Discussions, incl. FW Lawvere & others incl PT Johnstone post- 1st talk — Examples of toposes

0:00 Thank you very much. Thank you for your attention. I invite you to keep your breath and let everything flow in the same way as you do now, and that way you're going to be able to do a lot of things, and then you'll be able to do a lot of things, and then you'll be able to do a lot of things, and then you'll be able to do a lot of things, and then you'll be able to do it. This is a mystical view of the world, a new one, and it's probably a new one, because it's a non-mathematical version of the question, to the fact that it is, it's not a new one, it's not a new one, it's not a new one, it's not a new one, it's not a new one, it's not a new one, it's not a new one, it's not a new one,

2:30 The conditions of the standardized part of the answers from the previous lecture are not the same as the actual part of the answers. Then, once you have the general knowledge of the answers, you can take advantage of it to find answers on your own. We hope that you guys have a good evening and I hope that you have a good evening and I hope that you have a good evening and I hope that you have a good evening and I hope that you have a good evening and I hope that you have a good evening The goal is to answer the questions that are asked in the lecture, which are required in the lecture, which are required in the lecture. So, the goal is to answer the questions that are asked in the lecture, which are required in the lecture, which are required in the lecture. And there is the same interpretation, but defined, or better, for example, by the scientists, not to interfere, you know, with time, and then you move it back, and you start to give it a break, you know, and now it's pretty bad. But the second thing is that, you know, there are some, you know, from the start, you know, you know, you know, you know, you know, you know, you know, you know, you know, you know,

5:00 So, you see, you have the right and the wrong, which is better than A, and the E is not equal to the E of A, which is better than the right, which is not equal to the E of A, which is better than the E of A, which is not equal to the E of A, which is better than the E of A, which is not equal to the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is better than the E of A, which is In the red line, we have a professor, in the red line, a professor, a professor, a professor, The three main subjects are physics, mathematics, geometry, algebra, mathematics, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, You see, each thing that happens in one phase of that course is a part of the idea of the future, which is a part of the idea of the past. All of these are important points to consider when you want to study mathematics, physics, and geometry. The quality of the terms defined is by now on this tool. You need to define it on this tool. You need to define it like a machine-measure. Like you do in your environment. And it's on this thing that you like to do. Like machine-measures. And you've got to create it. You've got to know how to do it. You need to create the equations, so to say. So it's not all... All of these terms are used in the course of the course of the course of the course of the course of

7:30 So basically, if you have two opposing maps, then you can look at it from a different point of view. So I'm just predicting that the map may be B. There are two typical ways in which you have, you have Cartesian products, and every man has a graph. You just take the graph, and you disturb the objects any time you shoot them on the graph, but the other man still does it. But he utilizes it by himself, he uses the code graph instead. So, you take A plus B, and then you have two co-grabbers that hold as long as they can, and they're all connected, so you've got a co-grabber that goes from A to B, B to A, B.

10:00 Yeah. Okay, that's right. Notice that it's basically encrypted. The surface line is equivalent, but the sum is being matched or connected with those B's. Yeah, yeah, yeah. So, so, so this factorization problem that I was asking for, the critical issue is the category of cool products, what does that have to do with it? No, no, I said that. Yeah, you have to. But mostly the results are, we've noticed that they've been rapid, sufficient sets, and I think that's something general. They've degenerated to something too complicated to do. The general description I've got is that in any, um, say, that could be a cool generator, you know, in any, um, There are no objects, particularly not, but it's implied by some other techniques that I've got, that the subject is a very reformed, regenerative family. I'm not particularly aware of many things on the way, but it's pretty general. There's plenty of things that the person may never want, so we just sort of want everything. And this obviously extends to that. It doesn't, it doesn't either extend or be extended by the graphic example where it sort of, you know, regenerates in non-graphic software, so it's quite, couldn't quite see any, it couldn't quite be exactly prepared in that way.

12:30 I don't actually, I don't actually know a nice description of the graphic topos, you know, the graphic, no graphic category, but a bit of, like, surjection, surjection. Well, I'm happy to report that we actually used a graphic moment like this. We have a very important use in the well-established field of mathematics that I just found out about by a different name, which is something that's been studied for years by common-tourist, common-grain, common-tourist, hyper-claim arrangements. But it turns out that that's an example of such a graphic moment. I thought I was just, you know, looking at it. I just found that out. I haven't done anything with it, but at least it's encouraged me. I wanted to start with that. I thought that for me was something like this, just looking on this and looking at the computer. Yes, yes, yes. Then I thought maybe I should check my books and see who's writing the books for me. I was going to check that for you. Well, there's a lot more about the practical purpose of other people. Thank you for your attention.

15:00 If I'm saying something, I don't want anybody to pick it up. Yes, yes. Richard Wood. Thank you. You're welcome. Thank you.