From Klein to Kan / Q&A / Grothendieck's Universe

0:00 He does both. And then in 57, he does both. In the second zone, and I think this is very interesting, we have very simple conditions that we think satisfy in order to do homology. Now, we're trying to do something really simple. We have, and this is straight from the paper, for every topological space, you have to be able to take your product. For any topological structure and you have to have projection and, of course, projection. He defines what he is in a homotropic. I'm sorry, he says, what do we need? So, it's purely abstract. C is a category, one, that's my intention. The identity factor and E is the identity natural factor. A homotropic in C is a factor that you call... So, it's a functor together with three natural transformations, and here you see, like, concepts for, and this is for getting, okay, and you're not, you're not doing more than that, in a sense. You're not defining a new, complicated, you're getting a lot of stuff, and in that paper, he's to do afterwards.

2:30 It's basically to show, if you take C, space is the functor, it's just product, it's a unit interval, and cubicles, they go to complex. And then defining the product there. And then you can add a homotopy in that case. And then he does the same. And he does it in many parts of the paper. By showing particular cases you bring in that will give the possibility of doing homo directly in those. So something on the category S that he mentioned about may be extended to a homotopy. What he said may be extended to a kind of homotopy of the whole category S. So this is five, this is the half. Finding a counter, the infinity, our main tool will be what we call an extension, which is in a certain sense the dual and the greater than zero.

5:00 There exists in a natural way one-to-one correspondence between the subdivisions. That's what you find in that paper, which is a longer paper. Now that paper is published in the fall of 19- the editor's fall of 19- you know, it's dead, he has it.

7:30 But he said this also. The Symmetrical Approximation Theorem may be journalized to complete the MySymmetricalComp. Select K and L, V, then every continuous map from the geometric realization of K, homotopic with the subdivision, he's proving it for, but he's also using this correspondence for it. Using the answer in Poincare, Poincare's S, D, and X, a dual theorem may be obtained, which involves, is essentially because of this, as far as homotopic theory, behave like. So, they are. In fact, we're in 1950. In 1958, so this is the year of the public. The whole work was done from the spring of 1950. It's work already contained. So he publishes two papers. The first is involving Euclid's device, which is complexive, and Atron's punctures in the transactions.

10:00 It is clear that the paper of Atron's puncture is written so that you can actually see what is going on in the public. Why? He actually has one main theory, and then that. I've been working with my categories previously, and then some more. Let me just say a few things about that paper. There are a whole lot of new ideas in that paper. For instance, in that paper, it is defined the whole paper. For instance, a

12:30 CSS is now defined as a developed individual. Natural transformations may be constructed, which involve TSS conflicts. Several of the functors and natural transformations of penindus are well known. A new such functor will be considered. So here's a sort of very rough sketch of what we get there. Pan extensions are of course defined in the paper. Let's see if he can agree with this. He doesn't say so. And then what he said, which is a very general situation,

15:00 The algebra of mathings. When you see him moving towards purely combinatorial homomorphism, this of course was pulling him into space, behavior, and that type of practice. So the paper, as I said, was required in order to make sense of the paper on factors and qualities. In the very last section, so it became autonomous while he was writing. In particular, the connection with limits and coordinates became clear while he was writing. So the paper, in chapter one you have the basic definition as we know it.

17:30 There are no examples. There is chapter two where there are all answers. We're fairly aware of the generality in the general view. It's all given in full generality. So there's a reference to a lot of them, many references to them. And then Eilenberg is the new one. Captain Eilenberg says, speaking of blockchain, he says that it is a very interesting paper to find. Signage duology between the tensor product and the last pointer column, while both pointers are defined by a count.

20:00 As I said, they are used to prove its positive. In the paper, no connection is made to universal morphology. So it's clear that you have, for instance, a chord line applied in its book. It's an eponymous of science victims. If the dictum is true, then it is the punctures that can categorize it. If the progression was not here, there are massive new punctures and they're false. And then you have a different quote, quite different from the mannion. Symmetries of a geometric object are traditionally described by a convalescent proof, which often is an object of the same geometric class. Of course, such symmetries are only a particular type of morphism, so that climbs along a program made by the general mechanics. Whatever that means. It's not very clear and neither is Witten, so I'm going to tell you how to do it right. So, I'll try some test different theories. Finding it upon a deep basis is how to prove the problem of this space capture. Fundamental properties of this category theory is to be put by athoid vectors, not through the fundamental properties of the space.

22:30 Athoid vectors do elementary vectors, or non-trivial, what is that? Transformation rules to provide a unified, pure balance between geometries. So it is possible to compare geometries to show how they are related and extend to new geometries. What accurate punctures are available if it is possible to identify and unify three different mathematical principles to show how mathematical concepts are related and, again, building. I would plan with something that is coming, I think, called motion of transmission, which is at the right level to the accurate puncture. But then he asked me, I don't know. He's very happy.

25:00 Thank you very much indeed, Jean-Pierre. Extremely clear and a great deal of work done in Nostradt's topology where results are actually not allocated back to topology. And I want to say that because of how this method is allowed for certain programs in topology, But there's very powerful, very hard to match, where conceptually it's very satisfactory to understand how it does things when it comes to actually computing. It's difficult. So it's fantastic where it's conceptually very satisfactory to compute.

27:30 About a couple of these, it was not that much less.

30:00 Well, among the policies, there are many examples that I actually can't use. Yeah, I think he was just representing something he could say all the same thing, but not really in the language of mathematics. And then the idea of representative of time, so it's a merging of these ideas of choice. I'm just kind of asking, is there something in the Agon Center there, another kind of approach? I doubt it. You might represent... You can say all the same things, but without talking about the answers. But I read about the space system. I read that, um, that, um, I'll just take my time to do some references. Yeah, let's go over that. Yeah, I'll do some references. What, what is, what here is that?

32:30 And you refer to something you couldn't comment on, but I heard in the earlier question about the exact results, because it goes back to Venni and Plotera and the electromagnetic field that he gives us, And also on the quality of the space. I think it's very important to clear and prove by year-round that in some sense the content is very poor, but the quality, the quantities are very good. I'll ask you one question first. When did you first become aware of the Android market? And also when did you first, I mean, did you immediately see its conceptual organization and the things you talked about in your dialectic effect? Okay, okay. You have to know everything, comrade, don't you?

35:00 I know. I'll tell you this. Yeah, yeah. When I got there, I was out there. There was some comment from a student that said, I'm the fool. After I had been a student, I think the actual assumption is God. So, I went to see him and he said, Well, some student told me that actually the thing I was doing was God. That's the first I had figured out. And then he says, Like the board you want, don't you? You suddenly realize you've been spent. Thank you very much for a great talk and very interesting discussion.