Conversations incl M Wright, FW Lawvere et al on stasis & identity maps

0:00 So, um, to hear the last talk, I've got to mention the origin of this unique lifting of factorization. The paper originally written for the 60th birthday of Walter Noll, with the, uh, doing has to do, in a way, with relativity of different parallel time flows, generally. In fact, for me, it was just, the only way I could do something was to assume this very strong condition, but as you see in Marta and so on, they have shown, in a way, it's necessary. Yeah, yeah, yeah. I'm afraid that's a paper of yours. I'm not studied. I ought to do so, particularly since the motivation. I must have it. I'm pretty sure I do have it. The one that starts off with the example about the bus stops in Greece. I remember. Is that the walls or not? Yeah. You might use the same problem. Well, I don't. But I can see now where the interval of living comes from. Well, do tell people. What you thought of it. It's really interesting way of thinking of it, Jack. I mean, he seemed to assume that it was, it was sort of well-known. I didn't see it. Well, I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know.

12:30 I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I didn't really know what his notation was, he had come to a change of chronological algebra, any at all outside it. The triangle thing just being, so I'd not come across that notation, but I'd come across a comparable notation for the orthogonal as in the way you have the little... What is this orthogonality that you're saying? So, you've got two morphisms, you say that they're orthogonal, if whenever you've got... They're parallel morphisms or arbitrary? No, arbitrary morphisms. They're orthogonal if whenever you've got a square with one of your morphisms on the left and the other on the right. Yeah, parallel to each other, going down the left and down the right. Then you can fill in the diagonals as well. So for instance, a typical example is epi and mono. Yeah, that's the classic example. Exactly. It's just choice. If you've got, um, I don't think that's choice. Well, no, but epi-mono factorization gives you choice, no? It's equivalent to choice, epi-mono factorization. No, no, no. In sets? Yes. It's not. Epi-mono factorization you don't need. Well, you've got that in a topos. Um, ah, yes, no, true, so no, no, no, no, you're right, no. No, you're right, it doesn't actually imply choices, it's slightly weaker. Certainly the other way around, choice certainly implies every monofactorization, yes. I'm sorry, that's what I should have said. For any commuting square with your first thing on the left and your second thing on the right, you can fill in the diagonal. So if you have a commuting square, you just compose the arrows, obviously. The other diagonal. Sorry, the commuting square goes like this. Oh, I see. Sorry. Sorry, yes. Okay, right. Yeah, I see. And those are the morphs that are orthogonal. Yeah, you're saying morphs are orthogonal. And so then, similarly, you can say two classes of morphs are orthogonal if everything on the left hand side is orthogonal to everything on the right hand side.

15:00 Does that coincide with more conventional motions of orthogonality? Not in any particularly specific way. Why do they call it that? Is that the only sort of geometrical interpretation of that term? Yes, because you get... Yes, it is, insofar as the way you use it, insofar as the way you use it, it's more of a confluence in Bechtel's other products or something, because you use it generally to get a Galois connection on classical morphisms. So I can take this classical morphism and then look at everything that's orthogonal to all of those. And then we can look at everything from the fourth problem to all of those, which will include the original morphism of the first and the general morphism of the whole. And so that's sort of what he's looking at with these clusters and these others. And I think he was thinking about Galois connection that actually suggested this notion in the first place. I'm so sorry, Michael Wright. Very nice to meet you. I think we have just met, as it were, just to kind of say hello to each other at one of the, one sort of previous category. But yeah, I'm pretty sure that it was, on the historic bit, that it was precisely thinking about Galois connections as, you know, the very early, well, really, about the time the Adjoin-Planck theorem was discovered, when people were thinking a lot about Galois connections, that suggested this way of thinking about fill-in, you know, filling in diagrams from the diagonal. Well, me too. Of course, I'm just a historian anyway, as you know, not a working mathematician. It makes it all the more fascinating, too. It should be good.