You didn't quite make this claim; the part about finitary theories just suggests it. However, I don't think it's quite true that algebraic theories of unbounded arity always provide a category monadic over Set. There is no free complete Boolean algebra functor, for example (I learnt this from Johnstone's Stone Spaces).
You are absolutely right; the class of algebraic theories encompassed by monads is not quite everything, precisely because of ugly things happening when there's a proper class of operations in the mix (rendering "free algebras" impossible to construct in general, like you say). I didn't want to go too much into it, but in hindsight I really should have haha Everything goes through fine for accessible monads and subclasses thereof, though!
Monads are like the trusty pipe connectors we use to keep water flowing smoothly through a complex plumbing system. They handle all the messy joints so we can focus on getting the job done right.
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.
Three yellow books and like 5 of these algebra shitpost videos later and I still have no idea what category theory is useful for. I have not seen a single use for it, or seen it provide any insight that we do not already have via other methods. Category theory truly is the most useless branch of mathematics. Yeah it provides a clean way to express reoccurring structures in different areas of math, but it's not necessary.
@pendragon7600 While I'm not an expert by any means, Category Theory actually has a fair few applications. It can make proofs significantly easier (ironically, since it's so cooked). It can also take an existing proof and apply it to something entirely different if there are isomorphisms. You can't do that in other fields because there's always some kind of difficulty making specific edge cases (eg, is it continuous? Point wise or uniformly? May be valid for both if separate circumstances are present). CT doesn't look at specifics, so those edge cases are not there. That said, you do end up working in a more abstract or general setting, which can make it difficult to actually perform a more explicit application. CT is actually emerging in a lot of other fields including biochemistry and machine learning. They provide tools to make assertions about truths that otherwise would be difficult in the less general sense. In my case, it has made me a significantly better programmer and mathematician. I can reason about problems in ways some of my peers cannot. I also suspect CT will have more impact in the future, give the rising popularity of functional programming and the demand for things such as dependent types. Computers programming languages are, in a way, their own algebra. Hence, developing those new technologies at the edge of the field is very difficult. That is where CT and similar fields have applications.
@@pendragon7600 I am electrical engeneer and I have some undrstaning of this content. I can confirm, that this idaes helped me to tune my brain into being able to map high level problems into other domains. I was also able to solve some hard and novel problems with help of this. Instead of studing properties of your objects that you work with, category theory presents general tools to work with all sorts of objects. Yes, you can solve your problem wihtout category theory, however undrstanding it gives you much better insights into domain. Category theory teaches you about higher levels of abstractions that is usefull and foces you to start thinking in different way. CT is kind of mix of math and philosophy.
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.
That’s where I’m at right now I know there’s so much context to explore in my math journey I feel like it’s building my intuition for later on when I can go oohhhh now that makes sense and go back to reexplore content and concepts just with deeper understanding
Relatable. I'm three years into my degree and I missed feeling stupid. Feeling like you know everything is both dangerous and boring, it's nice knowing that there's so much more to math that I haven't figured out yet :)
I'm a first year computer science student and I like to watch your videos because they remind me to never get cocky because I barely understand anything, and it shows me that I still have so much to learn
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"
isn't the circular definition of things just an extension of the Yoneda Lemma? We understand the properties of each object by understanding how it relates to all the other objects ('doing X'), we don't need to know a universal construction (a definition or motivation) for it.
chose to watch this while sick in bed and for some reason this was the first time that category theory actually clicked for me, 2 years after getting out of academia... bet the weakened immune system was key
I am taking linear algebra right now, and honestly this video and format has helped clear up so much misunderstanding I have and also puts it in such an amazingly formal way. Man do I wish I could pick your brain.
I come from years of programming and I just have to say this is how I wish I was taught math I find it easier to perceptualize along with navigating different problem spaces.
I had trouble getting my head around finitary functors until I heard this characterization (in the Adamek et al book Algebraic Theories): a finitary functor is exactly a quotient of a polynomial functor. And this makes the connection with algebraic theories really clear! You can think of polynomial functors as signatures of the theory: it’s just a family of sets indexed by the natural numbers, i.e the set of operation symbols of each arity. Now glue some of them together (naturally) and you have a finitary monad/algebraic theory. (To get infinitary algebraic theories, just have a set of operation symbols for each cardinality)
Great video with some very good explanations and insights! As a grad student myself, I love the obscure inside jokes (I could totally be the guy at 1:03 treating HA as the bible; I laughed for 10 minutes straight)
Curious what you think of higher inductive types, which is the same concept but in the context of homotopy type theory. It has a more abstract interpretation of equality, meaning that objects like the circle or the integers can also be modeled similar to algebraic theories.
Hahaha I am studying at the ENS Rue d'Ulm and that first joke is very true. Even in prépa, in first year teachers excpected us to know everything about algebra before the start of the year even though the theories surrounding it were never studied in highschool! Good video otherwise
7:55 the following is something i've been wondering, related to this issue of inequalities. is it consistent for an algebraic theory to require that, in each of its models, its specified operations are all distinct? what if function extentionality is relaxed? (!) the reason i ask this has to do with the so-called "field of one element". every "model" of this "field" that i have seen actually has two elements, and for good reason: assuming extentionality, there is exactly one possible operation of arity 2 on a set of one element! but, if one relaxes extentionality, then it is consistent to assume that there are two unequal operations (+ and ×) on a set of one element. unless there are other troubles that i'm not seeing, this should allow the set of one element to be a zero object in the category of fields. (maybe this trick can be replicated classically by 'tagging' the operations by the set of two elements.) one potential issue that i can see is that, so far, it seems nothing excludes modelling these operations with larger sets but where + and × still do the same thing. but i'm not sure this is fatal: maybe it requires an additional axiom, but (i believe) it should be possible to ensure that whenever 0=1 then also x=y for all elements. some day i may try to formalize this in Agda but i'm too busy to attempt now...
idk man. The representable functors are solving systems of equations. So I'm pretty sure algebra is solving systems of equations by Yoneda lemma. Anything more general is just looking at different types of equations.
I like the video, as part of me is arguing that algebras are more important than types in 2024. This said, this video feels more like a reference than a teaching. I am not saying that is a bad thing.
Thank you algorithm for introducing me to your channel through this video. I'm a freshman math major and seriously hope to one day follow along at a 100% clip. Great quality : )
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.
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.
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
Im not trying to be a hater, Im just curious Why does category theory exist? Like, what problem is it trying to solve? What is its purpose? An analogy I would give is, something like topology tries to generalise the idea of open sets. What does category theory aim to achieve and why should anyone care?
Not a hater at all, it's a natural question. It's hard to give a comprehensive answer, given that it comes up in an assortment of fields nowadays, but historically it proved useful as a medium to formalise concepts in algebraic topology (think: homology theory) and algebraic geometry (think: sheaves). In these contexts, often the objects you are interested in are incredibly complicated to reason with, and general nonsense tools from category theory helps to sift out what parts of the theory are "formal / free". In these fields where you spend a lot of time studying the interplay of many different objects, category theory can really give you a leg up, if not at least as a very consistent and general framework. Perhaps a more extendable answer is that category theory gives you tools for defining objects based on how they're meant to behave (i.e., via universal properties), rather than fussing over how to go about constructing an object with the desired properties. This kind of angle allows you to "invent" substitutes for objects that provably can't exist as well (analogous to introducing complex numbers to resolve algebraic equations that are otherwise insoluble). The main example I have in mind for this is algebraic stacks.
0:44 lol...what about sigma-algebras in analysis?...Lol...if that's what they called them, lol...that Borel-set stuff, lol, don't quite remember the details...that's yet another meaning "algebra" can have, lol...and linear algebra, I guess, lol, but perhaps not really entirely distinct...
is this waht programmers do in graduate IT courses?? as a grad math student im kinda jealous because all of it seems fun but holy shit it's gibberish on top of gibberish written in latex
Algebra is when you are using symbol manipulation to determine the values of unknown variables in expressions. The proper term for anything downstream of Galois is "heresy".
One day I hope in one of your videos you can include a meme about science communicators on youtube that claim that 'theory' means (something to the effect of) "a model of reality that is empirically falsifiable and matches observations within a degree of statistical significance and maximal 'parsimony' and also has 'predictive power'". Erm, where do Category Theory, Proof Theory, Type Theory, Model Theory, Set Theory fit into that definition? Shaking my head my head!
@@quantumsoul3495 it is covered extensively in the work of Karl Popper. but anyway that doesn't matter since 'theory' does not mean 'empirical science theory'
a theory is simply a collection of formal statements (some definitions also include that it must be consistent and transitively closed under entailment)
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.
20/10 joke with zero mathematical content. I'd very much see this joke being made on this channel if the creator didn't always see the need to give meaning to everything so as not to fall into absurdism.
👎👎👎Watch via RUclips 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 !!!! 👎👎👎
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.
You didn't quite make this claim; the part about finitary theories just suggests it. However, I don't think it's quite true that algebraic theories of unbounded arity always provide a category monadic over Set. There is no free complete Boolean algebra functor, for example (I learnt this from Johnstone's Stone Spaces).
You are absolutely right; the class of algebraic theories encompassed by monads is not quite everything, precisely because of ugly things happening when there's a proper class of operations in the mix (rendering "free algebras" impossible to construct in general, like you say). I didn't want to go too much into it, but in hindsight I really should have haha
Everything goes through fine for accessible monads and subclasses thereof, though!
Why am i... why am i subscribed? I'm a plumber.
One could say that category theory is the plumbing of abstract algebra. 🤔
it's a me! Mario!
Would be cool if my plumber would explain things in terms of algebra.
Monads are like the trusty pipe connectors we use to keep water flowing smoothly through a complex plumbing system. They handle all the messy joints so we can focus on getting the job done right.
I don't know anything.
Totally and utterly incomprehensible to my freshman math undergrad brain.
Thank you.
real
same
If you don't mind me asking, which college do you study in? If you're uncomfortable with sharing that, no worries
Just gotta say that I relate
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.
Three yellow books and like 5 of these algebra shitpost videos later and I still have no idea what category theory is useful for. I have not seen a single use for it, or seen it provide any insight that we do not already have via other methods. Category theory truly is the most useless branch of mathematics. Yeah it provides a clean way to express reoccurring structures in different areas of math, but it's not necessary.
@pendragon7600 While I'm not an expert by any means, Category Theory actually has a fair few applications.
It can make proofs significantly easier (ironically, since it's so cooked). It can also take an existing proof and apply it to something entirely different if there are isomorphisms. You can't do that in other fields because there's always some kind of difficulty making specific edge cases (eg, is it continuous? Point wise or uniformly? May be valid for both if separate circumstances are present). CT doesn't look at specifics, so those edge cases are not there. That said, you do end up working in a more abstract or general setting, which can make it difficult to actually perform a more explicit application.
CT is actually emerging in a lot of other fields including biochemistry and machine learning. They provide tools to make assertions about truths that otherwise would be difficult in the less general sense.
In my case, it has made me a significantly better programmer and mathematician. I can reason about problems in ways some of my peers cannot.
I also suspect CT will have more impact in the future, give the rising popularity of functional programming and the demand for things such as dependent types. Computers programming languages are, in a way, their own algebra. Hence, developing those new technologies at the edge of the field is very difficult. That is where CT and similar fields have applications.
@@pendragon7600
t. has never designed programming languages
@@pendragon7600 I am electrical engeneer and I have some undrstaning of this content. I can confirm, that this idaes helped me to tune my brain into being able to map high level problems into other domains. I was also able to solve some hard and novel problems with help of this. Instead of studing properties of your objects that you work with, category theory presents general tools to work with all sorts of objects. Yes, you can solve your problem wihtout category theory, however undrstanding it gives you much better insights into domain. Category theory teaches you about higher levels of abstractions that is usefull and foces you to start thinking in different way. CT is kind of mix of math and philosophy.
@@pendragon7600 Expressing reoccurring ideas is exactly what math is about. None of it is "necessary" to do anything else, CT is not different
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.
That’s where I’m at right now I know there’s so much context to explore in my math journey I feel like it’s building my intuition for later on when I can go oohhhh now that makes sense and go back to reexplore content and concepts just with deeper understanding
Relatable. I'm three years into my degree and I missed feeling stupid. Feeling like you know everything is both dangerous and boring, it's nice knowing that there's so much more to math that I haven't figured out yet :)
@@fleefie y'all stopped feeling stupid? I still feel like I domt know anything and I have a masters degree
I'm a first year computer science student
and I like to watch your videos because they remind me to never get cocky because I barely understand anything, and it shows me that I still have so much to learn
I love your videos on very basic and intuitive topics like algebra and limits. It really helps with my homework!!
My brother in christ, we have very different ideas of "basic" and "intuitive"
@@FilupEilenberg-Moore algebras are a fairly elementary subject in category theory.
They help me co-pe with the fact my monad left to get some ffee and cohas returned
devilish post
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"
A tensor is something that transforms like a tensor
A vector is an element of a vector space
@@ondrejsvihnos2311 vector space is where you put all your vectors
@@ondrejsvihnos2311 that's the only correct definition.
isn't the circular definition of things just an extension of the Yoneda Lemma? We understand the properties of each object by understanding how it relates to all the other objects ('doing X'), we don't need to know a universal construction (a definition or motivation) for it.
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.
It is a testament to your clarity of explanation that it only took 2 watch throughs for a mere computer scientist to understand the subject matter.
“Impressive very nice, Let’s see Paul Allen’s algebra”
but whats a co-algebra? [vsauce music hits]
thanks again for giving me 22 minutes of not understanding a single word
obviously it's a coaction cofrom a comonad coto its coobject!
I'm here for the jokes. I don't understand anything
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!)
I'm a huge Universal Algebra fan and seeing this video in my feed warmed my heart so much
just wait till the coalgebra video
hi keith
chose to watch this while sick in bed and for some reason this was the first time that category theory actually clicked for me, 2 years after getting out of academia... bet the weakened immune system was key
Only sick people understand category theory
I am taking linear algebra right now, and honestly this video and format has helped clear up so much misunderstanding I have and also puts it in such an amazingly formal way. Man do I wish I could pick your brain.
I come from years of programming and I just have to say this is how I wish I was taught math I find it easier to perceptualize along with navigating different problem spaces.
Category Theory dominating every math subjects.
Conceptually subsuming is more appropriate
Its just one way to describe objects
@@berlinisvictoriousOOP mentioned???
@@redpepper74 Mentioned what?
@@berlinisvictorious Object-Oriented Programming, I’m just being silly don’t worry about it lol
I had trouble getting my head around finitary functors until I heard this characterization (in the Adamek et al book Algebraic Theories): a finitary functor is exactly a quotient of a polynomial functor. And this makes the connection with algebraic theories really clear! You can think of polynomial functors as signatures of the theory: it’s just a family of sets indexed by the natural numbers, i.e the set of operation symbols of each arity. Now glue some of them together (naturally) and you have a finitary monad/algebraic theory. (To get infinitary algebraic theories, just have a set of operation symbols for each cardinality)
Great video with some very good explanations and insights! As a grad student myself, I love the obscure inside jokes (I could totally be the guy at 1:03 treating HA as the bible; I laughed for 10 minutes straight)
Monad in theology is so much more easier istg
Video has 9 dislikes: 4 from finitarians and 5 from those who didn't learn anything new.
This moves so fast I'm going to have to watch this on 1x speed aren't I.
Curious what you think of higher inductive types, which is the same concept but in the context of homotopy type theory. It has a more abstract interpretation of equality, meaning that objects like the circle or the integers can also be modeled similar to algebraic theories.
Hahaha I am studying at the ENS Rue d'Ulm and that first joke is very true. Even in prépa, in first year teachers excpected us to know everything about algebra before the start of the year even though the theories surrounding it were never studied in highschool!
Good video otherwise
*What is* an algebra?
*vsauce music intensifies*
7:55 the following is something i've been wondering, related to this issue of inequalities.
is it consistent for an algebraic theory to require that, in each of its models, its specified operations are all distinct? what if function extentionality is relaxed? (!)
the reason i ask this has to do with the so-called "field of one element". every "model" of this "field" that i have seen actually has two elements, and for good reason: assuming extentionality, there is exactly one possible operation of arity 2 on a set of one element!
but, if one relaxes extentionality, then it is consistent to assume that there are two unequal operations (+ and ×) on a set of one element. unless there are other troubles that i'm not seeing, this should allow the set of one element to be a zero object in the category of fields. (maybe this trick can be replicated classically by 'tagging' the operations by the set of two elements.)
one potential issue that i can see is that, so far, it seems nothing excludes modelling these operations with larger sets but where + and × still do the same thing. but i'm not sure this is fatal: maybe it requires an additional axiom, but (i believe) it should be possible to ensure that whenever 0=1 then also x=y for all elements.
some day i may try to formalize this in Agda but i'm too busy to attempt now...
idk man. The representable functors are solving systems of equations. So I'm pretty sure algebra is solving systems of equations by Yoneda lemma. Anything more general is just looking at different types of equations.
wake up babe, sheag just dropped
what the fuck
I like the video, as part of me is arguing that algebras are more important than types in 2024. This said, this video feels more like a reference than a teaching. I am not saying that is a bad thing.
Thank you algorithm for introducing me to your channel through this video. I'm a freshman math major and seriously hope to one day follow along at a 100% clip. Great quality : )
My mind blew at 13:47 .
Lovely to see some (categorified) universal algebra here ❤
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.
Awesome video! Any recommended literature on categorical treatment of universal algebras / combinatorial algebra in general?
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.
@@anselmschuelerthanks
@@anselmschueler oh that is true, thank you!
0:14 Is a great start…
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 ;)
Great video! I am genuinely surprised I was able to keep up with this :)
What a nice video, I’m glad RUclips recommended it to me
Just watch the video in a direction orthogonal to the timeline to cancel out your misunderstandings
Wrong. Actually, algebra is when you solve for x. Hope that helps.
algebras are groups, rings, modules. coalgebras are automata, or transition systems in general.
Me studying the basics of abstract algebra in my spare time because it's fun: Ohhh, this is the tongue. And the worm is the whole thing.
was waiting for that finite thing :)
5:37 You interrupted the gamer, how rude! (Joke, he was just spamming the complain button on his controller.)
I'm not used to understanding more than 50% of these videos, so that's something new.
1:48 Nice infinity symbol!
I liked your fancy words
14:11 Didn‘t expect to see Steven He on your channel.
0:38 I am collecting the most ridiculous moments of the video.
Im not trying to be a hater, Im just curious
Why does category theory exist? Like, what problem is it trying to solve? What is its purpose? An analogy I would give is, something like topology tries to generalise the idea of open sets. What does category theory aim to achieve and why should anyone care?
Not a hater at all, it's a natural question.
It's hard to give a comprehensive answer, given that it comes up in an assortment of fields nowadays, but historically it proved useful as a medium to formalise concepts in algebraic topology (think: homology theory) and algebraic geometry (think: sheaves). In these contexts, often the objects you are interested in are incredibly complicated to reason with, and general nonsense tools from category theory helps to sift out what parts of the theory are "formal / free". In these fields where you spend a lot of time studying the interplay of many different objects, category theory can really give you a leg up, if not at least as a very consistent and general framework.
Perhaps a more extendable answer is that category theory gives you tools for defining objects based on how they're meant to behave (i.e., via universal properties), rather than fussing over how to go about constructing an object with the desired properties. This kind of angle allows you to "invent" substitutes for objects that provably can't exist as well (analogous to introducing complex numbers to resolve algebraic equations that are otherwise insoluble). The main example I have in mind for this is algebraic stacks.
@@SheafificationOfG alright, this kind of actually makes sense. Thank you for the answer!
i was NOT expecting the dunkey reference
Final year of math undergrad, abstract algebra will be right after christmas. Each (g+)+ video i watch makes me more excited and terrified 😅
You're gonna love it!
0:44 lol...what about sigma-algebras in analysis?...Lol...if that's what they called them, lol...that Borel-set stuff, lol, don't quite remember the details...that's yet another meaning "algebra" can have, lol...and linear algebra, I guess, lol, but perhaps not really entirely distinct...
Love love love the trainman reference
I cant believe youbare actually stringing coherent traisn of though in here
Any good book on category theory to start ?
is this waht programmers do in graduate IT courses?? as a grad math student im kinda jealous because all of it seems fun but holy shit it's gibberish on top of gibberish written in latex
@18:42
I cant understand why that doesnt build a complete order ? Can we exhibit a suplattice ?
the set of three elements a,b,c with a minimal and b and c unrelated
12:15 The memes are getting funnier every second!
It's 6 am I have slept 1 hour, I have a strong fever and I don't understand anything. Still feeling great
Love your video 😊!!!
Algebra is when you are using symbol manipulation to determine the values of unknown variables in expressions. The proper term for anything downstream of Galois is "heresy".
9:01 Greek, let‘s go!
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).
First one I actually understood
11:11 Nice flashback…
One day I hope in one of your videos you can include a meme about science communicators on youtube that claim that 'theory' means (something to the effect of) "a model of reality that is empirically falsifiable and matches observations within a degree of statistical significance and maximal 'parsimony' and also has 'predictive power'". Erm, where do Category Theory, Proof Theory, Type Theory, Model Theory, Set Theory fit into that definition? Shaking my head my head!
on a serious note it really does bother me when the distinction between theory and model is understated or totally confused
How would you define empircal science theories?
@@quantumsoul3495 it is covered extensively in the work of Karl Popper.
but anyway that doesn't matter since 'theory' does not mean 'empirical science theory'
a theory is simply a collection of formal statements (some definitions also include that it must be consistent and transitively closed under entailment)
A NEW SHEAFIFICATION OF G VIDEO OMG
3:49 That face looks familiar… (Okay, it‘s obvious.)
I can’t believe I wasn’t subscribed until today, sorry man
The shitpost type images with abstract definitions are hilarious. It would be nice if you left the references you use in the description.
12:08 History of humanity in a nutshell:
14:16 i think you're missing an H in your code
One the oneand, I can't believe you read that closely enough (I sure didn't).
On the other hand, what's enterprise code without a few typos ;)
@SheafificationOfG actually working code
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).
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.
awesome!
the joke about école normale supérieure got me 🤣🤣🤣🤣🤣 (the people there are literal aliens and they terrify me)
Why woukd anybody be into this?
12:28 wait now im into this
Just in case anyone is confused, a monad is a burrito 🌯
The real fans are the one who will generate other fans?
Why am I watching this video when I don't know high school math? Is that why I am poor?
Understanding all of this also makes one poor 😭
@SheafificationOfG Oh, no! 😂
Now do coalgebra :)
Who even cares about non-operadic algebras? Why would someone ever need to consider groups instead of monoids?
Algebra is just co-analysis duhhh
20/10 joke with zero mathematical content. I'd very much see this joke being made on this channel if the creator didn't always see the need to give meaning to everything so as not to fall into absurdism.
@@PrScandium There is a sense in which algebra is dual to geometry though. Although geometry isn't analysis
But Acero... G
I am so lost
5:21 I win, bye-bye
Ab Solüt Sinema
Titillating
12:41
I dont understand any of your video.category theory just goes beyond my mind
first
First
How am I suppose to ever understand this?
That's the neat part.
okbuddyphd
I dare you to translates every propositions and semantics in the bible into logical forms
👎👎👎Watch via RUclips 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.
You are so rude to the creator, if you don’t enjoy it don’t watch, no one is forcing you. We don’t need another Karen