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.
I am about to start teaching algebra 2 and all of this incomprehensible to me and that scares me. More scary is I know I am more qualified than the vast majority of high school math teachers.
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)
How is this the best combination of humor and math and interesting discussion I have ever seen on UA-cam? It's like you were generated by a super advanced AI commissioned by the YT algorithm to feed my dopamine hungry brain.
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)
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 : )
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...
@Mella-h7cright, one has to consider the larger class of "essentially algebraic theories". but anyway the only reason i brought up fields was as motivation for the question asked.
I foolishly took rings and fields in my final semester of uni (I'm not even in math idk how I got there either), the first 30 seconds of this video were alarming enough to fully wake me up at 2am.
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.
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.
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.
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.
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
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.
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...
It should be required (unless you're working with non-unital rings or something), though I suppose the correct answer is: "it depends on the intended applications."
My high-school teacher called himself an "algebraist." One of the other teachers asked him what he would call high school algebra. He didn't even know. All to say, don't get a man with a doctorate in pure math to teach algebra 1, he will make you teach the class.
I often lament being trapped on Planet Stupid. This video corrected that bias. There are oases of dedicated intelligence. That I am living in the desert is acceptable now.
14:48 This is where i got lost. Can we get some concrete example of what operations corresponding to elements of the free algebra means in practice? For example, how do we get the + and 0 operations from a list monad using this method?
Since monoids are typically noncommutative, I'll use multiplicative terms. The 1 comes from the unique element of T({}) [representing the empty list]. The product operation comes from the pair (x, y) in T({x, y}). Although these ops are enough to characterise a monoid, the free algebras provide several other operations. For example, the element (x, y, y) in T({x, y}) represents a binary operation that sends (a, b) to ab^2
@@SheafificationOfG thanks! though it seems like i might have gotten lost a bit earlier. I haven't completely figured out what T({x,y}) is. maybe I need to rewatch the episode on monads to understand this part
@asdfghyter it's the free algebra generated by the set {x, y}; that is, all formal expressions you can write down with x and y as variables. For the list monad, it's the set of lists whose elements are just x and y.
For some reason your video made a freshman like me interesting in Category Theory, even though I do not understand most of them. May someone suggest some books so that beginner can begin with?
In complete honesty, I would recommend building an algebraic toolkit first and foremost (you know: groups, rings, modules, rep theory). It's too easy to get lost in abstract nonsense without any real bearings on the mathematical underpinnings. Formally speaking, category theory can be taught without any of these, but people who do that tend to have a really shallow understanding of category theory.
My masters' thesis was about compact Hausdorff spaces as algebras - there is the compactification monad after all 😎There are uncountably many operations though so finitarians cancelled me on Twitter (profit)
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.
I think it was in this video or maybe it was another but I am sure you were and still are the author. Anyway topic was how proofs theories lemmas definitions and so forth have to evolve due to appearance of previously unknown, undiscovered things in math. And even for the odd monster or two. This forces math things to bloat as proofs theories lemmas definitions etc accommodate rare but important things which is mere preparation of groundwork to call upon you as a learned math person to influence professional math by: 1 - when a monster requires a complete rewrite can it not be added as an addendum, extension, important revision, ... to the theory, proof, lemma, definitions ... Reasoning: it does no harm for a theory to show its historical (hysterical?) development in time. Fror example: Windows 3. MS Windows 3 Revision is part of human lifecycle and software teaches how quickly some revisions are required So Algebra of Things (original) might be Algebra of Things (Revised 1984) Algebras of Things (Revised 1984, 2026, ... ) I am absolutely convinced you see the pattern here, how effective it is, how it respects rights and status of originators and addemdumers everywheres. Of course for some very very serious changes these may need a panel of experts to meet, discuss and implement changes and adaptions accordingly. Maybe we can call these peer group reviews and call upon existing bodies of learned mathematicians to implement these things on beahlf of professional standards, adherence to good standards and so forth My main motivation in bringing this to your (professional?) attentions is to somehow maintain simplicity or originating concepts while allowing for additional developments which may have minor or major or both consequences on how math is done, edjimikated proliferated and researched Yours sincerely Dumbledore (only kidding - it is me really)
comming from model theory… this is a really weird convention to me. sure, the infinitary operations and proper classes (or even uncountable) languages are scary, but they can be dealt with. the weird thing is not being able to negate things and so other boolean combinations. the fact that you can’t axiomatize fields in your theories is kind of a wild restriction. they are kind of the most natural algebraic object for me (maybe after the group). well, i’m sure a couple of videos from now will be “what is a co-algebra”, and that could answer my worries. but still. kinda weird, but cool.
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
Get books and watch yt videos. Pretty much everything you need is free online. This is how I taught myself. Khan academy (the real one) is a great place to start if you need to brush up on trig and calculus.
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.
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!
@@SheafificationOfG Bro what is you both talking about?
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.
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
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
@@sahibjotsingh8238 Same
I am about to start teaching algebra 2 and all of this incomprehensible to me and that scares me. More scary is I know I am more qualified than the vast majority of high school math teachers.
“Impressive very nice, Let’s see Paul Allen’s algebra”
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 think I found my people 😻
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'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.
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 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
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!
Ah, yes, "cofrom" and "coto", also known as "to" and "from" respectively
@@vftdan common mistake, 'cofrom' is actual 'co-un-not-anti-to'. it has slightly different mathematical properties to 'to'.
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.
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
hi Keith
Wrong. Actually, algebra is when you solve for x. Hope that helps.
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
@@SheafificationOfG *pulls out the sickos meme*
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'm here for the jokes. I don't understand anything
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)
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!)
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
How is this the best combination of humor and math and interesting discussion I have ever seen on UA-cam? It's like you were generated by a super advanced AI commissioned by the YT algorithm to feed my dopamine hungry brain.
This moves so fast I'm going to have to watch this on 1x speed aren't I.
*What is* an algebra?
*vsauce music intensifies*
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)
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 : )
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...
@Mella-h7cright, one has to consider the larger class of "essentially algebraic theories". but anyway the only reason i brought up fields was as motivation for the question asked.
I foolishly took rings and fields in my final semester of uni (I'm not even in math idk how I got there either), the first 30 seconds of this video were alarming enough to fully wake me up at 2am.
0:14 Is a great start…
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.
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!
Video has 9 dislikes: 4 from finitarians and 5 from those who didn't learn anything new.
wake up babe, sheag just dropped
what the fuck
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.
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.
I’m looking forward to watching anything involving exact sequences anytime at all, whenever you are ready 😊
Thank you for giving me more info so I can hate on algebra more effectively
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 :)
Lovely to see some (categorified) universal algebra here ❤
Monad in theology is so much more easier istg
Just watch the video in a direction orthogonal to the timeline to cancel out your misunderstandings
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.
i was NOT expecting the dunkey reference
My mind blew at 13:47 .
I'm not used to understanding more than 50% of these videos, so that's something new.
was waiting for that finite thing :)
I may not know what a co-limit is, ***but im at my limit*** (wonderful video, thank you sir)
1:55
this meme made me laugh way more than i expected it to lmao
What a nice video, I’m glad UA-cam recommended it to me
14:11 Didn‘t expect to see Steven He on your channel.
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...
0:38 I am collecting the most ridiculous moments of the video.
9:15 Is it necessary that a ring homomorphism sends 1 to 1? Sometimes when I've seen it defined this isn't required
It should be required (unless you're working with non-unital rings or something), though I suppose the correct answer is: "it depends on the intended applications."
Me: can we have Acerola?
Mom: we have Acerola at home
Acerola at home:
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.
Why woukd anybody be into this?
12:28 wait now im into this
My high-school teacher called himself an "algebraist." One of the other teachers asked him what he would call high school algebra. He didn't even know.
All to say, don't get a man with a doctorate in pure math to teach algebra 1, he will make you teach the class.
It's 6 am I have slept 1 hour, I have a strong fever and I don't understand anything. Still feeling great
I often lament being trapped on Planet Stupid. This video corrected that bias. There are oases of dedicated intelligence. That I am living in the desert is acceptable now.
@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
14:48 This is where i got lost. Can we get some concrete example of what operations corresponding to elements of the free algebra means in practice? For example, how do we get the + and 0 operations from a list monad using this method?
Since monoids are typically noncommutative, I'll use multiplicative terms.
The 1 comes from the unique element of T({}) [representing the empty list].
The product operation comes from the pair (x, y) in T({x, y}).
Although these ops are enough to characterise a monoid, the free algebras provide several other operations.
For example, the element (x, y, y) in T({x, y}) represents a binary operation that sends (a, b) to ab^2
@@SheafificationOfG thanks! though it seems like i might have gotten lost a bit earlier. I haven't completely figured out what T({x,y}) is. maybe I need to rewatch the episode on monads to understand this part
@asdfghyter it's the free algebra generated by the set {x, y}; that is, all formal expressions you can write down with x and y as variables.
For the list monad, it's the set of lists whose elements are just x and y.
For some reason your video made a freshman like me interesting in Category Theory, even though I do not understand most of them. May someone suggest some books so that beginner can begin with?
In complete honesty, I would recommend building an algebraic toolkit first and foremost (you know: groups, rings, modules, rep theory). It's too easy to get lost in abstract nonsense without any real bearings on the mathematical underpinnings.
Formally speaking, category theory can be taught without any of these, but people who do that tend to have a really shallow understanding of category theory.
I studied abstract algebra and I thought I knew algebra quite well, but this…
1:48 Nice infinity symbol!
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!
Awesome video! Any recommended literature on categorical treatment of universal algebras / combinatorial algebra in general?
I followed up to 5:44. Now my brain needs a break, I'll be back another day 😅.
I liked your fancy words
are u gonna do gabriel ulmer duality next? :D
My masters' thesis was about compact Hausdorff spaces as algebras - there is the compactification monad after all 😎There are uncountably many operations though so finitarians cancelled me on Twitter (profit)
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!
as kenny said and meant it: we ain't talkin bout nuthin
So what you're saying is... I should drop out of college and get a job in construction?
I think it was in this video or maybe it was another but I am sure you were and still are the author.
Anyway topic was how proofs theories lemmas definitions and so forth have to evolve due to appearance of previously unknown, undiscovered things in math. And even for the odd monster or two.
This forces math things to bloat as proofs theories lemmas definitions etc accommodate rare but important things which is mere preparation of groundwork to call upon you as a learned math person to influence professional math by:
1 - when a monster requires a complete rewrite can it not be added as an addendum, extension, important revision, ... to the theory, proof, lemma, definitions ...
Reasoning: it does no harm for a theory to show its historical (hysterical?) development in time. Fror example: Windows 3. MS Windows 3
Revision is part of human lifecycle and software teaches how quickly some revisions are required
So Algebra of Things (original) might be Algebra of Things (Revised 1984) Algebras of Things (Revised 1984, 2026, ... )
I am absolutely convinced you see the pattern here, how effective it is, how it respects rights and status of originators and addemdumers everywheres.
Of course for some very very serious changes these may need a panel of experts to meet, discuss and implement changes and adaptions accordingly. Maybe we can call these peer group reviews and call upon existing bodies of learned mathematicians to implement these things on beahlf of professional standards, adherence to good standards and so forth
My main motivation in bringing this to your (professional?) attentions is to somehow maintain simplicity or originating concepts while allowing for additional developments which may have minor or major or both consequences on how math is done, edjimikated proliferated and researched
Yours sincerely Dumbledore (only kidding - it is me really)
Any good book on category theory to start ?
I cant believe youbare actually stringing coherent traisn of though in here
12:08 History of humanity in a nutshell:
5:37 You interrupted the gamer, how rude! (Joke, he was just spamming the complain button on his controller.)
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).
the math jokes in this one were off the charts
comming from model theory… this is a really weird convention to me. sure, the infinitary operations and proper classes (or even uncountable) languages are scary, but they can be dealt with.
the weird thing is not being able to negate things and so other boolean combinations. the fact that you can’t axiomatize fields in your theories is kind of a wild restriction. they are kind of the most natural algebraic object for me (maybe after the group).
well, i’m sure a couple of videos from now will be “what is a co-algebra”, and that could answer my worries. but still.
kinda weird, but cool.
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).
11:11 Nice flashback…
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
idk what niche this guy is targeting, but i want to be a part of it. How do i begin to learn this much math?
Go to math class at your school, university, and get a phd in abstract algebra
Get books and watch yt videos. Pretty much everything you need is free online. This is how I taught myself. Khan academy (the real one) is a great place to start if you need to brush up on trig and calculus.
12:15 The memes are getting funnier every second!
Love your video 😊!!!
First one I actually understood
Love love love the trainman reference
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
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".
3:49 That face looks familiar… (Okay, it‘s obvious.)
That's weird, I thought a monad was a burrito. Am I confused?
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
the joke about école normale supérieure got me 🤣🤣🤣🤣🤣 (the people there are literal aliens and they terrify me)
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.
1:20 homeless kung fu theme from tiktok in my head (yes i'm brainrot
I can’t believe I wasn’t subscribed until today, sorry man