Locally small Cartesian functors / Variance vs. volition

0:00 In this presentation, I'm going to do an extension and for the second one, an imposition. I'm going to do something like this at the end of the lecture. That's what I'm interested in, this structure here. It's a presentation of a N plus one category because when you think of a N category, which I don't assume, I don't assume that it's the dimension of N, of C and E. When C is a N-category, of course, the extension and the new quotient are of dimension N plus 1. When I say a N-category, it doesn't mean much, it simply means that there is no cell of dimension above N, we can't really talk about it. These are the tools I have in my hands. So now, a categorical presentation, I call them omega presentations too. These are presentations that take place dimension by dimension. As I said, I'm going to present 10 polygraphic presentations. I don't have much time, so I'm not going to make fun of the fact that there are 10 polygraphic presentations. These are the ones where there will only be extensions. The 10 polygraphic presentations are the ones where I will do a series of things. Increase, quotient. Increase, quotient. Increase each time by one dimension. This gives us a series of categories and even a series of numbers, of 10 increases. Notations increase, this is the index, but it is therefore a sequence like that, and this sequence begins to increase. For the reason I gave earlier, but also for more and more reasons, because I would have liked it to decrease. With the following conditions, C-1, I imagine that there is a category that has a dimension of minus 1, this is not the cells, the cells are when you put the 10 below, the height is a nth category, it does not mean anything, it is the nth category of the following, on the dimension there is a nth, I need that, I need not to say that I am not an academic.

2:30 This is all there is to it. Apart from that, then, CN plus 1 must be linked to the previous CN by the following relation. The extension of CN plus 1 is just the quotient of the second increase. These are the important relations that we will discuss later. So, when there are no... when the signs are trivial, it leads to what I call a presentation, and I now need a relative presentation, relative to a category that is there, and it is simply the same thing as that, but the difference is that it is minus 1, there is no 1. I talk about art and I make my construction from the dimensions. I add points, I add arrows, I make a quotient, etc. This is what I call a presentation of the final result. Obviously, it assumes that all these categories are comparable to each other, and in fact, they immerse themselves in each other. And these plunges are monomorphisms, they only make the structure bigger, and the category that interests me the most is the limit, the inductive frequency of these different monomorphisms.

5:00 I'm going to move on to the cones, but the cones are ultimately very difficult to describe, it's a rather complicated object, but above all, what is complicated is to describe the monad that I proposed earlier, to describe it entirely, that is to say, in unity. There was a moment when I thought to myself, it's stupid to do the monad. If I guess what the algebras are, you'll have the monad right away. But indeed, the algebras, I'll start here, with the sets of sets, are the generalizations of the notion of initial objects. So, that's what I call it. Gamma is monad. So here, I'm going to suggest that you change the name a little bit during the game. It's a monad on the 4th of January. This monad is the root of the algebraic structure. I'm going to define it by its laws, its operations, and its equations. Its operations... I'm not sure because it's... I'm going to call it X, but it's better to call it C. It's C, it's the name of the equation.

7:30 The operations are of the following form. They take each cm. In other words, a single cell is attached to a single dimension cell. Equations start from zero. Equations are the following. Oh yes, geometry and dimensions. Instead of writing s of x and t of x, we propose to write x1 and x plus. In this case, the x-axis is reduced. For example, x is a red cell, and I reduce it by a unit, for example, it's called a x-axis, n-1, but we can also reduce it by a lower unit, which I use for the sub-sources. So this is the source of dimension m, the dimension of x being here, x being n. At this point, this is of dimension n+. Here, the result is delta x plus n-1. and so on, and so forth, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, I'm not going to spend too much time on these formulas, just to show you what they look like more or less, but I'm going to give you...

10:00 Key terms may include a lot of dimensions and the identity of details. So, these are the stages of my lecture. So, I have the lecture. I will go through it right away. I will tell you later how you will see it. I will tell you later how you will see it. I will tell you later how you will see it. I will tell you later how you will see it. I will tell you later how you will see it. I will tell you later how you will see it. There are elements that are not expressed, but the interest is to describe them. So, their description, well, in fact, I'm going to give you a little description first. It's a definition, either I have a part of the whole of the cells, I say it's a system of generators. It is written as a combination of elements of G by identity and composition, by source-cells, identity and composition, and that each cell has only one culture of its own. So what happens is that when C is a generator system G, this is a generator system that has a specific energy and G plus two. Because... I don't know if you can hear me. Five minutes? Okay. So, I have, for example, C here. I make the chrono there. Each cell that is here will be found here, by the state that is there. And then for each cell, there will be a cell of dimension... of dimension plus 1 and minus 1, which is... which is located there. And that's why it doubles. You will have here the double of the generator. In other words, if I start from zero, which has nothing at all as a generator,

12:30 if I have one, it is in the dimension of minus one. So it's not very, very, very good. Here, there is a rapid, we have it in the first step. It's huge, isn't it? There is a cell here. There is a cell that I have here, minus one. It is virtual, that is to say, it is in the dimension of minus one, but there is one anyway. So here, there are two. Then for O. So for O. These are the parts of N. The cells of the poor organ will take this form P0, P1, Pi with an N here, and P0, P1, and so on. N gave us P0 plus 1. We went from these objects to the higher dimensions, where the zero is in some way... These constructions, if you combine them with the equations that are there, you will obtain a way of calculating the 5-plexes. I call them omega-3-plexes, as it has not been proven. He himself did it up to the 5th dimension, which does not count. It contains half a page of calculations. I asked a computer scientist to go further and the machine came out very quickly with four or five pages of calculations, absolutely indisputable, but the internal structure of these orientations is only a few very simple forms.

15:00 Thank you very much. When the category becomes two categories, it gives a kind of initial object, the axis. If we have two categories, an initial structure is given one by one, as for any other object x, the arrow, there is an arrow here which is not unique, but which has been chosen, like this, and which has the following property, that each time I give myself another arrow here, I also have a unique arrow, like this, like this. But let's say I can't tell you the initials of the science-fiction and the generalizations of 5.0.

17:30 And the paper that I put in the web, it certainly talks about two non-discrete localic generalizations of 5.0. But it is too much to give here the two, and one of them I already spoke about in Florence. So, I'm only going to speak about one non-discrete generalization. So, this is now this part, point. That's the discrete 5-0.

20:00 I am going to work with topos as if they were generalised bases, that is, topos has many aspects and they all intervene, but it is always one predominant one that guides one's work. So I will think of the topos as an arbitrary base, but you are very free to think of this as such. And there is a structural morphism given, a geometric morphism, which makes this a grotesque topos over this, in other words, sheeps on a site inside this base. Because of some assumptions that one makes on the geometric morphisms, so for example, we all know that in one assumption there is to be a locally connected geometric morphism, which means that the geometric morphism There is a term that I join here, which is a term that does taking connected components. In fact, in the applications, that's what it is. So when one has this, there is a very high zero, it's just taking connected components. So the canonical factorization of such an error Here, the slice topos taking the connected components of the top element of the terminal object in this topos, which is the same as any of the space, so the connected components of the whole space.

22:30 In topo-stereo that means that the inverse image is fully faithful, but it's in general a sort of a bad notion, it's just that in this case it's not just an algebra, of course, and a method. So this is a factorization, a unique factorization, that has been there since time immemorial in this drawing. But the question is, but this is not the question that motivated what I'm going to say today, which is what I'm going to tell you about, but the question, this goes out, you know, this doesn't go out, would be if one deals with an arbitrary geometric models, well, I would say almost arbitrary, as were then these conditions. If you had to put any conditions, there would be too much of the truth. That's what I would mean by almost. Almost half of the truth? Almost half of the truth. He says, or William Donovan says, or really he does. In this case, do something that one wants to do sometimes.

25:00 In many constructions in which Y0 occurs, they discrete Y0. So the motivation for how we approached this problem was for something much bigger than this, and it should be more nourishing. It is saddening to think about what the completion process is, but I have to repeat it a little bit, because this is how the solution to the other problem arose, that he was of loss, more rationally, of some problem. On what? Say that x is r3, that he would take a divisible logarithm, but we assume a logarithm connected everywhere. But it was easier that way, now you can understand why. The inverse image part of the connected and the space is connected there. So, for example, the complement of a knot in R3 is obviously, the inclusion into R3 is pure.

27:30 Because you cannot, if you take a connected here, you will not get disconnected. And the component in R2, you embed this in R2, that is not pure, because in the symmetry of the connected, that means the circle will get this one. The notion of pure is a natural learning topology. So we wanted to see this as, say, okay, so I'm going to take the universal, locally trivial covering, or some locally trivial covering here, locally trivial. I have to say, more generally, I'm gratified. In particular, I'm happy to be a part of the documentary. It's complicated in such a way that this is a sort of invariant notion that no one wants to deal with. He invented the notion of a spread in order to do this, because spreads should include not just gratified coverings, branches, but also folds. The equal symbols of opens appear, the connected components of equal symbols of opens here generate the topology up here. That's what a spread is. Tell me what you know. The point of x minus 1 is a 1 of u. Take all such, and take all the connected components. So the alpha is equal to minus 0 of this. Atiyah, so these alphas generated with topology and it was completely transportable, very, very straightforward.

30:00 I must say that, so one has a section connected, there is something from G to S, actually from the side, from the side of G. All of this takes the connected components of the averse image of U. You can see that we are thinking of the spread. This G of U is a covariant functor. It's really a co-sheet. It induces a vibration. And I can come here again. This gives rise to a geometric morphism here. And we can take the pullback. Pullback y, pullback y. And there is a natural geometric morphism there.

32:30 This is the connected components of the representable hc. So the clear c-alpha logic is that c-alpha is simply the alpha which is easily rendered into a sub-object of one. And there is a use for this. This is what I put back. This is a complete spread in sense of forms. The site is generated precisely by construction and it is complete in the sense that it is the only one that is complete in the sense that it is the only one that is complete in the sense that it is the only one that is complete in the sense that it is the only one that is complete in the sense that it is the only one that is complete in the sense that it is the only one that is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that it is complete in the sense that In other words, four genres of components converge. And this block here, in the sense, this is the thing that we connected, and that's pure in the sense that I explained before. So that is the connection. Now, Hooke was very aware that it was too restrictive. So already, in his paper in 1947, he starts by saying, wait a minute, I mean, we could... We try to get rid of local components by replacing anthropological notions by logical notions, for example, why do we work with connected components if we can work with complemented sub-objects? And how do we spread that generated by the complements or the complemented sub-objects of English languages? Then pure would mean that if pure molecules were to use the inverse image and have an objection between the analysis of complemented sub-objects of an object and those of the inverse image of an object.

35:00 So instead of only the component, complemented sub-objects. Okay, so you can replace pure, you can replace spread, and then he even suggested, then what do we do of the spread? Two terms of components converge. What do we do instead of components? So, in the world, let's use quasi-components, quasi-components of quantity. After all, I mean, the quasi-component of space would be the intersection, one way of looking at it, of all complements or complements. Well, sophistication, I think of it as a filter of complements or complements. These are all elements which is inaccessible by other joints. And that suggests, I don't know if you realize what I just said, but that seems like even points of your power. I don't know, that's just the things. So anyway, he couldn't do it, Coffey couldn't do it, because working with quasi-components was extremely complicated. They don't work as well as components, they're all very strict, so you have a component less than or equal to another component that is actually the same, but you don't have a quasi-component. They are less included in another one, and they may not be the same, since it is already an organ, so there is a world of people working on space, there is a world of people working on how, and so on. And therefore, as the great one has been called cesarean topology, fewer has been already redefined, but does it work? I mean, now we have a pure, complete web, in some sense. Functorization of an arbitrary geometric model is impervious to the way in which those are connected. Well, following the criteria of the Gauls, Lusso, Michael, he had written, in Hellenism in 1963,

37:30 the constructions that Fogg had suggested, that fewer was not enough. Classy components can be used anywhere, and here more or less the same, but with a very complicated topology to make it work, and here replaced this by a condition called condition B, which in topology means something I'm not going to explain, because we have a condition that resembles it, but our whole approach is different anyway, because it was impossible, well, I don't know about impossible, but after 10 months of work. And many, many ways to find them. But the way Lenin talks was so easy to do, the Bible paper is not translatable, just the Bible. So we did something else. What we did was this. We looked at the original one, and we... So in order to get the topos, we needed a version of the Bible. And you want to begin and assume that this is locally connected because you want to analyze it. And then after we analyze it, we get rid of that.

40:00 That's what we do. So we see that the object here would be an alpha, a connected component. And we can denote it like this. The connected component of time. All of these are connected components, but connected components are alternate, so we can put it as C U, where U is an alternate sub-object of C, and by this you could conclude that the second dash is defined and was classified by 2, by 2 upstairs. Component, instead of by the big omega, by the two. So, this H is definable, not in the sense of equal, but definable in the sense of classified, up there, by two. You see there is an inclusion of the Y with the H. This is an inclusion, and here we can forget and get down to C. So this we have put the middle house in between here for a reason. The reason is trying to get rid of Y. The Y will not exist when F is not locally connected. So we have to find a time to see how to get rid of it. So first of all let's see now, we can analyze the white pullback. And here this H, there is an H here which is... Which is just the sine lower star or f upper star of the omega. If we look at what this is, we have the sine of omega s, omega s is 2,

42:30 we have the sine upper star of hc into f of omega s, which is 2, f upper star of 2. So in other words, assuming that this together with f upper star of 2, Classified, I mean, in other words, it's just a small sum even there, which is, this would give me, this would be the characteristic morphism of the definable component, the sub-object of scientists' dialogue with each other, and you. So, in other words, this age is related in a foreign way. For example, if we were in sets, and these were really two, this would consist of all the complemented sub-objects of one plus the zero, which is also complemented with zero. But again, you don't need the zero to generate, so you take that one away and you just have the complemented sub-objects of one. It's the same age, but the difference is what they did. Okay, so we know that here we want to get the G because it's the same, the same pullback and everything works because we know it does, we already did it before, this is pure, that's because it's red and so on. Okay, so now, why, how do we do this? Let's start talking about what's the problem on the finals that give rise. Look at that and have a look. Two diagrams of the same kind. It is very easy to describe because it is induced by something in Y and K.

45:00 Anyway, I gave you the results. This field Y, you can look into the picture that I put in the web to see what reality is there. This field Y, because it is induced by a component in Y and K, it seems that in H it is going to be covering Provided it contains all the objects, all the morphisms with domain in here, and I mean the fact that we are using locally connected and they generate and they connect, so we get the following lemma. Rule 5 is the purpose of the range, which doesn't mention components, for the property generated by the seas or the euston point, so the fine walls, the providing the fine walls, the sea, So that the supremum, our system of one, gives the whole of these. These are the seeds that come out. I mean, these are the seeds that come out, as well as the freedom, the freedom of this, and say, well, you know, forget about all this. Consider the focus of, this is a little bit of a distraction, as is the focus here generally. If it were not a connector, we would get the same thing. But here, we are not using components anywhere.

47:30 Alright, I mean, are we getting the right thing if we do this? Well, it is so simple. But it is. So, now it's not really. There's a complex spread. There are no conditions. A, A. One of those does... It should have been worse to make the view of the oligarchic theory, so now this is the oligarchic theory. I need to know that it's an oligarchic theory for those who know it. Obviously, everyone says everyone is an oligarchic theory, so forget the conditions. Actually, it's something that we call somewhere else a definable dominance. That the definables form a dominance in the sense, say, of Rossellini or Domenico. So the final was not false. What is the statement of the Hilbert sandwich? And that classifies the Hopper's term too. So now we have this, the only question now is then, what should this row be? Because the previous one was pure, we were being reconnected. And so the pure, even in the logical sense, would induce a digestion between the lines of... Complimented things here, complimented things there. But now, with everything changed, this must be more complicated. And in fact, this is what was more difficult to identify. So, we used the word hypercure to describe... So, I would just say, well, we have a unique partialization with an AA domain. And the role in Hilbert fuel. The idea of Hilbert fuel means the following. Suppose you have this role. What is the type of fuel?

50:00 But it's not just a matter of the direction between the definables. It's more than that. It has to be in the coverings. Of course, the way that Z has been defined is a little technical, but let me just be a little bit formal here and say that Reflects, but reflects not in such a diverse way. It really isn't creates either, but we still are in how to describe it informally. Formally I can do it in a minute, but... Reflects are definable, to be run, definable objects. I can see already that, yes, I can see already that it is easier to tell you the formal definition. So, it is the following. Halperture means, for every something of the form, but we want to say that this is a family of preferable sculptures and so on. So, this is, I use this little dot to indicate that I am wrong. But as a family, so that is indexed by some set A. In such a way, you can say, you can just have some, a single one. See, I'm making a picture, you know, well, let me put a one in here to make it simple. You can make it yourself by a family, but I'm making all that. So this is, this curve fits. In other words, it's a, this, I can think of this as... The co-product of some PAs can think of this as the co-product of CAs for any A single one, and so it is for use, not a product.

52:30 This is what defines the sub-object of some. But now, D over star of A is the subtraction mark for E, D over star of B is the format, in such a way that if this is and this is 2, such that rho of this, such that rho is not applied to 2, is 1. So, in other words, it counts. This is a more complicated situation, and hyper-pure implies pure, so the confidence is not true. For example, the inclusion of the positive reals into the topology sine curve by the function sine 1 over x is pure, but it is not, so it is not pure. There are other applications connected in place. So, with that, we have the theorem that everything with the volumes of arbitrary numbers can be factored in on this perfect binary square, but it is a very long one. It's a pretty good construction. And when everything is locally connected, it gives this 100% of the idea why and hopefully it's what we had before. So where are the quasi-components in all of this? That's the last thing I want to say.

55:00 So, because Michael had used a very contrived topology on the sets of quasi-components of the space. Let's look at the particular case where E is F is Z. So I'm applying this construction to F into Z, Z to the Z. And I look at this as defined by the topology on Arrhenius 2, I know, but I'm still the first. I use F, but it's Z, okay? So now what we have here is Z. So, it's VOA and here it's X. Now, this X must now be locating over the S. So, it's a surface of G on your power. So, I have here a hypercure followed by a complete spread. The point of the locale, where first of all we look at an x, is immediately that it is a zero-dimensional, a top-history. All of x consists of all the sub-objects of one, or an x, which are joints of both the negative ones. Which means that it is zero-dimensional. As I said, it is the point of the tag in the locale. Complementary mathematics, for one, is accessible by drawings, but that's a possible component, so the points of this are the possible components of that.

57:30 In other words, we have the micro-construction in a way that we would never have had in a video. So, there are many, many questions that arise on this, but I wait until you have them for me, if not, I can list six. Because this is new and, obviously, there are many things that are lost in the passage from the corrective component to just the logical aspect, so that's very interesting. Well, that is the... So, we have done a lot of work with mathematics. Now, doing this is a little out of power, but I think that there should be. This is the next step. That's one of my questions. Yes. The second is... In this situation that we had before, there is a topos classifying all of its spreads with locally connected domains. Is there a topos classifying all of its spreads with locally connected domains? My feeling is no, but, well, I mean, I have any other questions like that. But the question comes again, because Foxean stuff must have happened. I think we'll do it. So now it's a matter of fortune that this is a little bit too recent, but obviously. I think right now nobody can do it. Maybe you can.