Evening Talk (contd.)

0:00 Richard was saying to me tonight that he particularly wanted to ask you about, to do with your space and quantity paper. He said about Newton, he said that when he read it to him, he thought about those eight and nine students. In addition to their armor, wielded these enormously heavy axes. Really? Really? It's a, it's, it's, you, it's, you pull, I think you pull, it's a light, it's a pull switch, it's a thing, a pull switch. Actually, I have a suggestion to make, I don't know if anybody else would, uh, want to take me up on it. There are four of us, I don't know about you, I'm absolutely exhausted, I can, um, if we go for a taxi, do you like to go back in a taxi and split it four ways? It will only work out about a pound and fifty. I don't know, I mean, if you feel fresh and you still want to walk. No, no, no, no, it's just a problem. Well, it wouldn't be more than two pounds of those, would it? No taxi in Bristol can find this house. Well, if we stand out on the street, they should. But you can just walk down the street. Okay. You know where we've gone to get taxis? I've taken you by there many times. I normally would. I'm so tired at the moment, I can hardly remember. I have no name. Walk down past the restaurant on the other side of the street. Well, go down where we've just come from. And then into the next block. and it's halfway along and it's a taxi okay you mean going down towards the um towards the triangle because we're going that way as we'll walk anyway because by the time we get to triangle we'll be halfway back no it's just it's literally three minutes walk down the street here yeah but which street there if we're on the junction of four streets i'm sorry i'm i'm sorry john i'm just too tired i can't think you have to go down the way in the direction of the restaurant where we just came from okay that's fine this side of the street rather than on the restaurant side Okay, let's think about doing that. Oh, I know, you mean it's the little taxi. No, I know what he's talking about. It's a cab fur. Yes, I was thinking he meant a cab rag. I do know where John means by that. Well, I'm in favour of that if it's okay with you. Oh, yes, yes, for me. Because I doubt whether it's going to be more than £1,000 or £2,000. And I don't know about you, I'm just feeling so tired now, I could really deal with turning in sleep. Okay.

2:30 Thanks, John. Thanks for the coffee, and thanks for a nice evening. But be careful, there's a... There's also an outjoined tensor in that. Really a category of topo-coulombs. And any topos you can tensor them, that's a topo. What the tensor actually is, is all pointers from another topo. The construction of topos is not very often mentioned. Yeah, yeah. It's pretty clear, it's quite a Jiro definition, preserved both modite limits and armature covenants and stuff. All these smugglers satisfy the same populations as in the co-domain, but with a given topos. So, it's again a topos. Ah, right. Ah, okay, okay. And in particular, there's a kind of... I call it Fourier transform, but maybe not, given any topos, you take its set value models, which is a category with filtered co-limits, now you take the filtered co-limits preserving punctures from that into sets again, you get a topos that maps to the original one. It's okay, it's going the right way, just cross that, that's how we cross every time we walk this way. Ah, that's very good, but is this construction mentioned somewhere in the literature? Well, I think I mentioned it in some of my other papers. Not in any books that I knew of. Ah, okay. Now that's very impressive because actually, yes, one is away. And then the properties of the resulting transformation between topos has been studied or not?

5:00 Well, studied a little bit in the sense that if you have a coherent topos to begin with, then the set value models are somehow sufficient. And therefore this could actually be an equivalence. You can drop us on the next corner. You can drop us on the next corner. Just round here on the left. If you just take us down to the bottom of the street there, that would be fine. It's just along there to the left. It's just along there to the left. Yeah, thank you very much. Hang on, no, no, this is, no, no, it's okay, just to get out, it's okay, we'll get out here. No, it's that way, we should have, yeah, we'll get out here. Yeah, what's that? Oh, that's very reasonable, it's only... There's one after each, I can do, yeah. It's okay if I can, yeah, not at all. 520, and he operated on it. Yeah, he's doing something. He's a crook, but there's no point in having a row with him. No, no, no, of course. Let me get this and then we can settle out with, okay? Right. So how much? 620, okay. Can you give me a change? Thanks. I can give you £1.20. Yes, hang on a second. Yes, I can give you £1.20. No problem. Hang on. It's okay. I can give you £1.20. You don't happen to have a pound, do you, Olivia? I have a pound. Yeah, that's okay, because then he can give me a change. That's very good. There you go. Thank you very much. Thank you. Thank you very much. Cheers. I think you are going to want three. Yeah, that's right. No, that's enough. No, no, not three, because it's only £1.50. No, no, no. Thank you. That's right. Yes. Cheers. It's okay. It's all taken care of. That's okay. Well, thanks very much. Cheers. That's great, thanks a lot. It's that way, isn't it? No, it's that way. It takes us back.

7:30 It was only £6 altogether, so it's £1.50. Well, £6.20. Of course you can. Well, of course you can, yes, heavens. You helped me out a few times already. Yeah, it's just down at the bottom there. Okay. Well, thanks for agreeing to that, because I really did feel in need of getting into bed. Did you not mention something... No, it's just a fascinating idea of a two-categorical generalization of set theory in that way. That's something that could be stated in a certain way, you know, I don't know if it's in the sea.

10:00 Yeah, just in terms of the balance. But it's an essence, you see, you can see it's balanced, evident. Hmm. Could you take us through that on the board tomorrow? That's another question, yes. And it's spinning tomorrow? It's spinning tomorrow, yes. Yes, I was going to just ask you, isn't this the description, I was just going to say, isn't this the, isn't this the construction that you hinted at or, well, you did more than hinted at, but you outlined the idea very briefly, I think, in the 70s, it was in the Bristol Logic paper, I think, isn't it? Yes, that's right. Yes, it is, so very appropriate. Yes. Only, you actually describe them as the further enlargement of the idea of a domain of variation, but one which is determined by two, by a relation between, no, a map between topos, rather than by a single... Yeah, as an example, given in a geometric morphism, the image as computed in this larger two category. Yeah. Oh, okay. Yes. You don't actually mention it being the kind of two category of... The mass between topos is, but it is the idea. I can see the idea. I was just going to ask you, is that the same idea you referred to in the Bristol colloquium paper? Yeah. The two papers I most wanted to ask you about, Ashley, during this... Those two, unfortunately, are not. I've been nagging Bill to make those the two that he adds next, if he considers.

12:30 Yeah. Bristol colloquium. Yeah, I think so. Yes, it is. I recall it. I recall it. I was hesitated to revive those two because there were some wrong ideas. You didn't mention the filtered co-limits in the construction, but it is the same construction, and I'm sure that you're referring to that. I must have mentioned it, so it's crucial. Well, yes. Okay, maybe I just forgot that part, but I do recall it being in that paper. And the Eilenberg-Feschrich paper, too. I realize, because they're so full of wonderful ideas, that there's bound to be a few that you've subsequently refined. Well, yes. It would be great actually to have them rewritten with the same sort of introduction that you did for the TAC version of your thesis. I think that would be a great project. Yes, yes. That's a possible... That would be a really... ...instead of putting them on my website. Yes, yes. And with that sort of extended introduction, looking back on what's happened since. There's demand right now, here. The demand is concentrated right here in this like quantity. Yes, you can safely, we will, we can also. You see, like any good quantity it has a support and you're looking there and the support is standing right here in this and that and that little bit of space over there. Good night, Olivia. Thank you for... Oh, yes, yes, yes, thank you, yeah. So see you at 9.30? Yeah. That's good. That's fantastic. Thanks for a great evening. It's lovely. Thanks a lot. Thank you. You see, there definitely is support in more than one sense for the idea. Let's work on that. Thank you both for an absolutely brilliant day. Oh, I have to go. Yes, you can't get in that way at night.