Yoneda lemma & topos of music

0:00 Yes, I really understood a lot, maybe even before the other one, he also did, he never spoke, I'm afraid, in English, but he, for example, it was the result obtained by him that if you think of his hypothesis Lobachevsky, that is to say the negation of the postulate, so let's say that the postulate, the fifth postulate, is equivalent. It is said that there are, for example, regular triangles so strong that we can't see them. That's what happened. You know, in the geometry of the Bajir, there are no... What do you mean by the alert? Alerts. Yes, yes, of course, that's right. Or you can say that there are only similar triangles. In other words, he did things, but he never resisted, yes. I said it before, I explain it in some way, but I think that even the deep reasons were that he was not at all satisfied with this idea of geometric pressure. For him, he thought of geometry as something, as a thing. If you want, I find it a bit like... I'm afraid to say this, but it's a bit dogmatic, this pluralism is dogmatic, because normally it's something of a liberal philosophy, I don't know what he says. He thought it was a real geometry, but then we have a lot of geometries that we want.

2:30 The first thing, I don't understand why, if you think that, I don't know, the geometry is ancient, in what sense do you think that the ground is real? It's also working on the foundations and thinking about how we can change the foundations. But that's not obvious either. Why? I think it's not obvious. But I think it's a matter of interpretation. I don't know how to say it. No, it's mathematics. Yes, but for interpretation, I think it's false. Because what does it mean to believe as soon as you say it? You see, we've had this problem. We were not satisfied with these postulates, and you know, you never know. But in the end, it worked. I've been listening for a long time, because if you think about this problem or this commentary, okay, it's... But there is this suicide of the 5th postulate who came to build all this, it was... No, no, I agree that... But you told me that it's... I believe that for a common postulate, there is a logical reason. The logical reason is that... But I mean, if... If there is equality, it proves that it is parallel. Let's say, the existence of parallel, it proves that without the fifth. Because there is a theorem, the angle, but here I'm talking about the line.

5:00 That is to say, if you take all the postulates ... Because the first four postulates are also verified. It's just that this situation never occurs. There is never ... But here it proves, that is to say ... I think that the revolutionary idea is to represent geometry, but here we can prove that it is the absolute geometry theorem, it's just that they have the same way of doing it.

7:30 You can... No, you have to look at all these things. It's reconstruction. It's funny. You would like to have an external lamp. Yes, that's it. But we don't use it, we don't use it. We're supposed to know that there's a lamp. But we shouldn't use it. Thank you for your attention.

10:00 Thank you. But I don't think that's the case here. That's more of a question of... Yes, yes, yes, yes. What I feel is that the absolute limit theorem is saying that if two triangles have equal length on the equal side, then they are similar. Yes, yes. We need two triangles. Oh, yes. I know that you can look at the projectives, but I don't know how to do it.

20:00 I have two people here. Yes, but at least for the Greeks, it was a treatment that was not at all scalable. It's not natural. No, but that's what I understand. Thank you for your attention and see you in the next video. Do you want to say what we are doing with this program? So, how do we define this program? Well, I don't know. If you say, if it is a counter, you see that this arm here can't be this one.

22:30 Can it? Yes, it can. This is a triangle, this is an external angle, and this is an external angle, and this is an external angle, and this is an external angle, and this is an external angle, and this is an external angle, and this is an external angle, and this is an external angle. Yes, this one is not interesting, but this one is more interesting. That's right. It doesn't work that way. Yes, it doesn't work that way. How do you use it? Well, if there are collective terms, it can't be true. It can't be true. There is a triangle determined by a longer angle. Because precisely... Exactly. Well, I'll keep that. If I always take the angle of the bar here, I can't say that it is not determined by an angle and the two lengths that belong to it at the same time. And I can move from the first one to the third one. So I don't know if it's different from what we've used before.

25:00 But if we... I think we can use it when we show... So is it really the same as the triangle? If we take an angle of... not of two angles. Yes, that's it. We can't take an angle of two angles that are not the same. It determines if you take the side of two angles, it determines if you take the side of two angles, it can be like that, it can be like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, but not like that, What I wanted to say is that if you say that it is unique, you can say that this kind of construction defines it. There are two lines crossed by the other, and if the angles are given, and let's say this distance is given, you can say what's going on. But here, you see, there is no... and it's, let's say, you have the necessary conditions that are sufficient. But if you don't assume that... Let's say, you see, a continuum of possibilities.

27:30 It's simple geometry. It's because it's Lobachev's geometry that's a little tricky, in the sense that... But there are, let's say, several geometries and groups, but... If you give one, one... Then everything is defined. It's just to say, in negation of the fifth postulate, or I will announce an existential... Not given, but just given. After that, we looked at the whole story as if we were just changing the constants, because it's not a constant, but a constant that we call the absolute angle. What I learned after I took the drawing, I read this article by Beltrami, who found that there is also something that is interesting that I do not know. You know that these models do not work only locally, and it is Hilbert who proved it. There is no surface to dive into, of course, in Turkey, everywhere, you see, a constant negative. But the second thing is that Beltrami, he looked at that, okay, yes, here is all this credence, and in truth, it is that certain things in the spiritual space. And it's interesting that, of course, I, that's it, but I have the impression that, of course, Gauss, he understood a little that, let's say, it's that... It goes on the surface, because his main work is called the research disk of curved surfaces, where he does not talk about geometrics, he talks about properties that do not depend, that do not have good plunging, but that are variable, not on the supplements.

30:00 I imagine a surface, and then you can bend, you can do things, but not this property which is infinite. But it's the same idea, and in principle, why do I think he was really wise? Because we can expect that he could say, here, an Euclidean is reduced to an Euclidean. Because he didn't have an example when he didn't reduce. On the other hand, this idea of intrinsic geometry was not used as an instrument, as a means of reduction, but rather Riemann, who was his student, proposed several things for his thesis, which he also did with this function. If you want to look at this text, because I re-read it, you know it, it's called the Hypothesis of Geometry. There are really many things, obviously not very developed even today. Let's say the first thing is that he speaks of quadratic forms, as the simplest and most obvious case, but also he says, maybe we should look at the forms. The fourth degree is called the Finsler geometry, and that's really already in it, but also at the end he says something very interesting, for example, that of course the main hypothesis of all these constructions is differentiability, but also he says at the moment that even the concept of language...

32:30 There is no sense in a small infinitesimal. There is something interesting. To develop these ideas, I have the hypothesis, but I think that we can try to think logically in a transect of topos in the same way. For me it is not conceptual, of course there are others. Technical difficulties. It's not obvious that everything we call logical can be easily internalized. It's really not obvious. There's a paper I read, for example, of someone who... There are people who work who have what is called universal logic today. It's a bit like the idea of comparing logical systems, but it's two things. They look at logic just as a structure, as a whole, but the idea is just... and that's really deductibility, deduction, you see, just look at the whole of the position and then... But another thing is to take the known logic, if it's informal, and look at the morphisms, what it preserves. And it was one of them to look at what is preserved under homomorphism. And this author really says that there are things that, from a logical point of view, are important.

35:00 But I'm not sure because all of this is not done in the context of topos, it's a bit simplistic. But yes, but he says afterwards that there are important things in logic that do not preserve... But we have to look more technically, I don't know if he's right or not. It's technical, but in a conceptual way, I think it's not easy either. But to make it easier, maybe we can try to think of internal geometry in a more radical way, in the following way. Of course, it's a big change of point of view, in the sense that, believe it or not, it's only one geometry, but we always think of geometry as a framework of possibilities. And you can even imagine that there is another possible world with different frameworks, different ways of building it, but what remains is a geometry like... But with the internal geometry, the idea of space really changes. It becomes a little relative, because in a simplistic way, you have something like space and you have things in it. But here, you can look at things also like space. Instead of two objects, you have three. You have variety, with the word global, you have a kind of neighborhood, and in the neighborhood, you have points, that is to say you have all the points.

37:30 And of course, what has also been passed, I think, through these limits, but that, I think, is something like that, it's a big change, but it still has these different possibilities, that is to say, what, that on the local scale, you still have everything. It's a bit like an atomic hypothesis, less complex than the big one, which of course can be, because it doesn't make any sense to think that, to use it, to think of the reasons, but... I had the habit of saying something like, what interests the... It's a bit like the structure, the topology of space-time. I said, no, you're exaggerating. Because he believed that I was talking about topology, point set topology, topology of properties of the genre, I don't know, separation... It's you who thinks, it's him who thinks. No, it's him who thought that I was talking about that. I don't know, local connexity or something like that. I told him, no, I completely agree to say that... As far as the universe is concerned, locally, I have a rule to measure distances. Well, it depends on where you are. If you have Sirius, you don't know. I don't know what the principle is. And also, if you fall more into the quantum realm, you don't have any rules either. But it's a bit like a project I'm trying to do. Obviously, it's atomism, but let's say, you know, Richard Feynman, he says,

40:00 In short, everyone has something else, it burns in a nuclear war or something like that, I would say one, it's the atomic hypothesis, but it works, but you see, on the other hand, I believe, it's a bit my attitude, that... Let's say, these are empirical things, but it's just a bit true in the same material way, you see, that there are molecules, there is the atom, and that is to say, it's a kind of way of explaining things, of thinking bigger in the term of smaller. It works, but there is no reason to think that it works everywhere. Why not think the opposite? Why can't it happen in certain situations? You see, it's not the universal equation. You just have to think differently. Why can't it happen that in certain situations the bigger is simpler than the smaller? You can understand how it's smaller in the sense that it's bigger. And you can even imagine an example, you see, because yes, we're talking about physics, but if, for example, we're talking about biology, you see, all the things that were criticized in modern physics in the 18th century are still in the framework of biology. One idea was that with Aristotle, I don't know, there are four kinds of causalities. It's a bit special, but the two main ones are effective causes, it's like one ball falls on the other, it's classic, but Aristotle himself imagined that the final causes are the most important of the causes. That is to say, the ultimate question is not who moves who, but the ultimate question is why, for what reason. Then, OK, we criticized everything, we rejected everything, we moved everything, but in biology it still works. We discover, it's my mom, I know my mom, we discover very small enzymes, you see, molecular mechanisms.

42:30 And the explanation in biology is just to say what this little enzyme is used for, and if you can find out how this little enzyme starts to help for DNA or something, that is to say its result. What I'm saying is that you can sometimes explain the smallest things in terms of the greatest, depending on the greatest, it's also just reality. Okay, it's another question, should it be accepted? But from a metaphysical and more abstract point of view, there is no reason to think that, a priori, the smallest is simpler than the largest, even if it works in some cases. But then, this is the second part of my argument, I believe that there is a lot of mathematics... They are just based on this kind of metaphysics. And even today, okay, no one believes in this atomism in a way that is too serious, that's true, but it's already a bit in there, in the concept, you see. And that's why it's interesting if you change... I have the impression that, indeed, there is a lot of mathematics that is being built... That is to say... We simply think that to understand a complicated object, it is necessary to cut it into simpler objects, and that cutting is, in a way, taking smaller things. We can make the empirical observation that it works, but there is something more fundamental to it, that it works for a reason. There are things that are elementary and things that are more... But why? Well, you see, it's already Leibniz who said that. Or Lenin too, something like that. Leibniz said that there are very small flies, insects, and he looked at the microscope, you see, there were also small animals and all that. But let's say contemporary physics, it doesn't give... How can we say that the simplest level becomes something more...

45:00 But you see, it worked well with chemistry, when we arrived at about 100 atoms. But 100 is already a little too much, 3, magic number, 100, but still. That is to say, at this level of scale, this program of reducing small, it works well, I would say. But after, I imagine, particle physics is a little ... I have the impression that it continues to work, even if the standard model is rather 56 or something like that, it's a lot. But the standard model is a little different, because the standard model is the theory of fields. The theory of fields, you do not explain things in the terms of atoms, that is to say, you propose ... You can define certain fields, and then you can discover. In other words, in some sense, field theory is not atomistic. But even if we talk about pure mathematics, I think it was really one of the motivations of category theory. Versus ensemble theory. Exactly not looking or defining mathematical objects in the sense of... You will not understand what is inside, but rather in the sense of certain external relationships. And then you can think, do you have such an external relationship, and what can be inside, what are the restrictions on what can be inside? And even in Saint-Gilles, there is a very good book, I tell you, it is Topology via Logic. And I also thought of you when you showed the monad, because he says, look, there are two ways to prove things. The theory of ensembles. If you want to find something, it's a little bit, you know, it's a little bit like that. You study this object's body, you talk about how it relaxes, how it... On the contrary, if you want, you think about...

47:30 You have to do two things, that there is no real reason to think that either you always have to fall on something ultimate at the bottom, or then some complete universe, you see. Instead of thinking that there is something like an atom or their universe, it's a bit like we can say in Kant too, it doesn't exist, the atom doesn't exist, and the universe doesn't exist, the universe doesn't exist, not in the sense that it exists, but in the sense that it exists. That too, there is no relativity, these three levels, let's say, they are in a rigid way, you see, you don't have your global structure, but you don't want...

50:00 What I think about the project is to be able to say that it is to look at the other points of view, to look at the global. It's a little bit, when you look at it like that, we don't get to the global relativistic in the fixed sense. I'm talking to the people in the room. That's exactly why this talk is so nice. Because you see, what I mean is, of course, you can say these points, these points are together. The theory of the whole allows you to look at each element as a whole, but at least you can have a name. It doesn't really have a geometric application, because of course if you say these points there, it's also variability, you can say that. But on the other hand, it's a bit like saying nothing, because it's a bit like having a geometry. How can this modernity be compared to this great modernity? By this very simple reason which is, I think, there is no...

52:30 ...is a morphism, it's an infinite difference. But it's not idiomatic. Ah, yes, yes. Of course, but it's all. You can't distinguish between the situation of science and science. Yes, yes, that's true. And the problem may be exactly in the logic. Why? Because, you see... In a way, what are the varieties? You see, in a way, when you say, I take all the points, it's a bit like a program.

55:00 The first thing is to generalize the emotions, to differentiate, to define the forces on the waveforms, in the case of... What's the relationship? I just think, what's the relationship? If it's not differentiable, that is to say, at least here, you don't have this... Sometimes octal. A piece of octal is as complicated as an octal. Geometry, non-commutative geometry, these two kinds of things. Yes, that's what I've learned, but I haven't read this lecture by Petito. The political school made an explanation, but I don't know. But anyway, it's the same thing. The second thing is that we can really go beyond this. At the beginning, you see, it was when you try to do the same approach as for geometry, and not precisely by analogy, but if you want, just continue, but let's say, do it in the topos, all that, and from some point of view, you can say it's a bit trivial, of course, we can take points, not in the ensemble, but topos, it's not a big deal, but on the other hand,

57:30 What I also find interesting, we talked about it before, you started doing something and I asked you why you didn't do it together, and you said to me, of course, you can always take a big topic and do it in a big way, it's true, and it's a bit trivial, but what's not trivial, I find, you see, here it's a bit the same thing, just imagine someone who thinks like... like Beltrani. He always thinks in a very Pleiadian way. He says to you, ah no, no, someone says to you, okay, I'll tell you, variety. Then he says, how do we build? He says, of course we take some space, Pleiadian or larger than Pleiadian, sometimes we take time. But the interesting idea is just to relativize this thing, you see. When we talk about variety, we don't really need this. It's all good. We can always use a variety, but it's not a necessity. There is no such thing as a need for a founder. Maybe it's the same in logic. Of course, you can say, I take another topos, a big topos, do something in it. But what's interesting, it's not even what you take, but even this possibility of relativity. What you can build, one puts you in the other, it's a bit like you can build a variety in the other, and if you take this kind of relativity seriously, you can just say that, I don't know, I don't need it, but it's not obvious, it's not obvious, because what really helps, but when we talk about, we don't know, I know, by the way, it's something not obvious, someone who always says ...

1:00:00 Yes, yes, that's exactly what I wanted to say. That's why this notion of variety seems a bit simple. It's exactly because, at least, you think of something when you start in the first frame. I'm trying to think of how we can radicalize this entity, let's say. Replace the variety by the topos of the beams on the variety. So look at the topos in question. Yes, that's exactly the kind of thing I'm thinking of. But you know, it would be important to articulate these ideas that, let's say... In a certain sense, as long as you can do exactly that, how do you start all these questions about the foundations, what do you start with? You see, again, it's the idea, not mathematical, if you want, methodological, but which is very important in mathematics. You always have to start with the foundations, or just build the foundations. You can start with nothing, but here or there, you can be... But it's not just, you see, I'm just saying, you just don't need the basics, it can't be serious, but more seriously, I think finding the frame when you can tell me, not only in the geometric sense, that there is no answer at the beginning, you see, after you build, but rather, it can work together with, look at it in a higher way, maybe it's an archive of Michael, you understand well, you have this idea, and it's still there.

1:02:30 In all the books, even in the Joneses, ah, I see, not that. Ah, ok, but it's already a bit technical. He's talking exactly about the question of sites. That is, when we define, when we define the site as a function of locales or of ensembles, for example, it's interesting that it's not really important that it's an algebraic structure of locales. You just have to see what you really like. Yes, I agree. Anyway, for me, a site is a category with a lot of information. I don't know the local ones very well. On the local ones, yes, you can look. Here, it's a bit complicated. But other than that, there is no non-standard, non-standard space. It's in the computer. Is it without the small categories? No, because, indeed, there...

1:05:00 That's it. SGA is building topologies. For example, the topologies of Courtenay, which are... What is it? The Geometric Algebra Seminar. I don't know it very well. I mean, obviously, you have to look at the things of Rotundi and all that. Yes, because in the book you don't find it, you always have definitions. But maybe in Johnson it's better, because I told you he tried. Anyway, I think it's very complete. But what interests me, you see, is just to understand how we can arrive at the concept of topos in a purely geometric way, without any classifier. In any case, a topos of Grotonik is not exactly the same as a topos of Leuvert-Tierney. A topos of? A topos, as we present it, is a topos of Leuvert-Tierney, but it's not exactly the same thing. But it falls, it satisfies the action. Yes, but in fact the reciprocal is quite true. But it's still not very far. In fact, what Mugrotendi introduced in Adi, in Dopos, is a category equivalent to the category of assembly beams on a site. And in fact Giraud showed that there was a purely categorical characterization of topos in this sense, which we call the topos of Grotonique. There are a number of works in Riyadh. And from there, he shows that there is a categorical characterization.

1:07:30 It is exactly a category that verifies this. It is here where he defines what is functor-logical, what in principle is conserved by the object. And also geometric morphism, which is more common, but by definition it's just what keeps the limit final. But then there are the consequences, really very, very strong. And it's not exactly the same thing, as you say, but almost. Yes, maybe, you know, if you could explain all this in a more precise way. Yes, these are things that I tried to learn a little. Not today, maybe. Maybe we should go there. It's 7.30. 7.30? Yes, maybe next time you can do... Or you could continue with... No, I don't want to. I don't want to talk about it, it's a bit of an obligation. Okay, okay, go ahead. Okay, I mean, really. Thank you for your attention and see you in the next lecture. Thank you for your attention.