2nd talk / contributed paper / contributed paper

0:00 And you can tell it's recording, the echoing light which indicates the sound. The most tricky thing is to say that this function is unidirectional. So you say that if f of s is e and f of s is e-prime, then e is equal to e-prime. And what do you write this? Well, you say it's true if e is equal to e-prime, and it's false if e is not equal to e-prime. Now, you have to recognize that the soup, the soup of the propositions follows through of the set, the subset of what is empty if it is difficult to define and is not if it is equally defined. There is somewhere in between if it does not. But this is not finite, it is not infinite, it is a set. So the axioms, for example, and the other one... I almost said that it's a subjection. You said it's true advice for anything that holds that subjection.

2:30 For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. For anything that holds that subjection. So it has come up in various forms, in different ways, but you can put yourself inside a topos and do this for any two objects in a topos. So, um, I mean, so, it has five topos. Bt would be a topos, and bt would correspond to copies in E from S to E. Did you want to say that that was every word that Fonker believed at all? Yeah, he did. He did? Thank you for your attention.

5:00 There is no mountain in a strong sense, so the compost is non-territorial in a strong sense. Even if an element of E perhaps is much bigger than the rest, but it corresponds to the number of carbon-collapsing steps here, for example. You can always move to another number of steps here, even if the carbon-collapsing steps are different. So we can do this. S will be of the form, S will be of the form B, but that's the canonical kinematics indexed by the toposkin of B, upper star of S0, and our BP, to emphasize that we're working over D, will be the topo of the topos over D, i.e. there is a map, D, D, P, over D, and that will have to be expressed by geometric morphisms between topos all over D, so it's complicated.

7:30 The people and the diagrams themselves, all diagrams, and then what's left, what I want to be sure is that the port doesn't get any larger than that, right? I want to show that when I sheepify this, if the port remains no larger, if it gets no bigger, that means I want to look at what maps do I have from here to here. And what I want to show you, this is math, that m will have to be greater than or equal to m. Okay, now, let's make that, before I do that, let's pull back here and call this thing, this is the suboptimum here, and let me assume now that it's not m. What are the non-trivial partial maps from h m into h m when m is less than m? I want to get those right, so what I want is the fact, no non-trivial, p of m. Now I know what this means. It means there's going to be a H cube here, and I have something like this, and so let me now write down a relevant property over here that moves us out. If you have Q to M to N, where N is what, less than N, then what I want to prove is, in fact, I'm going to do this.

10:00 I don't have to go very far. I want to show that there's a pair of maps here I'm calling F and Q. Going from Q plus 1 to Q by F is equal to this thing down here. But the same thing here. That is, it would go into the same thing here. Q plus 1 by F into Q into N is not equal. I'm not finished yet. I've got to tell you that. That graph, I'll tell you, is that H-M and therefore C for the patients embed into the logitions that they agree on. It suffices to show that H-M embeds into the plot here. If the maps from H-M and 2 jointly are jumping on it, it's clear I don't need more than a hundred numbers. And this gets right back to this point. I need two maps coming into H-M and that one here. Given, see, two different maps in the H-M. Well, I wanted a map into two that distinguishes finding the H-M and a single map into here, so that these are the same, and then use this fact that having the images of these two maps are at a point, therefore I have a partial map from H-M into two that separates those two things. Now, sheetify, once you sheetify every non-zero thing, it's objective.

12:30 All of these allow you to look at mathematics as something that can be referenced out in the public because it's easy to do on the main category, which turns out to be remarkably difficult. What I should say, however, is that there are some game-changes right here on the system. A few of them fill asleep properties on the category. The objects are the non-zero orbitals, and then these three things I wrote here. There's a little topological proof that tells you that you can prove that the number of maps is m minus one. That's m minus n. In Presto, in this example, it's exactly equal to that. But when you write something like this, one's very tempted to raise the up and down signs and say, let's take the sorry little category, abstract category now, the objects will be non-zero natural numbers, the maps will be strings of natural numbers. This is going, by the way, at a string of length from m to m. We'll be a string of length m minus m of natural numbers less than m, right, 0, 2. By catenating, that doesn't work because you're not, you might have things that are too large here. Well, frankly, the proof does prove that there are differences. Yes, there is. For example, you can show this, show that every object has a finite order, sorry, has a total order. And in the other one, you can show that there's a countable power of two, of two element sets.

15:00 So this time around, if you get that, each piece of J will have just two volumes, both of which you can fill inside, and it will tell you the case of the product. You have all these things, and when you're ordering on it, then you can pick the least choice out of them. It was my experience a long time ago, and what I came up with there was the common language cycles that are already done. Finally, though, and out of all things, the very first example I have, but this was not quite the first one. I mentioned two types of conversations with my example of that is, but the example I had actually was one more, and that's the model, so it's harder to explain what was going on. So that example was nice in that, you could say, you have a low order axiom, for example. Every low order family does not have to have a place. What does that mean? Translates from the following. So, everybody knows the projective, Friday, Friday, Friday, Friday, Friday, Friday. D is projective. I don't mean the full string, but D is the technology. Projective, as I said, you all know that that means that all the, at least, characters will be split, but let me describe it as follows. In every relation... Entire is the same as total, but you have a relation from the P to wherever else on the x. The relation is the same on a pair of maths. Entire is the same on a pair of maths. The entire is the same on a pair of maths. The entire is the same on a pair of maths. Let me say that C is choice if every relation to C contains that.

17:30 But it is certainly equivalent to the existence of what a trace function is in a practical sense. It is a universal entire relation targeted at the state. Namely, you take the power object of C and you cut it down to the non-energy things. And this thing has a function, contains a function, and everything does. A function contains a variable here, so let's do this. A couple of games you can play here. We're back on the choice here. The first example I have was an example with choice-inclined projection, the other way around. Projection-inclined choice. I'm not going to start with that. You didn't have projection choice. But you have, one way of stating it is that if you have a one-word example, it's not a defense. That seems to be new in science right now. Thanks. This was the thing, the interpretation of that, to the P-upper star of U of P. That's the subject of form X. So if you forget that second part, you get the model of things. This is called the model of P. So the model of P is the P-upper star of a model, of the universal model. Okay? I will try not to use the R0 or M blackboard. Also, to relate this situation to the previous example,

20:00 The key terms could be constructed in terms of t, a lot of this would be t, in terms of t, now it is the same thing as t inside of t, or my three-double plan, but my three-double plan, the theory, again the proposition of theory, is levels between two sets. The role of s is now played by gamma of bp, which was gamma of s, and the role of e is going to be played by bp, which is given by a propositional theory in the absolute sense of s. It's also given by a much simpler propositional theory in the relative sense, seen from bp. How could we see just a single object? Is that because you assume there's a single source? Yes. If not, you refine the theory in such a way that you use predicates to separate them out. Or you do a variant of this where you have lots of surjections, lots of restores.

22:30 Now I will make a little jump. By specifying purposes, the position of theories is the same thing as purposes of the form, changeable in power. Well, you could take as a definition of what is called the locale topos, the topos, which is the four sheaths on the side, where the sides are such that they're partially overset. And if you have a compositional theory, then this category sin t is such that the category will just be partially overset. There are many different ways of defining it. It's very much like shearing space, except that it's much better behaved if you work inside the topos. And also, you can see that it's a little different. When we look at this example of the locale of surjection, because we started out with the theory of surjections from S to E, and it should become known that E is much bigger than that of S, the space of the locale is more than that. The propositional theory is consistent. The locale is more trivial. So this is 50% of the propositional theory which is given by a locale in set, and this is also given by a propositional theory which is given by a locale in VP. And the fact that these junctions occurring in this theory are nomadic or inhabitant means that this thing is open subjection.

25:00 Now I translate everything into locales in vp, so that it's a locale inside the cohomals, and y in steps, so that the triangle, the diagram, this diagram, is the forward diagram. You have sheaves on the locale y, so that this equation, sheaves on the locale y, is the same as sheaves that are less than 3, and that will match with vp, and that will be p. Now, this is a sheet of X inside the sheet of X. I use very much one of the Alton-type contributory topos, but it's also used as a universe in which the group is guessed by a topo. I proved that this group, which was originally proved by Diokonescu working directly with, in this form, in the memoir I'm showing on TV,

27:30 The spirit. Spirit is probably an open subjection. She's of y, t, whereover she's of y is the same as she is inside e on x. And now if we say this is an open subjection, in terms of locale, t, it means that the locale went from x to the y subjection. So now this is purely between locale and t. I'm going to discuss a little bit of open maps for mathematics, possibly for emphasis, possibly in a fixed situation, axiomatically.

30:00 So I'm not going to give you a definition of open maps, but I'm going to give you a few properties. Axiomal prisms are open, so we have a subcategory comparing all axiomal prisms identities, where in a fullback, B, Y cross B, X, if X to B is open, so is Y cross B, X to Y... And all of my students have repeated the open subjections, and I reflected down the human open subjections, reflections about quotients, which we have seen in some form, quotients, subjections, y to x, is the time equalizer, yxy, yx, but it's kind of there, but I'll go...

32:30 This kind of equalizer, the universal, I think, deserves our full back. I can't remember what it's called. You know, X, the back of the geophane. So the whole value of this over B, in fact, is calculated as this over B, and I'm going to assume that there's A, and I get something like this, but you can write it in a simple form, and I can't have colors from it, like I said. And also, it is an equivalence relation of locales, as well.

35:00 All of these properties are enough to develop the descent theory, and in a very easy way, as I will show you. So, you can write a class of maths. Which you could call all of them up, and they have these properties that you can use for this lecture. But the fact is that these properties aren't as accurate as they were in previous. So I think none of this occurs, again, in the same place as we were in Latvian, and none of it occurs in the same place as we find in the Moribund ribbons for academic groups you'll find in the library. The purpose of sheaves in the locale is sheaves in the locale X. E to X, from the store, and these, locale maps, set E to X, open, hand, open line, this one is open as well, this is an entire map, we're only looking, we're only looking for a sheet, and I have to descend, and, and, locales over B, some locales, F sharp, to locales over X, and I want to somehow characterize the image of this, if you would give the locale over X,

37:30 If you're given a locale p over x, can you descend it to b? Descend it so that it goes forward or something like that. If this is the case, then p should be constant along the primes of this line here. And these constants here are expressed by some action. So, p over p to x in the image have an action, a constant,

40:00 So, what theta draws is the following. So, theta x e, so here is x and here is e, this is x, here is a little point x, here is a little y, over which you are given e, and they are over the same point b and b, which is pulled back here, which is b. You can move this b two points over x. So it's also, if you're a differential geometrist, it's also a little bit like a connection, and you can move the horizontal path. And this connection, if you want to keep it as a rule, it should be flat, so it should satisfy something like this. If you start with D over Z, then you move it to Y. We've got this point, z over fe, and then we move this z-point to x, which is not good here, and that should be equal to x times x times e, that should be the same as moving the last step from z to x. So I've solved that, z e is e f e over z.

42:30 So this is for the unit condition, and this is for the sensitivity condition. By defining categories, p over x, theta, and with respect, the two subs, e to the 5, we find the outward x in the diagram involving theta and theta to the 5, and then we want to find the equation of theta to the 5. Again, from the mentioned reference of the general theorem, the approach of an open subjection of a star, the term f of a star, induces Categories, locality, so if you say that it's the way you use that, it helps to have an initiative to try to come up with a bunch of stories. We can look at locality X, we can look at locality B, we can look at locality over X, we can look at shop, we can look at equipment, and then we can look at... I'm thinking about saying that, which you can easily figure out for yourself, that if you thought that there was a count of B, then there is a clinical substation.

45:00 I want to emphasize that it's really completely based on these actions for open maps, these properties of open maps, the particular direction of proportions. So we can look at this proof very closely, which is, I hope, is, for example, Fulham-X, for Solana-Rensselaer. I'm going to just jump now, but then we'll try it. It's going to be very smooth, because b is b, 3 times a b.

47:30 And you can see we have a pullback. It's going to be b, x equals 3, 3 prime, with a projection. You have problems like X equals B, X equals B, B, X equals B, X equals B prime, X equals B, and here are 2, 3, and 5, 1, 2, 3, and then a G, and now the star, or F, B, F, B prime. Thank you for your attention.

50:00 Okay, so you have these rows and these lots of pullbacks and multiple subjections. So the rows are equalizes. Rows are called equalizes. So if we combine these two facts, and that g is in the sketch we're going to get, and f is the number that makes these two squares commute, and these rows are called equalizes, any such g that makes these two squares commute will factor uniquely here. So, g factors it uniquely. That's all nice and easy, but when you prove that the assumption is subjective, it goes as follows. Theta in the center of x, and cone in the right position on the left of the right, x goes to e. Theta at pi 2 to e is the product of an ultimate subjection from x to b. Theta is very observable. All of these maps are open suggestions, so we can form the co-equalizer.

52:30 It fits over B because everything fits over B. If you like, you can also grab this other co-equalizer. This is over here, this is over here, you have two different diagrams. I hope I hold them by B, so here you get that. So Q is over B because we co-equalize it. There is a map from E to X cos BQ that's already in the diagram, so let me give this map a name, so this is Q, and this is F, and this is E to X cos BQ. In this rotation, it is H to the left. So now, to factor that in the opposite direction, this would be Q.

55:00 Well, you know that Q is a co-equalizer, and it remains a co-equalizer if you cross it with X, because it's stable. And then it becomes easy to construct a vector out of it. X cross B, X cross B, E. X cross B, E. X cross B, E. You can check the design of the wheels, possibly replacing this 1 by 2. So I'm using the topological Q. You get this method just with diagonal checks. The theta bar and h2 are mutually inverse by diagonal checks. You can actually execute it. Using 5 on 3 here, I thought I could increase the number, and then we missed out. So when it actually comes to 4. So this is the end of the proof. So it's easy to let me default the most. Show the suggestions. It's an open subject. It's an open subject. Okay, you have exactly the same proof. Completely. So this proof works in any category, whether you have a class or course.

57:30 What are the open suggestions? There can also be suggestions that Schar also induces an equivalence to the category of schism X, which is the equivalent standard. This is just proof. They are preserved on Kubeck and reflected on Kubeck in the open-surjection formulas of these two lectures. In my next lecture, I will start reinterpreting this and combining it with the theory about the open-surjection from a locale, reinterpreting this theory as a statement that every topos is equivalent to a locale. And then I will look at some variations on that.

1:00:00 The committee will allow me to speak here. Unfortunately, the set-free proof of existence and its junction properties will not work on a free talk, since the set completion of the talk may not be a talk in general. This was the second question Sartov was telling us about earlier, but Carbone was interested in finding a proof of Tiber at Kultas earlier without using Ruin over Ceres. I'll tell you a little bit more about the proof of cause and the structure of properties without using going over sets. So I'm going to run the same work. Again, I will tell you a little bit more later, but for reasons that I'll tell you later, it doesn't work. There are exact completions of topos that are a little bit similar to before.

1:02:30 You have the very big categories, you have forgetful time terms. There is a very big category of categories of finite limits, and that has now left the joint, but given a category of finite limits, provides you with a category inside it. I think that the joint exists for very general reasons, but we're going from the very, very explicit of finite tree construction, for the exact completion of a category, the embedding of a finite limit C into its exact completion. Full on Facebook can be characterized as deprojectics and the existence of disjunctions and you also know it very narrowed my problem without worrying about its implications. In 1995, let me tell you all that was known then about the problem. If you apply the exact completion to an exact category, you don't get the same category unless every F is placed.

1:05:00 If an exact category is your choice, then the exact completion is the same. In whichever area it splits, you exact complete it, you get a topos. If you don't exact complete that, you really, what you have done is adding all co-limits to your original category, so you get a pre-shift topos. G-sets are essentially small class of intercomposable objects. If you exact complete G-sets, pre-shifts on intercomposable is that you exact... There are some fancy graphs, and Karboni introduced this to Loewe here, who noticed that if you take the possible reflection of that category, you get a proper class. You get a non-Woodian topos, which is exactly what Woodian is.

1:07:30 Let me tell you about that.

1:10:00 Of course, this is far from definitive, because in particular cases it might not be simple to check whether topos has a generic proof or not. Say, we don't know if the effective topos has one. It was a very eerie conference, so I was very nervous and unhappy about it. I gave a talk, and then Peroni, whom I had never met before, approached me, and before he said... With me too, he said, tell me if the exact conclusion of the Serbian skitopos is a topos. He walked away. What I thought, and I'm curious about it, is that if you take a slice and you possibly reflect it, it's always small. So the argument here, that just here doesn't work, the exact conclusion of the Serbian skitopos is not a topos. I informed the number of people that were there at the conference. At my talk, I told them about him. Professor Witten answered, and he said, that's very nice, and the rest of the pre-ship talk was a total of empty things.

1:12:30 And he said, he also said that of possible interest on him, to see, as mentioned before, it's an essentially small category, and they noticed that the group was equally, essentially, the compulsive logics pre-ships on C. It's essentially small and these suggestions are turned out to be right because put your hands into the bridges and move around and this is equivalent to two carbonics problems. This is the bridge topos, which are topos. I decided to forget about it for a while. Sir, this is the topos. I didn't think about it. I let it rest. Okay, that's for bridges.

1:15:00 But once you put the topologists in here, and again, he was mostly interested in realizability, but he, of course, was annoyed by this fact that we still don't have a normal example. Also, well, one doesn't find exact completion to realizability, so it's important to carry on the research. He found out, he first observed the existence of smaller products. It is not unclear that he preserves the existence of a bound. So he suggested that it would be interesting to consider which would be, for a bounded, he did a bit of an arbitrary base, but let's retreat here to this, over Grotendieck topos whose exact completions, they have to have a bound. And he then proved, slightly simplifies,

1:17:30 The problem, right, because now the bound allows you to perform some new calculations, and so it suppresses me a lot. You notice that if E is the exact equation, if and only if E is atomic. Unfortunately, Peter explained to us during the last 12 hours what it means. The impression that one has is that this works because being locally connected makes E very similar to openly connected.

1:20:00 What position is a topos? There is no value. And let me state it. Before we state it, the difference between the composable, we already said it, you cannot, and it is an atom, if it has no entity, it doesn't have any other sub-object out of it.