Doctoral course in Category Theory, Lecture 3

0:00 There are five categories that represent a family of objects, I know, the product, which exists, of course, and the solution, which exists, an object, which I am called the product, the object, the object, and the property, the property, the property. So, for any object z, any family of mathematics, there is a z such that, no matter how democratic it is, we are impertinent.

2:30 And then, I leave it to your sagacity to write the enunciation of the universal problem. Take a family of objects in any category. So, she gave us one. Okay? The question is not to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, the question is to know if it exists or not, So, I'm going to do the... I'm going to look at what happens when we have a triangle. So, I take x, y, z, y, and z. I suppose that in the category, we can always do the products. I suppose that it exists. So, I change my rotation, because I don't want to. So, since the products exist, I look at the product of x and y. So it's an object, and I've learned it well from the lecture, and then now this, x product y, is an object in C, so I can consider this object and its product with z, and after q1, q2, I look in the following way, product y, product z, here I have q1 which goes in x product y, and here I have q2,

5:00 All of these terms can be used to define a specific type of object, for example, an object of the shape of an object. For example, an object of the shape of an object. For example, an object of the shape of an object. And then I consider, I suppose that I have three objects, F, I start to look at the situation of the product of the three objects here. So since in the category, I suppose, I have only two T, two Y, such as T, K, and two Y,

7:30 So, x produces y such that I take d, I take x produces y, I take q2 here, I take ue here, I take the matrix k, I take the xn and xn, and k, given that there is a minus u. When we look at all this, we deduce that we have q1. So, what did we do? We showed that x equals y, y equals z, p1 equals u1, p2 equals p2, q1 equals q1, q1 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q2 equals q1, q

10:00 P2 times Q1 times Q2 is the solution of the universal problem of the product of the three objects. And now we do exactly the same work by considering first the product of Y and then the product of A. We find that this object of three members is also the solution of the universal problem of the product of X, Y and Z. And by unicity of the solution of a universal problem, it means that these two elements, the product and the cell, are isomorphs by a single isomorphism. That these are two solutions of the same universal problem that are therefore isomorphs by a single isomorphism. And that's what we call the associativity of the product. So to make it clear, we have understood that this isomorphism is not identity. Thank you for your attention. Well, that's what I wanted to say, that is to say that between, well, here it is particularly obvious, I mean, it's at x, y, or z, so the thing is also x, y, or z, okay? No, no, no, I understand, I understood that there was a point in the demonstration, but I don't see that it would change if there were more. Well, what would change if there were more is that when you want to go from this object to this object, you would have a choice. In other words, starting from an element here, let's say there are elements in the dance floor, starting from an element here, if there were two seismographs, you choose one, it gives you a certain image. I choose another one, it gives me another image. And then we still have the job of comparing these two images. This is exactly like when you take a vector space, you take R2, it's cute R2, but we can all take completely different bases, and therefore a same element, we will describe it in a completely different way, while when there is a single isomorphism, there is a single choice of magnitude.

12:30 But precisely, since I know that there is a unique isomorphism, starting from an element here, I know exactly how to write it here. In fact, to say that there is a unique isomorphism is a way of speaking. Between these two objects, there is not a unique isomorphism, but there is a unique isomorphism that makes you wonder. And if I define a product of three objects or one object in a naive way as a solution to a universal problem, why? At the same time, for all objects, do I really hide something from these people? No, but if you say, I define, for example, imagine that you say, I define the product of each object by the universal problem, for this will be the solution. Yes, but with diagrams, but let's say that ... If I say I take the limit of this diagram of three arrows, what do you think? Yes, but wait, there you are... Yes, but do I really hide something? Yes, well, I mean, it's not that you hide something there, but it's the fact that you... What is the universal problem of the product of three objects and then demonstrate that in fact we can bring ourselves back to two choices of the product of two objects than to define directly like that, there would have been a reason for this one to be the same as this one. If I take three objects, I know that it defines a present isomorphism. That's exactly what I wanted to say. But I don't have that kind of sensitivity, I agree with you. Ah yes, that's the problem. And then it will depend on... But is it really vocabulary language or is there a deeper connection? No, but if you define, if you give a definition of what it is to produce three objects, there is no ambiguity, there is no pressure, and that's what I did, I took grand I but you take I equals 1, 2, 3, and then it's good, but after when you want to achieve it effectively, it is useful to know that in fact, as you define it intrinsically for a family that is good, you can come back to a family that is good.

15:00 But in the calculations, we always come back to Madelaine's pentagon, so it's a problem of parenthesis, okay? Maybe, I don't know, but I can make this parenthesis at a lower level, perhaps, if I think of the general definition, that is to say, but hey, if I pay more attention to theories, I don't know. So, instead of making a parenthesis in terms of product defined for the pair, I can make this parenthesis in terms of... Yes, but at this point, it's going to be a problem of coherence. Because from the moment you define the parenthesis, you have to be able to compare the different parenthesis. You always come across the problem of comparing different types of writing. So, by using this associativity, we say that a category contains finite products. But when we say finite products, it should mean that it contains empty products. Well, that's exactly what I was going to say. We arrive not at individuality, but simply in the writing of things, that is to say, to add that eventually the products live.

17:30 So, to make it simpler, we admit that when we say the product is finished, it is included on an object-based family. This is the final object of the category. This can be verified immediately on the Chinese diagram. So, before 1900, when we say that a category has the Chinese products, it means that any family of Chinese, incomprehensible, will be able to say that the category has Chinese objects. I remind you that Chinese objects are not free in general, but they are isomorphic, by a single isomorphic. This is the definition of the final object. Obviously, for the sums, it will be the initial object. I will still talk about that. So I take the product diagram again. At all layers, FG corresponds to a minic H. It is in the form of C, C, and in the form of C, Z.

20:00 At all H corresponds to a minic S and a minic C. The subject of the lecture will be the unification of H by the unification of Y by consequence, and you will hardly deserve that its function is the same. The unification of H by the unification of Y is valid, of course, for any kind of family, for any kind of object, because it is exactly the same thing. For any family of phi, there is a unification such as this one. So you have the product of the product of the product of the product of . And isomorphs, a man or a woman, protists. Beware that Y, F, so what do I have?

22:30 In fact, this defines the product of these computers. So, you can define the product of the smelters directly based on the number of smelters associated with the product, provided that we are going to aim precisely because it is necessary to pay attention, it is necessary that in the categories there are products, I have already specified that. So for now, that's actually the definition of this smelter product. I have not yet defined it by name. This is the definition of a particular type of factor because it means that if you have something at the base of an effect, it has to do with the ensembles. So this is the product of two ensembles. So the product of two ensembles is not very bad. The thing I wanted to draw your attention to is that, of course, this is generalized to be any kind of object. And so, we have man, sum over i, apartment, plastic, physics, fung, and precisely, we produce men and women. Okay? You have to be careful. That is to say, on the appropriate member, it is always the product. Okay? Good. So, precisely, we will come to what you were saying. We didn't say yesterday that the finished product comes together. In fact, the amounts, not the sum. Yes, if the product is finished, there is no problem. But here, it's a mess.

25:00 Well, then, the product that has just been finished is transmitted in the same way to the computer. Because, in fact, it is defined like that. If we look for two, it means what is the product of two factors. Like the category T, this must be a factor of C. If we have to define on objects and on arrows. So in fact, if we have two factors, we have f and g and in there we have a product. And we have seen that the product goes to the morphism. So the products that have been given circulate on the products in the filters. So you have to be a little more precise. There I talked about filter products. Now I'm going to talk about something else. I'm going to specify that it's something else because you might be confused. I'm going to talk about the notion of a filter that preserves the product. So, what we are going to see here is that we have a category C with products, so D, a kind of F that you would like to see, to tell you what it means that F preserves products.

27:30 We look at X, Y, Y belonging to Y, a family of objects of C, and since C has products, it means that this family is a product of C. To say that F preserves a product means that the F family here has a product, and that this product is... I would like to draw your attention to the fact that, from the beginning, we said that, for example, in the case of two objects, product of two objects, it is an object and two molecules, so it is not just an object. But we want the product in D to be the image of the object, the product, along with the morphisms, the image of the project. It's more restrictive than saying that there are only two products. So you let us exercise to find a contract with the object? Absolutely! We need the students to give us a familiar exercise. I know, I know! If a counter preserves the product of two objects and preserves the final objects, it preserves the final products, because the final object corresponds to the product of an empty family, so if it preserves the product of two objects by associativity, it will preserve the final products.

30:00 A family of non-families. And if they preserve the final objects, they preserve the family products. An example of a functor that preserves the products is the human functor. The human functor of the first variant. And as a corollary, the plungement of the data preserves the products. What I would like to do now... We talked about products, about co-products, we gave examples... But I would like to show that all this is in fact a particular case of what we call in a more general way the limits and the co-limits. When we talk about the limits and the co-limits, we must first introduce the notion of the diagram of the two definitions.

32:30 We consider these categories. And, as I have already pointed out several times, a category can be associated with the graph that is assigned to this category, okay? The top are the objects and the ends are the objects. Well, then, a diagram is the energy data of a diagram, of a graph I, in a diagram, which should be precise, of base I. This is the data of a graph and a morphism of a graph. So, let's take the first example, which is completely equal. If the graph i is a summit and not a stop, then a diagram is considered an object. The second example is i equals 1 2.

35:00 Let us reflect on this data. Now, if the Y-diagram is the data, it is the following diagram. So I will draw it like this. So what is it that this diagram defines on this graph? Well, it means that we are going to give ourselves three objects. The image of 1, I call it X. The image of 3, I call it Y. The image of 2, I call it A. And what are the objects of I? Is I also a graph? I is a graph. So I say there are three summaries and two stops. I take the example of the graph where there are three summaries and two stops, defined as this. That is to say, it's a sort of indexation, it's I. Yes, in fact, if you anticipate the future, I is what will index the diagrams where you will take the limit. If you take, for example, for I the diagram that corresponds to N, you will have the inductive limits and the projective limits.

37:30 It's just A, A, B, L. It's an abstraction of diagrams. I mean, on these examples, of course, you don't need this curve, but if you want to define diagrams in general in the C category, what does it mean that you index how objects and morphisms and relationships between morphisms and objects? You have to give a definition that will tell you exactly what the diagram you are looking at is. And morphisms, perhaps any, that is to say, in these cases it is also possible to do just... No, no, a morphism of drape, it has a precise definition. That means that at the top of the drape, it corresponds to the tops here, If you stop between two ends, it corresponds to a stop between the ends of the image. But still, we can kill an object. I don't know if we can just take X-A and it will be a morphism. But let's say, when you write, it's, I don't know, it's isomorphism. We can also take X-A and it will also be a morphism. Yes, of course. It is also possible to include dictionaries. And precisely, all of this, this definition, it encompasses all of this. So the idea of i can be done on the shapes, on the forms of all the diagrams that you are going to look at? Yes, absolutely. If you want, it is something that, from i, you will be able to describe all the possible graphs. There are two possible bases, i and c. It depends on the idea. You mean they have the same art? Yes, we can say it like that if we want. But in the end, I think what we need to do is to look at what we can do when we have two ideas.

40:00 So that was just to say that with this very simple example, we will find our physics. I've been reading texts that didn't include this definition, where they talk about families and things that are much heavier, but it's much more pleasant to work with that, because the echo comes right away with a point that is a degenerate diagram, it comes right away, and it makes things much easier. What is it? It's hard to understand what it is. The definitions are clear and we know exactly what it is. Well, now we're going to talk about the cone. Of course, we can emphasize that we say it's a semi-diagram if the grand diagram, the topological definition, is the same. So, in the course, we assume that we have a diagram in C. So, a basic code of this diagram in C is the data C of the equation C and of a family of matrices T, I, C, and V of I along the sum of I.

42:30 We can say that I belongs to O of the equation C. And so, if I have an array of i and j, the graph i, then what do we have? We have c, we have v of i, v of i, we have a morphism here that corresponds to this array, we have a morphism pi, a morphism z, of course, and we have a graph i. The difference would not justify introducing the notion of morphism between diagrams and then saying that a projective cone is a morphism of a constant diagram. Indeed, we can do it like that. Rather than defining the cones and then the morphisms of cones, introducing the morphisms of diagrams. To say that a cone is a morphism of a diagram between a constant diagram and the diagram itself. So, if I have a second cone, that is to say that I have a second object and a second family of morphisms, it is obviously this object that makes the object.

45:00 So, of course, all of this is defined by Morpheus and Céline, all of this is well composed, so we have a category, I would say the category of basic cones, the diagram, we are fixed. Yes, it's a good detail, but I think I understand the difference. If I simply say that the cones... It is given the family of C objects, I am not talking about diagrams, and the family of spamorphisms with these objects. And then I repeat, I give C, I give family like that, and everything comes together. That is to say, I quickly spoke at the beginning of this B-morphism. What does it change? In the concept, nothing changes. But what will change is the ease with which you will write all this. Take a sheet of paper and write the precise definition of what you just said. Without talking about graphs or morphisms. Of course it's doable. But I think it will be more difficult. In my opinion. The concept is just a way of describing the concept. I believe that we must always choose the choice that gives the most precise definitions. I agree, but I think there is even something more... it's not profound because it's... but let's not say... what I said is more ideal, it's more commonplace. Here it's special when we don't... The difference, the difference... I have defined the notion of graph as saying that there is a set of objects and a set of arches, so obviously it is a little more respective. Well, so here is the notion of cone in an alarm, and from there we define the name of liquid in a diagram.

47:30 A diagram of dances, dances, as we have heard. Ah yes, yes, yes, I saw it like that, I think it's very beautiful. It was not me. You had to choose. Example, I take an example, and it's not by chance that I had given you the examples of diagrams earlier. I take the diagram also defined by ... I have given myself two objects. What is a cone? It is to give oneself an object and two objects.

50:00 What does it mean that I have a final object, the category of cones? There is a single object. This diagram is the product of the x-axis. The product is a particular case of the limit. So, exercises have shown that the product of a family of any object is a limit. That's what it gives. The diagram defines the solution. And you can see, it's obviously redundant what I mean, since I showed before that the fibrous product is a particular case. You can see that now, if I take the other example I gave, the diagram that corresponds to this, you can see without pain that the limit of this diagram is the product and without going through the intermediate of the product. Here is the example of the V2 diagram.

52:30 What is a limit? It is an object of the category with only one plus or minus one. Well, now I'm going to give another example of a limit that corresponds to something that is produced. I look at the following diagram. That is, I take the graph I, I'm going to draw it. This is the graph I'm looking at. So, if I take a diagram of Basse-Marche, it means that in the conventional category, the diagram has two objects, two branches. So, what is the limit of this diagram in the category? It means that I must first define a cone. So, what is a cone? It is a morphism. I'm going to have a morphism of C in A, and then in such a way that I'm going to have a morphism of C in B, but in such a way that it overlaps with C in A. So, in fact, I just have to define this one, and then I'm going to impose that there is no other cone. Well, so a cone is this data, a little like that. A final object is this data, such that for any other cone, there is a single morphism like this one, which makes it the idea of A.

55:00 This is exactly the definition of equalizer, equalizer, equalizer, generalized core, so this is the equalizer or equalizer of Rxg. This is the general definition of core. What is the equalizer there? It's that. It's E plus the mantis? It's E and the mantis. That's why when we generally define a nucleus, a nucleus, contrary to what is done in the vectorial matrix, we say that the nucleus is a sub-vectorial space that makes it move on the other side. No, that's not it. The nucleus is the morphism. In fact, the inclusion. That's what the nucleus is. A defined morphism. In the definition of categories, or rather in the characterization of categories, a category or a diagram has a dense limit.

57:30 This implies, among other things, that any family remains an object, an object. In a way, what is the lack of complexity in this sense, when we only have the products or the products and not the products? The answer is given by the following theorem. A category of objects and any couple of quantities is complete. What could possibly be wrong for the complexity of a category when we only have the products? Well, they are the equalizers. There is only a cleft between C and E for all the Cs, such that there is a cleft on A and on B that makes the commutator fall. So, I will repeat what I said. It is to say that A and B, more precisely than F and G, have an indicator. It is to say that there exists an object E, a J of E in A, in such a way that for any object C, for any object, such as C and G, there is C. So, what is a cone?

1:00:00 Yes, because there is only one C-A-D arrow. By definition, what is a cone? Well, now we would have to dualize all this. To talk about collimites of a diagram, we need to define the notion of a cocoon, the notion of the morphism of a cocoon, and then the collimite of a diagram is an initial object of the category of cocoons under the diagram. There is also a problem of terminology. The limit is a generic word, but it is also the limits that you have described. We also call it the projective limit, we could say the limit of the left.