 Maybe I can say a few words. I'm totally unprepared to say something, but what I like about this colloquium is the connection between geometry and toposteria and logic, which we can see in the talks. And toposteria began in 1960 with the work of Grotten Dijk and his students, and it has started a new way of looking at many subjects, especially the connection between logic and toposteria. And this is something which I believe will continue in the future, as we can see with higher toposteria and possibly a multiple-type theory. So I think that the future is very bright for toposteria. Okay, so what I would like to say is the following. Grotten Dijk had written in Récolté-Somay that Toulais-Chevaux du Roi Vien-Dône-Tibois ensemble, and this was our purpose with Pierre and Olivia when we began to organize and then other organizers. But what I have in mind is the following. What struck me during this conference was the fact that people that normally would never meet and would never sort of listen to talks of the others did come together and that unifying reason for that was exactly the idea of the topos. And I find this really amazing because of the very different horizons from which the people were coming. I mean, of course, there is a motto piece theory. And it's, I mean, when I was reading books in algebraic topology and so on, I was struck by the fact that people were not using the language of topos, even though if they had used it, many, many properties that they were dealing with would have been far more conceptually understood than what they were doing. So this is very striking. And it's a typical feature which I cannot explain, but I will ask you the question, which I cannot explain why is there this snobbism, which means that people like to use their own words for a notion which already exists since, I don't know, 55 years, and where instead of using the word topos, they prefer to write it in their own language and not make the link with topos. I was very struck. I mean, it's not only in the topology. I mean, in many different topics, I have witnessed this feature and, but I have no explanation. And, I mean, the intent of, at first, when we tried to organize this meeting, the intent was exactly this. The intent was to change the way the word topos is considered by the mathematical community. It's not done. It's not finished. It's far from, because, for instance, Laurent was explaining me that I forgot whom it was who was writing, wanted to write a paper about topos, and he was advised to write instead a category of shifts, because the word itself of topos is bad and didn't make it. And I can tell you why. I can tell you why, because I was on the other side for many years. So, in other words, for many years, when I was hearing the word topos, the mental image that I had of a topos was a very strange topological space that I would never have to use, and, you know, which had, I mean, which didn't, which was not part of the mathematics which I like very much, which are concrete, where you, okay, you have lots of problems of analysis and so on and so forth. And it's only when I understood the fact that small categories naturally give rise to a topos that I said, this is wonderful. So, I mean, what I am saying is that among most mathematicians, those who don't accept the word topos are ignorant. Well, first of all, I was an ignorant for many years, and I mean, and very often in life, this is not only in mathematics, ignorance implies disdain. Namely, because they are ignorant, they prefer to react to the notion by disdain and by some kind of contempt, because they don't want to make the effort to learn. And the problem with topos theory is the following, is that it's not an effort which you can do easily. One has to find for each mathematician the door, the small door, that will, the back door, well, it could be the back door, it could be the front door, it could be anything, but what I hope is that this conference has exactly achieved this, namely that each of you at some point has been touched by, oh, wow, this is great, you know. Okay. Yeah, thank you. Well, of course, I totally share this view of things. In fact, we were all inspired by looking at what Grotendick himself wrote about the notion of topos in his text, Rekolt and Semay, which is a text of reflection, both mathematical and philosophical, on the sense of the concepts that Grotendick had introduced and developed in his career. And so in particular, what strikes about this text is the insistence on the unifying power of the notion of topos at many points. He repeats that toposes are unifying, they can allow to embrace both the continuous and the discrete to unify different areas of mathematics. So I think that now we are in a situation where toposphere is certainly sufficiently mature from a technical viewpoint to actually realize this dream of unifying different branches of mathematics with each other. In particular, I think that the notion of classifying topos, which was first introduced in the 70s, is particularly suitable for formalizing this idea of unification, for formalizing the phenomenon that two completely different looking mathematical theories might actually have the same mathematical content or strictly related mathematical content. And so I have tried to explain in my course that taking this point of view of classifying topos seriously can really lead to a great amount of surprising insights across different mathematical theories, but not just across different theories, even if you want to study just a single mathematical theory. The fact that the theory is a sort of living organism is not a dead thing. So it has its internal dynamics. So basically this internal dynamics of a theory translates into the fact that its classifying topos admits this multiple representations. And so by playing with that, you can really extract a lot of information about your theories that you would not see with alternative glasses, let's say. So there is this element of magic, if you want, of surprise that is hidden in toposis. And I'm sure that we are really at the very, very beginning of exploring the potential impact of all of this. So in particular, my hope is that with all the talks that we have had in this colloquium, to have really illustrated this wide-ranging impact of this notion, and especially the potential that is very, very far from being exhausted. Okay, so I guess for the moment, I dig the word. So I think that I can see a lot of people, the transformation that was made on Alencon, his own research work. And for me, for a few years, there have been a lot of conversations that I had with Olivia. That is, Alencon explained that for a long time, he thought that in a way, the toposis was not serious. And for those who concern me, me who comes from the algebraic geometry, for years and years, of course, I had read what Grotendic said in Recolté Semaille, but for me, the toposis was used to make the homological invariants in situations where ordinary topology was not enough. So here, I knew the etal homology, the crystalline homology, and the Grotendic's toposis. For me, when we were at that point, well, maybe I had heard that the logicians were also repairing the toposis, but as most mathematicians, I thought that logic was useless. So that's it. And I thought about it until, now it's been four years, when Olivia came to IHS as a visitor for a month, well, one day, she sent me a message asking if, well, here it is, she wanted to talk to me, and then she came, she began to explain to me both the logical elements and to talk to me about toposis in relation to this logic. At first, I had a lot of difficulty understanding it, but I remember that what, in fact, I was very quickly convinced and convinced by the theorem of the existence of the classifying toposis, that Olivia explained to me almost from the beginning, which for me was a total surprise. And since that time, in fact, I have talked about this theorem a lot of times, I put myself in charge, I realized that most of the people I put myself in charge have never heard of it, and that, well, in fact, it's a huge surprise for me that this theorem was not known more than the study of the classifying toposis and was very neglected for decades. So, I would like to explain why I was convinced. So, in fact, it's largely in relation, despite everything, with my experience as a geométragebrist and my work in the English program. That is to say that the English program gives itself to objects, to put in relation two completely different theories, which are, on the one hand, the automorph theory, the theory of representations and automorph functions, and on the other hand, the theory of the Galoisian representations. And until then, a result like that, a statement like that, appeared as a sort of miracle in mathematics. And so, I used to think of this program in England as the fact that two completely different mathematical theories told the same story. And you see that when I express myself in this way, I express myself in an imagined way, I don't express myself in a mathematical language. I express myself in a literary language, mathematical, suggestive, by evocation, but not at all in mathematical terms. However, what Olivia has taught me, by learning the simple notion of Topos classifying, is that this notion embodies mathematically the fact of going from a theory, defined in logical terms, defined by a logical presentation, to go from the logical presentation of a theory to its mathematical content, with the possibility that two logically very different theories, and related or equivalent mathematical contents. So, in a way, the fact that two mathematical theories tell the same story, thanks to the theory of the Topos classifying, is no longer a metaphor, that is to say, it embodies mathematically, it is something that can be studied. The mathematical content is embodied and embodied in a geometric object, the Topos, with all the intuitions associated with the geometry. On the other hand, of course, I had read SGA4, so I knew, roughly speaking, to make it clear that different sites can define equivalent Topos. But for me, it was just a remark. So, you see that, thanks to the theory of the Topos classifying, this remark meant that the fact that two theories have the same mathematical content, is something of current in mathematics. It's not an exception, it's not a miracle, it's something that happens all the time. And so, since that moment, in fact, I have become completely convinced of the interest, of the power of this theory. I think that we are only at the beginning of the study and the exploitation of the Topos. In a study, Grotendijk said that the two most important and the most profound notions that he introduced are the notion of the motif and the notion of the Topos. At the time when he was writing, the two notions were not very well studied, themselves without a lot of them, and since then, the situation has changed for the motifs. Today, people have understood that motifs are very important, but what concerns me, I suspect that the Topos are even more important than the motifs. In any case, it's the future that he will tell us about, but once again, I think that we are only at the beginning. So, in this colloquial, we have seen a certain number of approaches, of different parts of mathematics with the Topos, another thing that hit me a lot, that hit me particularly in this colloquial, is that when we have mathematical theories, even very simple, with a very simple language, very simple actions, thanks to the theory of, we immediately have a Topos associated, or this Topos can be defined by a very simple formula, but that even if the definition of the Topos, or the definition of the geometric theory of the first order, or the site, even if these definitions are very simple, the associated object is already of an absolutely extraordinary wealth. For example, it was visible in the exhibition of Café Katia Consani on his works with Alain Cône, where we had completely elementary Topos, very, very simple. We even want to say that the structures studied are simpler, or at least less rich than, for example, the structures that we are more used to, such as groups, rings, you see, there was just the addition and the relation of order, extremely elementary things, and already we see, for example, the calculation of points in such a Topos, that we find things that are astonishing and extremely rich. So when we have this wealth with also elementary ingredients, we really wonder if it can go to the end. And so again, my perception is that we are really at the very beginning of the exploration of the Topos, even if, well, Olivia has already written more than 1,000 pages about the Topos classifying, Katia Consani, Alain Cône also writes a lot, and then a lot of people who were given exposure, so there is already an important literature on the Topos, but I really think that once again, we are only at the beginning, and so for me, what this colloquium has done is to encourage the development of this extraordinary theory. Between you, I think it was a kind of wordplay on Topos in a very elementary sense. Obviously, this is the Topos de Grotendic. Does it mean that we have to consider the elementary Topos as a parent intended for his poverty? What can be the relations and the relative importance of the Topos de Grotendic and of the elementary Topos? Well, it may be some impressions, to write some impressions. It seems that the notion of the Topos elementary is very important, the fact that we can interpret the superior logic, maybe intuition, but still, the superior logic in a Topos is a fundamental fact, rather surprising because on the one hand, we have a purely geometric object, the Topos, the notion of the Topos, and on the other hand, there is the intuitionist logic invented by Brower, who was, I have to say, a topologist, but he is mostly known, in fact, in mathematics, he is mostly known for his theorem of fixed points of Brower, but logically, he has a very important contribution, which is the development of the intuitionism that has been considered as a somewhat exotic branch of logic for a long time. And the fact that we can all of a sudden realize that there is a close relationship between a side of the geometry and the intuitionism that emerged from a philosophy or a rather independent consideration, is a bit of a miracle. I see, I think that mathematics works from a certain number of miracles, and that is surprising. So, I haven't philosophized for too long on the idea of the miracle in mathematics, but I think we can use the word. And well, there is a potential that has already been explored, but there is still a whole school of categorical theories that has developed the elementary topos theory, and it has very important contributions. There are notions of topos that emerged from that, for example, the realizability of Martin Allen. It is a construction of a topos that is quite different from other branches. Not only the branches that allow the construction of topos, there are completely different logical constructions that are very surprising and still incomplete today. But in addition to that, there is the development of the topic theory. Now I realize that I speak French. Maybe I could switch to English for a while. So, now we see that there is something new happening. First, higher topos theory. When a theory is not that important, when a subject is not really productive, there is no important development in it. I mean, the fact that topos theory has given rise, because clearly the people who invented higher topos theory were directly inspired by topos theory. So, the fact that topos theory is evolving is a very good sign for the subject. It shows that topos theory is not only topos theory has Grottenzik conceived it, but maybe Grottenzik wanted to have higher topos theory too, because you can see it in his pursuing stack. Clearly, this is a dream of higher topos theory, that he was not able to create for various reasons. So, this fact on one side, higher topos theory, which is very important for geometry. If you read the work of Luri, it's not easy and it's very extensive. You can see that there are theorems there that are extremely general and very important for geometry. There is a school around Luri and Mike Hopkins of people, especially in the United States, developing higher geometry and higher topos theory and stable homotopy also, theory and stable categories, etc. It's all built on this idea. It's just amazing. There are many things that I did not understand in the past. For example, the constructions of the Tom space. Tom was here, Tom won the first with Grottenzik. There are new ways of understanding this construction. For the first time when I saw that, there is a paper by Gepner and others, I forget the name, but I'm sorry. There are three or four people where you have this new approach. When I saw the new definitions of the Tom space, you say, wow, it is so simple and it just captures everything. This is something I waited for 50 years and it's there now and it's very, very simple. On the other side, you have the logical side, which is also evolving with homotopy type theory. Homotopy type theory is maybe not yet... I mean, there is a program, I would say. Eventually, we can hope that it will become user-friendly. Can you imagine? You sit on your desk and you open and you start doing some proving things using this formal system. If it's user-friendly and powerful, it will be a revolution in mathematics. There is no doubt. I cannot imagine how big it will be because young people are going to use it. I forget all people who can't, like myself, who can't learn this stuff, but clearly young people are going to take it and prove things that the older people can't prove. Topos theory is closely connected to local theory and classical logic, geometric logic, topos, higher topos. It's all booming. Yeah, so I'm very optimistic about the future. Just to say a few words about the distinction between elementary and growth-indic, because from the point of view of unifying different mathematical theories with each other, there is a particularly important aspect of growth-indic-toposis as opposed to elementary-toposis, which is the generators. Basically, the essential difference between the two is the fact that when you have growth-indic-toposis, you have sites. Sites correspond to presentations of theories. So, basically, in some sense, and you don't have that for elementary-toposis, so this is very important for formalizing this duality between the way of describing mathematical theory and its mathematical content is incarnated by the classifying topos. So, I personally think that this duality between sites and the toposes or between axiomatic presentation of theories and their invariant incarnations provided by classifying toposes is of utmost importance, and this is also an aspect that, of course, could be also generalized to higher toposes. I didn't have the time to talk about that in my course, but certainly, since most of the essential features of one-dimensional toposphere carry on, at least some... at least we have analogues of the notion of site, etc. They carry on to higher toposes. We can certainly expect these unifying techniques to generalize. At the moment, we don't yet have a formal axiomatic definition of higher geometric logic, but it will surely come in the next years, even though we have a semantic understanding of it. Well, this is... Yeah, I mean, we really want... I mean, you can say... is the sort of analog of the internal language. Yes, yes, higher order. Yeah, of course, higher order internal language, but what we would like here really is to have higher geometric logic. So this still lacks, but I'm sure it will come. And so I think it's very important to investigate... I mean, I try to encourage people to investigate how topospheric invariants express in terms of different presentations of the topos, because I mean, this really formalizes the feeling of looking at something from different points of view. So it is very satisfactory to have this possibility of materializing in a precise mathematical object this intuitive feeling that you might have. So you might find yourself working in a certain situation and noticing certain vague analogies and some vague connections, and you wonder what is really behind all of that. And so what I would suggest doing in that situation is to try to identify a topos that captures very well the situation that you want to investigate. And then you will have one topos that represents one point of view, and then you will have another topos which represents another point of view, and then you will try to compare these two topos to see whether they are equivalent or related in another way. And if you are lucky to find that there are natural relationships, you might extract information from the comparison of these classifying topos that would hardly be visible by using alternative viewpoints. Really, I mean, I like to compare topos' theory a bit to genetics of mathematics in the sense that, I mean, when you do genetics, for example, genetics allows you to extract some information about individuals, for example, biological information about individuals that you cannot see at naked eye. So you really have to make an analysis of the DNA. Of course, it is not something as intuitive as surgery or traditional medicine. So you are in another situation, but the kind of insights that genetics can bring are hardly obtainable using different techniques. So for me, topos' theory is a bit like that in the sense that you can really get by studying topos' theoretic invariance and from the point of view of different presentation, different sites, you can really reveal hidden aspects of theories that you would have never imagined. At least this is the experience that I have matured myself with this concept. Yes, so... I mean, things a bit analog, but by supporting me on my experience of geometric algebra, I was able to train at the school of Grotendik, that is to say, like everyone else, I spent years studying the Schema. And by studying Grotendik's theory, I learned that... So it is a geometric theory, and I learned that there are actually two important things. The first is to forge tools and general methods of geometric space study. So, first, for example, the Zariski community, later the Etal community, the crystalline community, the most sophisticated things. So that is the first aspect. But in fact, there is another aspect that is also important, which may be the previous one, is to build interesting spaces. And in fact, sometimes we forget it, but Grotendik first forged very general methods to define spaces, which then came from the Schema. It is the main subject of its series of exhibitions at the Bourbaki Seminaire, the foundation of geometric geometry. And now, the essential theories of representability. Grotendik had this brilliant idea of defining spaces by functors and then to show that these functors are representable. Of course, for me who learned that, when Olivier told me about the post-classifier, it was the same type of result. As a geometric teacher, I was interested in the English program, that is to say that with the general methods of building spaces introduced by Grotendik and the general methods of space studies introduced by Grotendik, I was attached to the study of certain particular spaces, in particular in my case, the varietals of Stuka invented by Drenfeld. Precisely, the Stuka are defined by functors, which are themselves associated with the choice of productive groups and some other data. And then, of course, in algebraic geometry, people have also studied the varietals of Shimura, which are also defined by functors. So I was used to not to what objects are defined by functors, but in algebraic geometry, I had the experience that functors are really interesting. There are not so many, that is to say, I have just mentioned some of them, so the varietals of Shimura, before the modular curve and then their equivalence to Drenfeld, but for example, all the varietals of Shimura are not representable. It poses a big problem in the theory of numbers. So in algebraic geometry, we have a little impression that we lack objects. However, with this theoretical representation, we have a family of objects that are representable. So I had the impression that there is a family of spaces incomparable than what I had suspected. Of course, all theory is not interesting, so the topos are not all equally interesting, but it prevents that we have an interesting theory in mathematics for one reason or another. We have an associated topos, that is, an associated space. So it gives a family of spaces to study, absolutely huge. On the other hand, I come back to my experience, the Chitukas, the varietals of Shimura. So what makes these objects very rich is that they have a double representation. They are defined by their points, because they are defined first of all by a founder. So it means that their points, for example, the officer in a body are defined by a certain combinatory data associated with a productive group. So we will be able to reason about that. And on the other hand, as they are represented by the schemas, as we can apply to them the general techniques of the Geometry of Eury, we know that they have an etal and an erratic homology. And by putting the two together, by linking the points and the erratic homology. So the points are linked to the representation of the object by a schema. So by putting these two things together, that we can do thanks to a precise theorem that connects both of them, called the theorem of the fixed points of Rotendik-Levchette. It is by linking these two things that we can demonstrate, for example, of the English program, that is to say, very deeply denounced. So here, the results come from the same object, and represented in two different ways. And so, for me, it is really a reason to have been very quickly convinced both by the theorem of the existence of the classifying topos, which says that each time we have a theory, this theorem itself already links two things, because it links two fosters, and then the topos which will represent it. And then we learn that this topos, being known as the general properties of the topos, will always be represented in several different ways. And again, the potential appears immediately huge because of this experience that we have. So, in some way, the theory of the topos seems to me today as something where I find many elements of the theory of the schemas as I have learned, but with even greater possibility. For example, the last thing that struck me which also has a similarity is that the geometry of the schemas is the algebraic geometry. So, a geometry that is founded on the algebra, on polynomial equations, we use addition and multiplication, while the theory of the topos classifiers is founded on the logic. One thing that I understood little by little is that the logic is incomparable more flexible than the algebra. And I will give you an illustration, just to remind you of a remark I made in one of the courses of Olivia. So, it's about the Galois theory. So, of course, I learned the theory of Galoisian and Grotendic categories and, interestingly enough, I learned the theory of the Tanakian categories. But you see that in this formalism which is inspired by the algebra, we have two parallel theories and different. That is, if we want to classify linear objects, it's the theory of the Tanakian categories. If we want to classify non-linear objects, it's the theory of the Galoisian categories. So, the two theories are really one on the other. In fact, the theory of the Tanakian categories is the theory of the Galoisian categories. But they do not join these two parallel theories. However, with the theory of the topos classifiers, you can try to classify this type of geometric theory of the first order and ask that the topos classifier and the properties that we expect of a Galoisian theory. Technically, it's the fact of being a topos atomic with two values. But this property can also be produced for a linear theory than for a non-linear theory. It means that with this theory of the topos classifiers the dichotomy between the linear and non-linear disappears. You have the same theory, the same formalism that allows you to study the two. From the point of view of... I mean, it's something that interests me a lot because for me, the question of the specificity of linear structures is enormous. For example, in the duality of the English language it's about the representation of the topos and the Galoisian representation. In the announcements, the representations that we consider are linear representations. For me, it's a question that has been asked for years. Why linear? What does linear mean? I have the impression that to try to answer this question in a mathematical framework where the linear is not the priority. That is to say where the linear and non-linear live together and where we will be able to discover the specificity of the linear that allows this extraordinary peculiarity of the duality of the English language. Do we have to pass the microphone in the room or do the microphones suffice? We will do it with the microphones. May I say something? May I use the microphone? In fact, you don't need it. I would just like to amplify something that André said some minutes ago. André recalled, I think, if I understood correctly the origins of categorical logic and Brauers' intuitionistic mathematics also mentioned Brauers' impact on topology via the Brau-Fix-Point theorem and then went on to say that it's rather remarkable that Topos theory gave rise to both higher Topos theory as well as to well, and there's also this relation to homotopy type theory where people keep saying that it's expected that homotopy type theory is supposed to be the internal language of these infinity Topos. I would just like to amplify this point maybe the single person who has been pushing the unification of all four of these points mostly in the last years, which is Mike Schulman who has been doing some fantastic work in this direction. I would just like to speak about having this one model of homotopy type theory in simplicity sets and I think it's worthwhile to recognize that Mike has meanwhile produced two infinite families of models of pre-sheaf infinity Topos one of which is the one of accurate homotopy theory which goes into quite some research level modern research level territory where actual homotopy theories I think are more likely to pay attention and just to close the circle of ideas just in his latest article he considers cohesive homotopy type theory and gives a formal proof in their intuitionistic proof if you wish of Brouwer's fixed point theorem so in cohesive homotopy type theory it's possible to actually speak about the continuous circle as it's with its geometric structure as well as of the homotopy type as BZ, KZ1 and put these things together and using the proof he did earlier of the homotopy groups of spheres fixed point theorem in intuitionistic but now homotopy type theory I think that is a beautiful unification of this kind of history of Topos theory that has been happening and I just want to advertise this fact this is a beautiful article that I think unifies many of the aspects that have been mentioned in this conference called something like cohesive homotopy type theory and Brouwer's fixed point theorem that's all I wanted to say, thanks Brouwer I think all this seminar has convinced in case the need to be convinced of the importance of Topos theory so I am convinced but I want to fill a little bit of number 10 of my program there not because it is my program but because it's for me philosophically very important number 10 is what I called a plea for the language I'm not meaning the language of Topos I'm meaning the human language which is the greatest achievement of humanity which has taken thousands of years to build and I think that mathematics has benefitted from say physics, from astronomy from etc etc and maybe there is a possibility by looking at the language to define new important structures not necessarily Topos's I want to give just one example apart from the thing which I said which are very elementary namely the notion of which is in the language the notion of small which has a very big content in spite of some smiles that may arise or that will certainly arise the notion of small to take all its full power needs the notion of fiber categories and some deep consideration on fiber categories which are not necessarily Topos's or fiber categories over Topos I cannot and I couldn't talk about this notion here because it's much much long it would be much longer than the things which I said and yet of this mathematical development comes from better understanding just a better understanding of etc a better understanding of small and forces you to study some structures in fiber categories would take I don't know how long the language provided you don't violate just as I said if you violate the notion of etc if you look at it on sets etc we necessarily force you to invent new interesting and important structure that's what I mean by plea for the language not the language of Topo's or the internal language the language the human language it is the most fabulous creation of humanity and maybe it's time now we didn't have the tools but categories or generalized categories or etc etc might benefit from study of some fragments of the language if you have something to say it's time now if no one wants to say anything there's no problem we'll stop there and André you forgot now I remember yes what I wanted to say is something maybe a little strange I think we can do the Topo's tour and use it without being a specialist so we don't need you don't need to convert to a kind of religion which would be called the Topo's tour which some would be priests, etc etc it's just that you start to like maybe this notion that you use on the occasion in your work and well, like that it's just a tool think a little like the tech it's a tool it's useful maybe comparison is not the best but I think you have to see things like that yes let's say what André just said I think we would be wrong to believe even in the geometry the idea of the Topo's is enough to learn what nature presents us what I have in mind is the next thing is that Grotin says he's right that the idea of the Topo's unifies the discreet and the continuous it's the quantique and the formalism of the quantique which has this linearity that Laurent talked about has this extraordinary marvel which is that it unifies it allows the discreet the discreet operators to coexist with the continuous operators and there is a deep absolutely incredible in the quantique in the fact that there is an extraordinary phenomenon which is that the formalism of the quantique is extremely efficient to say what variability is what a variable is and it goes to such a point that now in Switzerland there are people who invented a generator of random numbers which is even if an attacker knew all the details of the system that produces these random numbers he would not be able to do the same and it would not be the case if we used an ordinary computer the system is the following it's a small LED lamp which is sent on a screen of photons and the generator of random numbers looks at where the photon arrives which is not reproducible by the principle of the uncertainty of Heisenberg and produces a random number so there is a the formalism of the quantique which is something wonderful it's another wonderful world that we can reduce everything to a random number I repeat what André said I mean, it contains a lot of things I think that it's also to repeat what he said it's also very important that not only us can use it as a mathematician that is to say that we are able to know that there is a random number behind such or such a situation but in fact I noticed a trivial thing you will excuse me for saying it to criticize the scientists by the fact that they always have the idea that something is true or false that is to say that for example you have a discussion on television you see politicians around a table they discuss a notion and we all have the idea very naive, very simple, simplistic that something that is discussed is true or false but the idea of utopos in its logical character in the fact that there is this omega which is the classifier which is an extraordinary potential of formalization which allows us not to have only the true and the false but to have the idea of a truce of a path towards the truth and that I think that if you want it's really extremely sad that people who would be serious of course not escrow, but that people who would be serious who would know philosophy who would know situations in the current world are able to adapt this wonderful tool to better understand which means that we can be in front of a room where there is a political discussion and we don't understand anything we don't understand anything if we have the idea that it's x or y is right and if x is right it's y is wrong etc things are much more subtle than that and we could make a great service to the language which made us a huge service we could make a great service to the language by trying to translate in understandable and usable this idea of utopos it is there, it exists the great danger of course is that there is a lot of escrow and for example I read books called the utopos of music but it's nonsense and in general these people are people who will try to impress others by a completely obscure mathematical language without having understood it so there will be an evolution I think it will be a very slow evolution I think it will take time before the utopos idea goes into the once it has done it we will be able to formulate things in a much more subtle and interesting way than by the real or the false as we usually do it might be precisely the link with the language and the utopos the point of view of Benabu on all the empirics for example it's a step in this direction maybe it will take time but the question of teaching utopos no no no it's not a teaching the teaching of utopos if the utopos should not be seen as nothing should be seen as the grail that responds to everything it's not that but the connections are very strong with a lot of domain maybe all the two I don't know in any case with a lot of domain of mathematics I invite you to teach the utopos at a relatively early level in the end no no no not the elementary so would you say a few words of your feelings about the teaching we can ask the question we have to answer now we will answer as Urs Scheiber in his exposure the title was pre-quantum pre-quantum so it's interesting as a title I'm happy that you put pre-quantum because the real mathematical mystery is maybe the quantum in what measure I think the utopos are a contribution to the classical that the real quantum spaces are always maybe ideas about it let's say if you want to try but let's say what convinced me of utopos is that if you want an extremely simple utopos that's what Laurent explained we can see as a point a non-commutative space and I found that I was astonished by that because what you have to understand is that spaces in which the notion of equality is more subtle I take an example when Penrose once made an exposure he had transparent and on those transparent he had the pavages of Penrose and he showed that two pavages we can see things that look as much as they want without being equal that's quantum that's quantum that means that the quantum thing this kind of fluctuation is present there is this bridge between quantum and something that can be formalized by an utopos but there is more in quantum because there are complex numbers there is a real question and it's true that in the Ursa it was in a pre-quantum state but it's still very dangerous because for example when I considered the non-commutative geometry I didn't call it quantum geometry it's very dangerous to consider that we found the quantum geometry non-commutative is another thing I think this is a very important point if I had had more time I might have been able to comment allow me to comment on this this is your tangent topos this is what goes into the quantum direction so as Maxim Konsevich has been amplifying for many years now is that one version of speaking about non-commutative has hence quantum spaces as regarding them as not topos infinity topos but a stable infinity categories or if you wish as triangulated but there is this passage these things are connected via the tangent topos so these stable infinity carriers the way that Maxim Konsevich has been considering as long as they are presentable which is what we assume they are sheaves of spectra so they do sit inside the tangent topos of the corresponding sheaves of infinity stacks and it is in this way I have this little note on the archive for quantization via linear or multi-pedal where I kind of try to explain how to go from the pre-quantum version via the tangent topos to the quantum version so I remember that in the discussion with Maxim a couple of weeks ago you said something about non-commutative non-geometry yeah, no, in fact I came from a different world for me kind of topos a little bit too satioretic for my taste yeah, one can think about stable categories things really parallel to topos as we get to the alco limits and finite limits and so on but it's kind of a little bit different game so enriched by different categories not sets, not by spaces, but spectra and it's linear, yeah but also if you do things about props I don't think it's kind of not natural to return to sets back to sets, yeah so it's kind of which symmetric monodal category in some sense has its own logic and its own word of topos as we can say but in general also I have to say that I may allow myself to be a bit to have opinionated I got convinced that notion of topos per se it's belong to the past and it's like a billion category it's of course a great tahoko paper but it's like notion of stable category and also the infinity topos it's much nice and all things I think will be better yeah so it maybe one can see this usual topos it's like heart of T structure on infinity topos in some cases but yes, yes, yes, yeah I understand but stable categories are simply than a billion categories in a quantitative case I stress commutation law in my talk but in the stable case you told me that finite limits and finite cool limits commutes it's not that there is a distributivity law you can do one or the other and that characterize stable categories it's an amazing things that one could have this kind of universe and two things commute but because they are in principle very different the one is I don't know it's very mysterious to me unfortunately we have to stop now because the colloquium is not ended precisely so thank you very much for everyone