Conversations on Extensionality Principles & Topological Spaces (contd.)

0:00 I think I understand, but is it possible to give an example, a kind of concrete example, Take three-dimensional solids and take, as the category of figures, the polygons, the category of polygons is sub-category. And it provides you with a category of figures out of which you can define any polyhedron. That's all. So you have an average of some kind. Right. Now, let's consider another category. Okay. Can I just ask a question about the, again, can you explain how this adequacy condition connects with this condition on the, well for instance you said here you could have an example of, well for instance. In the case of connected components... Well, if you have figures, you take connected components. This is only a particular case of this picture, because the figures here are spaces, connected spaces, or even general spaces in which you consider the connected components of each part of this connected part of the space, so... You define the maths on the co-domain which take account of the connected components of your source of figures, but then you could have different... I'm going to try to get a feel for the way that the, you know, the Cantorian, the typical Cardinal, fit into this picture as a restrictive...

2:30 The category of figures, yeah. Reduces to one object and one matter, any single thing with the identical matter. There is only one figure in the category of science. And the fact that it is adequate, it is, is, right, the extension of the principle. So, in this case, but on the other side, let's look at the, on the other side, it is not matched from a subcategory of figures which are free, in the sense that allows you to recover the structure of the objects, but let's consider a subcategory where any object can be matched to, okay? And of course in here you have a subcategory where any of the quantities. Here is the place where in the case that C is a topos, you find omega. It is an object such that if there is any two different maps between an object, you can recover their difference through characteristic maps of this object to omega. Hence the expression sub-object class. Yeah, because you take these two maps as different elements from X. And then you take the characteristic map here, these are two different maps, and why you obtain different, since these are mikes, you have different characteristic maps to omega, and so this will factorize in what is the identity, in different characteristic maps of this, so the L-direction says that you have a diegetic correspondence between mikes. Here you have this omega, but you can't... Hence this, as it were, both algebraization and geometrization of the logical. So here you have the geometric part. Now here you have the logical and algebraic part because you can idea not only omega

5:00 as the logical correspondent and the omega axiom over top of it, but also, say, the reals. Because you evaluate, this is a quantity function with real-world quantities, then you could have natural numbers here, and you use this category, subcategory, you know, as a sort of, say, test, you choose for different notions of conditional quantities, the proper... In order to identify aspects of the objects you wish to. So in this sense even the third value object is a quantity object. And so then it means that provided you have identified a category of figures, you have to identify the The proper category of quantities which reflect the properties of the figures. Right, so for each different category of figures, for each different C, you have to find the proper one. So this means that there is a lot of research to do here and therefore different categories where you can identify different subcategories of figures. And you have two different conditions of adequacy and co-adequacy because here, all what we said here in this case, we have to dualize because we have to map to the objects in the series parallel band and maps from the objects in the sub-tablet. And so in a sense, in a sense, an equilibrated category from the... Topological and algebraic point of view is a category where you have identified the subcategory of figures and the subcategory of quantities which, say, are consistent. And where the figures, as it were, the space determined by the figures acts as a dimension of creation for the quantities. Exactly, yeah. This kind of elaborates that remarkable field right back in the early 1975 paper about the nature of the method of variation of the case of, you know, it's important to place within this order, this revival of the notion of quantum geoscience, understandably.

7:30 The algebraic energy and many aspects of the domain of variation. That's very, very helpful. I'd love to have pretended that I followed every single word, but I think I got the gist of it. Oh, I'm sorry. I wanted to ask you one other quick question about fixed points, separators. Oh dear, I'll have to come back to Florence soon. You'll have to come back to Florence. That's a good idea. There's no way that you can give me a very quick glimpse of what's going on in the case of the fixed points being there. We'll go to the bar then. Okay, we'll go to the bar. And also, I apologise to Manuela for keeping you, again. No, because I probably... You don't think we're going to be locked in? No, I hope not. Well, actually, I'd rather like to be locked in, and I can learn a great deal tonight, but... Just let me get everything... Oh, hang on. Now what I'll do is to do the rest of the corrections. You've got the, well I've corrected the typos, actually I haven't corrected all of them, I've corrected about three quarters of them, and then I'll try and, I mean given the length of time it's going to take for Bill to write all these lectures you obviously don't have to have this in a great hurry, but I'll try and get it to you within the next two weeks. Now, seriously, you've given me the will to live again.