 It's the first time for us that we're trying to bring such a, I would say, a very theoretical theory to something more, I mean, something that we could at least master a bit and understand. So my first question basically is going to be more on a historical perspective, which I think is very important for me to understand, is I'm going to ask each one of you, why did you do, why did you start doing research on topos? Because it seems a bit as a research topic, which is not something that you would tackle in a master, I would say degree, and find that this is the thing I have to do. So maybe I can start with Daniel and then go around here. So Daniel, why did you start doing research on topos? Thank you. In fact, I don't consider it as really a research on topos, except for the last part, I would say a word on that. Because I have learned about topos, and I was very generative of the work in SGR 400, but I never thought to apply topos. As I said, the problem was to define the form of formation without knowing nothing about topos precisely at this moment, and it was almost evident that the equation of Shannon could have an interpretation in a homology somewhere. And so I started to define this framework, just to define a complex, like you show in your slides, except it's a bit more complicated. And the complex was almost correct, but of course strange with respect to many things because of this kind of indexing that I have not really functioned at all, whether it's a variable which plays a special role. I put an index, but at this moment I don't put an index, it was with another notation. And I realized that, because I already called it was locality, this hypothesis, I worked on the question at the moment, but it was too quick, I named locality. And I had these ideas that perhaps they had to do with topos, and in fact, by comparison, but exactly that. And that is to realize that there was a seed, a seed, in this case a trivial topology, but in fact the fact that refinement is topology is my idea. So after that you have a much more rich situation where you show the specific seed and filter. But it is this idea that you don't try, as in ordinary topology, to define open sets, but you define covering. So it is this idea of refinement. And so on, the sheaf and so on, and after that, of course I look more carefully at what was done. Especially because to compute, there was a suggestion in a GA4, by Spectral Secrets and so on, but in this case no, because it was not really a trash topology. And but after that, so it is as I start, just I constate, I realize that I was doing a computation of topos. So you were doing topos without knowing that you were doing topos? That's exactly how you started. That's exactly the very particular case, of course. And after that, it was the contrary, because we were told on this extension of trying to define a geometry for preparing the robotics. Here, yes, I wanted to apply the principle of topos, for example, to find that this idea of the intrinsic logic, or contextual logic, as it was done by an informatician, for example, is a good point in this case, because you always prepare your action for generating several kinds of actions. So as you expose in your talk, you have a really different level and localize a different place of what is the choice which has to be made, that you don't realize, I think. So they are really suspended to new kind of information. So that was totally aware that the logic of topos is doing that, that I proposed this idea. And the fact that when we do such action, in fact, we don't manage to control points. Perhaps you control some synergy at some moment, but it's even not evident, like what we do. So it's better to have no point, not necessarily to have points. If you have, it could be okay, not necessarily. Jean-Claude, why did you start doing topos? I'm afraid that it doesn't apply to me. I said that I'm afraid that it doesn't apply to me. He was forced to do topos. I have to say that the very few I've seen looks very interesting and I'm impressed by you guys because it's, I mean, I know some mathematicians who master very narrow band, very narrow band mathematics, but to master all this area, I mean, you have to master a lot of things. You have to be broadband and very deep. So I'm very, very impressed by you. So, Olivia? Well, I got into contact with topos when I was still a master student, actually. I had read things on my own. And in fact, the main, my first motivation was to combine logic and category theory, which were my main interests at the time, and of course also nowadays. In fact, what has always interested me the most is depth in mathematics and generality. So apparently the two are a bit incompatible, one might think intuitively, because when you do very abstract stuff, in some sense, it's hard to be deep. But in fact, for me, this is really the challenge that I set to myself, I mean, to try to do work that is both abstract and deep. And in fact, topos theory offered me tools to achieve this goal, and I'm more and more convinced about its fantastic qualities as a subject, which is interdisciplinary mathematics, very general, but also very deep, in the sense that it can make you see things that would be really invisible otherwise. So the methods of toposphere allow you to prove sometimes in a very quick and easy way things that when you look at them from other perspective, they look completely obscure, you just don't have any clue. And so I find this extremely fascinating. And so I could say that in some sense I fell in love with the subject almost immediately when I started learning about that. It was this feeling of depth and of generality was there from the very beginning, and then I went on and of course I tried to get my work more and more sophisticated, but always with the same aspiration. Okay, and Thierry? Yes, so I was working in constructive mathematics and I wanted to understand some papers of André Joyel, that was about topos theory, but I was unable to understand them and I really started to understand them in constructive mathematicians. Henri Lombardi explained to me this idea of dynamical algebra, so people that were doing computer algebra that wanted to compute with algebraic numbers, but without being able to decide if a polynomial was irreducible or not, and they were still able to compute with algebraic numbers and with a very algorithmically very interesting idea, so this idea of lazy computation, to be lazy, and so this was a really nice idea, and then I realized actually that was what André Joyel was doing, but it is formalism of topos theory, and that's really where I started to understand topos theory, and it can be relevant for computation. Okay, good. I mean, in your presentation nobody made the differences between topoi, toposes, topos. It's a huge controversy. Huge controversy. Huge controversy. So at the end, what is the common use? It really depends on the authors. I was influenced by my own supervisor, British man, and so toposes. Toposes, okay. But some other authors, very distinguished authors, including Jacob Leury, Ike Mordaik, they use topoi, so I mean, everything is acceptable, I think. Okay, so this goes now back to Daniel. Very good presentation on toposes and information, and it goes back to my question, which I wanted to ask you before, is we have the feeling that the topos framework revisits many of the concepts we knew, and what can it bring new besides just revisiting and having a new interpretation of basically information in the case, and let's take the example of information. In this case of the information quantity, the first thing it can give new is if there is a higher degree quantity, because you have a homology theory, so here we can identify rather easily the one-dimensional, but the higher-dimensional could also be interesting. He could say that we have looked for, at this moment, for invariant of one random variable and also the probabilistic model, but it could happen that you have invariant, which only if you look at, for example, three variables together or in some order. And we could be not definable for individual measurement, because the look at a configuration already of three variables, that's one up, and here it's more this point of view of homology. And also it can be the starting point to examine the relation between these quantities. For example, you can integrate what is called better energy, which could probably be also interpreted homologically in the context here. And in this case, yes, I know that one year or six months if you're working with another student, which is already different, that this algorithm which is important in machine learning and information, the belief propagation, has a homological interpretation, but with different complexes. In fact, it's the first time I see, you have one linear complex, which is like D star, and you have D, which is nonlinear, and satisfies this problem. If you combine both, you have a kind of heat operator, and this flow is the heat flow for this operator, which is discrete, and that is the interpretation. It is a discrete version of the equation in this situation of this graph, the complex belief propagation, in terms of the graph factor. The generic name is graph factor. So probably it's because you have this structure of kind of topos, the relation between topos, that you could hope to see this kind of... In fact, I had the intention to make the citation because it was the text suited by Kotendik and Verdi telling that, in some sense, every time you will have some notion of locality underlying a form, you could have a topos in some sense. Our problem is, do information as a form. But as a form of information, I discussed some time with a person coming from gestalt theory and he hopes that the figures, which are pregnant in gestalt, could be related to this kind of form. Other comments on what can it bring more than just revisiting concepts that we knew? Well, yes, of course. It can bring a lot, not just a little. So no, in fact, the point of view that toposis offers is really different from that of any particular field to which it is applied because it is... I mean, I personally regard it as a metamathematical subject in the sense that you see, thanks to toposis, you have this incredible dynamics of investigation so you can switch from one mathematical theory inside a given domain to another domain, a completely different context. You can make all sorts of bridges and any topos supports an infinite number of bridges. So you see, when you do these bridges, you realize a posteriori, when you get the results. I mean, it's always a good exercise to ask yourself, could I have obtained these results without toposis? And I always pose myself this question. And the answer is, even with the simplest invariance, very often you are not able to find the direct proofs of these results. So I have made a collection of some of these main applications in my habilitation thesis, so one can read about this. It gives also an idea of the generality of the notion of topos because there are applications in different mathematical areas. And in fact, when the results are relatively non-trivial in the sense that one starts with a morite equivalence which is non-trivial and one considers an invariant that is just not the simplest possible invariant, then one often gets results that cannot be achieved otherwise. And in fact, it's not just that once you get the result, there isn't or there is a proof. It's not just that. It's the creative power of toposes themselves which I find very striking, the fact that they guide your mathematical investigations. They make you understand, for instance, if a concept that you have introduced is in some sense modular, modular in the sense that it is possibly transferrable to other contexts, in the sense that you see if you are able to identify a topos and an invariant on this topos which corresponds to the problem you are interested in or to the notion you want to investigate, this is an indication of the fact that in some sense you are on the right track that your notion is good, in a sense, because it comes from a center. It is not a marginal notion. So you see sometimes one can be lost in introducing new notions or sometimes one introduces things which are not very canonical. When you let yourself be inspired by toposes, you make quite intelligent choices if you are able to hear the voice of things as Grotendig was saying. In some sense, you have to develop a certain sensitivity to understand what the toposes suggest to you, but once you get to that level of expertise, you realize that the input that you can provide is invaluable reading. OK. Do you hear any comments on this? No, no. Maybe, I mean the fact that the topos theory is connected to intuitionistic logic, topos theory comes really from geometry, algebraic geometry. The fact that it is connected to intuitionistic logic and intuitionistic logic is connected to computation is something that we have not understood yet. I mean, how come Grotendig came to intuitionistic logic? He was not interested in logic. That's really surprising, and I think there will be more connection with computation in this way which are deep, which are not yet understood, I think this will bring something. Coming back to Grotendig, is it true that there are many things still that we cannot read from his papers? From the whole number for Grotendig? Yes. Yes, from the whole amount that he wrote, there are still things that we still need to discover on what he wrote. Is it still the case? Well, as far as I know, they are not publicly available yet. I mean all these very late writings, but maybe Emmanuel will be able to say something more about that. There are two parts. There is one part that was given to Montpellier, and they are now available, so they have been numerized and you can access to the documents, but they are not so easy to read even formally because they can be handwritten. In fact, I tried myself to read one of these, but the writing was very, very late. Inside these documents, there were some who already circulated among the community, and then you have the so-called Lacerre documents, which are 70,000 pages, but with few mathematics, it's more related to the questions of good and evil. Yes, it's more like... But with his still his mind, they are not available. From your presentation, we have the feeling that the topos is like the theory of everything, the theory of everything. With topos, at least, that's how Laurent Lafort convinced me the first time in telling me with topos, just like 5G by the way, is the theory of everything in 5G with everything. My question is, what are the big challenges now that we have in terms of doing research in topos from your point of view, each one of you? Maybe Daniel? What are the big challenges now in topos? I prefer that you answer, because I'm not really a specialist of topos. What do you think we should do? I think, yes, from this thing which makes this point of view coming from high geometry, the fact that you look at sheets on the side, and this intrinsic logic, the fact that, for example, you can try to understand more classical objects. For example, a geometry. That is, a geometry was traditionally thought, as I said, that you have a group and you have a subset of groups which are complicated doing something. If you now do that in the context of topos, what it is? If you do, you consider manifolds and now you try to make Riemannian studio, Riemannian manifolds, for example, or contact structure in this setting. It makes the advantage of this logic that you don't know exactly where you are. It depends on the time. And you can be more precise. You can have a manifold, for example, which is fibre of another one. And when you don't see the vibration, you have a space of four dimensions, but when you look at the fibre, it has ten dimensions, for example, or eleven. And that, I think, for me, it's interesting to try to do that or to, if somebody does that, to integrate the fact that, in some sense, you have a larger scope than set. To replace a set. And you look at the usual traditional differential geometry, for example, in this context. Jean-Claude, you have, I know your answer, a book on topos for dummies. That's the big thing that needs to be done. Maybe because it looks really fantastic. The problem is not that it's so difficult, the problem is that when new buyers like me try to read something about it, in fact, we immediately realize that we cannot read anymore. And that's the big thing. And we need, probably, to develop some reading skills for that specializing this year. Maybe it comes from categories, I don't know. Olivier, the challenge is... Yeah, I think the big challenge is actually, as he was suggesting, to make the theory more user-friendly and to apply it. Because, of course, ideally, I think progress in mathematics should be motivated by applications, but should also be systematic at the theoretical level. Especially when you work in subjects such as toposphere, I think it is important to have a systematic mind when you do research. So not to be too preoccupied by applications, but also not to neglect them. So I think it's important to have a good balance between the two aspects. And, of course, there is a lot to do in connection with the development of the unifying power of toposystem mathematics. So, in fact, I was mentioning after my talk about this idea of the encyclopedia of invariants and their characterization. This would be actually very important because it would help the working mathematician identify good toposystems and building variants which relate to the questions he is interested in. Of course, it takes time, it takes some investment, and it requires also the existence of a community, the beginning is small, but hopefully larger and larger works on that. And toposphere has been quite a controversial subject in the past 40 years. I mean, it has been very much used in algebraic geometry, especially as far as comology is concerned. Also, in the motopy theory, there have been many developments, including the theory of higher toposes. So there have been some directions in which research has advanced very well, especially in relationship with solutions to specific problems. Even though still at the theoretical level, there remains fundamental problems also concerning the comological formalism. I mentioned the issue of the six-operation formalism that is still waiting for a unified topospheric treatment which allows one to recover all the special cases known in the literature. So this is also quite interesting challenge. So, yeah, there are many, many, many things to do, both theoretically and from the point of view of applications. One research field within toposphere that I hope to contribute myself to in the next years is the factorial development of model theory. So, because as I mentioned, the classifying toposes allow one to go beyond the study of set-based models of theories by replacing the study of these models with the study of the classifying topos and the universal model inside it, because after all, this generates everything which happens in the set-perative world. So it is clear, just by definition of the classifying topos, that this perspective is liable to bring a lot of insights and results in model theory. So it would be very interesting to pursue that line of research and reshape the foundations of model theory by using a topospheric outlook. More in general, I think that what it would be very good to have is a very open mind on the part of specialists in different fields to talk with topospherists and try to establish a common ground, because the most interesting applications arise when there is this communication between the specialists of the area, which of course have the best sensitivity possible for the field. And on the other hand, the topospherists that can bring these completely novel insights. So something which I have noticed in the past years is that sometimes communities are a bit closed-minded and they are not always open to talk to category theorists and even worse topospherists, because they are a bit scared, it must be said. The language is not very easy to master at the beginning, so I understand that there can be resistances, but people should be aware that really it's worth to do such an investment. Results will come, but one has to think in a long-term way, as Grotendig was thinking. I mean, he has always promoted mathematics, a very systematic development of mathematics, and toposphere is a subject that certainly deserves a systematic development, also in relationship with applications. So we hope that more and more specialists from different fields of mathematics will get in contact with people with topospheritic training and that this will stimulate more and more results. That's all we tried to do here. Maybe Daniel, and then I'll give you once more. I want to make a remark on the context of why starting from a geometry comes to logic. In fact, the movement was beginning before, for example, with McLean, Hellenberg, and Cardo, which introduced category theory, and they were motivated by algebraic topology, and of course the interplay between algebraic topology and algebraic geometry, so our number theory. And so topos are inscribed in that. For example, it is already in SGR4, the theorem that you have this equivalence here between some category in such and such property. Of course, it was not developed in the model setting, but it was developed in the terms of property of a category. To have limit, invest limit and so on is equivalent to this category because this group was very interested by all the categorical aspects. To develop category theory, there were ones that it is changing mathematics. And now it's evident that logic is very influenced by category also. So this is at a larger level. So topos participates to this larger modification of mathematics. Thierry, maybe? Yeah, nothing to add to Olivier said. I was thinking the same. Everything, okay. I have a last question, and then maybe I'll ask the people here if they have some questions. It's about implementing topos and programming topos from an engineering point of view. I mean, when you read papers of topos, of course you're quite surprised because there's only arrows. It's not the classical type of mathematical paper that you would read, obviously. You need to get. And the question is that do you think the actual platform of programming and language we have are tailored to be able to use topos from an engineering point of view in the way we program? Do we need a specific language for programming topos? Daniel, maybe? I don't know if you have a point of view on that. Jean-Claude, then? You can find, I think, some... At least for categories, you can find the libraries in Python or Sage, which apparently work pretty well. So I don't know if you can do topos, but as it is a category, maybe? I don't know. I mean, there is this notion of functional programming. It's a language like Haskell or Camel. And there, I mean, they are already using ideas from a category. Like the idea of monad is very important to actually represent side effect or computation with side effect. It's really nicely captured, basically. The radical idea of monads. So there are people already using ideas from category. So functional programming is not too far from... From the way topos are done. Okay, so maybe I can give the speech to the people over here. Are there questions around topos that were not so clear before? Okay, I think that was nearly the time I wanted, 35 minutes, 40 minutes. I'd like to thank you again for your excellent presentations. I think you'll be on YouTube, you have to know. As far as I know, it's even a better, I would say, advertisement of the disciplines on which you're working. And I hope we'll have all more chances to interact and collaborate with you guys. Thank you very much.