Sites & coverages / classifying toposes, descent. groupoid representations 1 / coffeebreak conversations (contd.)

0:00 I would like to write about all the axioms, but the idea is that if you have something that has points, you have segments, the structure is different. I think that two objects, one is called points and the other is called segments. And for the second one, we're trying to get it into two points. The two seconds that we're getting at the end, they're oriented seconds. And for any two points, there is a second computer computer. And if you can't put the computer, you cannot explain it. So for any two points, if you have a second like this and a second like this, you can provide it. Provide it, ah, this is, you know, this is false. It doesn't get ahead. That's the introduction to algebra and mathematics, and that's what I'm going to do next. And it explains various things, but I also need a little bit of clarification. For example, the fact that you don't have to go through all the initials. But aside from that, if you know these types of pronunciations, there's an important conclusion about each one. I'm going to have to sort of tie you up. The fact is that, from what I hear, it's indicative of simplicity, and if you take it in re-magnation, you'll have a little extra structure.

2:30 From this first finalism, you can see that the re-magnation aspect is so important. I have these fundamental questions, and I think I should well put them in an order. So, if you do football, do you find, can you find a structure to stick around? Which is the question of all of them. Now, the question now, that is the logical answer, given the meaning of the question. And to prove this, the answer is written by E. We can write, we can view it as a structure. A structure is a linear structure. The answer is that, first of all, it's not the only meaning.

5:00 It's the main structure. There is another thing that you can often use, and that's a few of Markaya's amazing ideas. Is there a book of Macaio Reyes in the, however, you mentioned that there is, of course, Valerie O'Rourke also plays with the books in the next part of this house, where we go around to go to the Grand New York Mall of the Bulls and fight against the great girls and actually discuss. Is there a book of Macaio Reyes? Maybe. Okay. People were surprised at how easy it was to come up with these.

7:30 The co-determinants of both may not be much different. You look at a model called C-J, where C has five adrenals, log-couple is in J. It's a very different kind of definition. It looks more like a definition. Now, if C is a co-determinant of both, set by a sigma structure, then we have to prove that sigma structures in the non-particles E I agree that the solution is incomplete and that it corresponds to the universal sigma structure. Now, the conceptual conclusion is that if you only need the objectives set, then you need to use the sigma structure as a set, which is a much more familiar goal. The goal is the objectives that we need to use. It's a bit rough because you need the sigma structure to be a coherent object. So this is one of the... So the one thing I thought was exactly...

10:00 So, again, you could write a theory of string theory for a linear order, but it's only right for a number of reasons. So, there is, for example, C, because you can interpret it in the cohomology of categories, and the question is, is there a characteristic in the cohomology of C?

12:30 Now, the first level of what we run into is that this definition has a mathematical definition in it, and it really only makes sense if people are familiar with how to do mathematics. So, all of us have, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh. When does P have the property that any professional can use it? If you look at it in English, it deserves. I don't know whether there are any... Well, there are some of the predilection theories. I don't know if there are any of the fixed theories. I think that's an example I can use in my program. But I think it's useful here. The theory is that the following is the sequence of all of them, and there may be other conditions. So, you can use the preservation theories to find the answer.

15:00 Finally, we'll go over the logic. I don't know what the theory is. It's a novel. I'm not an expert. It's a novel. It's a theory, an infinite theory, that has historically interwoven with the magic. This is true, this is true in the quantum theory. It's a whole lot of exercise. Okay, good. Time to read, time to read. How much time do we have left? Explain what the categorical model is. What is the categorical model? I mean, there would be many interpretations of that. You just started with the first one. You have to pay for that. I'm going to mask it. I'm going to restrict myself to special time theories, which have been supported by a lot of other theories, but I'm not going to talk about them. I'm going to restrict myself to a couple of genetic theories. It seems to me that genetic theories are pretty normal.

17:30 The question follows, so, say, this is going to be very interesting, let's go here, go up, all four of the center and two of the four of the two, and then we, as a matter of fact, we're going to call it four and four, and all of that, five x, applies, so we jump to this notion of, if you just talk about five to eight quarters, which I will just define instead. This is just an inclusion of the object in question 5, but we can say that the object in question 5 is covered by the projection, by the feminine object, in terms of human color.

20:00 So this is the reason that this works. The international student shall build a track, which are called spin-t, spin-tactical-type-t, such that, well, it has, it has, it has, spin-t at five meters, spin-t at five meters, spin-t at five meters, spin-t at five meters, spin-t at five meters, spin-t Now, one of the opiates, phi of x, is phi coherent, but it is not the same expression.

22:30 The errors in this analysis are called coherently definable functions, no, coherently definable pre-suitable functions, and the errors are generated by the axioms, So what I mean by the final function is that if you have Y, let's assume that the sequence is not physical economics, and that the sequence is not variable function, then you can look at it as it is represented by a formula called X, Y, from here. So here is all the type and class, always appearing in terms of the function of the set of attributes set by the class, set by the class, set by the class. For example, for all that, this class is very very difficult. What class? Class. What class? Class. What class? What class? Chi and Rho represent the same error, where pi is pi over x, so we know that there exists a new type of chi, and there's also a new type of Rho, which has to be set by pi over x and chi over x.

25:00 And then the projections for an action are not so much. It's more like not to know than not to say. The predictions that we made of science and science are very, very, very, very, very, very, very, very, very, very, very, very, So, in my next lecture, I will take this as a common point and I will prove that there are special terms in mathematics theory, such as propositional theory, that are not very good at all.

27:30 And I will prove that every improvisational theory, every mathematical theory, has a conservative extension by propositional theories. It's not very common, right? But it's true. And another way of saying it, and I'll report this to our students, so the next class on the subject, from here on out, we're going to proceed with the intent of the 14th act. This is not a series that I know about, but it's a very good one. This is the IRCAM, and also organized by Moreno. Yes, yes, yes, yes. You know, I think the person who organized the one to pick on the mouse was, um, it's, it's François-Nicolas. Thank you for your attention.