Morning discussions / fragments of lunch conversations

5:00 At the same time, if you want to scroll up to the...

7:30 That's a very good one. The pizza keyboard on A is my favorite. That's a very good one. I recommend that. I think you might find it weird, it depends, but I'll tell you what, I'll get one as well, and if you don't think it's enough for two, then we'll split it between two and three, which will definitely be enough. So you'll get a key term? Yeah, it's my favourite anyway. No, I'm quite familiar. Rose? Can we not go for a little week? I'm very fond of Rose. Oh, up here. Yeah, yeah, fine. Which is far the pale last. Okay. Chocobo. Capitalism.

10:00 Capitalism. Exactly. And then half until his death. That's right. At the moment, it seems. I presume you also have to do house. I mean, have you finished?

12:30 Yes, I still have to do quite a bit in the house. I have to do a lot. I need to do a lot in the house still. I made it when I started, but I'm not quite sure, because we don't have the beads going back that far, and certainly until 30 years, you know, covered the world, which is why you've got those steps going up, because that was actually an alleyway, that was a public right, in a way. But they were paying interest. No, no, no, they must have put the... Yes, they did, that was that... ...beam, I mean, yes, that's why it was... This is the little road train for the tourists. They just have a kind of guided common train that takes you around from here.

15:00 And so you can define then exactly here which objects have these fibers and not by which. So in other words, it's easily done in the case of recessive graphs where you're saying that the, you know, map has its three fibers. No, it's, you know, it's just relativization. It's just, it's just relativization of the scale around it. And sure, I've found that every reflexive graph is sort of a... And so with the cook of this, you've got to do a series of questions like this, and you do this to the loop. You've got this one point and one arrow, which is not that. So the sheaves on that... Subtitles by the Amara.org community

17:30 Yeah, it's kind of like, I call it, I call it the base of D. The center face of D is two opposed arrows. You're getting actually the bipartite graphs in the usual sense. What is the... Bipartite graphs are those where the points can be divided into two parts, where all nine identity arrows go between the parts. I see. You never have any. Yeah, yeah, yeah. So you have these two discrete libraries. Yeah, yeah, yeah. Two parts are discrete over this space. Yeah, yeah, yeah, yeah. Yeah, so generalization is bifarcite, is bifarcite, is b equals a minus b, and that's the area that we've got. That's all pretty well spelled out. But then you see, hey, this, this, this, this is pretty simple, and I'm happy with it. It only supports my, my set of inputs, because, because how do you actually control this thing if it's another feature? You've got delta 1 plus B. You take that category and you simply force all the idempotents in it to be identities. And then the pre-sheaf on that is this B5 type thing. This is a case where it's kind of like, it's like when you take a general continuous map and you take the sheaf of germs of sections. This is like taking the sheaf of germs. Gruden-Dick talks, when Gruden-Dick's exercising, he talks about this small topos, you know, it could be both a subtopos as well as a quotient topos. All objects will be there.

20:00 And in his setup, it's always a subtopos, but it's a very special kind. It's a subtopos where the inclusion is essential, and even better, the left adjoint preserves products. It's not necessarily a left adjoint, it's one of the most important. So if it would happen that it was also left exact, then it would be a quotient map too, there's this smaller topos can be retracted by geometric morphisms in both directions, but the typical case is it's just this very special kind of subtopos, better than the central, whereas in my construction, you see, it's sort of coming out naturally as a quotient, it might not be... In other words, what I just said is an idea you can apply immediately to any pre-sheave topos. Take pre-sheaves on C. C is a small category. You take an example B, as such, and you have bold C slash B. We'll just kill all the interpotents, get another category, and call the pre-sheaves on that and the sheaves on B. This is very simple-minded and in some sense totally combinatorial. It's not necessarily clear that this should be considered as a paradigm, but in any case, there seems to be a lot of information in it, and it's interesting that we can get that effect right here on the interface. As I remarked there, it's very strange because you... The original category C, in the case of reflexive graphs, consists entirely of n-impotence. Well, that's good. You want the category of sets to come out, the discrete thing to come out over one point. If the sheave's on one point, it ought to be the discrete part of the big propolis. That's kind of a normalization condition. Any attempt to flush this out is unsatisfying. And you kill all the idempotents. You've got an awful lot left. You've got nothing with idempotents. You relativize, kill them. There's a lot left. It's kind of like this process of taking the triangles.

22:30 This is an aspect of what you had in mind in your remark in Boudoir about the story of idempotence having only just begun to be explored. How did you get into the pathway? Ah, through the gateway. In fact, just where we've come from, from that little square where you park the car, the gate is just behind you. The ticket office is just on the left and you just walk across that gate. So it's just round there? Yeah, you just go back through the gateway. You know, at the top of the road, the one we just walked through. No, I didn't. We drove. No, okay. Well, the one that you and I drove through this morning, when we drove to catch Mimi. And, you know, on the little square where you parked the car just now, just behind you is the gateway. And it's open, I think, until six o'clock every day. Yeah, it's well worth it. Who has the key? Oh, not you. Oh, good, okay. There we are. That's great. Okay, I'll see you later, Griff. God bless you. Thanks, Mimi. Thanks very much. We'll see you later. Yeah, see you. Well, will you come down here and join us? Yeah, that's great. Well, I should think probably around that time, actually. I should think probably about six o'clock. Lovely. Do you need... Yes, there you are. Of course. And if you give me the key, I'll just put the key back in the door because that's easily... That's the one place it's not going to get lost. So, we'll just leave it there. Okay, we're going to start about a quarter past, which, well, in fact, let's give ourselves five minutes. For some reason it didn't properly shut down. But the thing is, it does. I mean, it says to avoid this. Well, I did. That's what I did. But it's done this before. I don't know what's going on here at the moment. It runs every time it's scanning it. That's absolutely weird. But it did it this morning, but then it wasn't able to log on. What is it that we can't reconnect? I don't know. I don't think I've ever been to this. Yeah, this I also understand. Yeah, this notion in these setups of somehow defining groups of automorphisms,

25:00 there's something that's pulled up in our model of minimality. And this also triggers something, right? Yes, it does. They would trigger their own morphism and lose power. Yeah, yeah, yeah. Thank you very much for your attention and I look forward to hearing from you again soon. I have one, but you don't keep it, you just take the five, oh no, you need two. Does anybody have a five? I can show you why you're not on the ground. I should want to have you in an office on the Kediakic course here. It's one of the Kediakic courses. Well yes, in France my advantage is the Kediakic course, just at the top of the street, to the right. Oh, that's my advantage. I'll get it. Absolutely, I will take care of it. Thank you for your attention. Any covering of a line by regular open sets must necessarily have at least one infinite component. So there's sort of pushing problems off of infinity in that kind of a way. And that helps. I think, in fact, if you didn't have that, then this would not, this phenomenon would not occur. It's not just the students. Oh, it's what you're using.

27:30 Well, that's the amount of smoothness that remains in this law. Okay. Yes, that looks like a very interesting paper, actually. May I make a copy of that, Bill, at some stage? That one I definitely haven't seen. No, no, no, no, no, no, no, no, no. Okay, it's been mentioned that we were, this time, the session we were about to start now was to have been devoted to the topic of work between roughly 1950 and 1960, that is to say, examination of how his early work in functional analysis, related to his subsequent work in algebraic geometry and the expansion of his vision for the description of structures throughout mathematics. However, before we commence on that topic, it has been pointed out that in this morning's discussion we didn't say, we said very little about the role of Saunders-McLean and that it might be an idea just to have perhaps a brief treatment of that topic first. Who would like to lead off on the subject of Saunders-McLean? It was a rhetorical question Bill, it was a rhetorical question.