I wouldn't say it's different from other theories, exactly. It can be used as an overarching theory to describe the others, but it is also a theory in its own right. You can see the complete axiomatisation of categories at 8:53, and there's no mention of what the "objects" are; they can be categories if you want, or they could just be vertices in a graph. So there's a category of categories, where category theory appears both as a big framework and as just another object, and it gets self-referential. But you have to restrict it a bit if you want to avoid Russell's Paradox. (That's if you're working in set theory - if you're in type theory then I don't think Russell's Paradox is an issue.)
@@mikemondano3624 I agree. Syntax and semantics are not common words in math I believe. The "structure" I meant here is also more in theoretical computer science perspective, from my understanding that is nothing concrete relevant to practice can be defined in computer means. I borrow syntax and semantics to math theory, which of course do not justify the similar quality of theories in math.
@@davidknipe4113 there are set-theoretic frameworks (different from ZFC) which avoid Russell's paradox by using the notion of classes (like in von Neumann-Bernays-Gödel set theory (NBG)), without needing the common approach of working under Grothendieck universes and small categories.
I’ve watched like 6 videos claiming to explain this and you’re the first to actually do it in a way that’s actually comprehensible for people who aren’t already pursuing a math phd. 11/10 explanation. Also I love your description at 9:30 “…we encode it into a common template and then we carry out a bunch of reasoning and discovery using the affordances of that representation.”
People pursuing a PhD may be more likely to have no knowledge of this. Every degree means a narrowing of interest and expertise, not a broadening to the entire field.
@@mikemondano3624 I'm not sure how you took that away from my comment. Is it completely reasonable to assume PhD's have at least an undergrad's experience and will generally have an easier time understanding new mathematical systems without a dumbed down explanation designed for laymen. I can change my comment to say grad student if that satisfies your pedantry.
@@HuntersHunter Many people assume that a PhD means having a complete understanding of a subject. I wanted to point out that it is not cumulative as most secondary school subjects are. Many PhD's have completely forgotten early learning in their subjects and even grad students sometimes struggle correcting undergrad papers. Many PhD's are ill-equipped to take in new systems. It really is an individual matter depending on interest and background. Mathematics is vast.
Whenever I watch these kinds of videos, I feel like they always get to the same point but never go further as to how it is actually useful to use categories in practice.
It helped me to watch a somewhat complicated tutorial on the application of CT in epidemiology in Julia. Although it does requires an understanding of some basics of CT and its syntax.
@@AlexBerg1 To expand a bit on what siquod said, and give you (or anyone else who's interested) a starting point to Google / read up on more, what you're getting at is something called the 𝘊𝘶𝘳𝘳𝘺-𝘏𝘰𝘸𝘢𝘳𝘥-𝘓𝘢𝘮𝘣𝘦𝘬 isomorphism. This is by necessity extraordinarily simplified, but as a super quick overview: Folks already familiar with Type Theory will likely have at least heard of the Curry-Howard isomorphism (or correspondence). To recap/summarize: this is an isomorphism between a (purely functional) programming language - nominally the typed lambda calculus - and a fragment of intuitionistic propositional logic, where there is an equivalence between programs on the one hand and proofs on the other. That is: given types T₁, T₂ and T₃ in your programming language, and a function (or program) 𝘧 that takes T₁ and T₂ as input. returning T₃, there are equivalent propositions P₁, P₂ and P₃ in the intuitionistic logic such that there is a proof 𝓡 that you can derive P₃ from P₁ and P₂. And thus an isomorphism between {T₁, T₂, T₃, 𝘧: (T₁,T₂) ↦ T₃} ⇔ {P₁, P₂, P₃, 𝓡: P₁∧P₂ ⊢ P₃} Curry-Howard-Lambek extends this idea by demonstrating that there is a category 𝓒 (nominally a cartesian closed category; meaning it has useful things like sums and products etc.) added to the isomorphism. So there is a 3-way isomorphism between the language, the logic and this category. And in 𝓒, the objects (nodes) are, as siquod mentions, types; and the maps (arrows) between them are functions from one type to another. So our function 𝘧 and proof 𝓡 correspond to an arrow from the node corresponding to (T₁×T₂) to the node corresponding to (T₃). (As an aside, the presence of the product in that starting object is why we need it to be a cartesian closed category.) Again, that's simplified. There's lots of other stuff going on in the category, such as arrows corresponding to lambda abstractions and β-evaluations... and ooh! polymorphisms! Et cetera. But that's an entry point if you want to look it up.
Good going. Best intro video on Category Theory, so far - does give me a clear high level understanding what category theory is, which Eugenia Cheng's summary encapsulates nicely. Love how you proceed from examples to abstractions, and your recursive 3d "bookshelf" diagram of a what a mathematical theory is. Excellent visual design. There are two main conceptual gaps that I want to see filled in: motivate the directed graph representation of a theory better with examples, and give a more explicit example of all that is included in a whole theory by giving a simple example. It makes me think that category compensates for a mess left by mathematicians for centuries - the lack of an explicit notation for representing an entire theory.
What a beautiful talk. I enjoyed it as well as the visual aspects. For some reason, the audio cuts off abruptly and then comes back but that is just a very minor detail. All in all, good work. Thanks!
While everything you have said is true, and historically category theory clearly evolved from the seeds of more elementary mathematics, I think it is more apropos to think of category theory as the theory of composition. Category theory encodes/codifies the most basic rules/axioms/ideas of how to compose things from building blocks. Objects our our basic building blocks which we can then combine them in to various structures. While the details are a little more complex to make it all work out, the fundamental idea is that of "blocks"(objects) and "glue"(morphisms and composition). Since pretty much everything in the universe is things that are composed of other things this means that category theory is essentially universal. It studies/abstracts/makes explicit the very thing that exists in almost everything. With some thought and experience composition seems to be the most basic, universal, and easily understood idea that we humans have... in fact, it is likely impossible for anyone to think of doing anything that is not based on some form of composition. Because of this, category theory sort of went at the meat and potatoes of what it means to do "math" with stuff... which even includes eating the meat and potatoes. It's amazing that such "simple theory" can be so powerful but it's similar to how the atomic theory is so powerful. Simple but universal ideas are the most powerful. So the video and Paul explain things more from the historical and learning view but, in fact, the categorical view is from the category being the seed or atomic idea giving rise to everything else. E.g., atoms form to make substances, it is not substances that decompose to form atoms. Although the concept might be dual in some sense there seems to be an asymmetry that actually exists. The main issue is that we naturally will learn category theory from the top down(or bottom up if you use Paul's perspective) and so we always have this view available, but the hard part is learning to see it from the other perspective, that is, as category theory being a generative element/seed that gives rise to (almost) all the complex structures we have in math, science, thinking, and society. Category theory is best though, at least one way, as the theory of how to compose/combine arbitrary things. Turns out one can combine arbitrary things in all kinds of meaningful ways.
@@RafaelRocha-po2yi Glad it meant something to someone ;) Be aware many details are left out, of course.... But like many thing sin life when we are first learning it, we use a different mechanism to get our bearings and then after we have learned it things have a different feel. It's like using training wheels as a kid to learn to rid a bike. It's not how we ride a bike but it helps us get going much quicker with less pain. Ideally we would learn category theory from the ground up, from day one before anything else that would obscure our true understanding of it.... but that is now how we humans work. We have to use building blocks to learn too.... And category theory being a theory of building blocks is too abstract to be teachable... but from someone that has studied category theory for a while it all seems obvious and as if it could easily be understood... and yet it can't.... or maybe in some sense it is and if we make it in our journey of learning about "category theory" eventually we arrive at "Categories for the working mathematical" after many years of working out categorical examples like learning how to ride a bike, learning to speak, learning how to add, etc. For me, category theory is a language that expresses the concept of how to concretely think about composition. It makes composition about as tangible as it can be(at least from everything I have ever experienced). Of course I don't think I could have gotten there first without learning algebra, calculus, arithmetic, etc. All those provided concepts that could be used to provide examples to say "Hey, see that, that thing right there... NO! not that, this thing over here... That is category theory!". Of course it was being screamed at me my entire life but only when I was ready could the training wheels come off(only for, if the past is anything like the future for another type of training wheels to have to come off). Probably a great way for the average person who doesn't know much about math but understands at least some basic physics and chemistry is to treat category theory is a sort of analogy theory about the world of structure that atomic physics is for the material world.
@@kodfkdleepd2876 While you are insightful in describing how category theory operates, I believe the video’s approach is much better for explainingt the idea of Category Theory. All theories need to have a motivating Semantics and an operational Syntax for humans to grasp what’s going on here. The way he laid out the semantic motivations of category theory made a lot of sense to me. Meanwhile, although I have read a fair bit on category theory, to the degree of delving some ways into a text book on elementary Category Theory, I still flounder at grasping abstractly what composition is about. I know it doesn’t mean just the combining of objects, but it is a generalization of the function composition that exists in group theory. It’s a powerful concept, but as an introduction to the subject, it overwhelms rather than clarifies.
@@abj136 I'm not denying that what you have said but there is a difference between learning and understanding. Learning is precisely non-understand which is why we have to learn to understand(else there would be no need for learning). I'm sure you've experienced the difference between learning something and then understanding it. The goal, of course, is understanding. I'm sure you've also experienced "understanding" something without understand something. E.g., understanding motivating examples of something without really feeling you understand the big picture. E.g., "I get that but I still feel like I'm missing something". Generally when learning something it takes time for all the individual concepts to sort of mesh together enabling one to start to grasp the true depth of the subject mater. This is true at all levels too(just as much as learning at theorem or topic as much as it is all of mathematics. Learning is fractal(it is a recursive process)). Lawvere has a very good book - Conceptual Mathematics - on what really is going on in that it approaches the subject matter not first principles. It may not immediately make sense on how it ties everything together but with a little bit of time and maybe a reread it will open up all of mathematics. When I read it I saw concepts that took me years to discover laid out(of course I have hindsight but I've not seen any other books approach it like this and all category theory books start from category theory rather than the foundations). It's actually extremely easy stuff. Lawvere develops category theory from function theory in a way that allows you to see why category theory goes to the lengths it does and it gives parallel structures. It is a book that teaches rather than simply presents the theory. It's far from a perfect book but is one of those rare books that actually explains mathematics in a meaningful way(although it is entirely possible that a novice may be just as much as confused by it as any other book I imagine one would get much more out of reading it first that say Categories for the Working Mathematician). I've also found most "introductions to subjects" that attempts to give a layman some understanding actually completely fails and generally only confuses them except they think they have some actual understanding. Trying to condense such complex theories down in to short presentations actually does the opposite. Of course it might motivate one to put more time to learn the subject matter(and maybe that is the real goal), in the long run it is counterproductive because it creates a crutch that most people will use for a long time and slow their learning down. They think they have some deep understanding and then try to mush everything they learn afterwards in to that understanding so they can feel like they continually understand. Category theory is one of the simplest and most easily understood theories excluding it's complex jargon. Why is this? Because we all use category theory all the time. It's how we think. Of course that could also make it difficult to learn since on one hand it is describing and encoding how we think(or how things work) but using a somewhat meaningless nomenclature and description to do so. E.g., the theory is very alien to how we think but the thing the theory describes is extremely natural. This "duality" may be what makes things complex to learn because the meat and potatoes of the theory is obscured by the language used to present the theory and the examples given that partially obscure since they must bridge the gap using some other subject matter. E..g, category theory has nothing to do with topology, group theory, etc... and yet you will find these examples abound in trying to explain category theory. Oh sure, you say it makes sense... but the point is that they can obscure rather than help in some cases. Usually for a novice it hurts rather than helps because it actually reduces their focus on category theory, the very thing they need to focus on to see and understand, by presenting them with something else. Magicians use this trick all the time but educators shouldn't. That is, obviously every group is a category but not every category is a group. So when we try to learn category from the perspective of group theory we are trying to stuff something more general in to a place it doesn't fit. It sorta works and sorta doesn't. What it does is both. It helps us understand but also hurts our understanding just as a crutch helps us but hurts us in that we might rely on it too much. Ideally we would learn category theory first(first set theory and logic of course) then group theory. But we don't learn optimally and we don't discover optimally so it is what it is but if we can understand the inefficiencies it helps us progress faster.
This was incredibly helpful. Thanks so much for taking the time to make this. The visuals were terrific and so was your explanation. I would love to see more of these in the future too. I've been trying to wrap my head around category theory for a while and it's been so abstract that I can't bring the definitions and diagrams into focus, but this definitely helps frame it very well. I am always on the lookout for videos/books where someone works through an application of category theory and explains the goals and benefits of individual problems they are solving or discoveries they are trying to deduce. If you know of any resources like that (or plan to make any), I'd love to see them.
I love the visuals! This is very much what I see in my head when I'm working in category theory. I was going to ask about your graphics stack, but I see you answered it---highlighted here for others with the same question: “ I do my 3D graphics in Cinema 4D and everything else in After Effects. I found all of the individual topological models for free on 3D printing websites!”
I've been trying to understand category theory from time to time searching in youtube or wikipedia and never succeeded, until I found this video. Excellent explanation.
Thank you. this is a short but comprehensive explanation, in-turn with the proper axiom in a written and visual format that is direct and to the point. Thank you teacher
Really great presentation, but the obvious next question is how do we actually use/manipulate these graphs? There's a lot of (infinite) connections going on! What kind of math do you get by building very simple categories? Is there something distinct about categories with only a finite number of connections? What kind of math would even have so few connections? Can we use already existing math to analyze categories? Like graph theory or topology? Or are they not powerful enough? Can we operate on one or more categories? And of course, how would I learn any of this?
this video felt nice & graspable; other videos about cat theory were confusing; but (even this would have been 30% more confusing) if (i didnt have current level of understanding about group theory) not to imply that even other vids would be easy with my current knowledge
4:50 perhaps the motivation for directed graphs are that we are interested how things interact with one another rather than the definition of the thing itself.
My understanding is this. The DAG is just what happens when you map everything within the domain to everything it can possibly be mapped to. In CS, if I had a circle object, and I wanted to map it to other objects, I’d store that relation in a dictionary/hashmap. If you did that for every object inside that, and repeated until you covered all possible values within that group, the data structure you’d have created is a DAG. It just so happens to end up being a DAG.
@@Isaac-zs3db thats right about all possible relations with other things. But its not acyclic. You can have loops in a category. Isomorphisms is one i can think of; two arrows between two objects forming a loop. Categories are transitive directed multigraphs with loops, not DAGs.
Thomas Hangerford introuduces Categories in last chapter of his Algebra.This gives the impress that category is some sort of structure similar to groups,rngs and etc.. .
Directed graphs are a great way of visualizing mathematical categories. I think if you wanted to escape that idea, you can try to represent categories with morphisms (each morphism being a relationship between 2 sets: can be a non-Euclidean line segment, a graph line, nodes, or a map. Morphisms are types of mappings: we typically use object streams with our mappings in comp sci. There are also concrete map structures / systems.
Nathan Jacobson in his Basic Algebra one utulizes set theory and no mention of category;but he starts Basic Algebra two with categoris.this gives impress there are certain things which can be dealt with no category however after some point we need categoris.
A very nice presentation, very thoughtful and careful, with excellent animations and style. I worry about one particularity of the presentation though: is there anything said here about category theory, at this level of generality, that doesn't also describe mathematical logic? We have a theory of groups, a theory of vector spaces, even a theory of topological spaces if you set things up correctly. Their classes of models are all groups, all vector spaces, all topological spaces, et cetera. My worry is basically just, it would be nice to have a motivation of category theory in particular distinct from the motivations of say, first-order logic. One way to motivate the move to category theory in particular is to say looking at classes of models isn't enough, we need finer individuation, more structure, etc., we need relations between structures. Maybe that's just my view because I'm a logician and view things from that lens. Anyway, excellent video! Thanks for sharing it!
By Curry-Howard-Lambek, intuistionistic logic is isomorphic to the cartesian closed categories, and linear logic is isomorphic to closed monoidal categories. Seems the category theory tries to be the most general. I have a hunch if you try to find a logic corresponding to all categories, you probably can't prove much as you removed too many axioms. All you have are Identity (if you have a term, that proves it exists) and Composition (if you have two proofs they can combine to a bigger one). Thus category theory becomes the building blocks to make more logics: you'd need to add something by choosing a particular category, defining its objects & morphisms.
I wonder if another way to see the difference is in the emphasis on maps *between* different categories/logical systems. Representation theory studies functors from groups G, each considered as a 1-object category, to the category of vector spaces. Being able to consider the category of all such functors-all such maps between logical theories-as itself a category is something I think is not emphasized or well-developed in logic (as far as I know).
It seems to me like a lot of very smart people were bored in the US. I'm deeply against infinite thinking and geometry seems pretty pretty out of space for me. Can we make Plotino great again?
Great presentation! It's amazing how much insight/motivation for the purpose and thinking-style of category theory you were able to pack into 10 min. Can I ask, what software did you use to create the 3d visualizations for the category of topological spaces?
Thanks Nick! I do my 3D graphics in Cinema 4D and everything else in After Effects. I found all of the individual topological models for free on 3D printing websites!
Great video! Very well explained, it's understandable even for a layman. On a side note, an anyone tell me the name of the topological space on the bottom right at 5:28?
At 3:06 you mention the structure of mathematical theories is to be about something which is a semantics and then it has a syntax. The semantics of group theory is symmetry. My question is what would I have to learn in order to create my own semantics and syntax for anything I find interesting? Also is there some limit to the type of objects that would be considered mathematical? So if I want to create a semantics about symmetry that would constitute a mathematical theory but if I did the same for the concept of love that wouldn’t be mathematical. So is there some range of objects mathematics can theorize about? I really like the idea of being able to create my own semantics and syntax for anything I find interesting. Even if I want to reason about symmetry or some other object/concept that already has an existing theory or branch of mathematics, it would be great to be able to know how to build a mathematical theory for it.
Category theory lets you make custom theories. It gives you certain guard rails, ones that are satisfied for a great deal of other mathematical theories. But yeah: the days of being forced to play someone else's mathematical games are over; you can make your own!
You would probably find fascinating Conway’s construction of Surreal Numbers as it was done as a recursive exercise of inventing syntax and semantics for things.
To build a mathematical theory for something, you first need to identify the *operations* which can act on the objects. Can you "combine" objects to generate other objects? Are there unary operations, which derive an object from a single object? After that, identify the *properties* of operations: "combining" two objects is commutative? Is it associative? If we apply a unary operation twice, do we get the original object back (i.e. is it an involution)? From these operations and properties/laws, we can construct theorems (consequences of those laws). The branch of mathematics that most directly studies such theories is Abstract Algebra.
you don't have rigid definition of the rules and sequences for generating different categories, so there are infinite possible categories for the same object
Thanks for this great video. What resource (book, tutorial, course) would you recommend to start learning this subject? Having some under grade math knowledge, is it too advance for me?
I recommend "Category Theory for Programmers" by Bartosz Milewski (both the book as well as his youtube lectures), and the textbook "Category Theory" by Steve Awodey.
@epgui I've already watched Bartosz's RU-vid lectures, but they felt somewhat hand-wavy. I'll take a look at Awodey's book. Up to now, I've read/watched many introductions about the topic, but when trying to take the next step and learn how to use Category Theory in more concrete scenarios and daily design, the resources I've found are too difficult for me. I'm trying to think in these terms to actually apply them, but I can't go further than the canonical examples. I'm looking for some resource between hello word and abstract academic magic. Thanks for answering
@@codesur Yes, I think the Bartosz resources are a little bit more hand-wavy by design. I think they are helpful for gaining some practical intuition, but if you want to really understand the topic (even just the basics) you will probably need multiple resources, and a large number of examples. You'll probably need to embrace both the applied and purely theoretical perspectives as well. (PS: I am still early in my CT learning journey). I found Awodey to be much more formal, but still approachable.
@@codesur One of my biggest sources of confusion in the beginning was that monads and functors in CT are slightly different than monads and functors in Haskell. I would say that they are slightly different things, and that Haskell took inspiration from CT, rather than directly implementing it 1:1 (I am not sure if CT is even computable, so it makes sense for the link between the two to be imperfect). It took me too long to figure this out... I hope this helps!
@@epgui I'll definitely give Awodey's book a try. PS: I've heard that Haskell is more inspired by than based on Category Theory. I remember someone on /r/ProgrammingLanguages laughing about the idea of Haskell being CT-based, but I'm still trying to grasp the difference between Haskell functors and CT functors.
It is amazing to find out just how little I know about everything! So, Dancstep is an interesting last name btw. Any relation to the Dancestep family or the Dubstep offspring, no relations to The Offspring of course. But you are pretty fly for a white guy!
Isn't _"Category theory is the mathematics of mathematics"_ a cyclic-dependency which injects an imagination void into the mind of the reader/listener/receiver?
This is beyond my mathematical education, but that's OK. I have not got any further than my own category theory, which states that just because the human mind can create categories, that does not oblige reality to fit within any of them.
@@semioticapocalypse9774 Semiotics and semasiology are parts of semantics. Meaning precedes signification. "Category" is a word with many meanings and the one in math simply signifies membership of a subgroup.
@@mikemondano3624 Not according to Saussurean semiotics, nor according to contemporary Russia/Tartu school of semiotics. Meaning is bound in signification, via connection of signifier and signified. Same with Peirce's three-part model. Not sure who (besides you) says that meaning PRECEDES signification. Both semiotics and linguistics utilize semantics as a sub-branch, not the other way around.
So you just take any existing math theory and generate a bunch of neat gifs to represent the different parts of it then insert infinity arrows between all the different parts and you are doing category theory. Can you get paid for doing it?
Revisited: Dude tries to define categories to his friends... proceeds to define everything But categories, except for how they are related. (Great video on abstract algebra. I didn't get a clear understanding of categories from your video, because you DODGE the topic.)
In the beginning was the Word, and the Word was with God, and the Word was God. 2He was with God in the beginning. 3Through him all things were made; without him nothing was made that has been made. (John 1:1-3)
1:30 aahhh the same symmetry thing. i wonder why not the much more graspable "linear algebra" like approach is used? i.e. semi-grp -> monoid -> group -> abelion grp (e&oe)
... and then saying: so far we did all this with numbers u already were familiar with, but this doesnt have to be numbers. so, it can be transformations of an object.
