Discussion - Panel: What is the role of Topos in Information and Communication Technologies?
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00:00
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Transcript: English(auto-generated)
00:15
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...
00:25
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, because 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
00:42
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? If I don't consider it is really a research on topos, except for the last part, I would say a word on that.
01:04
Because I have heard about the topos, and I was very amniotic of the work in J.R. Foro-Hundtra, but I never thought to apply topos. As I said, the problem was to define the form of a mission without knowing nothing about the topos precisely at this moment.
01:26
And it was almost evident that the equation of Shannon could have an interpretation in a cohomology somewhere. And so I started to define this framework, just to define a complex, like you show in your slides,
01:45
except it's a bit more complicated. And the complex was almost correct. But of course, it was strange with respect to many things, because of this kind of indexing that I don't really function, that always there is a variable which plays a special role. I put an index, but at this moment, I don't put an index.
02:06
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 didn't have locality.
02:25
And I had this idea that perhaps it had to do with topos, and in fact, by comparison, it's like exactly that, that is to realize that there was a seed, a site, in this case a trivial topology, but in fact, the fact that refinement is a topology is my idea.
02:46
So after that, you have a much more rich situation, where you choose a specific seed and filter. But it is this idea that you don't try, as in ordinary topology, to define open sets,
03:04
but you define covering. So it is this idea of refinement. And this sheaf and so on. And after that, of course, I looked more carefully at what was done, especially because to compute, there was a suggestion in GR4, by spectral sequence and so on,
03:27
but in this case, no, because it was not really a church comology. But after that, it is as I started, I realized that I was doing a computation of a topos.
03:46
So you were doing topos without knowing that you were doing topos? That's how you started. That's exactly that, in a very particular case, of course. And after that, it is more to the contrary, that's because we bet on this extension of
04:05
to try to define a geometry for preparing... robotics and anyhow. Yes, I wanted to apply the principle of a topos, for example, the fact that this
04:23
into this logic, or contextual logic, as it was done by a politician, 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 really different
04:46
level and localized a different place of what is the choice which has to be made, that you don't realize it. So there 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.
05:09
And the fact that when we do the action, in fact, we don't manage to control points. Perhaps we control some synergy at some moment, but it's even not evident, like what we do.
05:25
So it's better to have no point, not necessarily to have point. If you have, it could be okay. Not necessarily. So no success, no... Jean-Claude, why did you start doing topos?
05:43
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. No, 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
06:06
band mathematics, but to master all this area, I mean, you have to master a lot of things. You have to be broad bound and very deep. So I'm very, very impressed by you.
06:22
So, Olivia? Well, I got into contact with topos when I was still a master's student, actually. I had read things on my own. And in fact, my first motivation was to combine logic and category theory, which were my main interests at the time. And of course, also
06:47
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,
07:02
because when you do very abstract stuff, in some sense, it's harder to be deep. But in fact, for me, this is really the challenge that I set to myself. I mean, try to do work that is both abstract and deep. And in fact, toposphere offered me tools
07:21
to achieve this goal. And I'm the 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.
07:42
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 that this extremely fascinating. And so I could
08:02
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,
08:22
but always with the same aspiration. Okay. And Thierry? Yes. So I was working constructive mathematics and I wanted to understand some papers of Andre Joyal. That was about toposteory, but I was unable to understand them. And
08:43
I really started to understand them. So a constructive mathematician, Henri Lombardi, he 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
09:03
decide if a polynomial was irreducible or not. And they were still able to compute with algebraic numbers. And with very algorithmically very interesting ideas. So this idea of lazy computation, to be lazy. And so this was a really nice idea. And then I realized actually
09:24
that that was what Andre Joyal was doing, this formalism of toposteory. And that's really where I started to understand actually toposteory. And it can be relevant for computation. Okay, good. I mean, in your presentation, nobody made the differences between
09:41
topoe, toposes, topos. Huge controversy. So at the end, what is the common use? It really depends on the authors. I was influenced by my own supervisor,
10:00
British man, and so toposis. Toposis, okay. But some other authors, very distinguished authors, including Jacob Lurey, Nieke Moradzic, they use topoi. So I mean, everything is acceptable, I think. Okay. So this goes now back to Danielle. Very good presentation on toposes and information. And it goes back to my question,
10:26
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
10:44
information. In this case, of the information quantity, first thing it can be new is if there is a Eiger degree quantity, because you have
11:00
cohomology theory. So here we can identify as easily the one dimensional, but the Eiger could also be interesting. He could say that we have looked for, at this moment, for invariant of one under variable, under some probabilistic model. But it could happen that
11:27
you have invariant which are only, if you look at, for example, three variable together, or in some order. And we could be not definable through individual measurement, because the
11:45
look at the configuration already of 3D variable, that's one hope. And here it's more this point of view of cohomology. And also it can be the starting point to
12:02
examine the relation between these quantities. For example, you can integrate, this was asked, the energy, so you have this what is called beta energy, which could probably be also interpreted homologically, but in a context here. And in this case, as I know,
12:25
since one year, six months, working with another student, which is only different, that this algorithm, which is important in machine learning information, belief propagation,
12:41
has a cohomological interpretation, but with a different complex. 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 non-linear and satisfies this. And if you combine both, you have a kind of heat operator,
13:03
and this flow is the heat flow for this operator, which is discrete, and that is the interpretation. It is a discrete version of heat equation in this situation of what is called graph, context, belief propagation, yes.
13:28
A graph factor. The generic name is graph factor. So probably it's because you have this structure of kind of topos, or relation between topos, that you could hope to see this
13:45
kind of thing. In fact, I had the intention to make this situation of, it was the text or thing by Kortendek and Verdieu, telling that, in some sense, every time you will have
14:00
some notion of locality underlying a form, you could have a topos in some sense. Our problem is, do information as a form? That's part of it. But what are the forms of information? Like I discussed on that with Pearson coming from gestalt theory,
14:25
and he hoped that the figure, for example, 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,
14:47
not just a little. So no, in fact, the point of view that topos is offering is really different from that of any particular field to which it is applied, because it is,
15:02
I mean, I personally regard it as a meta-mathematical subject in the sense that, you see, thanks to topos, 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
15:25
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,
15:40
could I have obtained these results without topos? 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 rehabilitation thesis, so one can read about
16:05
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 morette equivalence,
16:24
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.
16:44
It's not just that, it's the creative power of topos 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,
17:01
modular in the sense that it is possibly transferable 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,
17:22
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
17:45
by topos, you make quite intelligent choices if you are able to hear the voice of things, as Grothendieck was saying. I mean, in some sense, you have to develop a certain sensitivity
18:02
to understand what a topos suggests to you. But once you get to that level of expertise, you realize that the input they can provide is invaluable. Okay. Do you have any comments on this?
18:20
No, no. Maybe, I mean, the fact that 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, Grothendieck came to
18:42
intuitionistic logic. He was not interested in logic. That's really surprising. I think, I mean, there will be more connection with computation in this way, which are deep, which are not yet understood. I think this will bring something. I mean, coming back to Grothendieck, is it true that there are many things still that
19:02
we cannot read from his papers, from the whole number for Grothendieck? 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?
19:20
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. I mean, I'm not aware. I mean, there are two parts. There is one part who was given to Montpellier, and they are now available. So they have been numerized,
19:43
and you can access to the documents. But they are not so easy to read, even formally, because they can be handwritten. Yeah, in fact, I tried myself to read one of these, but the writing was very, very little. Inside these documents, there were some who were already circulated among the community.
20:06
And then you have the so-called Lasser documents, which are 70,000 pages, but with few mathematics. It's more related to the questions of good and evil.
20:24
Yes, it is more related. But with still his mind, they are not available. From your presentation, we have the feeling that the topos is like the theory of everything.
20:42
La theory du tou. With Topo, at least, that's how Laurent Lafford 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
21:03
do 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. Or what do you think we should do? I think, yes, from this point of view coming from high geometry,
21:26
the fact that you look at sheets on the other side, and this interesting logic, the fact that, for example, you can try to understand more classical objects. For example, geometry. That is, the geometry was traditionally thought,
21:45
as I said, that you have a group, and you have a subset of groups which are conjugated 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, the study of Riemannian manifolds, for example,
22:06
or contact structure, in this setting. So that 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.
22:22
You can have a manifold, for example, which is fiber under Riemann, and when you don't see the vibration, you have a space of four dimensions, but when you look at the fiber, it has ten dimensions, or eleven. For me, it's interesting to try to do that, or to,
22:42
if somebody does that. To integrate the fact that, in some sense, you have a larger score than set, which it replaces set, and you look at the usual traditional differential geometry, for example, in this context.
23:03
Good. 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, and the problem is not that it's so difficult, I think. The problem is that
23:26
when the new bias, like me, tried to read something about it, in fact, we immediately realized that we cannot read anymore, and that's the big thing, and we need probably to develop
23:41
some reading skills for that, especially in this. Maybe it comes from categories, I don't know. Olivia, the challenges? Yeah, I think the big challenge is actually, as I was suggesting, to make the theory more user-friendly, and to apply it. Because, of course, ideally, I think progress in mathematics
24:04
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,
24:25
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 toposys in mathematics. So, in fact,
24:46
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 toposys and good invariants which relate to the questions he is
25:06
interested in. Of course, it takes time, it takes some investment, and it requires also the existence of a community, at the beginning small, but hopefully larger and larger works on that.
25:22
Toposphere has been quite a controversial subject in the past 40 years. It has been very much used in algebraic geometry, especially as far as cohomology is concerned. Also in homotopy theory there have been many developments, including the theory of higher toposys. So there have been some directions in which research has advanced very well,
25:49
especially in relationship with solutions to specific problems. Even though still at the theoretical level there remains fundamental problems also concerning the cohomological
26:02
formalism. I mentioned the issue of the six-operation formalism that is still waiting for a unified topospheoretic treatment, which allows one to recover all the special cases known in the literature. So this is also quite an interesting challenge.
26:26
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
26:44
in the next years is the functorial development of model theory. Because as I mentioned, the classifying toposys 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
27:02
model inside it, because after all this generates everything which happens in the set-theoretic 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
27:24
interesting to pursue that line of research and reshape the foundations of model theory by using a topospheretic outlook. More in general, I think that what would be very good to have is a very open mind on the part of specialists in different fields to talk with topospherists
27:48
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
28:04
possible for the field. And on the other hand, the toposphere is that it can bring these completely novel insights. So something which I have noticed in the past years is that sometimes communities are a bit close-minded and they are not always open to talk to category
28:25
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.
28:45
Results will come, but one has to think in a long-term way, as Grothendieck was thinking, has always promoted mathematics, a very systematic development of mathematics, and
29:02
toposphere is a subject that certainly deserves 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 topospheratic training and that this will
29:23
stimulate more and more results. Okay, that's what we tried to do here. Maybe Danielle and then I'll give you once more. Just to remark on the context why starting from algebraic geometry it comes to logic. In fact, the movement was begun before, for example with
29:43
Maclean, Eilenberg, and Cardo, which introduced category theory and they were motivated by topology, and of course the interplay between algebraic topology and algebraic geometry, or number theory, and so topos are inscribed in that. For example, it is already in
30:08
SGF4, the theorem that you have this equivalence between some category and such and such property. Of course, it was not developed in the model setting, but it was developed in the terms of
30:23
property of a category, to have a limit, an inverse limit, and so on. It is equivalent to the search category because this group was very interested by all the categorical aspects. To develop
30:41
category theory, they were aware that it is changing mathematics, and now it's evident that logic is becoming very influenced by category also. So this is at a larger level. So topos don't participate to this larger modification of mathematics.
31:09
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.
31:26
I mean, when you read papers of topos, of course, you're quite surprised because there's all the 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
31:43
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.
32:03
You can find, I think, 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 and the language
32:26
like Haskell or KAML. And they are already using ideas from a category theory. The idea of monad is very important to actually represent side effect or computation with
32:41
side effect. It's really nicely captured by the theoretical idea of monads. So there are already people using ideas from a category theory. So functional programming is not too far from... From the way topos are done. OK, so maybe I can give the speech to the people over here. Are there questions around
33:06
topos that were not so clear before? OK, I think that was nearly the time I wanted,
33:21
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. So it's even a better, I would say, advertisement of the disciplines on which you're working. And I hope we'll have more chances to interact and collaborate with you guys. Thank you very much.