Euler, Maxwell, Grothendieck & mathematical representation of cohesion of space (contd.)

0:00 ... with the property that the lower one is what they call U, topos in abstract sense. They're both categories that have function spaces and sub-object classifiers, but this one has more. It has split ephes, and it has only two sub-objects in one, almost set theory. So that, so the Grotendieck toposes are just that sort of thing, additionally structured by a pair of... But then of course we should always remember to go to the other precisely mathematical but philosophical discovery, namely we should always look at the relative case as well. So after all, it's not only the case where S is sort of modeling Cantor's ideas, but even a more general kind of topos could also be the base that is contrasted with another one as being itself less cohesive. In a way, this is the solution to the problem of points in algebraic geometry, because if the base field is not algebraically closed, then the base topos should just be bar sheaves on fields. Then everything is good. The base is blue, the underlying preserves. For example, an algebraic group is still a group in that topos, even though it's not when you go all the way down to sets. In this general process of going from the big, cohesive topos to the particular topos, In this case, the topos of variable sets on an object, you find that this smaller topos picks up the structure. It's sort of a reflection of the fact that it was born inside this big topos and we plucked it out so it's no longer merely a Brittenite case of that kind, but it has an internal structure like the chiefs of local race.

2:30 So anyway, it's useful to relativize even in a familiar subject like geometry. Not always use just the abstract sets. So my axiomatic theory refers to such a contrast. And of course we call these functors by zero points. So we have simple ways of measuring every cohesion space. Namely look at its Cantorian points or look at its set of components and there is a comparison between these two. Green space X and E is a canonical map because of the agileness. It says every point is in a certain component. I refer to pi zero as the components, but of course I mean these are sort of indices for the components. What I usually call the actual components or subspaces, points there to get by. Okay, so anyway, by equality, I mean a functor, equality of space, I mean a functor that goes to domain D, which is over the same S, which has also two adjoints, there's also an inclusion of two adjoints, except this is a quality term. So these two are equal. It doesn't analog with pi zero and points, but Q is compatible with one or the other.

5:00 So Q is compatible with pi zero, usually not with points. I call Q an extensive quality. Maybe with the same D or another one, we can also consider intensive quality, which is, again, a functor to a D, and where D is a quality type, also with points instead of with isosceles. This idea of something over S with two adjoints of the same, you could call it also an abstract quality, an abstract quality type or abstract quality, but a particular quality is a functor to that. From the situation where you have a genuine opposition, possibly, probably, between the two sides, but which you have compatibility in one sense or the other, so you have two kinds of, now, also for any possible deed. Among these qualities, or the toposes I want to study, there are actually canonical examples. There's a canonical extensive quality and a canonical intensive quality. Probably these two are adjoins, but I haven't been able to find exactly the universe inside which is the universal property. So, in other words, given E and S, then among all the categories that are in some sense on S, that's the problem of what sense, exactly.

7:30 But among all those, there are those which are where the adjoin string collapses, like that, and there are the functions which are compatible with pi zero. Well, somehow universal among those... There is the canonical extension of quality. And that must be what? That's got to be homotopy type. Homotopy type is such a thing. The two functors agree because when you make the homotopy category, what were components become points. Maps from one are just components. In good cases, in classical case and in finite cases, although not in the combinatorial which gave rise to a lot of these, You can recover the components of the space, of course, from its homotopy type. On the other hand, among the intensive qualities, again, I'm morally sure what the canonical one is. I mean, once you have one, you have lots of others by composing with other factors, right? So it's not that there's only one. There can be a preferred one. Well, you look at the part where this is true. Look at the part where this map is an isomorphism. When I say part, of course, that's a subcategory, but it turns out that it, under the axioms that I have, it turns out that it is, that this is a central geometric dwarfism. This is the topos again, and it's still over s.

10:00 The two things that are supposed to be equal in equality have become equal. That's the universal way of making them equal, right? Yeah, that's right. And in order to get this well-behaved, I assume what I call the no-still-and-nots, which is that this canonical map for all x is epimorphic. The canonical map is there just because I have a full inclusion in two adjoints, but I can require that it be epic. So that says, intuitively that says every component contains at least one point. These may be defined as a recovery of those I've mentioned, because it's not said to be a split epi, but just an epimorphism. So under the assumption of this weak Nostradamus, then the subcategory of E, where the map is an isomorphism, not merely an epi, is a topos, it's not a subtopos, it's not hardly ever a subtopos, it's a quotient topos, but an essential quotient topos. And of course it's equality. So this essential morphism is equality, too, which I call the canonical intensive equality for any topos, any cohesive topos, which includes the Nostromatism. I haven't told you the infinitesimal setting for all this, and this is all arising in analytic and differential and has a great geometry in a certain way, which is... Which is born from, it's in a decimal, so I'll detail, but let me just, let me just, let's consider the spectrum of the dual numbers. That space is sort of the same, sort of the same thing in all of the examples, even though the examples are vastly different.

12:30 I've already construed that they are different, but the particular space is basically the same sort of description. Instead, the Cantorian negation of cohesion that's relevant here proceeds in that way, that we define S to be topos of cohesion, which have the property that they have no tangent vector. Again, every tangent vector is trivial. So there's a canonical diagonal map from any space into any space to some power. Consider those for which that map is an isomorphism for this particular choice of D. This is starting from topos having an object that behaves like this one. Doesn't have to be that one, but behaves like that one. Then, remarkable things follow that this S is itself a topos automatically. It has a reflection which preserves products automatically. It's actually rather like the categories of category of categories. There's a reflection that preserves products automatically and all sorts of other desirable properties. So this is an excellent setting for any such category. It's an excellent setting for partial differential equations in mathematical physics. They can all be defined there in a unique way. So that's the kind of infinitesimal picture. And then you may ask, well, what is then the canonical intensive quality for such faces? So it's the following kind of thing, which I'll illustrate with the much simpler combinatorial examples of reflexive graphs. Here, of course, S turns out to be the totally trivial in the abstract sense. The points are what you think they are.

15:00 Proponents are what you think they are. The arbitrary graph has components. You can see them if you draw a picture of the graph. Now those two functions are clear. What is the canonical intensive quality? Reflection graphs for which points? Components. That's clear. Those are graphs which consist entirely of loops. Well, that seems even more abstract. Where are we going here? It's plugged in. This is plugged in to the starting point. So we have the two adjoints. So then we have the adjoints which... We cover the points, so every graph is assigned, oh, well that's easy, the category, the graph, which just consists of the given dots and the given loops that throws away the rest. So I'm asking you to think of this as superheated material. I started with a sample, which has particles and interactions. I expand it or superheat it or, you know, whichever you keep constant. You expand it so much... That the interactions can be neglected. Well, you still have something. You still know things like how many quantum numbers of such and such a kind there are at every point. On this example, there's only one number, the number of loops. But it's easy to make up more complicated combinatorial examples for lots of information. But it's only the things that live at each point because you've made it rarefied. I say that a good name for the canonical intensive, the substance of a general algebra, is what you see if you turn it into a gas, as Avogadro did.

17:30 And then, you know, when we say that, you know, this stuff here is H2O, we are completely neglecting the interaction between, I'm simply saying at each point we can count so many. So that suggests that the other adjoint now, which itself is an extensive quality, we should call it, no, sorry, that's not right, form is the homotopy type. And these graphs all have homotopy types. In fact, all homotopy types are realized by graphs. There's another quality in which we define the maps between objects as the components of the function space, as Huravitz and others. So that's another, that's a real extension, canonical extension. Again, if the other one really is universal, then there should be a map between them, and I don't know whether that's true or not. Whether it's the homotopy type, you could recover the following thing. Maybe you can. Can you recover the non-extensive quality, which is the adjoint to the intensive one, from doing the homotopy type? It seems like maybe you could. It seems like there should be a universal extensive quality anyway. Maybe homotopy theorists haven't quite found it yet. I don't know. But, you know, there should be something like that. But concretely, what is the canonical, the adjoint to the canonical intensive qualities?

20:00 Even in the case of graphs, instead of superheating or supercool, we take this graph, thought of as sort of a sample of material again, we supercool it, compress it, and what do we get? Well, instead of lots of molecules that don't interact, we put the interactions back in, but now we get only a few super molecules, and their atomic numbers, or whatever, are partly coming from the mere interactions you had before. There are those that are there intrinsically, but there are the additional ones. So, I'm not sure whether we keep pressure constant or volume constant, so as a substance you can think of it as an avocado quality. But then the adjoining to that is the Bose version. Now, so you see, it depends on the components of the graph. But it's much more than that. Because essentially, you get a new graph which has only one point in each component. So all the trivial loops have been amalgamated. Let's say you start with a graph and it's already connected. That's the simplest case. It's already connected. You compress it by taking the trivial loops at every point and amalgamating them, so now you get again a graph which is all loops, but it's the one which is universal for maps from your graph to one that has only loops, the superheated version. And just as with a map from points to components, whenever you have this adjoint situation, you of course have a map from Avogadro to Bose, God rest their souls. Or, you know, canonical intensive quality or substance into the adjoint of the canonical intensive quality, which happens to be the adjoint of substance, which is not the adjoint, the adjoint of the adjoint. The adjoint in the middle is just the thing that unites the two of them. It's this business of super molecules. And there's a canonical map between them. The same kind of canonical map that goes from points to points.

22:30 This is precisely the information. What am I trying to say? In other words, knowing the superheated picture and the supercooled picture, neither one of them tells you everything about the graph, of course, but you could, and this is like you do experiments, see if the two ends of the spectrum, due to the actual nature of the sample that you have, you'll find out something, but you can't reconstruct it. Well, there's more information that you have, which may help to reconstruct. It may still not be sufficient, but definitely is more information, namely the canonical map between the two, which is due to the fact that you started out with the real thing in between, which did have interaction. Here you have a thing which is just all points and loops, and it goes over to another thing like that, but as I said before, this canonical map tells you which of these electrons, or whatever you call them here, We're really originally just interactions. They're virtual, they're virtual particles, virtual particles, virtual atomic things, things that are not in the image of that cronic collapse. So that's additional information which might be useful in trying to reconstruct the actual object. Here's where the lore of agile encounters gets a little bit stymied, because you might think, well, obviously we can consider a category whose objects are maps of We certainly have a functor there, which is to take the triple map points of the Avogadro aspect of the substance, the Bose aspect of the substance, and the map between them.

25:00 That certainly can be put together into one functor. So you say, ah, now we can reconstruct the universe from our experiments. We just take the adjoint. But unfortunately, this doesn't have an adjoint. It's constructed out of two parts, one of which is very much on the right, one's very much on the left, and they are interacting with each other, but the thing put together, well, it doesn't at least preserve products, but it doesn't preserve equalizers, I guess, and on the other hand, it doesn't preserve sums either, so it's sort of interesting. For example... Take an example from algebraic. So what if you have an algebraic group? Well, that's an algebraic structure based on the product. So if you look at it in this sense, the extents of that, you will still get a group, which in fact means two Lie algebras and a homomorphism between them. One of the Lie algebras being sort of trivial for this. I'm going to explain the infinitesimal part. There are infinitesimal spaces, non-trivial spaces, but they're entirely infinitesimal, so the reflection of, for example, any space has an infinitesimal core and the infinitesimal core of a Lie group is another kind of group which still has the same Lie algebra, although it has become spread out. We get partial information that preserves global algebraic structure, but not even category. I don't think I should quit sometime. I was going to say, I'm going to have to...

27:30 Do you want to draw to a conclusion in the next five minutes? Yeah, okay. This is it. I mean, I've already actually published something about the Internet Cousins case. Maybe this background will... Questions? Could you say a little bit... Tracking back to the presentation, to the point about the epimonic cancellation to the Nolte-Koenigsegg, how this relates to the weight. I mean, there is a connection there, is there? Something like simplicial sets, the epimorphisms of the models, and then that gives rise to degeneracies when the sequential theory. So I should have said that at least. While most of these E's of interest, it seems, themselves have no central idempotence, they must have some idempotence in the site, because they're not, I mean, they're putting the opposition, I'm making an approximation of the opposition between two different kinds of topos, generalized spaces that parameterize variables of street sets on the one hand, and cohesive topos on the other hand, both over S, over D. The first kind always have sites with no inputs, at least. That's more general than the cancellation one, but I pray for this.

30:00 All the examples of the basic idea of degeneracy seems to be fundamental. The fact that figures can have degenerate figures, or they might be degenerate in a sense, you know, a figure is called degenerate. X is a figure in capital X of shape A. Well, if there exists a split, that'd be on it. So if X factors across, then you say X is degenerate. And if it doesn't factor across any split ethics, I wouldn't even have to mention this, it's fixed under a non-trivial good influence on its domain. If it's non-degenerate, it means it doesn't factor through. So it seems that somehow this possibility is inherent in the notion of the pieces, as little as it sounds. For example, consider the projection of any non-trivial projection map. If there exist maps, then by means of graphs you have projections. And so, if you say take the projection from R3 to R2, and you have some quantity, some figure and other quantities, well, you could ask yourself, does it really depend on those three variables? Maybe it only depends on two of the variables. That's the potentiality of being degenerate. Although it seems like only a corner of the whole edifice of topology, it more and more seems to me like it's quite fundamental, that it's almost the same. Now clearly you can have an adequacy of such A's, even though they're globally no central, no idempotence.

32:30 So that's not the general thing I said before. If you think of all the examples, those reflexive graphs, by its very nature, consist basically of two hidden components. Take any arrow and take either end. That's a degeneracy. So the geometrical structure, the way that the substance and form behave and everything flows out of this possibility of having a degeneracy. All the examples. As soon as a site needs to have products... You might find a site that consists of only one object, but one object can be a square of itself, so it's a little bit unclear. It's always unclear if you're going to try to translate a property into a site. Actually, could you probably say a little bit more about this idea of extensive and intensive problem of motivation, because somehow it sounds a little bit strange probably, I need to understand. You know, category of points is extension, so that would be called extensive, so somehow you describe it kind of point by point, you think, in terms of category of points, and then exactly you put it, you cool it just on one thing, or probably it's not that important, that's why you call it extension. The idea of intensive is better understood, right? It always means intensive quantities vary on a certain domain as functions do, so the idea that a variable quantity on X is just a map from action to real, that's an intensive, because it depends on points, or even generalized points, whatever, but I mean, it depends on points, that's the intent.

35:00 The space itself, what kind of quantity is it? Here is what kind of a function, q, is it? Does it depend just on... Or by contrast, consider functions as measures and distributions, which are extensive in their way of quantities. But again, the idea there is it depends on the space as a whole. A measure gives you values that apply to the space as a whole. Now, extending this ideal parallel for qualities to the pi zero, to be able to extract from a graph what is the number of components, you have to consider the graph as a whole. The number of components is merely a feature of its extent, but it is a feature that it can be extracted from. Extensive quality, leaving aside the homotopy, which isn't there too, but it is in this, that you have to have, you have to think of the graph as a whole, well, first of all, to see how many components it has, which means how many points it will have after you've cooled it, because all the things that really do interact with collapse into one point, but the thing that doesn't consist only of points, it has all these huge number of loops. But except you've identified two points if they are in the same, the points as such, if they're in the same component. So again, you need to know about all the extensive interactions in order to know which points to identify. So I think it does agree. But does it also agree with the kind of traditional, say, idea of set as extension of concepts, say? Well, no, extension and extensive are maybe different. I don't know. There probably is a relation between the two. I'm not very good on philosophy. I mean, there's a relation, probably, but it's a very different thing. So the extension of a property is the part of the universe where the property is true, yes.

37:30 Because there is a sense of that, I think, kind of where you think you just say it. Instead of thinking of, I don't know, a concept, whatever, a group, you just follow these things and kind of imagine a class, you know, that's something. These ideas have been long discredited. Yeah, yeah. Within a given set, within a given universe, currents in physics, currents are extensive. Perseverance forms are intensive. Intensive variable quantities tend to be representable. And, well, this is maybe a kind of representability because this L is another topos in contrast to the homotopy types, which are still Cartesian closed, but another kind of. Also, another point. You just, you mentioned, say, that growth and ectoposis would be always in your sense, right? Yeah, over U. In fact, if you look at SGA4, really they never mention topos. They always talk about U- topos. Is that right? Yeah, I understand this present suggestion is kind of, I don't know, torn back to this idea about avoiding taking this U as kind of fixed, but then you have this relational picture of, like, two E, S, so... No. The U, in other words, the U topos, look in the glossary of my book with Rosebrook on sets for mathematics, and say, well, a U topos is this and this, however... Using both Diggs' principle, if not his original definition, we can talk about utopos, where U itself is in a libertarian utopos, and it's perfectly sensible to do so, and moreover, there are many examples where, well, really, it's much more agreeable properties of the functions, like pi-zero preserving products is very crucial, but it isn't true if you take the direct limit and go all the way down to...

40:00 So somehow, I don't know, that does relate, say, for what Squall and did for set theory, when you say, okay, you can't just say set is one object, you should also relativize to some kind of meter, set, and it's a relational thing. You can consider models of set theory, which of course are precisely special purposes, for example. See, the idea that epimorphism is split, I should have said that explicitly. This is the most common... This is the expression of Cantor's idea of negating cohesion. Because if the objects had cohesion, the map would preserve it and not every epi would split, you see. So this is the common everyday manifestation of Cantor's negation of cohesion, simply the axiom of choice. That was my question about how the axiom of choice falls into place in this geometric setting. On the point about intensive and extensive quantity types, as it were, I understand those are also tend to be strongly connected respectively with covariant and contravariant functoriality into, maps into and out of domains. Well, no, no, no. Could you say a little bit more about the connection between the covariant and the contravariant functoriality? We've kind of talked about that, haven't we? Yeah, I mean, relax. Obviously the guy has a plan. Yeah, yes, I'm sorry. Well, I'm not going to ask you to give an entire second talk, although personally I'd like to hear it. Anyway, there's a traditional, let's go back, there's a traditional discussion of extents in philosophy, goes way back, I don't know how far, but anyway, it does correspond to some kind of intuitive distinction made out that all quantities are intents of this, somehow, this is somehow the idea of perception, you see this, this, and this.

42:30 One at a time, there are emotions, feelings, and so you somehow, and since only feelings exist, therefore everything is, his argument is going bad, just as, on the other hand, on the other hand, Hegel said there's no difference, he couldn't see the difference between intensive and extensive, which is odd, actually maybe I didn't read far enough, maybe he comes back to this later, but at one point he says you can't see any difference. Well the reason he couldn't see any difference is because he was considering constant quantities. In any case, it got out of the hands of the philosophers when Maxwell used the terms intensive and extensive extensively, and again, oddly enough, in physics, in courses on thermodynamics, people talk about this all the time, but not in any other part of physics or other part of mathematics, even though, according to me, the basic opposition between the two is in all aspects of physics. And so I analyze it as modes of variation. You know, what is extensive or extensive in the way of quantity is mode of variation. There are variable quantities. If constant quantities, you can't see the difference, okay? But variable quantities, you certainly can. And intensive or contravariant. Well, variable quantities are analyzed as varying over certain domains. Domains are objects in a category that you can change by a morphism. And if you change the domain by morphism, the intensive quantities pull back. They pull back, and they pull back in usually in a homomorphic way, so they preserve whatever algebraic operations. Contravariant homomorphisms, and we transform them, are the earmark of the intensive mode of variation. So this applies not only to functions in the obvious way, but also to differential forms. Or even to cohomology classes. Even though cohomology classes tend to vanish on points, they're nonetheless very good variable quantities, and they pull back. Now, on the other hand, I analyze the excess of quantities, like mass and energy and entropy and thermodynamical. All these kinds of things are extensive.

45:00 Well, these vary covariately. So when you change the domain space, they push forward. And they typically push forward only linearly, not homomorphically. They might preserve co-multiplication, but they don't preserve multiplication. And in particular, they are typically not just mapped with a certain domain, in the way that intensive are just mapped with a certain co-domain. These are just a certain kind of domain. Because they're not figures. In other words, the relationship between figures and extensive quantities is typically Xerox delta. Many kinds of extensive quantities. For example, a typical kind is linear functional or linear math with values in some module. So this is the Rietz paradigm for an example of an extensive quantity type, depending on v, the distance of x. These are the measures or the distributions of, depending on the various exact... In the smooth topos, as I was talking about, these should be thought of as the distributions of compact support, and if V happens to be just R itself, then there will be a canonical map from X to 0 to octozen. So in particular, any figure, say point if you like, fine, but any figure, little x, give rise to...

47:30 But this is not, this is not an isomorph. Typically this is a linear object. This is a non-linear object in a non-linear space. This is a linear space. It's underlying non-linear. So it's not simple. At that point, the simple-minded dualization doesn't work. It's not, of course, a mass from a particular thing on a covariance vector, but not a kind of quantity. It's the fact that it's quantity that, at least intuitively, you'd think. Well, I think in that case, we should once again thank the speaker for an unprecedentedly rich exposition and a wonderful seminar. I hope not unprecedentedly tiring. Stretching, certainly. I know I can see you're not. Why don't I just get you a very stiff body, before you at least let me get you that. What? A stiff, how do you say, room. A very stiff room. With your lemon. Little by little, each time I understand. Little by little. I didn't understand the question.

50:00 I saw the review in mathematics by someone. You don't know the review? So maybe he made it up. He's saying, he's trying to give me a brief history of categorical physics. Of course, he's praising your book. And so I'm saying that... Be prepared for... Of course, it seemed, and if I can just say, and it's incorrectness on stilts to say that structures are all there is in this sense, this completely vapid and empty notion of structure, this sort of ontological structure. Yeah, because you're going to die of hell. I'm still starting a head cold as well. Yeah, he's starting a stinking head cold right now. No, he has got a bit of a cold. I mean, when I was with him having dinner last night, I didn't tell. He's beginning to get a quite heavy cold. Have you received some information about something in London organised by... Yes, organised by Bob. Well, actually, organised by Andreas Doring and Chris Highsham, I think. For the 9th of January. Yes, I have. Yes, and in fact I've already replied to it, saying that there seem to be an awful lot of players on the field.

52:30 Come on guys, before you catch cold. Bonjour. Nous sommes quatre. Then, of course, you've got the variations of each of the different states. Well, the things that vary are the same. The things that vary, again, are the same. The base states are the same. The life is the same. The things that vary are the same. The things that vary are the same. In the case of pure discreteness, though, and probably in the case of a constant set, there's always a chance of a construction error. But the point is that you're modeling the discreteness within the variation.

1:25:00 And, of course, another Phrygian notion of objects.

1:27:30 ... have a theory that decides pushing all the way down. And that thing is very wrong, some mathematicians. So that's the dream. Yes, so you end up like Phrygo, wanting, you know, insisting that... ... have some basic level of... Yeah. ... for myself to present... Yeah, yeah. Or there are contexts in which it clearly doesn't matter, other contexts in which it doesn't matter, because then you have to bring some deepening, because then you have to... No. And this is the whole problem. Then you're blaming the real world in front of you. Exactly. And then the Frege philosophy, which is that until you've done the ultimate deepening, you know, the once and for all deepening, because it has to come out of the one true logical theory of things, the global top element is a crazy idea. It's certainly going to land you in mysticism.

1:30:00 Of course, his book was from Trago de Gaulle. Why do we assume that there's an isomorphism?

1:40:00 I think he did, I think one... I would investigate that very carefully, and that's how he led. I think it is pretty clear that he did come to a theological position. But I don't think it was the thought that did. I mean, he did end up writing about, you know, the footsteps of God, so...

1:47:30 Oh, I did get the point, I did get the point, I did get the point, it's precisely the point I tried to get across to John Mabry when he was arguing about this, I was making that point.