motivation and curiosity to learn more about formal mathematical logic. Socratica does not produce a monologue nobody can follow. YOU TALK TOO MUCH !!!! 👎👎👎
whats the point if he doesnt talk too much
Sounds like a you problem. Git gud.
i AM NOT reading allat 💀💀💀
Different types of learning material appeal to different types of people. This style might not be for everyone (I'm not even sure it's the most effective for me) but that doesn't mean nobody can get value out of this. Perhaps you could have simply recommended Socratica as an alternative for people who might have trouble following this video? That would have still potentially helped those interested to learn more, without coming across as rude to the creator and those who do enjoy these videos.
👍👍👍Watch via UA-cam the course ABSTRACT ALGEBRA with Socratica and enjoy the competence of Socratica. She knows how to teach so that students generate motivation and curiosity to learn more about formal mathematical logic. Socratica does not produce a monologue nobody can follow. YOU TALK TOO MUCH !!!! 👎👎👎
Why am i... why am i subscribed? I'm a plumber.
One could say that category theory is the plumbing of abstract algebra. 🤔
A guy once told me that "one is doing algebra" when you are working with an analogue of the 1st and 4th isomorphism theorems and also "it looks like you are doing algebra"
I love your videos on very basic and intuitive topics like algebra and limits. It really helps with my homework!!
👎👎👎 What a stupid video !!! 👎👎👎
@@crigsbe bro is THE hater
First
first
hawktuah by hawk
Real
Hello, this video could actually be legit, solving every logical problems if you make another video about "The Hardest Logic Puzzle Ever" by the American philosopher and mathematician George Boolos.
Wtf is monad? Is it from Monadology by Leibniz?
I was able to calculate 10 millionth term in 0.9s while the naive(while loop one) took 680 seconds
3:55 made my day for some reason
how 2 go even faster: import an existing list of fibonacci numbers and go forward from there using fastest algorithm
There is an error with multiplicity 2 at 1:58
I must be blind. What are the errors?
@SheafificationOfG No, I misread 😢 so I made the error
@@mrl9418 haha all good then!
It's not part of the axioms, but it seems you also assumed that If a=b, and c=d, then a+c=b+d. Necessary to do stuff like adding a number to both sides Or is it a consequence of the other axioms? I can't see it. I'm only asking because a core part of the video is that 2=0
Yeah I didn't go all the way with the axiomatisation because it gets a bit too overboard imho to mention that all operations provided are well-defined *functions* of their arguments. But you are right, strictly speaking.
Love the video! I was impressed by your addition of "Ad Hoc 2". Nothing escapes brainrot.
I didn't understand 4th life lesson can someone explain? 19:17
5:30 hawk tuah
14:03 This image is just too funny. But seriously, what is a QUADRUPLE integral doing there?
Very nice vid thank you :D I'm new to category theory and while I understand the yoneda lemma superfically I don't have good intuition for it. Could someone elaborate on the statement G makes at 1:52, about how all objects are set objects? I don't see how this follows from the yoneda lemma but it seems like it could be a helpful fact to think about general categories.
Glad you liked the vid! Essentially, this is an interpretation of how the Yoneda embedding is fully faithful, or that objects are uniquely determined up to isomorphism by their representable presheaf. Basically, we can think of an object X as a "set" by taking "element" to mean "map S -> X" (where S is some other object). Allowing this object S to vary gives you precisely the presheaf represented by X, and Yoneda implies that this is enough information to determine X. This perspective allows for a very down-to-earth understanding of universal properties of limits, which I explain in my Kan Academy video 🙂
I find this a bit hard to follow, I just wish there were some arrows or something to direct my attention.
I was watching a video on powers of matrices, ua-cam.com/video/pSXQmxSSvZI/v-deo.html Rather than matrices you could rewrite your step function as (x',y') = (x(x+2y), x'-(2x-y)y) (1,1)->(3,2)->(21,13)->(987,610)...
With zero optiization in assembly i get to 4e6 in approximately 0.001 second
i counderstood this
I just watched another video [studying with alex YT channel] that explains a monad as having mainly three features to it: 1. Wrapping function - a function that wraps a value to a monad 2. Wrapping type - a type that wraps a value 3. Run function - a function operates on the wrapped type He then went on to mention that an Option[Maybe type] type is an example of a monad, that represents a maybe state in a type and provides various monadic functions to work with the type. An example from RUST and C++'s version of an option type contains monadic functions like or_else, unwrap, and_then etc. And from this video i learn that a monad is a function that manages (external)state which is fundamentally the same explanation as the above. From what i have gathered, a monad is just a state wrapper, why the complicated explanations then?
I'm not much of a functional programmer. The concept originates in math, where the rich structure is used in other ways (this is explained more in the full video). There's nothing "complex" about monads; in the context of programming, it's best to think of it as a design pattern for encapsulating a certain class of side effects. The axioms of a monad basically describe preconditions that need to be satisfied for a monad to be realised, as not just any wrapper yields a monad. The point, though, is that once you *have* a monad, you can take advantage of its design pattern to produce compositional and functional code that manages state in a fairly elegant way.
@SheafificationOfG okay, well said. Awesome short, oh and I'm not much of a functional programmer too, mostly dabble with C++ and other languages of interests :)
I got confused between nu and v lol. I paused for so long... v_ν
I'm too co-co to understand this.
interesting, in python I implemented the iterative log(n) version, which is like the matrix version, and I get fib(5000000) in a little less than a seg... I might try those other versions to see how it does...
Dear G, I love your videos!
Thank you so much!