Discussions, incl. FW Lawvere, C McLarty, L Corry, M Wright (contd.)

0:00 You can ask your technique about this question of sectional gravity, the point we're making just now is whether it's conditional or not. If you only have a connection, and then you take a plane through this space, well, that connection tells you how to do parallel transport of vectors. But that parallel transport is likely to take a tangent vector in the plane. You're going to transport a longer curve in the plane, a tangent vector in the plane, and that connection is likely to rotate an outer plane. So that connection is not in the first instance. It's a connection on the plane. It doesn't give you power or transport in that plane. You think, well, we'll transport it, and if it comes out, we'll project it back, but a connection doesn't give you orthogonal projection. It doesn't tell you how to project it back. Now, again, it's possible that when you work it all out, it does the first order. I just don't know. But on the face of it, it's not obvious that it connects on the ambient. Whereas a metric, it gives you a length for every vector. If it doesn't have a space, well, then let them both be on the plane. It's a metric on the plane. To a little closer approximation, a connection tells you what the geodesics are, but the geodesics may not stay in the plane. So you don't know what are geodesics on the plane. Now again, if you have a metric, you start over again. You don't take the geodesics and project them. You use the metric and use metric and get the geodesics here. But if you don't have a connection or geodesics or something on the plane, you don't know what the curvature is on that plane. If you're going to try to look at the total curvature of a point by looking at the sexual curvature on all planes through it, I'm not sure you can do that with a connection. Maybe you can. So, as you say, you can answer questions sectionally as well. Yeah, a metric gives spectral curvatures. It's a very useful way to approach it.

2:30 Yes, I see the point. Are you out? You're deemed to have. Projecting. Yeah. Projecting. The fact that orthogonality is a weaker structure, orthogonality is a weaker structure than the method. It's weaker than the method. I mean, my approximation is always, it's a metric up to scalar multiplication. It's interesting to think of these things. They're both structures on the category. In other words, you can think of the vector space with a metric, of course, but then the natural morphisms are, you know... In that context people consider arbitrary linear maps as more present. So really what it means is that you up the equivalence. You've got a category that's as equivalent as a category to the category of vector cases, but it has an extra structure. Namely, to every map, a reverse map.

5:00 Satisfying. I find it hard to think of it that way, but every time I have thought of it, I come back to it. We're quite accustomed to the idea that a set of products might be an additional structure on a category, or that a given modem adds an additional structure to it. On the other hand, so often things are intrinsic to the category that you forget that sometimes there's an additional structure needed on the category. So really, the category prevalence of the category is actually a little less prevalent. It's kind of like a C-star category, because there's a star operating on the maps. What's an involution of interchanges of main and full-main, from to where it's an isomorphism of a category with its opposite, which preserves its identities. Then you see that the metric aspect comes out of that because if you take f composed of that star, or f star composed of f, either one will have the same trace. That trace is the... In general, you could take F and G star to get to the inner product, to do things as the trace of F star G. Now you say, what's trace? Well, there's an intrinsic theory. What is trace? Well, it's like Kate there, you know. There's a universal object which assigns to everything. It's the co-in. The science of every anamorphism, you've got this thing, okay, and through every anamorphism you assign a norm to that thing, but in such a way that if you have any two opposed maps, then the two anamorphisms that you get from it could have the same value. So it's universal with respect to that. And that makes sense in any case. If you're talking about linear categories, you would want this to be a linear correspondence, if you could take it also away.

7:30 The key point is that opposed, given any two proposed maps, the composites would have the same trace. And we get this equivalence relation on the but, the oh, and the no. That's where the inner product has its value. It's good and enormous in the world, but it has to be done. This discussion should go on in every tangent space in particular, just in my review across them. Moreover, have a factorization in the category. You know, take the star of the epic and the star of the body and look at the details of that. You have to, you can deduce the concept of our side analysis. We certainly are postulating it to some degree. And we'll show you the same thing. I'm just impressed you've come across a mathematical physicist who actually knew what the cost of the axiom was. Well, this guy, we were talking about this guy, Torres Muller, who spoke at this conference in December last year. He's a young guy, he doesn't know. Thank you for watching. It was making quite a large waves through different sciences, particularly in the context of the way that one should do general relativity if you want to generate the right...

10:00 That's very interesting. Obviously I don't have the technical background. Send it to me too? Sure, sure, sure. It's nice to see general relativity and all the figurines together. Yes? Well, there wasn't much explicit category theory in his presentation. That may have been because his audience was the audience it was, and he thought that bringing in too much category theory, as it were, him naked among the skies might frighten him off, and hurt. Well, we're almost in Paris, aren't we? Yes, we're almost in Paris. John and Mimi are still snoozing away, aren't they? Really? Yeah. Well, we'll get their bags off and wait, don't we? Yes, I was thinking about that. So, there's no danger in that, right? No, but they may get a little bit late, a little bit late, finding their stuff, even though... Well, I'm not going to go to Paris right now. Thank you for your attention.