My buddy has a PhD in pure mathematics and says.. "Studying category theory is like eating your vegetables." Not sure what that means but it has never left my mind.
@@mastershooter64 yeah the most ironic part of the comment is that the video is an introduction, if you don’t know nothing about a topic you watch an introduction, it’s just logical
The problem I have always had with category theory is the existence of a forgetful functor from the category of versions of myself who understand things about category theory to the category of versions of me who don't
This is okay, because there is a free functor Learn from that category of versions of yourself that do not understand category theory to the category of versions of you that understand category theory, making the diagram commute.
@@jsmdnq That's the one thing I never seem to forget about category theory, and I regularly make internal jokes to myself about categorical dual notions of every day objects and activities. Jokes I can share with precisely nobody because almost nobody would understand them, and most of the people who would understand them would not find them funny.
Category theory is awesome, it feels like the most 'human' thing ever: "Yeah so we have this thing called 'maths' which we use to simplify the world and make connections between different phenomena by putting everything in neat little boxes. ...so then we started wondering whether or not we could make a system of bigger boxes to help sort our existing boxes and find connections, which we were able to do. ...and then when we tried to sort the bigger boxes we realised they already explained themselves!" "Wow! So you can sort anything then?" "Nope, there are still things we can think of that don't fit into the system." "..."
@@chri-k Imma need a minute to process this Even the best phone has a limited battery = all phones have limited battery Even the worst phone has a limited battery = all phones have limited battery Even the best phone has a limited battery = Even the worst phone has a limited battery phone = A a limited battery = x Even the best A has x = Even the worst A has x = All A has x
This is actually the most understandable introduction to category theory I've ever seen. True, I'm somewhat familiar to it already, but explanations like that somehow make much more intuitive sense than "monad is a monoid in the category of endofunctors" stuff. Good job!
A teacher in my class of Type Theory in computer science degree, while explaining Category Theory he was doing a lot of little parenthesis about things that seemd unrelated and he said: "What i'm explaining now is like we're going into a journey, and i can't resist to stop and enjoy the landscape and describing how beautiful it is to you."
They are meant to exist to serve Google with our information. But his work is great, because video uploaders like him are what draws people to Google's information collection platform.
What an excellent introduction to the subject! I wouldn't have believed that in 26 minutes someone could explain (in terms a beginner could perhaps understand) all the way to equivalences & natural transforms, but you sure did. Very jealous of your production values too. Cheers mate, bravo. Your comments about adjunctions & Yoneda have tempted me to want to try and intuitively motivate them. So here's another explanation to add to the pile of unhelpful, vague descriptions you've clearly encountered: - Adjunctions are the category theorist's version of minimization/maximization problems. Of course, the adjunction between free & forgetful functors is the paradigm example: the functor F:Set - > Mon solves a minimization problem -- it gives the least solution, subject to a particular constraint. The "least" bit here is the universal mapping property of the free monoid, which is the definition of adjunction applied to this example: F(X) doesn't have any more structure than it has to. The "constraint" we have to "solve" is that, for a given set X, we want X to embed into U(F(X)). So F(X) is the least monoid whose underlying set contains a copy of X. On the other hand, right adjoints solve maximization problems. Consider how the category of groupoids is a coreflective subcategory of Cat, that is, that the inclusion functor Grpd -> Cat has a right adjoint. This functor takes a category C to its "core groupoid" core(C), which consists of all the objects of C but only isomorphisms. This solves a maximization problem: core(C) is the largest groupoid which is a subcategory of C. Other category-theoretic concepts which also perform a minimization/maximization function (e.g. equalizers give the largest subobject equalizing two morphisms) are all instances of adjunctions. - I like to think of the Yoneda Lemma as a category-theoretic version of Euler's Identity (that e^it = cos t + i sin t). Doing algebraic manipulations with cosines directly sucks because cos is not an algebraic object. Hence all the annoying "trig identities" that precalculus students have to memorize and later forget. This is a serious issue in fields like electrical engineering, where they have to do stuff like cos(2t) and stuff all the time (e.g. doubling the frequency of a wave function). So what do they do? Use complex numbers! Instead of working with cos(t), they'll work with e^it, which is super nicely behaved algebraically (because exponentiation _is_ algebraic). Then, at the end of the day, they take the real part of the answer, and it was like they were working with the cosines all along. A pretty neat trick -- indeed it's (basically) the reason complex numbers were invented in the first place -- and it all works because of Euler's identity. The Yoneda Lemma plays the same role in category theory: often a category you're working with will be kinda crappy (e.g. not having (co)limits). But the category Prsh(C) of presheaves on C is a very nice category indeed: it has all limits, colimits, exponential object, subobject classifier, etc. The Yoneda embedding allows you to embed C into Prsh(C) in a nice way, and do all your algebra in Prsh(C). If the solution you get (e.g. of taking a limit) is "real" (representable), then a standard Yoneda Lemma argument says that the solution (e.g. limit) existed in C all along. So Yoneda basically allows you to have imaginary (co)limits/exponentials/whatever, which might turn out to be real all along. Very nifty, and very useful. Hope that helps, and hope you make more videos about category theory!
What a wonderful insight. I'm a math graduate student tackling with an exam on category theory right now, I'll try to keep in mind those points of view studying the subject. Thanks!
Commenting before viewing, but hopefully the Yoneda explanation finally works for me. Yet to find it explained to my learning style, and I legit have a Phd in math.
i dont think the respected property here is his strong self-confidence, but rather his true mathematical curiosity - that he wants to know the amswer no matter what. (i don't know which is the one you referred to in your comment)
the reality of mathematics is that most papers are read by virtually no one. if someone finds an error in your work, its usually a net positive. firstly, the work usually can mostly be salvaged, next, it means someone actually cared about your work which is always nice.
Also, a math PhD dissertation ain't like handing in a homework assignment. You can be sure he's poured over every detail of it - many times. We're all human, so mistakes are always possible, but by the time you're sophisticated enough to pass your quals and take all those grad courses, your confidence and capability in checking your own work will be enormous, and justified.
My favourite description of equivalence of categories comes from Awodey who, in his book on the subject, says: 'One can think of equivalence of categories as “isomorphism up to isomorphism”.'
Don’t know if you’ll see this, but I actually go to Bath university as well, and after watching your 27 facts video, asked one of my lecturers about category theory as I knew that he studied it at PhD. That lecturer happened to be the very Thomas Cottrell that supervised you! Funny how the world works
Thank you, I finally understand the difference between isomorphism and equivalence. I’m a hobbyist Haskell programmer so the nitty gritty details of category theory aren’t usually required, so all I really knew was (endo)functors, natural transformations, and handwaving of isomorphisms as ‘these types can be converted into each other without loss of information’. Still trying to wrap my head around as adjunctions and representable functors though.
Imo, you can't really grasp them without knowing some very important examples of them. If you learn moduli spaces, you will 200% understand representable functors. Adjoints are a bit less clear to me too -- sometimes my mental picture is "something with the flavour of free generation" (e.g. forgetful functor GROUP->SET and "take the freely generated group" SET-> GROUP functor are adjoints), sometimes it's something a bit less abstract which unfortunately idk how to get across (the main example for me is pushforward and pullback of sheaves, but I assume that's too specific).
I have always been thinking that category is similar to groups but have different rules. But in the video said the most epic thing that I've ever heard in math: "let's take a special case of category - sets"
adjunctions are actually quite simple, if you have two functors L : C D : R then you essentially have a representation of C in D (via L) and of D in C (via R) the adjunction then just tells you that for every c in C and d in D you have a natural isomorphism D(Lc, d) ~ C(c, Rd) which essentially means that whatever relations you find between objects in the image of L and the category D at large are "mirrored" as relations between objects in the category C at large and those in the image of R. The quintessential example is the Tensor - Hom adjunction in abelian groups where it tells you that (a b, c) ~ (a, Hom(b, c)) which essentially tells you that providing a bilinear map from the product abelian group (a X b) to c is the same as for each a providing a linear map from b to c which is not that suprising after all that once you "fix" one of the variables in a bilinear map you get a linear map on the second variable.
So, from my understanding, applying a category theory on a subject is like: 1. Learning how to say the same thing in multiple languages. 2. Observing how each language uses its own methods to convey the same thing 3. Using that knowledge to learn how to convey information better, not only in your own language, but also in the others. 4. Taking that experience with you to the future. So that whenever you stumble upon a new language, by just saying the same thing in that one as well, you would already have a deeper understanding of it.
Physicist with a little passion for abstract math here. Well done! I was very confused about why I would be seeing them when learning about differentiable manifolds when I thought Categories would just be ultra-abstract things for algebraists. Subscribed!
I'm making a presentation at uni next week about this exact topic, so this is incredibly good timing I've wanted a gentle undergrad introduction to category theory for years now, so it's pretty funny that I find it now that I've taken matters into my own hands and started learning it formally lol
This is gold! Thanks! I am doing a PhD in engineering simulation and I am trying to formalize the wizardry I do to my sets of eigenvectors before building my reduced order models. I am not from a pure math background so videos like this one are invaluable to kickstart my understanding of these math branches :)
this video gave me a bird's eye view of my lifetime math adventure. From 1st grade maths until I graduated college and any other math I encountered. Different math rules, new kinds of numbers but all feel somewhat the same. That sameness was explained by this video.
i absorbed very little of this but i think if i was more awake or had a better grasp on maths this would've made a lot of sense. maybe. i think your video is good
That was some pretty cool dense mind blowing shit I feel I've barely understood but still I've enjoyed. It's always nice to have new videos from you Oliver
I'll always remember that when I was studying, in the 1990s, I borrowed a book about category theory at my college's library. The card inside showed that nobody had borrowed that book since 1978
Wow, great video! I've heard some very difficult math lectures on category theory, so I never really understood it very well. But this video is excellent because you give several useful concrete examples, which greatly helped me to understand what's going on.
Hey, statistician here who's been really loving learning about CT, and your funny video was actually quite helpful. Thanks for this one which is perhaps more helpful... perhaps. Thanks for the delightful and informative content ✨
Adjoint functors make sense if you think of them this way: The category C (dom of left adjoint and codomain of right adjoint) is the category of interest The category D is a category where you can manipulate stuff and use it to "predict outcomes" The left adjoint is like encoding a vague object (or morphism due to functoriality) of C The right adjoint is like decoding a well understood object (or morphism due to functoriality) of D The unit shows where you land if you immediately decode something after encoding it The co-unit shows where encoding something you have decoded lands after immediately encoding it again The triangle laws show how functoriality and naturality work in this framework and the homset isomorphism shows that there is a bijective correspondence between manipulations of encoded objects to yield certain outcomes/predictions and processes from "unencoded" objects to decoded ones. So in a sense, adjuntions are like a version of substituting morphisms of C with an easily encodable domain, with processes of D with an easily decodable co-domain. It is somehow like predicting the weather using theories instead of waiting for the phenomena to happen.
Thanks a lot! I was wondering how much priority it should take in my self-study of maths, and understanding that categories are a certain simple mathematical structure that happens to be useful to encode some aspects of maths, but doesn't easily deal with *everything*, is super helpful.
The moment where you introduced the category of categories (21:00) was such a sublime experience, it might well be the happiest math-related moment in my life so far. Thank you for this wonderful video:)
Your video comes at the perfect time. I haven't been reading category theory per se, but instead a translation of Gödel's work on undecidable propositions. Even though this is unrelated, I came out from watching this with a greater sense of understanding. Thanks!
I studied pure maths and we only quite briefly touched this topic as preparation of module and ring theory. I am still amazed how you can use the general definitions and see them play out in specific ones. Very interesting introduction, thanks a lot for the effort. :)
I gotta say, I learned more from this video about category theory than from my (failed) course on category theory in uni. Though I was a lot less diligent studen then than I am now. Great video, thanks for doing it!
the biggest lie youtube told me is that this video has the duration of 26 minutes and 19 seconds, I've spent more than half an hour watching this and I'm only at 18:02 >:(
I'm doing a masters degree in september in Algebra, geometry and number thoery and I have to know the basics of category theory and your video helped me know that I guessed right the fact that you could have a "category" category (not in those terms but the idea is equivalent). That makes me feel like I got what it takes to success in getting my PhD
This was the most fun I’ve experienced from a TH-cam video in a while! You’ve done something casual and very fun, and I’m looking forward to looking through the links you’ve included to learn more
This is by far the best introduction to category theory I have seen (and I have seen quite a bit of those). Brilliant! Sorry any mistakes, my english is a work in progress.
This is a fantastic video, wow! Props to you dude, took a subject that is generally INCREDIBLY boring and unintuitive, and turned it into an engaging introduction where I finally felt I could follow along. Clearly a lot of work went into this, I'm looking forward to seeing what else you come up with in the future!
You still should start with group theory tho... That's like, maybe not kindergarten stuff, but high-school-ish. Can be understood even with relatively little prowess in the abstract. Category on the other hand, is post-grad stuff, you better be prepared for a war of attrition with it. ^^
I had just finished my a levels a few weeks ago and will be studying maths as my major. This is my first time learning something about category theory. The video is wonderful and quite comprehensible to me. Thank you for such a wonderful video and I hope I’ll be able to explore more through this channel👍
The only case where adjunction really obviously makes sense to me is the fact that currying/uncurrying form an adjunction (since (X * Y) -> Z and X -> (Y -> Z) are basically the same thing). In all other cases my eyes start glazing over as the nLab page functs me into the (Category of Pain)^op and my brain becomes naturally isomorphic to strawberry icecream mmmmmmmmmmmmmmmmmmmmm
nLab is a dangerous place... you can go there to look up a concept you know and understand, and after reading their exposition of it realise you understand it no longer.
Since I started school for electrical engineering, I felt like the math classes were leaving something out. There’s something I’m looking for, like some kind of explanation of math that my teachers haven’t given me. In my mind I could barely explain to myself what I was looking for. I knew it had to do with boxes of logic. If you start with AB CD for example, there would be two different boxes of explaining this backwards CD AB and DC BA. See how the two different things can be seen as two different ideas based on how big the chunks are? Those are like boxes to me. I want all of math organized in boxes. Keep in mind I’m not even sure if I’m expressing the real thing on my mind. But I’m looking. I didn’t completely understand the video, but I’m drawn to it, and every once and a while in the video my curiosity was satisfied. If you could make a video, in MUCH MUCH more detail, I’d appreciate it. It seems like every five seconds of this video could be expanded into another 2 hours. Thanks for the video, as is though.
Probably the best introduction video to category theory out there. Please elaborate on the 'downsides' of category theory that you alluded to in part 2 video
wow that rock paper scissors example really clicked for me. now i get how important associativity is for keeping structure simple and consistent. if i invented rock paper scissors i’d be pretty upset about it not generalizing to larger groups of people
@@Zxv975 This is how I understood it: You know that paper beats rock, rock beats scissors and scissors beat paper. Now let's make a binary operation x that takes two players as input and produces the winner as output, then we can write the following: rock x paper = paper x rock = paper (meaning, in a game of rock and paper, or a game of paper and rock, paper wins); paper x scissors = scissors x paper = scissors; scissors x rock = rock x scissors = rock. Now, how do you calculate rock x paper x scissors? Assuming the operation is associative, you can go about it in two different ways: (rock x paper) x scissors = paper x scissors = scissors; rock x (paper x scissors) = rock x scissors = rock. But as you can see they produce different results, so the associative law doesn't hold, and the operation is non-associative.
I love the way you think about this!! So many interesting analogies. I've taken classes on category theory, read countless Wikipedia articles, and lots of TH-cam, and this video helped me get the point the best way I've ever seen
Thank you very much for that excellent explanation! Several times I had to stop or repeat parts of the video to read thoroughly the symbols or to think over what I had heard. But just this is the advantage of videos compared to lessons. I studied math over forty years ago (in Germany) but we never had this topic. Yet I know isomorphisms from group theory ... It's easier to understand category theory if you are familiar with some examples.
Category theory is a unification of all of mathematics because it abstracts the structure of mathematics in to a common language. It is like a universal language. It would be analogous to spoken languages. If you learned a "master language" it would allow you to speak and understand all other spoken languages. It's not actually this because no such master language could exist as it would essentially have to be all languages and category theory doesn't actually let you understand other mathematics. What it does though is provide a common abstract interface so all those different mathematical areas can be translated in to a common language and then one can see how the various different areas relate to each other and see the deeper structural aspects. It turns out that almost all mathematics(maybe all) uses basic common ideas and ways of thinking which one wouldn't know otherwise. Category theory is extremely powerful because it lets one see the underlying machinery. It, say, is analogous to how understanding that all matter is made of atoms. From the atomic perspective it unifies all things and things that might have seemed different is just one variation on the atomic theme. Category theory is difficult at first because one doesn't realize the concepts it presents are "universal"(unless they have already learned a large amount of mathematics). Like anything one just has to learn it to see it's use.
At 6:29, I like to emphasize how nontrivial this assumption is. A direct flight from Athens to Berlin and a direct flight from Berlin to Cairo is not enough, by itself, to guarantee a direct flight from Athens to Cairo (even if there is one in fact).
I can‘t resist to ask for more. Very instructive video. I think I can say I came a little bit closer to the spirit of category theory. I watched a lot about this topic and seeing a lot of explanations. The main issue for me regarding CT is that it is almost always everywhere explained the same way regardless of the known problems that one faces with it. But you go in the right direction.
My Understanding of Category Theory got better after I studied "Abstract Algebra" and Group theory. Many courses on Category theory state that you need only basic algebra to understand Category theory, which I am not sure.
I understand that Eugenia Cheng is an expert in category theory and has managed to apply it to many useful aspects of society, which has recently sparked a great deal of interest for me in the subject.
Thankyou thankyou thankyou! Thanks to the coconut video I became able to understand the maths in an advanced physics video course, it's really nice to know more details of it.
This video is very entertaining. If someone wants an example of how people sound when writing to look smart rather than simply say they are entertaining, entertainment which allows themselves to get to the grand details, feel free to read below. I think gluons, which help to assemble quarks, which make up protons, neutrons, and electrons, which make up traditional usable material such as hydrogen and water, have mass defined by the pressure inside and outside of whatever balanced system they are in, radioactive decay bound to happen in the system they are in, and how well the gluon orbits its quarks. In this hypothetical or theoretical model, quarks are composed of super-quarkinos -- 7 each, and gluons are composed of super-gluoninos -- 14 each; 2 hooks, and 12 load-bearing threads. 2 of these 'threads' are most likely stuck together in the middle of the gluon. I also think that super-gluoninos hold charges, such that they aren't ever quite equal to simply positive or negative but are octonion charges or sedonian charges. This is why I think there is a most massive reasonable atom in terms of proton and neutron count and this is why I think said biggest possible reasonable atom using our current model of non-anti-matter safe assemblage atoms has a half-life and radiation so harsh, that when an atom of it undergoes radioactive decay, it ejects a gluon or super-gluonino or gluonino -- and directly. There is probably a mathematical group or category or both that can classify this theory or hypothesis in more detail. So yeah, kinda a theory dump, but also a congratulations at the same time.
This has been a nice follow-up to the notorious nut video, haha. If categories alone are still somewhat understandable, categories of categories completely stump my intuition.
I remember when the internet was all about stupid cat videos (and pr0n). This gory cat video isn't what I signed up for! My head hurts. Serious now, that was heavy stuff. You may think this is trivial, Oliver, since this your thesis is faaaaaaaaaar more obscure, but to those of us casually interested in math, it was really hard. Anyway, you have a great voice. Keep doing what you do, please!
Finally some light in this confusing "generalisation" of sets. I was trying to get a taste of what the hell Groethendiek 's deal was, but it's even higher bird's eye on even this. Whatever "getting stuck" is the path to understanding mathematics.
I'm a physicist whose pure maths only got as far as elementary group theory, so as far as I can make out category theory is a sort of generalisation of group theory, as group theory is a kind of generalisation of set theory. Which allows us to explore the interconnectedness of all things. Sort of. So you can translate a problem in number theory or linear algebra into a problem in topology or complex analysis, solve it there, and then translate the answer back to solve the original problem. Kind of like Wiles' proof of Fermat's Last Theorem. Sort of. I think... Anyway, it makes more sense than "constructor theory"...
Group theory is not a generalization of set theory though. Group theory is its own kind of special fuckery, I can only recommend 3Blue1Brown's video on the 196,883-dimensional monster.
Well up to and including functors I was mining value. Once you started talking about natural transformation, and equivalence, absolutely no idea. Overall a nice introduction to major concepts and the ideas they represent.
I finally finished watching all the videos that are over a half hour long And that I care about. So I just got this video on my page. Still haven't seen the coconut nut video, my introduction to you was the 5-hour long mass extinction debate.
At 19:58, I certainly understand why you omit going from Sets to free groups, but for the interested viewer, it's really quite simple, beautiful, and for me at least, fascinating. As always, thanks for your consideration.
I'm impressed with you having enough balls to stop yourself from talking about elements of object without authomatically saying something like "given that \mathcal{C} is concrete", or talking about "category of categories" without mentioning "small"
My most favorite example of a category is parametric functions that are differentiable both in their argument and in their parameters, aka neural networks.
Great indtroduction! Next video: natural transformations, limits and colimits, and the Yoneda lemma? Perhaps Adjuncionts? (Which would probably would require a video of its own?)\ Great job, looking forward for more!
This popped up in my feed, and I thought "category theory, that's the thing Tom Leinster works on, okay I'll watch this" (his partner was in my department at work, and he worked in the same building as me somewhere else). And his is the first name that comes up here!
The thing you said about the scaffolding That exact thing is said about mathematics _themselves_ : That mathematics is scaffolding with which you construct reasoning, remove the scaffolding and the construct remains self-standing hopefully with fewer bricks
Honestly, I think the only reason why I get Category Equivalence is because I've already gone through all of these emotions when learning about P & NP from my CS degre and the concept of NP Complete, which is basically "this problem is just this other problem written differently."
back when the video came out I became so intrigued that I ended up studying category theory just to make the mum joke at the end with more conviction ps: not that this matter but my introduction to your chanel was Pitagoras broke music
In my yet incomplete theory of cognition, certain mathematical structures arise once and again in different contexts varying from physical to sensory, from sensory to conceptual, from conceptual to imaginary, from imaginary (but plausible) to fictitious, from conceptual to absurd or sound, from neuronal to neural. Category theory is not my forte but, thanks to this introduction, now I know where to look up and what to do in my next step.
The image _[shown here at __23:17__]_ appears to be a diagram related to category theory or a similar abstract mathematical concept. It shows two diagrams, potentially commutative diagrams, which are used in category theory to represent objects and morphisms (functions) between them. The left diagram has four objects \( A_1, A_2, A_3, A_4 \) with morphisms \( f \) and \( g \) mapping to objects \( B_1 \) and \( B_2 \) respectively, which then map to object \( C \) through morphism \( h \). The right diagram seems to involve tensor products of the objects \( A_1, A_2, A_3, A_4 \) and the associated morphisms. If you have questions about the specifics of this diagram or the concepts it represents, feel free to ask _[ChatGPT]_
3:50 The funny thing is, I paused when you mentioned Twelve Angry Men, because I'd argue the setting of Twelve Angry Men is very important, even if it's not visible right away. It's a very American story about how Americans of that time period would see the importance of the rule of their laws. It's punctuated with an immigrant reminding another man how important it is for Americans to value their independence to form their own opinions. The whole point of Twelve Angry Men as a progression from conflict to resolution is that they all try to act civilly and self-importantly at first, until through this very American depiction of argumentation their every implicit bias is laid bare. It's a trial but the ones passing judgment are themselves being judged. It's very much peak Americana in a very optimistic light, and that's part of the joy of watching that movie; you get to feel the importance of a society that takes the concept of legalism seriously, across class, across age, and across vast differences of life experience. I wonder how telling that will be in the revealed importance of Category Theory as I continue watching.
You know, as a non mathematician*, I found this video very helpful in understanding this subject. *My degree is in molecular biology. (My hobby is trying to understand mathematics)
My buddy has a PhD in pure mathematics and says.. "Studying category theory is like eating your vegetables." Not sure what that means but it has never left my mind.
It means you’ll live healthy
probably "you'll hate it but it's good for you"
you don't want to but it's good for you
Noether and algebraic suggestions would be my guess.
I guess studying group theory is like eating someone else's vegetables then.
Sure I'll watch a half-hour video on category theory, a thing about which I know nothing about.
Yeah you got me there
Well, it’s an introduction
I took a graduate course in algebraic topology in 1980 from which I learned nothing except to avoid category theory like the plague.
yes...that's usually why people watch these kinds of videos...to yk learn about things that they know nothing about...
@@mastershooter64 yeah the most ironic part of the comment is that the video is an introduction, if you don’t know nothing about a topic you watch an introduction, it’s just logical
The problem I have always had with category theory is the existence of a forgetful functor from the category of versions of myself who understand things about category theory to the category of versions of me who don't
This is okay, because there is a free functor Learn from that category of versions of yourself that do not understand category theory to the category of versions of you that understand category theory, making the diagram commute.
@@leptogenesis3558 It's not because my adjoint is alway co-operating on my co-self to forget my freeness.
@@jsmdnq That's the one thing I never seem to forget about category theory, and I regularly make internal jokes to myself about categorical dual notions of every day objects and activities. Jokes I can share with precisely nobody because almost nobody would understand them, and most of the people who would understand them would not find them funny.
@@timseguine2 It's that comonad inside you that is conflicting because you haven't dualized the universal limit.
Doesn't that all make you isomorphic between you and yourself?
Category theory is awesome, it feels like the most 'human' thing ever:
"Yeah so we have this thing called 'maths' which we use to simplify the world and make connections between different phenomena by putting everything in neat little boxes.
...so then we started wondering whether or not we could make a system of bigger boxes to help sort our existing boxes and find connections, which we were able to do.
...and then when we tried to sort the bigger boxes we realised they already explained themselves!"
"Wow! So you can sort anything then?"
"Nope, there are still things we can think of that don't fit into the system."
"..."
Then let’s create more systems that together encompass everything and can be sorted into boxes
Even the best library has an "other" section :)
@@Soken50 unrelated, but it sort-of interesting how “Even the worst…has…” and
“Even the best…has…” essentially mean the exact same thing
@@chri-k Imma need a minute to process this
Even the best phone has a limited battery = all phones have limited battery
Even the worst phone has a limited battery = all phones have limited battery
Even the best phone has a limited battery = Even the worst phone has a limited battery
phone = A
a limited battery = x
Even the best A has x =
Even the worst A has x =
All A has x
@@chri-k no. "even the worst" is used for positive qualities, whereas "even the best" is used for negative qualities.
This is actually the most understandable introduction to category theory I've ever seen. True, I'm somewhat familiar to it already, but explanations like that somehow make much more intuitive sense than "monad is a monoid in the category of endofunctors" stuff. Good job!
Everybody asks "What is a monad?". Nobody asks "How is a monad?". :
Well, it's understandable if you already have a grasp on it, but otherwise it's not.
@@marcus3d I only had classes of linear algebra and it's still somewhat understandable
I'm glad someone else hates that description of monads with a burning passion.
@@imacds LMAOOOO
For anyone curious, the motivation behind doing category theory is so you don't have to do any actual math.
Sounds like you got it.
care to elaborate?
@@sharif1306 category theory isn't actual math
@@danielyuan9862 what is it then? Alchemy?
Sounds like my kind of maths.
A teacher in my class of Type Theory in computer science degree, while explaining Category Theory he was doing a lot of little parenthesis about things that seemd unrelated and he said:
"What i'm explaining now is like we're going into a journey, and i can't resist to stop and enjoy the landscape and describing how beautiful it is to you."
After watching just a third of the video I must say I indeed learned a bit, and I am studying category theory for almost a year
This is why platforms like TH-cam are meant to exist. Love your work
They are meant to exist to serve Google with our information. But his work is great, because video uploaders like him are what draws people to Google's information collection platform.
What an excellent introduction to the subject! I wouldn't have believed that in 26 minutes someone could explain (in terms a beginner could perhaps understand) all the way to equivalences & natural transforms, but you sure did. Very jealous of your production values too. Cheers mate, bravo.
Your comments about adjunctions & Yoneda have tempted me to want to try and intuitively motivate them. So here's another explanation to add to the pile of unhelpful, vague descriptions you've clearly encountered:
- Adjunctions are the category theorist's version of minimization/maximization problems. Of course, the adjunction between free & forgetful functors is the paradigm example: the functor F:Set - > Mon solves a minimization problem -- it gives the least solution, subject to a particular constraint. The "least" bit here is the universal mapping property of the free monoid, which is the definition of adjunction applied to this example: F(X) doesn't have any more structure than it has to. The "constraint" we have to "solve" is that, for a given set X, we want X to embed into U(F(X)). So F(X) is the least monoid whose underlying set contains a copy of X. On the other hand, right adjoints solve maximization problems. Consider how the category of groupoids is a coreflective subcategory of Cat, that is, that the inclusion functor Grpd -> Cat has a right adjoint. This functor takes a category C to its "core groupoid" core(C), which consists of all the objects of C but only isomorphisms. This solves a maximization problem: core(C) is the largest groupoid which is a subcategory of C. Other category-theoretic concepts which also perform a minimization/maximization function (e.g. equalizers give the largest subobject equalizing two morphisms) are all instances of adjunctions.
- I like to think of the Yoneda Lemma as a category-theoretic version of Euler's Identity (that e^it = cos t + i sin t). Doing algebraic manipulations with cosines directly sucks because cos is not an algebraic object. Hence all the annoying "trig identities" that precalculus students have to memorize and later forget. This is a serious issue in fields like electrical engineering, where they have to do stuff like cos(2t) and stuff all the time (e.g. doubling the frequency of a wave function). So what do they do? Use complex numbers! Instead of working with cos(t), they'll work with e^it, which is super nicely behaved algebraically (because exponentiation _is_ algebraic). Then, at the end of the day, they take the real part of the answer, and it was like they were working with the cosines all along. A pretty neat trick -- indeed it's (basically) the reason complex numbers were invented in the first place -- and it all works because of Euler's identity. The Yoneda Lemma plays the same role in category theory: often a category you're working with will be kinda crappy (e.g. not having (co)limits). But the category Prsh(C) of presheaves on C is a very nice category indeed: it has all limits, colimits, exponential object, subobject classifier, etc. The Yoneda embedding allows you to embed C into Prsh(C) in a nice way, and do all your algebra in Prsh(C). If the solution you get (e.g. of taking a limit) is "real" (representable), then a standard Yoneda Lemma argument says that the solution (e.g. limit) existed in C all along. So Yoneda basically allows you to have imaginary (co)limits/exponentials/whatever, which might turn out to be real all along. Very nifty, and very useful.
Hope that helps, and hope you make more videos about category theory!
Thanks
my hero
What a wonderful insight.
I'm a math graduate student tackling with an exam on category theory right now, I'll try to keep in mind those points of view studying the subject.
Thanks!
Commenting before viewing, but hopefully the Yoneda explanation finally works for me. Yet to find it explained to my learning style, and I legit have a Phd in math.
This is a very friendly introduction to the Yoneda lemma! Thanks.
The adjoint introduction reminded me of what I've read on nLab, and also works well.
Imagine having the balls to share your dissertation publicly before knowing that it's right... big respect
i dont think the respected property here is his strong self-confidence, but rather his true mathematical curiosity - that he wants to know the amswer no matter what. (i don't know which is the one you referred to in your comment)
the reality of mathematics is that most papers are read by virtually no one. if someone finds an error in your work, its usually a net positive. firstly, the work usually can mostly be salvaged, next, it means someone actually cared about your work which is always nice.
Also, a math PhD dissertation ain't like handing in a homework assignment. You can be sure he's poured over every detail of it - many times.
We're all human, so mistakes are always possible, but by the time you're sophisticated enough to pass your quals and take all those grad courses, your confidence and capability in checking your own work will be enormous, and justified.
@@mathboy8188 A PhD dissertation, sure. But this is a master's dissertation. In fact, I'm just putting the final touches on my own :)
That's what you do when you write a dissertation. You put it out to be judged.
Thanks for the notice ! I now feel obliged to continue my CT series …
@@PefectPiePlace2 Yes, but I named my channel before knowing this proof assistant. And once I knew, I added an extra "s"
Yes please!
My favourite description of equivalence of categories comes from Awodey who, in his book on the subject, says: 'One can think of equivalence of categories as
“isomorphism up to isomorphism”.'
And isomorphic means equal up to an isomorphism :)
I love that book! really helped me grasp the subject as well as I do model theory finally
Don’t know if you’ll see this, but I actually go to Bath university as well, and after watching your 27 facts video, asked one of my lecturers about category theory as I knew that he studied it at PhD. That lecturer happened to be the very Thomas Cottrell that supervised you! Funny how the world works
Thank you, I finally understand the difference between isomorphism and equivalence. I’m a hobbyist Haskell programmer so the nitty gritty details of category theory aren’t usually required, so all I really knew was (endo)functors, natural transformations, and handwaving of isomorphisms as ‘these types can be converted into each other without loss of information’. Still trying to wrap my head around as adjunctions and representable functors though.
Imo, you can't really grasp them without knowing some very important examples of them. If you learn moduli spaces, you will 200% understand representable functors. Adjoints are a bit less clear to me too -- sometimes my mental picture is "something with the flavour of free generation" (e.g. forgetful functor GROUP->SET and "take the freely generated group" SET-> GROUP functor are adjoints), sometimes it's something a bit less abstract which unfortunately idk how to get across (the main example for me is pushforward and pullback of sheaves, but I assume that's too specific).
What do you use Haskell for?
chat is this real
@@tcookiem WTF YOU FOUND ME
I have always been thinking that category is similar to groups but have different rules. But in the video said the most epic thing that I've ever heard in math: "let's take a special case of category - sets"
adjunctions are actually quite simple, if you have two functors L : C D : R then you essentially have a representation of C in D (via L) and of D in C (via R) the adjunction then just tells you that for every c in C and d in D you have a natural isomorphism D(Lc, d) ~ C(c, Rd) which essentially means that whatever relations you find between objects in the image of L and the category D at large are "mirrored" as relations between objects in the category C at large and those in the image of R.
The quintessential example is the Tensor - Hom adjunction in abelian groups where it tells you that (a b, c) ~ (a, Hom(b, c)) which essentially tells you that providing a bilinear map from the product abelian group (a X b) to c is the same as for each a providing a linear map from b to c which is not that suprising after all that once you "fix" one of the variables in a bilinear map you get a linear map on the second variable.
So, from my understanding, applying a category theory on a subject is like:
1. Learning how to say the same thing in multiple languages.
2. Observing how each language uses its own methods to convey the same thing
3. Using that knowledge to learn how to convey information better, not only in your own language, but also in the others.
4. Taking that experience with you to the future. So that whenever you stumble upon a new language, by just saying the same thing in that one as well, you would already have a deeper understanding of it.
Good analogy
Physicist with a little passion for abstract math here. Well done! I was very confused about why I would be seeing them when learning about differentiable manifolds when I thought Categories would just be ultra-abstract things for algebraists.
Subscribed!
Congratulations on finishing your master's!
I'm making a presentation at uni next week about this exact topic, so this is incredibly good timing
I've wanted a gentle undergrad introduction to category theory for years now, so it's pretty funny that I find it now that I've taken matters into my own hands and started learning it formally lol
This is gold! Thanks! I am doing a PhD in engineering simulation and I am trying to formalize the wizardry I do to my sets of eigenvectors before building my reduced order models. I am not from a pure math background so videos like this one are invaluable to kickstart my understanding of these math branches :)
this video gave me a bird's eye view of my lifetime math adventure. From 1st grade maths until I graduated college and any other math I encountered. Different math rules, new kinds of numbers but all feel somewhat the same. That sameness was explained by this video.
i absorbed very little of this but i think if i was more awake or had a better grasp on maths this would've made a lot of sense. maybe. i think your video is good
That was some pretty cool dense mind blowing shit I feel I've barely understood but still I've enjoyed. It's always nice to have new videos from you Oliver
I'll always remember that when I was studying, in the 1990s, I borrowed a book about category theory at my college's library. The card inside showed that nobody had borrowed that book since 1978
"cathegory theory is not a religion"
*dramatic music*
"HERE WE BEGIN TO GLIMPSE THE TRUE POWER OF CATHEGORY THEORY"
Wow, great video! I've heard some very difficult math lectures on category theory, so I never really understood it very well. But this video is excellent because you give several useful concrete examples, which greatly helped me to understand what's going on.
Hey, statistician here who's been really loving learning about CT, and your funny video was actually quite helpful. Thanks for this one which is perhaps more helpful... perhaps. Thanks for the delightful and informative content ✨
Adjoint functors make sense if you think of them this way:
The category C (dom of left adjoint and codomain of right adjoint) is the category of interest
The category D is a category where you can manipulate stuff and use it to "predict outcomes"
The left adjoint is like encoding a vague object (or morphism due to functoriality) of C
The right adjoint is like decoding a well understood object (or morphism due to functoriality) of D
The unit shows where you land if you immediately decode something after encoding it
The co-unit shows where encoding something you have decoded lands after immediately encoding it again
The triangle laws show how functoriality and naturality work in this framework and the homset isomorphism shows that there is a bijective correspondence between manipulations of encoded objects to yield certain outcomes/predictions and processes from "unencoded" objects to decoded ones.
So in a sense, adjuntions are like a version of substituting morphisms of C with an easily encodable domain, with processes of D with an easily decodable co-domain. It is somehow like predicting the weather using theories instead of waiting for the phenomena to happen.
I was thinking exactly the same thing... in totally different words
@@popularmisconception1 So there is intuition behind adjunctions after all!!
Thanks a lot! I was wondering how much priority it should take in my self-study of maths, and understanding that categories are a certain simple mathematical structure that happens to be useful to encode some aspects of maths, but doesn't easily deal with *everything*, is super helpful.
The moment where you introduced the category of categories (21:00) was such a sublime experience, it might well be the happiest math-related moment in my life so far. Thank you for this wonderful video:)
Categories remind me of classes in programming. Good video!
Your video comes at the perfect time. I haven't been reading category theory per se, but instead a translation of Gödel's work on undecidable propositions. Even though this is unrelated, I came out from watching this with a greater sense of understanding. Thanks!
I studied pure maths and we only quite briefly touched this topic as preparation of module and ring theory. I am still amazed how you can use the general definitions and see them play out in specific ones. Very interesting introduction, thanks a lot for the effort. :)
I gotta say, I learned more from this video about category theory than from my (failed) course on category theory in uni. Though I was a lot less diligent studen then than I am now. Great video, thanks for doing it!
3/10
didn't do a Morbius meme during the morphism section.
the biggest lie youtube told me is that this video has the duration of 26 minutes and 19 seconds, I've spent more than half an hour watching this and I'm only at 18:02 >:(
I'm doing a masters degree in september in Algebra, geometry and number thoery and I have to know the basics of category theory and your video helped me know that I guessed right the fact that you could have a "category" category (not in those terms but the idea is equivalent). That makes me feel like I got what it takes to success in getting my PhD
BTW the subscribtion was fast. Damn I love the way you express yourself
This was the most fun I’ve experienced from a TH-cam video in a while! You’ve done something casual and very fun, and I’m looking forward to looking through the links you’ve included to learn more
This is by far the best introduction to category theory I have seen (and I have seen quite a bit of those).
Brilliant!
Sorry any mistakes, my english is a work in progress.
This was a nice refresher on things I studied 20 years ago
This is very helpful! I'm reading some abstract AI papers right now, and just knowing what all the symbols mean makes it much more comprehensible!
This is a fantastic video, wow! Props to you dude, took a subject that is generally INCREDIBLY boring and unintuitive, and turned it into an engaging introduction where I finally felt I could follow along. Clearly a lot of work went into this, I'm looking forward to seeing what else you come up with in the future!
I've been trying to get into more abstract mathematics like group theory, and this is a great and intuitive explanation of category theory!
You still should start with group theory tho... That's like, maybe not kindergarten stuff, but high-school-ish. Can be understood even with relatively little prowess in the abstract. Category on the other hand, is post-grad stuff, you better be prepared for a war of attrition with it. ^^
I had just finished my a levels a few weeks ago and will be studying maths as my major. This is my first time learning something about category theory. The video is wonderful and quite comprehensible to me. Thank you for such a wonderful video and I hope I’ll be able to explore more through this channel👍
Man, I'm at the first year of my career, and you made me think in everything in such an abstract level. I fucking love it
The only case where adjunction really obviously makes sense to me is the fact that currying/uncurrying form an adjunction (since (X * Y) -> Z and X -> (Y -> Z) are basically the same thing).
In all other cases my eyes start glazing over as the nLab page functs me into the (Category of Pain)^op and my brain becomes naturally isomorphic to strawberry icecream mmmmmmmmmmmmmmmmmmmmm
Take a look at forgetful-free adjunctions. Constructions like free groups, free monoids, and free modules all arise from such adjunctions.
nLab is a dangerous place... you can go there to look up a concept you know and understand, and after reading their exposition of it realise you understand it no longer.
Is this object oriented mathematics?
Since I started school for electrical engineering, I felt like the math classes were leaving something out. There’s something I’m looking for, like some kind of explanation of math that my teachers haven’t given me. In my mind I could barely explain to myself what I was looking for. I knew it had to do with boxes of logic. If you start with AB CD for example, there would be two different boxes of explaining this backwards CD AB and DC BA. See how the two different things can be seen as two different ideas based on how big the chunks are? Those are like boxes to me. I want all of math organized in boxes. Keep in mind I’m not even sure if I’m expressing the real thing on my mind. But I’m looking. I didn’t completely understand the video, but I’m drawn to it, and every once and a while in the video my curiosity was satisfied. If you could make a video, in MUCH MUCH more detail, I’d appreciate it. It seems like every five seconds of this video could be expanded into another 2 hours. Thanks for the video, as is though.
Probably the best introduction video to category theory out there. Please elaborate on the 'downsides' of category theory that you alluded to in part 2 video
wow that rock paper scissors example really clicked for me. now i get how important associativity is for keeping structure simple and consistent.
if i invented rock paper scissors i’d be pretty upset about it not generalizing to larger groups of people
I'm struggling to understand it. Could you explain it?
@@Zxv975 This is how I understood it:
You know that paper beats rock, rock beats scissors and scissors beat paper. Now let's make a binary operation x that takes two players as input and produces the winner as output, then we can write the following:
rock x paper = paper x rock = paper (meaning, in a game of rock and paper, or a game of paper and rock, paper wins);
paper x scissors = scissors x paper = scissors;
scissors x rock = rock x scissors = rock.
Now, how do you calculate rock x paper x scissors? Assuming the operation is associative, you can go about it in two different ways:
(rock x paper) x scissors = paper x scissors = scissors;
rock x (paper x scissors) = rock x scissors = rock.
But as you can see they produce different results, so the associative law doesn't hold, and the operation is non-associative.
22:48 I think I was trying to label this concept about a year ago and I referred to it as "the negative space around no true Scotsman"
I love the way you think about this!! So many interesting analogies. I've taken classes on category theory, read countless Wikipedia articles, and lots of TH-cam, and this video helped me get the point the best way I've ever seen
Thank you very much for that excellent explanation! Several times I had to stop or repeat parts of the video to read thoroughly the symbols or to think over what I had heard. But just this is the advantage of videos compared to lessons. I studied math over forty years ago (in Germany) but we never had this topic. Yet I know isomorphisms from group theory ... It's easier to understand category theory if you are familiar with some examples.
i like that you chose to reprezent composition of 2 morphism as fade animation to the new object instead of drawing another line
Category theory is a unification of all of mathematics because it abstracts the structure of mathematics in to a common language. It is like a universal language. It would be analogous to spoken languages. If you learned a "master language" it would allow you to speak and understand all other spoken languages. It's not actually this because no such master language could exist as it would essentially have to be all languages and category theory doesn't actually let you understand other mathematics. What it does though is provide a common abstract interface so all those different mathematical areas can be translated in to a common language and then one can see how the various different areas relate to each other and see the deeper structural aspects. It turns out that almost all mathematics(maybe all) uses basic common ideas and ways of thinking which one wouldn't know otherwise. Category theory is extremely powerful because it lets one see the underlying machinery. It, say, is analogous to how understanding that all matter is made of atoms. From the atomic perspective it unifies all things and things that might have seemed different is just one variation on the atomic theme.
Category theory is difficult at first because one doesn't realize the concepts it presents are "universal"(unless they have already learned a large amount of mathematics). Like anything one just has to learn it to see it's use.
At 6:29, I like to emphasize how nontrivial this assumption is. A direct flight from Athens to Berlin and a direct flight from Berlin to Cairo is not enough, by itself, to guarantee a direct flight from Athens to Cairo (even if there is one in fact).
I can‘t resist to ask for more. Very instructive video. I think I can say I came a little bit closer to the spirit of category theory. I watched a lot about this topic and seeing a lot of explanations. The main issue for me regarding CT is that it is almost always everywhere explained the same way regardless of the known problems that one faces with it. But you go in the right direction.
thanks so much for the clear explanation dude, very nicely put together
As a math major and enthusiast, greatly enjoyed the approach here
Best intro ever, thank you so much for the karma I got on the Haskell subreddit for your last video. I hope you post this one there before I do.
Finally, the long-awaited sequel is here!
My Understanding of Category Theory got better after I studied "Abstract Algebra" and Group theory. Many courses on Category theory state that you need only basic algebra to understand Category theory, which I am not sure.
Would've been cool if this was in SoME2
I considered it, but I do have another idea or two I could try for that. Won't make any promises though.
@@OliverLugg SoME 4 coming up?
Congratulations on the MS thesis. No comment was on the math, but the writing seems clear, simple, and engaging.
I understand that Eugenia Cheng is an expert in category theory and has managed to apply it to many useful aspects of society, which has recently sparked a great deal of interest for me in the subject.
Thankyou thankyou thankyou! Thanks to the coconut video I became able to understand the maths in an advanced physics video course, it's really nice to know more details of it.
Congrats on your thesis being submitted and a very clear video on category theory to boot!
A+ use of the "they're the same picture" meme there
The best! Getting to the essence quickly is the sign of mastery. Thank you.
"Hopefully there are some bells ringing in your head right now..."
Yes... and I am 100% sure its the fire alarm.
Ohhh that's why coffee and donuts go so well together! 😮
Now let's make a funny joke video on complex differential geometry and then a serious follow up video!
This video is very entertaining. If someone wants an example of how people sound when writing to look smart rather than simply say they are entertaining, entertainment which allows themselves to get to the grand details, feel free to read below.
I think gluons, which help to assemble quarks, which make up protons, neutrons, and electrons, which make up traditional usable material such as hydrogen and water, have mass defined by the pressure inside and outside of whatever balanced system they are in, radioactive decay bound to happen in the system they are in, and how well the gluon orbits its quarks. In this hypothetical or theoretical model, quarks are composed of super-quarkinos -- 7 each, and gluons are composed of super-gluoninos -- 14 each; 2 hooks, and 12 load-bearing threads. 2 of these 'threads' are most likely stuck together in the middle of the gluon. I also think that super-gluoninos hold charges, such that they aren't ever quite equal to simply positive or negative but are octonion charges or sedonian charges. This is why I think there is a most massive reasonable atom in terms of proton and neutron count and this is why I think said biggest possible reasonable atom using our current model of non-anti-matter safe assemblage atoms has a half-life and radiation so harsh, that when an atom of it undergoes radioactive decay, it ejects a gluon or super-gluonino or gluonino -- and directly. There is probably a mathematical group or category or both that can classify this theory or hypothesis in more detail.
So yeah, kinda a theory dump, but also a congratulations at the same time.
This has been a nice follow-up to the notorious nut video, haha. If categories alone are still somewhat understandable, categories of categories completely stump my intuition.
I remember when the internet was all about stupid cat videos (and pr0n).
This gory cat video isn't what I signed up for! My head hurts.
Serious now, that was heavy stuff. You may think this is trivial, Oliver, since this your thesis is faaaaaaaaaar more obscure, but to those of us casually interested in math, it was really hard. Anyway, you have a great voice. Keep doing what you do, please!
Finally some light in this confusing "generalisation" of sets. I was trying to get a taste of what the hell Groethendiek 's deal was, but it's even higher bird's eye on even this. Whatever "getting stuck" is the path to understanding mathematics.
I'm a physicist whose pure maths only got as far as elementary group theory, so as far as I can make out category theory is a sort of generalisation of group theory, as group theory is a kind of generalisation of set theory. Which allows us to explore the interconnectedness of all things. Sort of. So you can translate a problem in number theory or linear algebra into a problem in topology or complex analysis, solve it there, and then translate the answer back to solve the original problem. Kind of like Wiles' proof of Fermat's Last Theorem. Sort of. I think...
Anyway, it makes more sense than "constructor theory"...
Group theory is not a generalization of set theory though. Group theory is its own kind of special fuckery, I can only recommend 3Blue1Brown's video on the 196,883-dimensional monster.
Well up to and including functors I was mining value. Once you started talking about natural transformation, and equivalence, absolutely no idea. Overall a nice introduction to major concepts and the ideas they represent.
I finally finished watching all the videos that are over a half hour long And that I care about. So I just got this video on my page. Still haven't seen the coconut nut video, my introduction to you was the 5-hour long mass extinction debate.
At 19:58, I certainly understand why you omit going from Sets to free groups, but for the interested viewer, it's really quite simple, beautiful, and for me at least, fascinating. As always, thanks for your consideration.
I'm impressed with you having enough balls to stop yourself from talking about elements of object without authomatically saying something like "given that \mathcal{C} is concrete", or talking about "category of categories" without mentioning "small"
My most favorite example of a category is parametric functions that are differentiable both in their argument and in their parameters, aka neural networks.
(One does have to phrase it in a certain way though to actually make them practical)
It is even a symmetric monoidal category, nice.
i've only ever taken a discrete math course and this seems pretty dope
Thank you so much for this! Hope to see more of such content on this channel
Great indtroduction! Next video: natural transformations, limits and colimits, and the Yoneda lemma? Perhaps Adjuncionts? (Which would probably would require a video of its own?)\
Great job, looking forward for more!
This popped up in my feed, and I thought "category theory, that's the thing Tom Leinster works on, okay I'll watch this" (his partner was in my department at work, and he worked in the same building as me somewhere else). And his is the first name that comes up here!
The thing you said about the scaffolding
That exact thing is said about mathematics _themselves_ :
That mathematics is scaffolding with which you construct reasoning, remove the scaffolding and the construct remains self-standing
hopefully with fewer bricks
Honestly, I think the only reason why I get Category Equivalence is because I've already gone through all of these emotions when learning about P & NP from my CS degre and the concept of NP Complete, which is basically "this problem is just this other problem written differently."
back when the video came out I became so intrigued that I ended up studying category theory just to make the mum joke at the end with more conviction
ps: not that this matter but my introduction to your chanel was Pitagoras broke music
Holy crap you actually did it you madlad! Also congrats on your channel starting to pick up steam!
The Terence Howard joke took it over the top. Well done.
In my yet incomplete theory of cognition, certain mathematical structures arise once and again in different contexts varying from physical to sensory, from sensory to conceptual, from conceptual to imaginary, from imaginary (but plausible) to fictitious, from conceptual to absurd or sound, from neuronal to neural. Category theory is not my forte but, thanks to this introduction, now I know where to look up and what to do in my next step.
The image _[shown here at __23:17__]_ appears to be a diagram related to category theory or a similar abstract mathematical concept. It shows two diagrams, potentially commutative diagrams, which are used in category theory to represent objects and morphisms (functions) between them. The left diagram has four objects \( A_1, A_2, A_3, A_4 \) with morphisms \( f \) and \( g \) mapping to objects \( B_1 \) and \( B_2 \) respectively, which then map to object \( C \) through morphism \( h \). The right diagram seems to involve tensor products of the objects \( A_1, A_2, A_3, A_4 \) and the associated morphisms. If you have questions about the specifics of this diagram or the concepts it represents, feel free to ask _[ChatGPT]_
found this only because I actually rewatch the joke video regularly, subbing
I watched the other video like 3 days ago so happy to get this one now x)
3:50 The funny thing is, I paused when you mentioned Twelve Angry Men, because I'd argue the setting of Twelve Angry Men is very important, even if it's not visible right away. It's a very American story about how Americans of that time period would see the importance of the rule of their laws. It's punctuated with an immigrant reminding another man how important it is for Americans to value their independence to form their own opinions. The whole point of Twelve Angry Men as a progression from conflict to resolution is that they all try to act civilly and self-importantly at first, until through this very American depiction of argumentation their every implicit bias is laid bare. It's a trial but the ones passing judgment are themselves being judged. It's very much peak Americana in a very optimistic light, and that's part of the joy of watching that movie; you get to feel the importance of a society that takes the concept of legalism seriously, across class, across age, and across vast differences of life experience.
I wonder how telling that will be in the revealed importance of Category Theory as I continue watching.
I love the fact the TOP considers all things w/ 1 hole equivalent😂
You know, as a non mathematician*, I found this video very helpful in understanding this subject.
*My degree is in molecular biology. (My hobby is trying to understand mathematics)