 So in the globular, in higher category theory, there are several points of view. Basically, they are the sumptuous point of view, which is the most well-known around the world. There is also others like the globular, or the obitopic, and others. Thank you. Sorry. So I'm going to speak about some advance, some progress in higher category theory from the globular point of view, and especially what should be an infinity to pulse in this globular sense. OK. So je commence en français. Sorry, I started in French. So I'm going to talk about the infinite categories with a globular geometry. There are two geometries. There is one that is in my head, and I just forgot that it is completely similar to the globular. It's the one that is, as soon as I remember, I'll tell you. And? Cubic. Cubic. Exactly, very good. Thank you. So the globular, the cubic in reality, if we understood the globular, the cubic, it's a reflection. So it's more a philosophy, there are difficult results that had to be found. Me, in my doctorate thesis, first I'm going to explain quickly, is that I managed to pursue Michael Matani's work. He defined the operation of the infinite categories. And I did the rest, that is, the operations of the infinite categories, the transformation of the weak, etc. All types of weak transformations. André Joayal, here, had helped us to correct the problem of contractility that has been done recently. And there is also a big conjecture, which is probably the biggest conjecture in theory of the globular categories. Is that the infinite category of globular of the infinite categories exist? And this problem is a problem that interested people as a result of these noise. And then probably with this very big problem. Then, in the conclusion of my thesis, I explain how to build operations for the weak adjunctions of the infinite categories. So the infinite categories of adjunctions are managed by the operations. It is a published work, but it is in the conclusion of my thesis where I give you the great ideas how to build them. From these adjunctions ideas, we immediately have the idea of the famous, there are immediately the limits and the collimits that come together with the delta founder. And then we do all this in a superior dimension. So we will see, in this talk, a notion of infinite limits, infinite collimits, and then, at this point, we will try to say what are the little theories. We will be inspired by a classic theory on the basis of the idea of a characterization to say again what must resemble the infinite tropos of Greta and the infinite entropos of Greta. So I will talk at the start of some superior structures. Then I will talk about the weak categories, the weak categories. So even if this problem does not fall, we start to have progress on it. And then I will talk about the infinite entropos and the infinite tropos. So for the superior structures, I want to quickly remind you about the people who do not know the global world. So we consider the weak category, which is subject to relationships. So it is a small category and in fact we take the opposite category and we take the three episodes above to combine what we call the global ensemble. I do not know what to say, I do not want to take it. So we consider then another small category which is also amusing. So we take the global category again, and then we add formal arrows that go in that direction and that go to what we will go to the opposite category. So we will do GR versus 0 OP and we take the three episodes above, we will come to the respective global ensemble. Then I will talk about infinite N ensemble global. So these are less known. So it is a work that was published in 2015 during the Altaque Journal. So it is a bit the structure which is in the infinite groups the infinite categories, the infinite categories and all that. In the global world it is a bit the structure which is a bit hidden behind. So an infinite N ensemble global is given by a couple. So X is an ensemble global. JMP are applications and which verify relationships. So they verify these two diagrams. In reality, if you look at these two diagrams, the two parts at the top are the same. It is at the bottom which makes the difference and which makes that we obtain these infinite N ensemble global. So these diagrams will in an infinite graph, an infinite ensemble global, they will reverse all the cells and all the dimensions and all types of depths. It is a bit simple to say. So we say that an ensemble infinite N global is an ensemble global equipped with an infinite structure. There are several. In my article, I invite you to read this article. Finally, for those who are interested in infinite N categories, what is interesting is that even for the infinite groups, for example, there are in fact different types of infinite zero structure possible. And so that's something quite curious. So we define the morphisms of infinite N ensemble global. So it's defined as an ensemble a morphism of an ensemble global and which does as it is of course what we call what I call the reversers. That is to say these JMP applications which can be used to define the opposite categories but the opposite categories. We do the opposite all types of cells at all levels. It's just a little parenthesis. So this category, we will write it like that. So then, I talk about the magma infinites so I'm still in the superior structures. I complexify these superior structures. At first, I talk about the superior structure. It's very simple. I enrich it, I put the reflexivity, I put an elementary structure. It becomes a little more complicated. Then I talk about the infinite N ensemble and now I talk about the infinite magma where we begin to add operations which are more commonly known. So it's composition laws which check position actions but that's it. Position actions are the following. So that means that the composition when we compose the cells tomorrow is composed and well-placed. Exactly. We imagine quite easily what it should be. Yes, of course. I do a descent. No. Because it means that there is a minute when we can consult. The morphisms of the infinite magma are morphisms. The global morphisms are morphisms which of course preserve these operations. So we have written the infinite magma for the category of the infinite magma. So now what is an infinite category? An infinite category is a reflexive magma which satisfies the action of associativity and the action of exchange. More other small actions which can intervene so reflexivity and then operations. It's a way where I like to define the infinite categories. So the morphisms of the infinite category preserve the structures which are classically we have infinite categories. I forgot to write the small infinite categories. So Yes, where the morphisms of this category are found in the strict infinite categories. It is the small mark which is not indicated because I will talk about the infinite category later. So the infinite categories I will not do in my talk what is an infinite category because I do not have the time but basically it is a released version and strict infinite categories. That is to say that the actions are more true of associativity are more true but they are true to a this coherence that is to say that the associativity there is not that equal to that it is that and weakly equal to that and a sort of weak equivalence or an infinite equivalence between these two quantities. Ideal for exchange actions etc. So this is a mark I like to do is that the infinite categories can also be seen as as the infinite categories which as the infinite categories finally there is no action this amount of action we will fill it with an infinite equivalence and then and it is filled in a coherent way and it is controlled this filling is controlled by the strict structure, the strict categories so there is a global opera that I call B0 on Glob, the category of the global ensemble which is the models that I use here for the infinite categories weak. I note this category so I give examples of superior structures other interesting so we can of course with the infinite magma make a lot of variations we can talk about the infinite magma we can talk about the infinite there is a strict category so for example the infinite 0 strict categories are the infinite group of the strict which have their importance in all this in the abstract model so there are also the algebraic models of infinite categories we say infinite categories but I add them to make the distinction to the strict called infinite categories we note like that so they are algebraic in the sense that they are algebraic for a disease so it is a work as I told you that was published in the TAC journal in 2015 so for example our models of infinite groups are defined as our models by definition are our infinite 0 categories so now I go to the second stage the infinite categories so before talking about that you have to talk about the infinite functions so I built in my thesis in 2008 then it was announced then published in 2011 then published in my thesis but then corrected so B1, B2, BN1 etc what are these objects so the B0 is the opérade of Batani which the algebraic models are the infinite categories of Batani B1 are opérades so the algebras are the infinite finite function B2 is so the opérade on globe to square so the algebras are the infinite natural transformation and so on so let's say that I didn't change the combination that I already had the problem was a problem of coherence that I and Andrea had to solve before Réram and then I had to do bad and finally I solved the problem a few months ago and then we can build a global object like this one in the superior opérades category global opérades so B0 is the opérade of Batani B1 is the opérade of the infinite function B2 is the opérade of the infinite natural transformation BN is the opérade which the algebras are a model for the infinite N transformation a small parenthesis what I say in my thesis and what I also say in my article is that when you replace Batani with another what I call strict contractility you come back to other opérades which will be the opérades of strict natural transformation etc so one of the most important conjectures as I said earlier is the category of the categories we know that they exist well, that's it model of the universe the problem of the infinite category of the infinite categories in the global world is a difficult problem it's a difficult problem in a simple world I believe that I believe in fact you have shown that all the infinite categories form an infinite category is that correct? yes, it is but it doesn't give an infinite category especially not yes, it gives an infinite category it also gives an infinite category yes, but what happens is that we associate a category and then we take what is one of these categories it's a bit ideal the construction is a bit ideal ok so, the conjecture is the following if we take the global object that I just defined we take the b0, b1, bn, bn etc the question is does it organize in a b0, b1 we go back to the point we have to see that the b0 must be seen as a point because these operators behave like topological balls it's funny we can have fun if we take the point the interval the disc etc to infinity we go back to the point that's the thing so we can re-formulate this conjecture like this the infinite category of infinite categories exists the global context so this conjecture it's true it's a nightmare it was a nightmare it became me thanks in fact I replaced this nightmare problem at the beginning including for Michael Matanou and many people a very precise problem I'm going to talk about preliminary how to re-form this conjecture so if we note the coin of infinite categories if we take a category that admits small balls small small balls for any object in it produces an operator called a morphism so it's an operator on omega 0 on the hand of omega 0 that is to say it's an operator with a cartesian morphism towards the hand of omega 0 so we go back to the new definitions in the world of these global operators so an object a global operator superior that I note W which means when I note W delta kappa it means that I have W0 which includes twice in W1 which includes twice in W2 etc like we see the point which injects itself in the disc in two ways so if I take such a global object so a global operator in a category that has small balls so I call it algebra so this global object is an algebra if the first operator is an operator above the infinite category strict and if we have there is an operator morphism of this first operator towards the operator of the bodies of morphism that's for the definition of algebra so global objects operatic then we say that an operator is fractal if there is a global object operatic which lives in a category of superior operators with the first operator in question W0 is Galar so I gave examples of fractal operators for example the four examples I gave in an article that was published again this year so I showed that the operator of the infinite which is something completely obvious but finally I showed that the operator of global objects is fractal it means that if we take all the global objects it forms a global assembly here it's obvious but it's as if I open a box with a big but in reality this big box will serve us by the following I demonstrate for example that the operator of the reflexive global objects is fractal the operator of the magma is fractal the infinite magma is fractal the operator of the infinite magma is reflexive is still fractal that is to say that if we take all the infinite, the category of the infinite magma is reflexive but if we take we keep the objects we remove all the morphisms we replace all by morphisms of the infinite magma reflexive transformations transformations etc etc we go back to the infinite magma reflexive that's curious so just for management the category of the graph the category of the graph is the same as the graph exactly and in addition it's so obvious I almost forgot to mention because the category of the graph we take the font and forget we forget the structure of the category and we go back to the graph so the category of the graph is the graph and the category of the two categories the category of the two graphs is the graph exactly there is the same thing for the magma and all that we show it in my article in the basis of dimension we can show everything with the operators we do the same I don't break it for example it doesn't allow us to show that all the two categories are strict form a three categories that's not the philosophy of the subject all the two categories form a two categories that's a small important point to note ok now the projection we have made a very strong simplification of the projection by reducing this difficult projection in a very precise technical problem is that the opérate is fractal for the global object that I built so the question to show the fractality there is a strategy to show that if I take the opérate of quantum of morphism associated with this global object if we manage to show that it is contractible and that it has a composition system it's won because it means that as the opérate of mathanie is initial in this category we add so if the opérate of mathanie is fractal look what happens we have this diagram at the top this diagram it expresses an action of opérate B0 on this in fact we have to look at the opérate which is important and the last one this opérate the opérate of endomorphism it takes all the algebras the algebras I wrote B0, B1, B2 etc so I can make algebras of B0 the category of algebras the category of algebras for B1 and so there are sources sources, sources, goals I take the object part in the category of together the big together and then I take the opérate of endomorphism well we have a morphism of the opérate of mathanie towards this famous opérate and it expresses exactly that the infinite category of infinite categories exists so we will note it like that simultaneously so now we will suppose that this infinite category of infinite categories exists in the global world to go a little further well now we will talk about the infinite about the big together in fact the idea so I start to talk about what we call the infinite adjunctions of the weak so in the conclusion of my thesis as I said earlier I explained what is an infinite adjunction of the weak I give a lot of precision it is a work that is not published I have not had the opportunity to do and then so so intuitively it is a superior version and released from the notion of the father of the adjunct founder so so more precisely there is the opérate that I call pH on the cartesian product of global in my thesis in the conclusion, the error I made I made a mistake in fact it is not the adjunction of the infinite categories it is another adjunction you have to forget this pair of adjunct founders for example I give you a very simple example we take a pair of adjunct founders we forget it in graph in square and then this founder forgets the adjunction on the left because the thing above is projectively what it knows so so an infinite founder can have an adjunct on the left not an adjunct on the left in the classical sense an infinite adjunct on the left or on the right now do not ask me what are the criteria that allow to show this kind of result we are not there yet so I consider very important it is like the category of founders I will define the infinite category of the infinite founders so these zero cells are the infinite founders between two infinite categories DC so the one cells are the infinite natural transformation and then we define the n cells of this infinite category by an adjunction so now we consider the data following it is defined as the usual data to define to say for example that a certain type of diagram in a category the limits are possible for this type of diagram and at that moment we reconstruct the data but in all dimensions so we assume that the data now so if it is to infinite small infinite if for all all the categories D up and delta an infinite adjoint to the right and it is to small and infinite co-complete little if it is an infinite if it is an infinite adjoint to the left to the left I go very fast so now I define the notion of the preservation of the infinite and the infinite and I say that the line this preservation is no longer an equality nor an isomorphism it is an infinite that goes into this what is an infinite that goes into this it is like an infini group now we consider the infinite so they are I take a small category and I take the category of the infinite co-complete little so we assume that we assume that there is a sub-infinite category of this category of this infinite category and it is reflexive what does that mean? well, thanks to the adjunction notion that I finished earlier I soon finished Stéphane so it means that the inclusion is an adjoint to the right in the weak sense and in the superior dimension I forgot an infinite here there and we assume that the air preserves the finite if we have this we can say that it is an infinite category of an infinite field so an infinite category which is weak and infinite and equivalent an infinite category of an infinite field is called a topos infinity so in fact I used the characterization here we know very well the purely categorical characterization which says that when we form a category of prefaces the inclusion is added to the left and it preserves the finite and this is in projection with the grotantic topologies and I completely forgot the grotantic topology here I directly used this purely categorical characterization to generalize it this is the definition thank you thank you so now we have defined the infinite models to a category so as I said now we go to the last the last thing and I finished in position so if we take an infinite category weak which exists which are defined as algebras for a disease which are weak infinite equivalent to an infinite category of an infinite field so there are examples of what I can call an infinite opposite of the grotantic that's all, I have nothing I simply replaced an infinite category by an infinite category I'm done thank you without necessarily entering the debate the right definition no no just a detail of bibliography, what is TAC? what is that? TAC is the theory and application of categories it's a journal what is there in Australia? no it's a very good journal of the Austrians what is it? what is it? what is it? it's a paper an Iranian newspaper a young Iranian newspaper on the theory of categories and structures I published my work my main work on the fractal operas because I was tired of projects that lasted years and these people had the great kindness of recognizing contracts and all their articles are online so they include Kamal Kondai very easily yes and Leroy tries to be integrated they have reason in the case of the infinite categories that is the infinite opposites that is, if there is a definition of the infinite opposites yes, of course simply if you take the definition that I just gave I don't give a definition I say that if we take the infinite categories in my sense what I defined in the journal TAC well it's an infinite of opposites if in a an infinite category of opposites like the left adjoin and the left adjoin is not the left adjoin it's an infinite weak left adjoin that preserves infinite finite limits let me know if you have an example with n equals 1 or n equals 0 or something if n equals 0 it's the opposite of TAC it's the same you mean in one dimension I don't understand I would like to know if you can get something that has the opposite of TAC with this definition but it doesn't work with the dimensions there is an issue the coherence is bad and in the end this definition is a natural extension of the opposite of TAC if we consider the definition of the category it should have the notion of the opposite the one opposite I don't know there is the Australian school to use a notion of cosmos cosmos it's a notion of the opposite but it's based on the category of small categories rather than on the categories would you like to know if you can get that for instance it's a very good question André I talked to Ross it's a crossbow because he was interested before that he didn't know and in fact Ross when he wanted to do that he wanted to extend what he did but also for the categories so some times it's rather the cosmos you have to see it as a definition of the two elementary opposites but I'm not talking about the two elementary opposites it's the two opposites exactly it would be good if you could illustrate your general definition by cutting and then blocking a little the model that it gives it would be nice to know that there are two two opposites ok it's just a suggestion if we have the examples we can start thinking it's quite easy of course if someone have a question there is my email so I'll let you go and I am looking for a job