Your clear style very much reminds me of a wonderful teacher i once knew who unfortunately passed away. One can observe this often in those who have mastered various forms of advanced logic whether mathematical or categorical, programmatic, philosophic, etc. A clarity and complexity. Concision, in a word, when surveying even a complex landscape or on varying scales, without sacrificing correctness or even "the essence of the matter". A natural fit for category theory!
i first stumbled upon your lecture on Projective Geometry few years ago and never stopped watching ever since. now the time has come for me to dig into category theory, and your teaching is the best i could hope for. THANK YOU !
I think with this I've found a series to watch this summer! Thank you very much for making this. It really shows that invested a lot thinking about how to intuitively describe things without any hand-waving or any terminology that would be obfuscating to beginners.
You are obviously right about category theory. Clearly it's very simplistic structure is that of composition. I would add though that what makes category theory and really any expressive theory powerful is it's recursion. if you think about it, almost everything we do is in some way recursive: Life itself(genetics/procreation), computer programs(e.g., 3D video games algorithms are recursive as is database traversal), learning, fractals, etc. Why is this recursive structure in category theory so powerful? Well, since objects can be categories essentially we get a hierarchy of composition. We get composition "across objects"(object to object) and then composition "through objects"(when those objects can be seen as categories). What this does then is allow for a very rich analysis that is effectively static. Once you understand a category "across objects" you effectively understand them through objects and this allows for "recursive understanding": Categories of Categories of Categories of Categories of . For example, suppose we write a category (o, m) where o is the objects and m the morphisms on those objects. Then ((o,m), M) is a category of a category where the morphisms M are on the objects (o, m) which are actually categories themselves. In the first case we call m "morphisms" but in the second case we call them "functors". If we have (((a,A),B),C) then we have a category of categories of categories. The morphisms in this case(C) are natural transformations. **This can continue indefinitely** and the power of recursion is that once we understand a category we understand categories of categories(the nomenclature changes but the structure remains the same). E.g., we can have products of natural transformations just as easily as we can of objects. Category theory doesn't care what the objects are, hence they can be anything as long as their usage can be composed. We use different naming conventions to distinguish the different levels and there is a "slight of hand" used so we can pretend we are not "looking in to objects"(we from the bottom up). Generally when we learn things that have levels to them we always have to use a different language to talk about the composed structure. The new language makes it seem like the structure is different but eventually we learn that it's just to distinguish hierarchy but the internal structure of the language is the same as what we previously learned. That is, the abstract language is the same. E.g., the abstract grammar of all languages is effectively the same even though many languages are very different. E.g., With category theory a natural transformation is a morphism AND a morphism is a natural transformation IF the object supports it. But what this enabled us to do is assume the object is really categorically structured(it turns out almost everything seems to be composed of other things which are composed of other things(recursion)) and so what we think are "hollow morphisms" are really very complicated morphisms(functors, natural transformations, and higher level morphisms). What I try to do when studying category theory is think of all these hierarchy concepts as the same abstract thing but that it makes explicit the hierarchical level we are working at. Natural transformations, say, are working at 4x zoom while morphisms are 1x and functors are 2x. We can see further in to the structure. As I mentioned earlier though we are working from the "bottom up" because our objects are suppose to be structureless(but they almost always have some structure in reality). The power of category then is that ultimately it creates a very simple language to talk about structure along the "horizontal"(with some about the vertical but ultimately isn't needed except to avoid ambiguity) that unifies the "vertical". This allows us to sort of reduce/compress complexity on to one plane. An analogy might be how learning genetics and biology lets one sort of compress the genealogical hierarchy of humanity down in to a single layer in terms of the structure of the human body. Every human is different and there is a hierarchy that includes a vast amount of information but in terms of the physiological structure of the human body we are all essentially the same. E.g., procreation develops along a vertical axis but DNA replicates the "same" structure at each stage. If we just learn one stage and know the vertical is the same we know the structure is the same for all stages(more or less). If you did not know anything about biology then you might not understand that the process vertically is the same at each step and so might come up with things like offspring are very different inside their bodies than their parents. You might create a very different language for each generation and never realize the vast similarities. Category theory lets us align all structurally identical objects analogous to, say, in group theory how we can quotient out similar structure so we can ultimately see the true differences. It's an extremely powerful language because of it's simplicity, it's use of composition(and decomposition), and it's use of recursion. By being simple it can represent wide variety of things, by using composition it makes explicit composition which also seems to be one of the underlying processes of the universe if not the process, and by using recursion it allows for infinite levels of patterns based on the core structure. The best way to understand the recursive part is to learn about fractals and how the basic structure manifests itself through the recursive process. The same happens for all recursive structures.
I have to agree with others' praise; this is indeed the most beginner-friendly category theory video course I have ever seen. Instead of just listing definitions, it explains the motivations behind them, making everything seem very natural. I hope to watch the entire playlist. Thank you very much for your effort!
fantastic, thank you. i have been trying to work through some more sophisticated material on category theory and i have frequently found myself lost. this course is an excellent introduction to build a strong foundation in category theory. i will be contributing to your patreon as a thank you.
Holy shit thank you! It never occurred to me to speed up lectures like this but actually yeah this is fantastic. (1.5x ended up working best for me lol.)
This is fantastic! Thank you so much for taking the time to make this. I've been reading about dynamical systems and bifurcation theory all night, and it popped into my head: "oh right, I meant to look into category theory. It seems to have relevance to the question of the foundations of mathematics, or so I have read." This is pretty much exactly what I was looking for.
Amazing lecture. Thank you for sharing these. When you last explained the final object example it really clicked for me why Category Theory is a thing. Gonna continue watching you're an amazing teacher.
We've defined a 'set' category, but our definition uses concepts from outside of category theory (elements in sets.) It seems likely to me you can avoid this, but it may be tedious. I'll make a start: Every object has a 'size'. For now I'll only consider finite sets, so the size is a natural number. Everything about the object and its relation to other objects is determined by their sizes. Consider an object B of size two. This object has four arrows to itself, which we will call id, f, g, h. [In square brackets, I give a concrete example for the set {0,1}.] id is the identity. [id(x)=x] f has the property that f.f=id [f(x)=1-x] g, h have the properties that g.g=g, h.h=h, f.h=h.f=g, f.g=g.f=h. [g(x)=0, h(x)=1]. Thus I have defined the arrows from an object to itself using only the language of category theory (for an object of size two.) Now consider an object C of size 3 [concrete example, set {a,b,c}] and arrows between this and our size two object, B. There are 8 arrows C->B, which I will call 'j' with some subscript, and 9 arrows B->C which I will call 'k' with some subscript. Then there exists a unique arrow j_0 with the property forall k:B->C, j_0.k=g. [j_0(x)=0, hence j_0(k(x))=0, hence j_0.k=g, where g(x)=0 from last paragraph.] Similarly there exists a unique j_7 with the property forall k:B->C, j_7.k=h. [j_7(x)=1] These j_0, j_7 are the only two 'constant functions' among the j_i. I expect that with lots of tedious effort, I could define all of the arrows between objects in the 'countable set' category, without reference to sets. There would need to be some recursive definitions (behaviour of a size n object defined in terms of a size n-1 object.) If you allow uncountable sets and various sizes of infinities, I expect it gets much more painful.
I love the way you explain things, and your hands gestures actually help (don't know why). You are an amazing teacher. It is quite a hard subject to grasp. I think showing a bit more examples would be helpful to just get a clearer, more practical view on the subject (as well as maybe some historical insights ?). Beside these small points, yours is the best course on category theory on RUclips.
Question about the category Set. In this category, the final object is the singleton set. Is there a single final object, or infinitely many? That is, is the set {1} the same object as the set {a}? And how does the answer extend to other categories? Is it meaningful to have a category with multiple objects that have identical properties (arrows), or do you always collapse those to a single object?
In Set there are infinitely many final objects. Many categories have multiple final objects, but any two final objects A and B will always be isomorphic. If you know the definition of isomorphic objects in category theory you can try and show this. I cover it soon, in a video or two within this playlist
isomorphic objects act similarly in category theory. One thing you can do is just take one copy of each non-isomorphic object. This gives you something called the skeleton of a category. But there is no need. Many constructions in category theory (like the idea of final objects) just define things "up to isomorphism ", but that is fine since most category theory ideas are only sensitive to form up to isomorphism anyway
This video is terrific. The one thing that confuses me about category theory is this: why even mention the identity arrows? I realize I'm missing something fundamental here, but it seems completely trivial to even mention that a thing has to essentially be itself. Is there some scenario where something doesn't have an identity arrow? If that was taken out of category theory altogether, what would happen?
Good question. Think about what arithmetic would be like without zero. Its helpful to say that some operations can be composed to yield a "do nothing" operation. Also, in the category Set (which has objects as sets, and arrows as functions), there is an identity function from a set to itself (leaving each element unaltered). It's a good idea to include it, so you represent all functions. And I'd say it's a good idea to highlight it because its special. You can think about when the composition of two functions equals the identity function. This is an interesting and important problem
Great video! I've searched for a few introductions to category theory videos, but this one is the most clear by far! I have one question, in the final example, why is it required that three arrows from the singleton set go back to the original set? Is it required within category theory that if there is any arrow from object A to B, every element of B gets its own arrow? If so, why?
I has been one year since your question, but I am going to answer it anyways for those who are interested. As I understand it (I am learning category theory too) the category of sets has all possibile sets as objects and all possibile functions between them as arrows. Therefore, by definition, there always will be functions from the terminal set to any other set and viceversa. I hope it helped :)
Here is a better way to think about category theory through a particular example: A final object is a final object through behavior. If an object only has morphisms into it then it **for all intents and purposes** is a final object. That is, if it walks like a duck, talks like a duck it **is** a duck. The point with category theory is that things are defined through their structure rather than arbitrary definitions. E.g., a singleton is set with one element. Ok, great! But this is a definition in terms of sets and it is not of the "acts like a duck". If you are in a category of something and some object(or set of objects or morphisms or whatever) has a particular structure within that category then it can only behave in a certain way. If we make a definition about something in terms of properties of that object then the definition will be restricted and less general. If another object in another category has the exact same structure then we won't realize they function both as "ducks". But if we define things in terms of how they are used or how they behave rather than what they are we get very general definitions. So it isn't remarkable that you can define things in such ways. It is that generally how we define things limits or ability to see related structure. if you define a duck as a biological creature that has DNA code XYZ then anything that doesn't have DNA code XYZ can't be a duck. This means a cartoon duck can't be a duck and has to be something else. This then makes the language complex. if we define a duck as something that behaves a certain way then both the cartoon duck and a biological duck are both ducks. If we have a category then within that category any final objects will have the same behaviors as any final object in any other category. We can then study final objects in an object independent way and this definition becomes extremely powerful since we do not get confused when the objects are different(e.g., not realizing a cartoon duck is a type of duck). This is what make category theory power. since it is studying abstract structure rather than specific objects and specific structure. Maybe a better way to see it is this: For most of human progress many independent people came up with intellectual understandings of many different things in the universe. They all used different languages, for the most part... but as people of different professions start to communicate they started to realize there were certain overlaps in things. The more they communicated the more they understood. But they still all used different "languages". Category theory is a language that attempts to be abstract enough, not using specific objects as a foundation to define things from but simply how they can be used, to be a language that exposes the structure of all structured things without obfuscating them by specifics. In a world were everyone learned category theory, communication would be very efficient and effective since what is communicated is pure structure. The universe itself is so structural that category theory is a much more natural language than say English or Chinese.
Very good video, it helped me a lot because I'm trying to learn functional programming and its explanation clarified many things. Thank you for explain this.
Some people have said they want to support my efforts to make educational videos and software, and so I have made a Patreon page www.patreon.com/richardsouthwell Any support would really help me produce more videos and software.
I have to ask. You seem like you can answer this. Is category theory better than set theory? Meaning, is it more abstract and able to deal with higher level math? I tried doing homological algebra with sets and I’ve been told that’s impossible just as an example. Are there things sets can do that categories can likewise?
A lot of category theory is made using sets, and in a sense I feel set theory and category theory can do what each other can do. But I feel you get a lot more intuition from category theory. Plus set theory rests on certain axoims, and there is a branch of category theory called topos theory which allows you to study vast generalizations set theory and logic.
@@RichardSouthwell and do you think it might be practicable / applicable in broad contexts? (functional programming aside) Can you break very complex natural transformations theoretically down to easier say "operators"?
Sets with functions are sort of like ways to collect many arrows together and to perform arrow compositions in parallel over those objects. I suppose that's one purpose of hom-sets.
@@RichardSouthwell thank you! Your videos are a great resource, extremely helpful and interesting. I’m working through them from the beginning a second time
Why are there exactly 3 arrows from the final object to the other objet at 55:05? Is it always true that the number of arrows between a final object and another object b corresponds to the cardinality of b?
Yes it is in the category Set. The three arrows correspond to the functions p,q and r from {1} to {a,b,c} with p(1)=a, q(1)=b and r(1)=c. These can be thought of as representing elements a,b and c respectively.
If F is a functor linking category C to category D, and G is a functor linking category C to category E (and assuming the functors are bijections), there is a unique natural transformation between functors F and G. This seems true to me. Is it?
For me the exposition of introductory Category Theory would be more betterer if I could see a demo of what would break if a feature was missing. For example the identity morphisms. Or alternatively, how do we bootstrap the idea of categories from a blank sheet of paper, given we want to be sure one system can do the work of another?
I have a feeling that I am missing something here. After trying to grasp the concept of a final object for the category "Set", having a set with just one element in be considered Final - kind of seems like an insufficient definition. If the category "Set" contains all of infinitely many possible "Sets" that there are, doesn't this also mean that there must be an infinite amount of "1 element" sets? What sort of properties apart from having a single unique arrow per any other set would there be? It is all way too general and abstract to even come up with an example that would be generally applicable.
@@Unaimend Glad to hear it. Feel free to write me if you have more questions etc. You can find my email here sites.google.com/site/richardsouthwell254/home/about
How can you say that category theory is more fundamental than logic and set theory when you are using logic and set theory to describe the basic definitions of category theory? For example, the definition uses "collection of objects" which comes from set theory, equality which comes from first-order logic, and seems to treat composition much like a relation between objects, also a notion from first-order logic. Wouldn't it be more accurate to say that category theory is built using some kind of set theory and logic, and can then be used to further generalize set theory and logic? Is there a *formal* definition of category theory? Is there a formal proof theory for category theory? If there is then category theory can be expressed in terms of a logic, but if not then it seems to me that category theory is not fully mature since it's underspecified.
@Calum Tatum You make a good point, but your point also contradicts the idea that category theory is more fundamental than set theory, which is what some people in the comments and the host in the video have claimed. You are right: neither is more fundamental than the other in a strictly objective sense.
@Calum Tatum This can be both an advantage and a disadvantage, though, at this is dependent on context and the application. It also should be noted that the Curry-Howard-Lambek correspondence is actually much weaker than the Curry-Howard isomorphism between propositional calculus and type theory. In fact, the formal grounding of topos theory is somewhat reliant on some ideas of type theory, although of course, types are not the primitive notion of topos theory. Regardless, there are advantages to topos theory and type theory over set theory, and it is true that they both can serve as foundations of mathematics without resting on set theory, which is a good thing to clarify, given that the OP claimed the contrary. However, I think the main point here is that in a similar fashion, these advantages do not and cannot meaningfully imply that category theory or type theory are somehow "more fundamental" than either, especially since the convenient foundation to choose is actually entirely subjective and dependent on the application. For most applications, the foundations of mathematics are actually themselves irrelevant, but for the applications in which they are relevant, there is not one foundation which is the most objectively convenient in every application without fail. I think this is the important takeaway.
It is a wonderful approach and pace for the presentations of the first concepts of category theory. By the sound of it Category Theory lays claim to a certain generality (I do not know anything about it). Is there any specific known (mathematical) structures/concepts which can not be represented in terms of Category Theory? Or does it provide a structure to embed all known mathematical structures? Not even sure if the question is a reasonable one...
A very reasonable question. It depends what you mean by "all known mathematical structures". But basically yes, you can do almost all maths starting with the category of categories. You can also generalize the standard set theoretic foundations of maths using topos theory (another branch of category theory). Many languages have this same capability of being able to discuss most of maths, but category theory is special because it is so minimalistic, and (in my opinion) reveals the true nature of mathematical objects
I would definitely definitely recommend "Conceptual Mathematics: A First Introduction to Categories" by Stephen Schanuel and William Lawvere. It is a masterpiece. No prerequisites, easy to start, available online and Lawvere is one of the masters of category theory. I do recommend working through the exercises though. I've also heard good things about "Cakes, Custard and Category Theory: Easy recipes for understanding complex maths" by Eugenia Cheng, but I think that is a more like a pop math book, whereas Lawvere really teaches you deep category theory starting from the very basics at and moving towards really profound stuff. I might go as far as to say Conceptual Mathematics is my favorite maths book, and I have a lot that I like.
@@RichardSouthwell, thank you very much. I have some graduate level math, but I'm a philosopher teaching physics, which is why I made the qualification about accessibility for non-professionals. Your recommendation looks fantastic for me, and I really appreciate your taking the time. I look forward to your other videos on the topic. PS. In case it interests you, I am taking up the subject, as I've been examining A.N. Whitehead's philosophy very closely. His mereotopology in "Process and Reality" is said to be situated in category-theoretic thought, as opposed to set-theoretic. Thanks again!
@@milliern In that case I think Lawvere's book should definitely suit you. I've heard a lot of good things about Whitehead, and am keen to learn more about philosophy and how category theory connects with it, so if you feel like it feel free to drop me an email at richardsouthwell254@gmail.com and we can exchange category theory and philosophy ideas.
In a relation you have a set of elements, some are related and some not. In a category you have a set of objects, some are joined by arrows and some not. But in a category you can compose arrows and you can have multiple arrows from A to B
@@cya3mdirl158Yes, you are definitely on to something. For example a group corresponds to a single object category where each arrow is invertible (that is each arrow is an isomorphism). But yes, it should become clearer as you get into it. Good luck
For the last part of the video, where you mentioned we can identify the number of objects in a set from the quantity of the arrows, I have a question that would it be possible that the set contains null objects that no arrows flow from or to it?
If A is a set with one element and B is a set with n elements then there are n functions/arrows from A to B. This is also true when n=0, in which case B is an empty set (there are no functions from a singleton set to an empty set). Hope this helps
I don't understand how you can have valid math in a monoid, as any composition is valid you could say 2o3=1 as 1: a -> a and 2: a->a and 3:a->a. What am I missing?
Very interesting. BUT. Looks like the "fundamental" concept of category depends upon two even more fundamental concepts, collection and equals. Does talking about collections instead of sets avoid Russell's paradox? Wouldn't different kinds of objects have different kinds of equality?
The issue is of Russel’s paradox is often sidestepped in category theory. Categories are divided into small categories and large categories. In small categories the objects form a set that cannot contain all sets due to Russel’s paradox. In a large category the objects form something called a proper class which can contain all sets but classes themselves are not members of other classes and thus the paradox is resolved. The word “collection” usually just refers to a class which may or may not be a set. I believe you are right that category relies on some notion of equality (often equivalence is enough) and collections with membership.
@@ViktorKronvall Victor, thanks for the replay. BUT. Call me a nit-picker. It seems to me that if category relies on some notion of equality, that notion of equality should be counted as part of the definition of category. Maybe you could use some sort of "special" equality symbol to mean a and b are equal in accordance with whatever notion of equality is appropriate for things like a and b, say A =c B. Not to be explicit about equality feels a bit like "hand-waving". They way 2 matrices are equal is different than the way 2 organisms would be "equal". Even if we stick with mathematical objects, there's no one size fits all equality relation that works for all of them.
Russ Hatton On locally small categories (categories where the hom-functor between two objects is a set, that is between every pair of objects we have a set of arrows) we don’t need any extra definition for equivalence since equality is defined for sets and equivalence is given by isomorphisms. An isomorphism f : A -> B in a category C is an arrow for which there exists an arrow g : B -> A such that g • f = id_A and f • g = id_B Since g•f is an element of the set C(A, A) we can verify whether it is the same element as id_A which also is an element of the set C(A, A). I don’t know how to introduce equality of morphism in categories that aren’t locally small but I’ve also yet to use such a category. In conclusion, you usually don’t talk about equality of objects as that leads to some issues. Instead, the notion of equivalence which is defined through isomorphism is used. For morphisms in locally small categories we can say whether morphisms are equal or not by the fact that they are elements of sets.
At 25:00 I'm not sure I'm following. Why are those the only four statements you can derive? Why aren't 1x1=0 or 0x0=1 also derivable? idA∘idA takes you from the same point to the same point as 0. I feel like I'm not understanding what makes two expressions equal in this system. The beginning of the made it sound like equality was having the same beginning and end, and hence g∘f was the arrow equivalent to following f and then g, but that's clearly not true.
Oh wait, I think I got it. Aside of the identity rules which you have to obey you can set every other g∘f pair to any outcome and the result will be a category, is that correct? A monoid is just an algebra? For example you showed that the category where you set 0∘0=0 represents binary multiplication, but if instead we we chose the category where 0∘0=1=idA we could say we have the category equivalent to XNOR? Is that correct? I honestly feel like I could be completely off.
There are different ways to talk about division. To see how it can be thought of as a non-associative binary operation, think of division (/) as an operation that takes in two positive reals, x,y that returns a positive real (x/y), which is obtain by dividing x by y. In this case you can see that (/) is not associative since (1/(1/2))=2 does not equal ((1/1)/2)=0.5.
I am calling the object of the category A. I did not name the category with a symbol. The interpretation of the object does not play a big role in the examples I go through. The idea is more to show how composing arrows from an object to itself can be used to model arithmetic.
@@strofikornego9408 The graph on the board shows two arrows from object A to itself, but only one such arrow is the identity arrow (the one labeled id_A = 1). The arrow labeled 0 is not an identity arrow. In general, an object can only have one identity arrow (as can be shown from the definition of a category). An identity arrow is a self loop with the property that pre-composing or post-composing it with another arrow gives the other arrow. The arrow labeled 0 is a self loop, but as you say, it is not an identity arrow in this case.
To see why there can only be one identity arrow of an object A, suppose i and j were both identity arrows of A. Then, by definition, (i after j) = i and (i after j) = j, so i = j.
How can the second arrow point to itlself but also not be an identity. Surely, any function that loops to itself is an identity. It is true that there can be only one identity function. And that means there can be only 1 self-loop for a state.
Categories are specific kinds of graphs. We are interested especially in compositions. That is, pairs of edges with a tip-tail joint. And for each such pair which are configured that way, we include an edge in the graph which goes from the initial vertex of the first edge to the final vertex of the second edge. This corresponds clearly to function composition. See 7:00 and onward. Another property of categories after composition is the existence of "identity" edges from each vertex to itself. But its important to sort of "lift your feet and fly" in a mental sense and cease to think of the vertices necessarily, or a graph. It becomes more natural to just think about the arrows, the edges, the "morphisms" as they are called.
Most of us have ADHD I guess... that's why despite not being mathematicians we come to realize there must be something called Category Theory.. just our ability to intuit patterns behind patterns.
@@jamiepellegrin2371 that sounds brilliant! I wonder whether taking semantic nets and forging them together with this kind of mathematics could make knowledge represenation easier in general
Your clear style very much reminds me of a wonderful teacher i once knew who unfortunately passed away. One can observe this often in those who have mastered various forms of advanced logic whether mathematical or categorical, programmatic, philosophic, etc. A clarity and complexity. Concision, in a word, when surveying even a complex landscape or on varying scales, without sacrificing correctness or even "the essence of the matter". A natural fit for category theory!
What a wonderful and encouraging thing to say. Thank you.
i first stumbled upon your lecture on Projective Geometry few years ago and never stopped watching ever since. now the time has come for me to dig into category theory, and your teaching is the best i could hope for. THANK YOU !
I think with this I've found a series to watch this summer! Thank you very much for making this. It really shows that invested a lot thinking about how to intuitively describe things without any hand-waving or any terminology that would be obfuscating to beginners.
This is the best introduction to category theory I've ever seen! Thank you!
You are obviously right about category theory. Clearly it's very simplistic structure is that of composition. I would add though that what makes category theory and really any expressive theory powerful is it's recursion. if you think about it, almost everything we do is in some way recursive: Life itself(genetics/procreation), computer programs(e.g., 3D video games algorithms are recursive as is database traversal), learning, fractals, etc.
Why is this recursive structure in category theory so powerful? Well, since objects can be categories essentially we get a hierarchy of composition. We get composition "across objects"(object to object) and then composition "through objects"(when those objects can be seen as categories). What this does then is allow for a very rich analysis that is effectively static. Once you understand a category "across objects" you effectively understand them through objects and this allows for "recursive understanding": Categories of Categories of Categories of Categories of .
For example, suppose we write a category (o, m) where o is the objects and m the morphisms on those objects. Then ((o,m), M) is a category of a category where the morphisms M are on the objects (o, m) which are actually categories themselves. In the first case we call m "morphisms" but in the second case we call them "functors". If we have (((a,A),B),C) then we have a category of categories of categories. The morphisms in this case(C) are natural transformations. **This can continue indefinitely** and the power of recursion is that once we understand a category we understand categories of categories(the nomenclature changes but the structure remains the same). E.g., we can have products of natural transformations just as easily as we can of objects. Category theory doesn't care what the objects are, hence they can be anything as long as their usage can be composed.
We use different naming conventions to distinguish the different levels and there is a "slight of hand" used so we can pretend we are not "looking in to objects"(we from the bottom up). Generally when we learn things that have levels to them we always have to use a different language to talk about the composed structure. The new language makes it seem like the structure is different but eventually we learn that it's just to distinguish hierarchy but the internal structure of the language is the same as what we previously learned. That is, the abstract language is the same. E.g., the abstract grammar of all languages is effectively the same even though many languages are very different.
E.g., With category theory a natural transformation is a morphism AND a morphism is a natural transformation IF the object supports it. But what this enabled us to do is assume the object is really categorically structured(it turns out almost everything seems to be composed of other things which are composed of other things(recursion)) and so what we think are "hollow morphisms" are really very complicated morphisms(functors, natural transformations, and higher level morphisms).
What I try to do when studying category theory is think of all these hierarchy concepts as the same abstract thing but that it makes explicit the hierarchical level we are working at. Natural transformations, say, are working at 4x zoom while morphisms are 1x and functors are 2x. We can see further in to the structure. As I mentioned earlier though we are working from the "bottom up" because our objects are suppose to be structureless(but they almost always have some structure in reality).
The power of category then is that ultimately it creates a very simple language to talk about structure along the "horizontal"(with some about the vertical but ultimately isn't needed except to avoid ambiguity) that unifies the "vertical". This allows us to sort of reduce/compress complexity on to one plane.
An analogy might be how learning genetics and biology lets one sort of compress the genealogical hierarchy of humanity down in to a single layer in terms of the structure of the human body. Every human is different and there is a hierarchy that includes a vast amount of information but in terms of the physiological structure of the human body we are all essentially the same. E.g., procreation develops along a vertical axis but DNA replicates the "same" structure at each stage. If we just learn one stage and know the vertical is the same we know the structure is the same for all stages(more or less). If you did not know anything about biology then you might not understand that the process vertically is the same at each step and so might come up with things like offspring are very different inside their bodies than their parents. You might create a very different language for each generation and never realize the vast similarities.
Category theory lets us align all structurally identical objects analogous to, say, in group theory how we can quotient out similar structure so we can ultimately see the true differences. It's an extremely powerful language because of it's simplicity, it's use of composition(and decomposition), and it's use of recursion. By being simple it can represent wide variety of things, by using composition it makes explicit composition which also seems to be one of the underlying processes of the universe if not the process, and by using recursion it allows for infinite levels of patterns based on the core structure. The best way to understand the recursive part is to learn about fractals and how the basic structure manifests itself through the recursive process. The same happens for all recursive structures.
You are an amazing teacher!! You have such a beautiful and clear way of explaining things that gets to the heart of the matter.
Very nice of you to say. It's so motivating to get encouraging comments like this.
I am very grateful to have found your videos, such a hidden gem. Thank you for sharing your valuable knowledge!
You are most welcome!
I have to agree with others' praise; this is indeed the most beginner-friendly category theory video course I have ever seen. Instead of just listing definitions, it explains the motivations behind them, making everything seem very natural. I hope to watch the entire playlist. Thank you very much for your effort!
Thank you! This RUclips channel is pure gold.
Amazing work, especially the part where you talk about the Category of sets. You make complex stuff look kind of easy :)
fantastic, thank you. i have been trying to work through some more sophisticated material on category theory and i have frequently found myself lost. this course is an excellent introduction to build a strong foundation in category theory. i will be contributing to your patreon as a thank you.
I saw quite a few videos this explains the concepts the best
For my fellow speed enthusiasts I found 1.75x speed was a good playback speed
I like the religious tempo better.
Holy shit thank you! It never occurred to me to speed up lectures like this but actually yeah this is fantastic. (1.5x ended up working best for me lol.)
It is really useful to record lectures at a leisurely pace, then everyone has the opportunity to run at the speed appropriate to them.
actually i would slow down.. but i stick to rewind every few minutes
Excellent intro to Category Theory. Concise; lucid; etc.
This is fantastic! Thank you so much for taking the time to make this. I've been reading about dynamical systems and bifurcation theory all night, and it popped into my head: "oh right, I meant to look into category theory. It seems to have relevance to the question of the foundations of mathematics, or so I have read." This is pretty much exactly what I was looking for.
Thank you so much! Finally an introduction for people who do not have a math background.
Amazing lecture. Thank you for sharing these. When you last explained the final object example it really clicked for me why Category Theory is a thing. Gonna continue watching you're an amazing teacher.
Richard, Great lecture series. Appreciated very much!
We've defined a 'set' category, but our definition uses concepts from outside of category theory (elements in sets.) It seems likely to me you can avoid this, but it may be tedious. I'll make a start:
Every object has a 'size'. For now I'll only consider finite sets, so the size is a natural number. Everything about the object and its relation to other objects is determined by their sizes.
Consider an object B of size two. This object has four arrows to itself, which we will call id, f, g, h. [In square brackets, I give a concrete example for the set {0,1}.]
id is the identity. [id(x)=x]
f has the property that f.f=id [f(x)=1-x]
g, h have the properties that g.g=g, h.h=h, f.h=h.f=g, f.g=g.f=h. [g(x)=0, h(x)=1].
Thus I have defined the arrows from an object to itself using only the language of category theory (for an object of size two.)
Now consider an object C of size 3 [concrete example, set {a,b,c}] and arrows between this and our size two object, B. There are 8 arrows C->B, which I will call 'j' with some subscript, and 9 arrows B->C which I will call 'k' with some subscript. Then there exists a unique arrow j_0 with the property
forall k:B->C, j_0.k=g. [j_0(x)=0, hence j_0(k(x))=0, hence j_0.k=g, where g(x)=0 from last paragraph.]
Similarly there exists a unique j_7 with the property
forall k:B->C, j_7.k=h. [j_7(x)=1]
These j_0, j_7 are the only two 'constant functions' among the j_i.
I expect that with lots of tedious effort, I could define all of the arrows between objects in the 'countable set' category, without reference to sets. There would need to be some recursive definitions (behaviour of a size n object defined in terms of a size n-1 object.) If you allow uncountable sets and various sizes of infinities, I expect it gets much more painful.
Brilliantly well explained - Bravo!
I love the way you explain things, and your hands gestures actually help (don't know why). You are an amazing teacher. It is quite a hard subject to grasp. I think showing a bit more examples would be helpful to just get a clearer, more practical view on the subject (as well as maybe some historical insights ?). Beside these small points, yours is the best course on category theory on RUclips.
Very nice lecture
Excellent videos, I'm learning a lot from them! Thanks a lot!
Brilliant
That was an amazing introduction. Thank you!
I feel like I can understand category theory now, this is awesome!
Brilliant
See you in >implying we can discuss mathematics, Oillie.
MWAHAHAHAHAH! (it's just me, keith)
Hey my dudeee!!!
We are the cool mathematicians of the internet lol
ahahaha
Question about the category Set. In this category, the final object is the singleton set. Is there a single final object, or infinitely many? That is, is the set {1} the same object as the set {a}? And how does the answer extend to other categories? Is it meaningful to have a category with multiple objects that have identical properties (arrows), or do you always collapse those to a single object?
In Set there are infinitely many final objects. Many categories have multiple final objects, but any two final objects A and B will always be isomorphic. If you know the definition of isomorphic objects in category theory you can try and show this. I cover it soon, in a video or two within this playlist
isomorphic objects act similarly in category theory. One thing you can do is just take one copy of each non-isomorphic object. This gives you something called the skeleton of a category. But there is no need. Many constructions in category theory (like the idea of final objects) just define things "up to isomorphism ", but that is fine since most category theory ideas are only sensitive to form up to isomorphism anyway
Thanks, that clears up my confusion perfectly! Watching video 2 now and it clarifies these things as well. Great series!
Great video, very clear exposition.
This video is terrific. The one thing that confuses me about category theory is this: why even mention the identity arrows? I realize I'm missing something fundamental here, but it seems completely trivial to even mention that a thing has to essentially be itself. Is there some scenario where something doesn't have an identity arrow? If that was taken out of category theory altogether, what would happen?
Good question. Think about what arithmetic would be like without zero. Its helpful to say that some operations can be composed to yield a "do nothing" operation. Also, in the category Set (which has objects as sets, and arrows as functions), there is an identity function from a set to itself (leaving each element unaltered). It's a good idea to include it, so you represent all functions. And I'd say it's a good idea to highlight it because its special. You can think about when the composition of two functions equals the identity function. This is an interesting and important problem
This is very helpful. Thank you so much!
Great video! I've searched for a few introductions to category theory videos, but this one is the most clear by far!
I have one question, in the final example, why is it required that three arrows from the singleton set go back to the original set? Is it required within category theory that if there is any arrow from object A to B, every element of B gets its own arrow? If so, why?
I has been one year since your question, but I am going to answer it anyways for those who are interested.
As I understand it (I am learning category theory too) the category of sets has all possibile sets as objects and all possibile functions between them as arrows. Therefore, by definition, there always will be functions from the terminal set to any other set and viceversa.
I hope it helped :)
Awesome video as always. Looking forward for the new one.
Thank you, I am having a lot of fun making these.
Here is a better way to think about category theory through a particular example: A final object is a final object through behavior. If an object only has morphisms into it then it **for all intents and purposes** is a final object. That is, if it walks like a duck, talks like a duck it **is** a duck.
The point with category theory is that things are defined through their structure rather than arbitrary definitions. E.g., a singleton is set with one element. Ok, great! But this is a definition in terms of sets and it is not of the "acts like a duck".
If you are in a category of something and some object(or set of objects or morphisms or whatever) has a particular structure within that category then it can only behave in a certain way. If we make a definition about something in terms of properties of that object then the definition will be restricted and less general. If another object in another category has the exact same structure then we won't realize they function both as "ducks". But if we define things in terms of how they are used or how they behave rather than what they are we get very general definitions.
So it isn't remarkable that you can define things in such ways. It is that generally how we define things limits or ability to see related structure. if you define a duck as a biological creature that has DNA code XYZ then anything that doesn't have DNA code XYZ can't be a duck. This means a cartoon duck can't be a duck and has to be something else. This then makes the language complex. if we define a duck as something that behaves a certain way then both the cartoon duck and a biological duck are both ducks.
If we have a category then within that category any final objects will have the same behaviors as any final object in any other category. We can then study final objects in an object independent way and this definition becomes extremely powerful since we do not get confused when the objects are different(e.g., not realizing a cartoon duck is a type of duck). This is what make category theory power. since it is studying abstract structure rather than specific objects and specific structure.
Maybe a better way to see it is this: For most of human progress many independent people came up with intellectual understandings of many different things in the universe. They all used different languages, for the most part... but as people of different professions start to communicate they started to realize there were certain overlaps in things. The more they communicated the more they understood. But they still all used different "languages". Category theory is a language that attempts to be abstract enough, not using specific objects as a foundation to define things from but simply how they can be used, to be a language that exposes the structure of all structured things without obfuscating them by specifics.
In a world were everyone learned category theory, communication would be very efficient and effective since what is communicated is pure structure. The universe itself is so structural that category theory is a much more natural language than say English or Chinese.
Very interesting and informative. Thank you very much for this video!
Really glad you like it. I've just uploaded the next video in this series too.
Well done!
Thanks so much for making this. Amazing overview, very good explanations!
You are very welcome
Very good video, it helped me a lot because I'm trying to learn functional programming and its explanation clarified many things. Thank you for explain this.
That is really good to hear. Good luck with the functional programming
Amazing video !!
Really glad you like it. I've just uploaded the next video in this series too.
5:26 - 5:28
I think the category theory is kickin in
👁👄👁
Great video man thank you so much
I was angry after watching your video. How are you not being discovered by more people? I am going to share your work at every chance possible
Thank you for this !
This is amazing
Very glad you like it!
Some people have said they want to support my efforts to make educational videos and software, and so I have made a Patreon page
www.patreon.com/richardsouthwell
Any support would really help me produce more videos and software.
Thanks so much!!!
I have to ask. You seem like you can answer this. Is category theory better than set theory? Meaning, is it more abstract and able to deal with higher level math? I tried doing homological algebra with sets and I’ve been told that’s impossible just as an example. Are there things sets can do that categories can likewise?
A lot of category theory is made using sets, and in a sense I feel set theory and category theory can do what each other can do. But I feel you get a lot more intuition from category theory. Plus set theory rests on certain axoims, and there is a branch of category theory called topos theory which allows you to study vast generalizations set theory and logic.
@@RichardSouthwell and do you think it might be practicable / applicable in broad contexts? (functional programming aside) Can you break very complex natural transformations theoretically down to easier say "operators"?
Who told you doing homological algebra with sets is impossible?
Sets with functions are sort of like ways to collect many arrows together and to perform arrow compositions in parallel over those objects. I suppose that's one purpose of hom-sets.
A very astute comment
@@RichardSouthwell thank you! Your videos are a great resource, extremely helpful and interesting. I’m working through them from the beginning a second time
Why are there exactly 3 arrows from the final object to the other objet at 55:05?
Is it always true that the number of arrows between a final object and another object b corresponds to the cardinality of b?
Yes it is in the category Set. The three arrows correspond to the functions p,q and r from {1} to {a,b,c} with p(1)=a, q(1)=b and r(1)=c. These can be thought of as representing elements a,b and c respectively.
If F is a functor linking category C to category D, and G is a functor linking category C to category E (and assuming the functors are bijections), there is a unique natural transformation between functors F and G. This seems true to me. Is it?
is this play list is good for CS student who study functional programming and want to learn about category theory?
So what would the object A @25:00 be? The set {0,1}?
How are all of the arrows identities at 24:40 if they don’t all return the same thing (1!=0)?
Marvelous thanks youuuuuuu :-)
Thanks for sharing! Wondering if you have any recommend textbooks to pair with this great course?
Lawvere Conceptual Mathematics
What is the order i which I should watch these "Category Theory for Beginners" videos?
For me the exposition of introductory Category Theory would be more betterer if I could see a demo of what would break if a feature was missing. For example the identity morphisms. Or alternatively, how do we bootstrap the idea of categories from a blank sheet of paper, given we want to be sure one system can do the work of another?
I have a feeling that I am missing something here. After trying to grasp the concept of a final object for the category "Set", having a set with just one element in be considered Final - kind of seems like an insufficient definition. If the category "Set" contains all of infinitely many possible "Sets" that there are, doesn't this also mean that there must be an infinite amount of "1 element" sets?
What sort of properties apart from having a single unique arrow per any other set would there be? It is all way too general and abstract to even come up with an example that would be generally applicable.
thanks! 🙏🙌❤️
In short it's making little diagrammes of what is a mental process. Have fun!
Thanks professor! Excellent motivation and introduction to the amazing world of "abstract nonsense". Haha
Great video, especially the example about sets of words, numbers, and booleans is really nice.
Thanks, I hope you enjoy the next video too
@@RichardSouthwell I am enjoying it right now.
@@Unaimend Glad to hear it. Feel free to write me if you have more questions etc. You can find my email here sites.google.com/site/richardsouthwell254/home/about
5 years ago I would say this is the stupidest field ever. Now, I desperately need it to clear my thoughts.
How can you say that category theory is more fundamental than logic and set theory when you are using logic and set theory to describe the basic definitions of category theory?
For example, the definition uses "collection of objects" which comes from set theory, equality which comes from first-order logic, and seems to treat composition much like a relation between objects, also a notion from first-order logic.
Wouldn't it be more accurate to say that category theory is built using some kind of set theory and logic, and can then be used to further generalize set theory and logic?
Is there a *formal* definition of category theory? Is there a formal proof theory for category theory? If there is then category theory can be expressed in terms of a logic, but if not then it seems to me that category theory is not fully mature since it's underspecified.
@Calum Tatum You make a good point, but your point also contradicts the idea that category theory is more fundamental than set theory, which is what some people in the comments and the host in the video have claimed. You are right: neither is more fundamental than the other in a strictly objective sense.
@Calum Tatum This can be both an advantage and a disadvantage, though, at this is dependent on context and the application. It also should be noted that the Curry-Howard-Lambek correspondence is actually much weaker than the Curry-Howard isomorphism between propositional calculus and type theory. In fact, the formal grounding of topos theory is somewhat reliant on some ideas of type theory, although of course, types are not the primitive notion of topos theory.
Regardless, there are advantages to topos theory and type theory over set theory, and it is true that they both can serve as foundations of mathematics without resting on set theory, which is a good thing to clarify, given that the OP claimed the contrary. However, I think the main point here is that in a similar fashion, these advantages do not and cannot meaningfully imply that category theory or type theory are somehow "more fundamental" than either, especially since the convenient foundation to choose is actually entirely subjective and dependent on the application. For most applications, the foundations of mathematics are actually themselves irrelevant, but for the applications in which they are relevant, there is not one foundation which is the most objectively convenient in every application without fail. I think this is the important takeaway.
It is a wonderful approach and pace for the presentations of the first concepts of category theory. By the sound of it Category Theory lays claim to a certain generality (I do not know anything about it). Is there any specific known (mathematical) structures/concepts which can not be represented in terms of Category Theory? Or does it provide a structure to embed all known mathematical structures? Not even sure if the question is a reasonable one...
A very reasonable question. It depends what you mean by "all known mathematical structures". But basically yes, you can do almost all maths starting with the category of categories. You can also generalize the standard set theoretic foundations of maths using topos theory (another branch of category theory). Many languages have this same capability of being able to discuss most of maths, but category theory is special because it is so minimalistic, and (in my opinion) reveals the true nature of mathematical objects
Any suggestions for a good intro book on category theory for non-professional mathematicians?
I would definitely definitely recommend "Conceptual Mathematics: A First Introduction to Categories" by Stephen Schanuel and William Lawvere. It is a masterpiece. No prerequisites, easy to start, available online and Lawvere is one of the masters of category theory. I do recommend working through the exercises though. I've also heard good things about "Cakes, Custard and Category Theory: Easy recipes for understanding complex maths" by Eugenia Cheng, but I think that is a more like a pop math book, whereas Lawvere really teaches you deep category theory starting from the very basics at and moving towards really profound stuff. I might go as far as to say Conceptual Mathematics is my favorite maths book, and I have a lot that I like.
@@RichardSouthwell, thank you very much. I have some graduate level math, but I'm a philosopher teaching physics, which is why I made the qualification about accessibility for non-professionals. Your recommendation looks fantastic for me, and I really appreciate your taking the time. I look forward to your other videos on the topic.
PS. In case it interests you, I am taking up the subject, as I've been examining A.N. Whitehead's philosophy very closely. His mereotopology in "Process and Reality" is said to be situated in category-theoretic thought, as opposed to set-theoretic.
Thanks again!
@@milliern In that case I think Lawvere's book should definitely suit you. I've heard a lot of good things about Whitehead, and am keen to learn more about philosophy and how category theory connects with it, so if you feel like it feel free to drop me an email at richardsouthwell254@gmail.com and we can exchange category theory and philosophy ideas.
What are the prerequisites for this series?
What’s a difference between relation and cathegory
In a relation you have a set of elements, some are related and some not. In a category you have a set of objects, some are joined by arrows and some not. But in a category you can compose arrows and you can have multiple arrows from A to B
@@RichardSouthwell Thanks. I see some common parts between algebraic structures (neutral element) and theory category but I am at very beginning.
@@cya3mdirl158Yes, you are definitely on to something. For example a group corresponds to a single object category where each arrow is invertible (that is each arrow is an isomorphism). But yes, it should become clearer as you get into it. Good luck
@@RichardSouthwell Thanks. ☺️ BTW Have you seen Bartosz Milewski lectures?
For the last part of the video, where you mentioned we can identify the number of objects in a set from the quantity of the arrows, I have a question that would it be possible that the set contains null objects that no arrows flow from or to it?
If A is a set with one element and B is a set with n elements then there are n functions/arrows from A to B. This is also true when n=0, in which case B is an empty set (there are no functions from a singleton set to an empty set). Hope this helps
8:10 g after f
I don't understand how you can have valid math in a monoid, as any composition is valid you could say 2o3=1 as 1: a -> a and 2: a->a and 3:a->a. What am I missing?
You can choose any elements/arrows and any composition rules you like, just so long as x(yz)=(xy)z and (id)x=x(id)=x for any x,y,z
24:22 🧐🧐🧐
Is 0×0=0
?!
How and since when?!
thank u
Very interesting. BUT. Looks like the "fundamental" concept of category depends upon two even more fundamental concepts, collection and equals. Does talking about collections instead of sets avoid Russell's paradox? Wouldn't different kinds of objects have different kinds of equality?
The issue is of Russel’s paradox is often sidestepped in category theory.
Categories are divided into small categories and large categories. In small categories the objects form a set that cannot contain all sets due to Russel’s paradox. In a large category the objects form something called a proper class which can contain all sets but classes themselves are not members of other classes and thus the paradox is resolved. The word “collection” usually just refers to a class which may or may not be a set.
I believe you are right that category relies on some notion of equality (often equivalence is enough) and collections with membership.
@@ViktorKronvall Victor, thanks for the replay. BUT. Call me a nit-picker. It seems to me that if category relies on some notion of equality, that notion of equality should be counted as part of the definition of category. Maybe you could use some sort of "special" equality symbol to mean a and b are equal in accordance with whatever notion of equality is appropriate for things like a and b, say A =c B. Not to be explicit about equality feels a bit like "hand-waving". They way 2 matrices are equal is different than the way 2 organisms would be "equal". Even if we stick with mathematical objects, there's no one size fits all equality relation that works for all of them.
Russ Hatton On locally small categories (categories where the hom-functor between two objects is a set, that is between every pair of objects we have a set of arrows) we don’t need any extra definition for equivalence since equality is defined for sets and equivalence is given by isomorphisms.
An isomorphism f : A -> B in a category C is an arrow for which there exists an arrow g : B -> A such that g • f = id_A and f • g = id_B
Since g•f is an element of the set C(A, A) we can verify whether it is the same element as id_A which also is an element of the set C(A, A). I don’t know how to introduce equality of morphism in categories that aren’t locally small but I’ve also yet to use such a category.
In conclusion, you usually don’t talk about equality of objects as that leads to some issues. Instead, the notion of equivalence which is defined through isomorphism is used. For morphisms in locally small categories we can say whether morphisms are equal or not by the fact that they are elements of sets.
@@ViktorKronvall Well put.
At 25:00 I'm not sure I'm following. Why are those the only four statements you can derive? Why aren't 1x1=0 or 0x0=1 also derivable? idA∘idA takes you from the same point to the same point as 0.
I feel like I'm not understanding what makes two expressions equal in this system. The beginning of the made it sound like equality was having the same beginning and end, and hence g∘f was the arrow equivalent to following f and then g, but that's clearly not true.
Oh wait, I think I got it. Aside of the identity rules which you have to obey you can set every other g∘f pair to any outcome and the result will be a category, is that correct? A monoid is just an algebra?
For example you showed that the category where you set 0∘0=0 represents binary multiplication, but if instead we we chose the category where 0∘0=1=idA we could say we have the category equivalent to XNOR?
Is that correct? I honestly feel like I could be completely off.
@@maksiiiskam2 XNOR as in computing!?
@@MS-il3ht XNOR as in the logical operation:
0 0 -> 1
0 1 -> 0
1 0 -> 0
1 1 -> 1
So we can identify a singleton. But what if we ar talking about a set with more than one element?
A set with n elements is identified by the feature that there are exactly n functions/arrows coming into it from any given singleton set
why is division not associative when you can express it through addition (as in CPUs) which *is* associative?
There are different ways to talk about division. To see how it can be thought of as a non-associative binary operation, think of division (/) as an operation that takes in two positive reals, x,y that returns a positive real (x/y), which is obtain by dividing x by y. In this case you can see that (/) is not associative since (1/(1/2))=2 does not equal ((1/1)/2)=0.5.
@@RichardSouthwell Thank you! I always learned that division is non-associative but also that CPUs do it by addition basically :) So I wondered...
22:30 if A is the categroy, what is the dot/object?
I am calling the object of the category A. I did not name the category with a symbol. The interpretation of the object does not play a big role in the examples I go through. The idea is more to show how composing arrows from an object to itself can be used to model arithmetic.
25:45 - how can zero be an indentity operation of multiplication???
In the multiplication example, the identity element is 1. In the addition example, the identity element is 0.
But the graph on the board at 25:45 shows 2 identity operations - 0 and 1 at the same time
@@strofikornego9408 The graph on the board shows two arrows from object A to itself, but only one such arrow is the identity arrow (the one labeled id_A = 1). The arrow labeled 0 is not an identity arrow. In general, an object can only have one identity arrow (as can be shown from the definition of a category). An identity arrow is a self loop with the property that pre-composing or post-composing it with another arrow gives the other arrow. The arrow labeled 0 is a self loop, but as you say, it is not an identity arrow in this case.
To see why there can only be one identity arrow of an object A, suppose i and j were both identity arrows of A. Then, by definition, (i after j) = i and (i after j) = j, so i = j.
How can the second arrow point to itlself but also not be an identity.
Surely, any function that loops to itself is an identity.
It is true that there can be only one identity function. And that means there can be only 1 self-loop for a state.
May I naively ask, where the definite difference to Graph Theory comes from?
Categories are specific kinds of graphs. We are interested especially in compositions. That is, pairs of edges with a tip-tail joint. And for each such pair which are configured that way, we include an edge in the graph which goes from the initial vertex of the first edge to the final vertex of the second edge. This corresponds clearly to function composition. See 7:00 and onward. Another property of categories after composition is the existence of "identity" edges from each vertex to itself. But its important to sort of "lift your feet and fly" in a mental sense and cease to think of the vertices necessarily, or a graph. It becomes more natural to just think about the arrows, the edges, the "morphisms" as they are called.
Most of us have ADHD I guess... that's why despite not being mathematicians we come to realize there must be something called Category Theory.. just our ability to intuit patterns behind patterns.
Well done! If I may, you are rather soft-spoken could you speak louder or bring the mic closer to you in your content moving forward?
Can't 'arrows' be called something more meaningful like 'relationships'?
Usually they're called morphisms!
2x
Constructor theory brought me here
Me too! How deep in the gloryful swamp are you yet!? :-)
@@MS-il3ht i went deep for a while but ive pulled away. thank you for this comment i am going to dive back in.
ruclips.net/video/W-AHNn4eGKQ/видео.html this is interesting. im so excited about category theory of knowledge
@@jamiepellegrin2371 that sounds brilliant! I wonder whether taking semantic nets and forging them together with this kind of mathematics could make knowledge represenation easier in general
@@MS-il3ht I think so. im kind of hoping a unified theory of praxeology/economics and psychology shakes out
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