CJ Isham & Andreas Döring's topos-theoretic framework for physics (contd.)

0:00 Yes, I don't know. If you want, it would be a connection if we were on a reality. Attention, rappelle-toi quand même que chaque objet, c'est censé être un ensemble causal. Donc, bon, tu as raison, ça joue le rôle d'une connexion, puisque c'est défini, c'est bien défini, finalement, au niveau infinitésimal, puisque ça, ça représente ton vecteur. Donc, quand tu passes d'un objet à l'objet à côté, c'est effectivement infinitésimal, If it's reading the fibers that are at the infinitesimals, it's exactly what it is. In my opinion, it's only the second approach where the use of categories is justified, if it really gives a good physique, of course. You can say the ones that are recent. No, no, no, but here, when you redefine it. Because in the first case, with all these, how do you say, aero fields, but there's nothing new. without the notion of category. It's something that gêne a little bit, because you're obligated to reduce this monoid, but why not start with this monoid? Maybe I don't see it. You can reduce monoid without talking about category. Oh, no. No, it's fine. So, let's talk about what you said. But at the same time, but at the same time, the way he introduced it is very artificial. What I haven't really understood, but in that measure... You're right, I've given you the solution. Instead of talking about Fibril, you need to talk about Préfeso, because that's really the language catégorical. In the second paper, which I haven't looked at, he applied it to the postmo ensemble. It's interesting because this paper of Topos, the approach is from the other side, it has another motivation, this logic motivation. Maybe we can obtain the same thing from the inside.

2:30 What do you mean from the inside? Here, we don't think of giving a logical cadre. Not at all. But the category of the prefaceo is always on topos. That's a theory. D'accord, d'accord. And then... Well, it would be possible... Effectivement, Monsieur... I was convinced that we had started talking about this. We talked about the other articles, but we didn't have been very far. However, of course, there are also some limits of this approach because they talk about a small category, and maybe it's a sort of protection, we can say it's infinite, it's physical, we can do it. The fact that this category is small is to define the inner product, I think. It's important for that, but... I mean, in physicality, it's not the impression that we're going to be able to envision other things than small categories. But no, because to my opinion, small and large, it's maybe a little wrong language. It's important that, for example, if you take a small category and you say that the limit exists, it's cartesian, then it's trivialized. It's to say that there is no interesting notion of small category, but of course the limit, which is not the case here. We can say that a small category is something even trivial, in fact. I don't know. Well, that's for you. Any other remarks? Well, it's possible to say that this quantification has an intérêt pretty and limited. It's about finding a way to get rid of operations. It's to say that, obviously, everything that is happening inside, in the ensemble causal, it doesn't have anything to do.

5:00 The flash, it's that. So, in fact, we don't really know the space-temps, whatever it is, we don't know, we don't know, we don't know, we don't know, we don't know a sense to the history, if we follow his language, the history is defined on the same page. To say that this history corresponds to the space-temps, it's a guarantee because it's in the ensemble causal that there should be a certain structure that would allow us to see how all that corresponds to our vision habituelle. It's to say that the paper solves a problem that exists without a doubt, the problem of building in the mathematics, the same proso, for example, we take an ensemble proso and we add a plus, so how to link all that, etc. So it's a kind of story consistent, story coherent, on the same proso. But the other problem, for example, is what we do with an ensemble proso, which corresponds to our notions habituelles. And the answer to Ticham, that these are the stories on the ensemble photo which, they-mêmes, correspond to the space-temps, in a certain way, I don't know. A space-temps quantique? Yes, in the sense that we don't look at the internal structure of this ensemble causal, it is completely arbitrary. Wait, you don't have an ensemble causal, you have the collection of all the ensemble causal. Yes, I have a category, but if I have a small category, like in these examples with 5 elements, I have exactly the same thing as if I have a category of an ensemble poseau very complex.

7:30 In fact, I believe we can respond to this objection in a very general way. Maybe we will need a large category for that, but in fact, what is the interior of objects? I think it's part and not complete, it's to say that you can express all this internal structure in terms of fleches that come from there. But in general, even more general. I just want to say that instead of looking at the ensemble of fleches that come from there, You can look at the dual ensemble, the ensemble that comes in, and that is what gives us the notion of interne. There is no other notion of interne. And in this sense, if you have a category of objects such as the Cozo, you don't need to say something more, which is already in the categories. It is a structure of the internal object, even without saying what is the object. But if you have all the flèches defined and you make the difference between the composition or the composition or the non, if you do not forget this way, you have a whole structure of the internal object. What is important is that for Echam, it is what we can say. At one moment, he said in another page of the Cotard, that we can not look at the flash LX, but the flash inverse, that is the flash that has given the domain. So, probably, we could do the same thing with it. and for him it's a whole story that he says that it remains at a level of abstraction such that we don't see any liens. Every time that Surkine does his ensemble proso, he tries to build a lien with the operators so here we are really on a meta level, there is no question to take an example of this and to look at what are the operators, I don't know, the volume, etc. Here we are at a level, Picham 3 is sufficient, but I'm not sure because I'm not sure that this kind of analysis is sufficient

10:00 Yes, I'm sure that it's not enough, but I think it offers a framework. Yes, it offers a framework, but it's already good... Well, obviously, there isn't a result, so there are operators who... More or less, Rolly and Crouseau are... There are questions about it that we can ask, why it's simple, why it's simple, why it's simple, why it's simple, why it's simple... You could ask if the spin networks are going to enter in this case, or if the spin forms are going to enter in this case. Is it possible that the spin forms can be considered as the version quantique of the COSET? Well, I don't know, but I have a question. I think that's what he did, because in this case, he has a group that is... Yes, he is an opérateur of creation and deanimization. Well, you're right. If you want to use it as an analogy, it doesn't show that formally, this would give us the theory of quantum change. That would be interesting. Yes, I agree. But I think that, according to the operators, it's not the operators of compression and annihilation in the sense of the theory of quantum change. Well, it's not obvious. I haven't managed to find the answer to this question. In fact, I think I'm going to ask a question. If things are similar, not perfect, but the most standard and common in the category. For example, of course, there is an interest, we can look at the category given as an algebraic object. Yes, we can forget all the complexities between objects, etc. this notion of monoidal category. For example, we talked earlier about the external products. And this is something that is made from a mathematical point of view,

12:30 a little more coherent. Introducing this kind of monoid, etc., I don't know, but... It's not that he introduced it, it's already there, because of a certain way, you can see that what he said in saying that the flesh reminds more or less the impulse. Well, when you think about it, it's true, it's an analogy. And finally, the flesh, it's not enough, because it's not enough, it's forced to generalize the flesh and it forms naturally a mono. This monoid, you have to be there, and you have to tell... There is the section 5 of the second paper where he tries to do the theory of calculus. I don't know. I think there is a standard approach of calculus, is that when we go down to the standard theory, we have a lot of things from the beginning. I think that the standard form of the operator of the creation in the space of Pop is a little bit too presupposed. I would say that, when you do the standard theory of the standard theory, There, you have plenty of presuppos. So if you apply them to your category vision, do you find the space of Fock, for example? I don't see anything else, a priori, but it's surely a big job of showing them. So in this section 5, he takes R3, he writes the formalist with R3, and then at a moment he says, and we are not sure that such thing becomes the habit of the practice of the practice of the practice of the practice of the practice of the practice. So if he says that... In general, everything he says is just the point of view of the calculation. In any case, I didn't see any insuffisances, even in general, he does all the steps.

15:00 It's quite easy to read, but it's at least one of the biggest interests of you. Yes, after the question of Moukowsky, if for a European space, it will be the same thing for Moukowsky. I think it will work. No, because what happens is that every time we introduce the LFR on something, that is, we look at the L2 on L3, we look at the L2 on the Minkowski, that comes with plenty of structures. We already see how things interact with L2. In L2, I see a lot of structures. I'm not sure. I don't know. Yes, yes, for sure. The principle of the geometry is that, in the algebra of functions, you have to code the structure of your variety on which are defined as functions. So, yes, you're right. But it's something that you need to exploit. It's not algebra. That's why, what I'm most worried about, is that when we introduce them, we put something in hand that comes from hand, I don't know. But I'm sure that passing by the Prefesso, if it can be done in a natural way and more categorical way, it's good. No, no, but the history of the professor, as you saw in the papers with Doring, for him, now, it's become a bit more affinable. For him, the catégories, the topography, it's... Yes, but... Well, I... The question I ask is, is that I adhere to it or not? No, but the professor on the algebra, it's not... Justement, we are here, we could decide that the question of whether we adhere to it or not is perhaps one of the important questions. In general, you can say that if you are interested in the categories of causes of one side, and on the other side, you don't look at the relationship with Elberts or the algebraic operator on Elberts,

17:30 then you have to look at the categories of factors between these two categories. That's absolutely general. I'll start a little because I'm not going to read the article. I feel like there is an advantage in the fact of not taking into account the structures of the morphisms of an object on itself, because the operators point of view do not change the nature of the particle. So if the structure is described by the morphisms of the double, in this case, we have an evolution of the same particles, I think it's an advantage. But they don't describe the internal structure of the particle. There it doesn't come very well. The interest of the categories is that there are the identity and there are the isomorphism. That's to say, what defines the internal structure of an object is the whole of the morphism. but it's all the morphisms that come inside. It's not just morphisms. Yes, it's just in the sense that the object has no structure interne. Everything is in the flesh. But what he does with the flesh, he takes the flesh of this type, a range and he built this structure of the operation and integration. So, for him, in general, it's all that there is in the level of the structure. There is no more. There is no more. There is no more. There is no more. There is no more. There is no more. There is no more. to find the structure of an object. There's no one. But I wonder if it's enough to be contented with this kind of analysis, if there's no more to say, more to say about these things. But he makes himself a remark. I think we need to go further. Because if it was enough,

20:00 it would mean that On aurait pu simplifier énormément au lieu de... Il y a quand même dans les espaces causaux une richesse énorme. Si on avait abandonné, si en gros on avait remplacé toutes les flèches qui vont du même objet au même objet par une seule, on aurait perdu énormément et ça devient complètement trivial. Il y a quand même des remarques un peu bizarres dans sa part. Par exemple, il dit dans ce papier 2, quand il analyse la théorie critique, il dit « Note that although the objects have no internal structure, it is still of course true, it is still of course true that the Hilbert space L2 on R3N, L2 sur R3N, carries a representation of the equilibrium group E3N. » So, yes, of course, there is no structure with objects, but still, L2, we know it, there is the rock that he has, so, he also... That's why, to start by an space of Huber, it comes with a lot of things, because L2 on R3, L2 on Minkowski, and then, immediately, we have everything that comes with. There, he has still a Hilbert space almost natural. The functions are natural, it's a vector space, they are in a product. The only thing that would not be natural, it's a matrix. How to find a matrix? It's not at all obvious to find a matrix, a matrix, I mean, which will operate on all the spaces. I don't remember, but there was an educator who showed an adjunction with a relationship with an adjunction in the category. At this point, there is no need of a metric. It could be, in this case, define... Yes, an adjunction without passing by an adjunction. It's an adjunction a little bit elargie, but it was just done in the geometry projective and the work that these students did. I think that could be fantastic. It's interesting to take part of this work for René Littard.

22:30 I don't know, I don't know. You've heard the news about René? René, he was here at home, he married his mother. But he was here at home. Yes, of course. Just a remark, also general, maybe, you know, there is still a little problem epistemological, because there is a problem of what we expect as a theory, as a good model. And maybe here, with the categories, it's not absolutely the same. It's maybe something else. Because normally, we're looking for something compact, like a group, a law, that we can apply in several situations, and it always works. Something general in that sense. If you take a situation, something like a initial condition, or a situation given, It's a general problem that gives something that corresponds to the equation. I don't see it like that. It's true in classical physics. But it's true for quantification. And not for quantification, because if you don't have enough time, it's not. But with the category abstract, there is another idea that we need to generate, rather than to say something absolutely abstract that applies in every situation possible, we need to find constructions that can generate, in some way, all the other. For example, when we talk about the sémantique ponctorielle, we have a theory, which is something like a generative model, not abstract, which gives you a category of models, something like that. Here I think there is a tendency to reduce, to take everything in a general way, like a monoid. It is to say that you take all these flashes together. Instead of thinking in a composition morphism by morphism, they think... It is quite obvious that you will not say that the theory of quantification is the theory of the catheories. You will be obligated to specify things. What I found interesting is that if you take the most general category,

25:00 you already have something that looks like a certain element of the structure of the quantum physics. So, of course, you have to go further, you have to specify, but I already see it as an encouragement. So after, what do you need to restrain your categories, or what do you need to add? I don't think that the construction of this film is the right, but it is certain that you need to add something, or restrain it in a way or another, so that among the enormous number of categories, you consider those who will have the opportunity to represent the music. You could rappel un truc, I could add to it. A quel niveau ça vient quand on parle de l'ensemble causal ? L'ensemble causal ? Là, on l'a oublié. C'est ça, oui. Non, non, mais si tu veux... Est-ce que c'est pas en s'appuyant là-dessus ? Bien sûr, c'est ça. Ça, si tu veux... Dans son intro, il dit que la motivation de tout ça, ça va être de pouvoir quantifier la catégorie des ensembles causaux, So, quantifier a category. So, first paper, how to quantify a category. Second paper, which we haven't discussed today. In particular, the category of Sandcuso. So, I agree. It's just the ensemble. Yes, it's just the ensemble. But it's maybe... Yes, of course. Yes, yes, but the problem is that this approach is, here, it talks about a small theory in general. It just needs to be done with the ensemble. After, in the paper 2, it announces the mission of everyone to go see the ensemble pose, and every time it starts already from a structure established by others, and then he established the analogies between that and that language. So, he will be able to look at the same language as well as the analogies.

27:30 He will be able to look at the paper 2. I think it's worth a lot to remove the paper 2, and maybe even look at the light of the new paper with Doreen. Doreen? It's not the same thing. There are the professors, so if we... It's interesting. But in Dohring, it says that there are many things here. No, but in Dohring, it's rather... It doesn't quantify the category of the ensemble pose, it will rather quantify one ensemble pose. Dandering? Dandering is a good thing. They are looking for value and composition. The characteristics are already there. Yes, that's interesting. To obtain it more physically. Because here, you see, the non-realists are there from the beginning. Here, in this paper. Yes, but in my opinion, when you do the physical, you will be able to do the physical. Or something else, I don't know. Yes, that's the point of debate. Yes, that's the point of debate. But in the physical, you have the measurements that are at the moment of the real. But at the limit, I think that if you have the real, but you replace the real by the rational, You can even replace it by the complex, without a doubt. That doesn't seem to be very important. There remains the freedom of an ensemble that you can choose. You can do a physics on the rationnel, or a physics on the entier, or a physics on the PADIC, you can do that. on the physics on the fractions, pardon, on the matrices, that has been studied, it's what makes Lincoln. I think that the intérêt of the category is that there is such a generality that if you have defined a physics that works with the real, you can perhaps keep the same structure in replacing the real by the matrices or the entiers or the rationnels,

30:00 And that gives you a new theory, a new theory, which has no doubt a pertinence. Maybe that the quantum physics is that. In a certain case, replace R by the matrix. And you can already interpret it like that for a lot of problems. I think that's it. Well, I think that the quantum action, which is the base of the quantic, is just the application of a morphism. We can't divide the morphism in two parties, in two moieties. And so the morphism identity, you can't make a half. So somehow it corresponds to the quantum of action. Then the fact of creating a story and saying that there is a particular story who, in the course of a story, has regularly its identity and its identity is certainly something very complex. Yes, all of that is canon. This is the vision canonic of things, that is to discover action. We are not in a game of technicality. No, d'accord, mais par rapport au matriciel, je pense que la seule... La question, justement, c'est que tout ça, ça s'applique au cas où on ne sait pas du tout quel est le sens du matriciel. Ça veut dire qu'il y a une action indivisible, et le fait qu'il y ait des morphismes qui soient indivisibles est quand même un bon point de départ en disant qu'il y a plus de valeur qu'une approche matricielle. When you say that it corresponds in a way to the impulsion or the way to the person. It's to say that we can't do less than croix. We can create something by croix, but it's not divisible. In fact, there are also other strategies possible. They talk about morphism, structure, preserve. It is to say that if you think of morphism in this way, you have a certain limit of application of categories. Well, it is reasonable, but to go a little further, they don't think that morphism should preserve some structures, but you have to just try to build what is called space. But that, he talked about it at the beginning, but he didn't use it.

32:30 Because there is something that protects the structure, but what is the structure of what? The structure of causal, of causality, of causal. So for that, it is necessary to have the causal and the characters of causal that protect the structure. In all this, it is necessary to have the lights and objects of the particular category. In the case of concrete, we will take the lights. But the other approach that is interesting is just to try to take this category that we can give, and try to build what is happening with the category. It's well done. There are good propositions in this sense. And if you think like that, you can find structure, money, dalle, whatever you want. But it's just... it's a bit more constructible. The next session is... It's the 16th, it's on Monday. It's John Bell. This is John Bell, yes. And, in fact, he will look at how we can build something like a German man in terms of flesh. It's interesting because in fact, we can change the concept, not just reconstruct it. The question is, do we continue the session on these articles? After John Deere? After John Deere? Yes, I agree.