Euler, Maxwell, Grothendieck & mathematical representation of cohesion of space

0:00 In spite of having started, as Mike said, as a lecturer on technicians in my academic career, I think what I'm saying is of interest for foundations, although I have some ideas about it, I will not talk about quantities like metrics and atlantic connections and energies and so forth, but rather of what should be... The nature of the environment in which these quantities are defined, or since they are variable quantities, and since we objectively visualize variable quantities as varying over a certain domain space, what is the nature of these spaces over which physical geometrical quantities could vary? I'll say immediately that this variation of quantity can be either of intensive or extensive nature. Intensive has been analyzed a long time by analysts. Extensive has been, although it played a role in philosophy, has been neglected, very much neglected due to the use of Hilbert space and so forth. Appetitions tended to try to get away from these extensively variable quantities by reinterpreting them as intensive quantities in one way or another. Distributions as, quote, generalized functions, unquote, which have had an extremely pernicious effect. Despite the great virtue of distributions per se, they are extensive quantities, basically. So the reference to Maxwell, of course he was talking about many different things and made this observation clearly motivated by his experimental

2:30 But theoretical concepts, namely levels of precision, can be cranked up and screwed up as needed. It sounds like an engineer's conception, but really I think it does spring from the very nature of knowledge, because our knowledge, theorizing and so forth, is always involved sharply. The sharply dialectical feature of neglecting those things that are neglectable and concentrating on a few features which are irrelevant to understanding the certain phenomena seems that so-called fundamental physics ignores this principle and pretends that somehow or another they're going to try and be one final level. But the actual experience is that there is no final level and can always be finished. So even theories of numerical analysis, let's put it on that level, in other words, there is a, my son who is a physics graduate student, a teacher, When he was very young, he made a computer program to describe the motion of a satellite around the Earth, but when he viewed on the screen the satellite's motion, he found that it always fell into the Earth and said it's continuing.

5:00 Now, of course, this program in itself has nothing to do with satellites, with motion, it's just a subjective reflection into electronics or whatever. Because we give that interpretation. But nonetheless, I said, well, you forgot to program in the conservation of energy. In other words, even though the thing is totally numerical, it has nothing to do with physics per se, one can still insist that the computation preserves energy. And then, of course, the satellite doesn't fall into the picture. So the point about that is that even that is a certain level of fear, I think. And again, it can be done in a serious, useful way or not. The argument has been deflected except a few features, philosophies. So this basic dialectic of the origin of knowledge in a way dictates that we will always have a procedure. So, therefore, if we're talking about... I could say the plenum or the ether. People say ether doesn't exist anymore. On the other hand, they constantly use it. They just say it has much different properties than where it mattered 150 years ago. So if there's some kind of plenum or ether, or maybe ether is a kind of matter that moves in the plenum, whatever, the nature of this plenum, this space, this ether, At least it should be cohesive. It has to have some kind of cohesion. Otherwise, although that has been denied recently in the Scientific American, where we read that space and time really are discrete without any evidence, what we observe, of course, in physics is that in the continuous world there are all sorts of very interesting discrete manifestations or aspects. It's a, quote, principle of quantum mechanics, unquote, that overwhelmingly space and time...

7:30 There is no coherent mathematical theory of that that has ever been offered, as far as I know. Perhaps some of you know better that there are coherent theories that give me a valid level of precision, but I think that to say that it's the ultimate level, that's completely... So with that preface, what I'm saying is that I want to describe the kind of categories of space which might arise at different levels. Fortunately, we have tools for doing that because of work done 50 years ago, on one hand by Daniel Kahn, who crystallized the notion of adjoint functor out of his lifelong study, which continues even now, of that level of cohesion known as combinatorial topology, or combinatorial homotopy theory. Pull out this notion of adjoint function which turned out to apply to everything. Maybe this could be compared actually, this thought of this, could be compared with the fact that within the very narrow world of constructivist philosophy and mathematics, the idea that concepts are generated by pure thought and you go from one stage to the next and so forth, from that idea of moving thought came hiding out. It itself has nothing at all to do with that context, but it turns out to be descriptive of the nature of truth in all kinds of situations, which vary in a more physically suggestive way, a geometrical way, and not just in a subjective, idealist way. Anyway, to come back to Kant, the tool that he provided was immediately taken up very quickly by many people, Roden, Dietz, Merle, Bleat. His friends and myself and many many many many other people. It was really a flowering of category theory made possible by Adjuvite founders and by that other great inventor 50 years ago, namely Grunbeek's Tohoku Godemar, which was my introduction to him.

10:00 Because, although I won't speak directly today about the AB5 abelian categories of Rotary, they do come out immediately from the setting that I proposed, basically as the linear objects in a non-linear setting. So, I'm not going to talk in detail about abelian categories or even dual abelian categories or anything like that, but rather... The specific kind of nonlinear background in which it seems that in most cases such things live. I think one could say, I don't have any direct quotation about, it seems that Grotendieck himself in the space of two or three years realized this, that the actual home of these eugenic objects, nonlinear, but at least the fundamental investigation about the nonlinear environments for these linearizations. Now, yeah, I should say that one has somehow the idea that nonlinear is more complicated than linear. But actually, conceptually, linear is an additional structure. Now, I think tomorrow I will talk about some other kinds of cohesion having to do with the monological, vector space, ambinatorial topology, and so on. Of course, the kind of topos of cohesion that we have studied the most since 1967 have been of a more of a smooth nature. The worlds of algebraic geometry, analytic geometry, and smooth geometry, the infinity geometry, are the typical examples of which I want to give a common treatment.

12:30 For example, this is something that's never been investigated, although I proposed it for you, that just as one talks about field extensions and all that, for solving polynomial equations, in the same sort of way, one can, there is a whole range between algebraic geometry and smooth geometry, say, or even algebraic geometry and analytic geometry, which rejoins both functions. It's clearly an important example of the big difference between analytic and smooth. So you could adjoin bump functions to the algebraic theory of rings and obtain an algebraic theory whose corresponding large should again be a tokos of the same kind that I'm describing, but perhaps of interest from that point of view of finding the minimum context in which certain differential equations have a solution. The problem of finding environments where differential equations have solutions has been studied a lot under the name of differential algebra. One should point out that it's in some sense inadequate because it talks about derivations, things that satisfy Leibniz's rule and so on. But differentiation equally, at least equally, involves the chain rule. And differentiation has certain properties with respect to composition. It's surely just as important in extending the realm of solutions of differential equations as does mere multiplication of products. So I think rather than just plain rings, the real environment for differential algorithms should be certain kinds of small categories, like the theory of rings.

15:00 There's a range, and those are three well-known kinds of examples. Of course, there are many profitably studied for many years, precisely different versions of algebraic geometry and so forth, and analytic geometry. Generically, there are three types of well-known examples. It may be necessary still to mention and underline the fact that these kind of topos that I'm talking about These are quite a distinct kind from the toposes of sheaves on a topological space, actions of the group, or the Petit et al. topos of a scheme, for example, all of which have the character individually of generalized space. A lot of the attempts to popularize misleading disinformation, All of this suggested that toposes were invented in order to generalize spaces in that way, but this is distinctly wrong if we look already at Bergdeg's presentations in the Cartan seminar of 1960 about analytic spaces, the category of analytic spaces. He treats there as a topos, which is definitely not of the generalized space kind. It's rather the category of all analytic space, generically speaking. You have the word topos, I guess, in 1960, but analogous toposes in algebraic geometry, studied by many people, were in fact called topos. So there is the grove, the risky, and the grovely tall, and so on. And, of course, by the... Sometimes seven years had passed and it came around to my attempting to apply D'Avril's version of Ruben D's theory into continuum mechanics. Of course, I knew that it should be, but the word was used then. But one has to... I've been trying to make explicit for a long time the distinction between these two classes, which I philosophically describe roughly as...

17:30 Topos is a cohesion and topos is a pure cohesion and pure variation, with the general case being somehow a mixture of the two or a combination of the two. The pure variation, the original example was sheaves of sets on a topological space. There the idea was it's a variable discrete set varying over the topological space as a domain. Something even simpler is g-sets, where g is a group. Again, discrete sets vary, but in a different way, varying along the group. Now, actually, algebraic topology, as we know it today, arose in the 30s from the realization that these two examples really are part of the same, Maths, too, is fundamental group. If you take actions of the group and sheaves on the space, there's a morphism that co-closes there. And if you pull back the point, you get the universal covering space. So the dialectic between the covering spaces and portions of the fundamental group arises precisely from a diagram in one category. Certainly belong to the same. Now, Grothendieck wanted to try to unite these two, and successfully united these two by his concept of étendus, did I pronounce that correctly? The étendus were defined as toposes which are locally logical spaces. There's a well-defined sense in which a topos can be locally something, something, and so there's a... A basic example is that a group is locally in topological space, namely a point. It's locally a point. When people talk about groups acting on points and on a point, it's not really a joke, because it makes sense.

20:00 That's sort of the problem. Now, actually, Grudendieck made a mistake in that what the generality of a group is. He thought they all had something to do with each other. Sites which are consistent monomorphs are necessarily invertible. Studies, for example, the discrete dynamical systems, which are actions of the additive monoid of natural numbers, a category because of cancellation of addition, and that topos is locally a certain topological space. Or an even simpler example is, I call them irreflexive, namely the idea of a graph as a set of arrows and a set of dots. If one considers that category, it's a topos, but it's a topos of the pure, of the variation kind. In fact, it's an A times U. In fact, it's locally isomorphic to the topological space. Three points. You see, this is, the sheaves on this space are really just diagrams of this sort, because you have global sections over each of those two opens. You have sections and you have restrictions. Diagrams of this sort, the same thing, and this, so this maps to the sets, I use S for sets, graphs, in that way, graphs, anytime there was a quotient slope of the ice I'm working to, in a way that the graphs are obtained from starting from this space and then amalgamating these two points, but without amalgamating the arrows.

22:30 Oh, and this thing, of course, also comes up all the time in the following way, is the quotient of the real line, where you have a positive and non-negative and non-positive, there's two open sets, so you can collapse the line into three points, very important in physical measurements and so forth. Can you obtain something without this topological structure? This is a category here. There's no species on that category, and that category, there's no compositions, and also, by the way, epimorphism. This is, of course, very important in physics, to know whether a certain quantity is greater than or less than something, or equal to. And abstracting the reals in that way, you arrive at this space, and then taking the quotient space of that, you get this space. You can also fit into this very nicely because if you interpret a graph, the special graphs where one of these maps is an isomorphic, so you put together the identity, you end up with a system that can be viewed as a graph from now you go on and now you go on.

25:00 What exactly is the role of that viewpoint, two points that opens? The fact is that a sheaf on this three-point space, a sheaf of sets, involves a global set of global sections, whatever that is, and a set of sections on one open and a set of sections on the other, and two restriction maps. The whole has to be restricted to each. Now, if you...actually the space has a total of five open sets, but if you impose the usual... There's a numbering condition on that. You find that if you only look at these basic open steps, then the chief condition is nothing. So any diagram of that sort is a chief on the space. An extended Sierpinski space. Two basic open steps. And if you want to present it as a hiding algebra, then the two generators are disjoint. I mean, that's the axiom. It has a lot to do with ergodic theory because, you know, you look at invariant sets and measure-preserving transformations, if you want to look at the, sorry, you want to look at the orbit space as a measure-preserving transformation, well, if you think of it as a classical space, it disappears, but if you think of it as topos, it's very non-trivial, so there's analysis in any topos. As I want to emphasize, so a particular analysis in the Etan-Du that results in the quotient of a measurable action, probably, you know, the analysis of that topos probably contains most of what's known about Ergodic theory, no matter of working out, tabulating that.

27:30 So as I said, it turned out that the spaces which are locally, the topos which are locally spaces, will modulo the existence of points. Spaces don't have points, that's more or less a triviality, but if we ignore the question whether points exist or not, a Rotenberg topos is locally a locale, which means roughly locally a topological space, if and only if it has a site which consists entirely of monomorphisms. It's easy to see that not only the pre-sheaf toposes, but any sheaf topos based on a site consisting of monomorphisms. This is an aton-do, because if you just take the destroyed sum of all those monos, it becomes a poset, you see, the basic idea of a locale is that you have a site which is actually a poset. The converse, I conjectured the converse a long time ago, but it took a while for Koch and Murdoch to prove it. The point is, it's natural to forget, and I always forget it, the site need not be sub-canonical. So any etendue has a site consisting of monomorphisms, maybe not a site which is fully included in the topos itself. It sheaves on, it's a subtopos of one which is based on, of course, in a way a more significant generalization. Groups are pretty significant. For a generalized space, it was the petit etal toposes which were more exciting, more interesting, more useful.

30:00 These are all true of A-conjecture. So the T-etan-toposis are not etandu, but they do belong to another class which was actually studied later by Peter Johnstone. Again, he approached it, the conceptual approach was, well, there ought to be an adequate number of objects which are separable. To say that an object is separable means that, of course, it has a diagonal map, but that there should also exist... And finally, an actual complement, a sub-object of the square, which is detachable in the sense that if you take the sum as an inclusion map, and if you take the coproduct, that will of course induce a map toward A cross A, and you require that that be an isomorphism. Most of the time, in a cohesive category, of course this is not at all typical. If it were always true, you wouldn't be very cohesive. You'd be Boolean. But if you, the diagonal is well defined. There is in a topos a complement, a hiding complement, but the union of the two is not the whole, it falls short of being the whole, but a special kind of object, a separable one, it comes from the, by the way, this is also called decidable by logicians, but rather that's a subjective version of it. You can decide if two things are equal, right? But objectively it was separable. In traditional algebra, polynomials that don't have repeated roots, the roots are separable. If you do have repeated roots, then you won't have this simple implementation, there'll be something missing. Alright, so the idea of having an adequacy of these, let's say within a given category,

32:30 which has sums and products, we can isolate these separable objects. Exceptable objects. And then the whole category, let's say this is a subcategory, a larger category of objects that are more, in general, more cohesive, then this will give rise to an adjoint pair, studied by Kahn 15 years ago, expounded very beautifully in Gabriel and Ziesmann's introduction, that a small... A small category inside a co-complete category automatically gives rise to an adjoint pair, like this, but not like this. The requirement of adequacy is that this should be full and faithful. So to repeat, small category inside co-complete category gives rise to an adjoint pair, connecting the co-complete category with the topos and preces on the small category, but Isbell proposed the study of Special situation, when is A adequate? It's adequate if and only if this is called a phase. The general notion of cohesion that X gives you, while not at all entirely separable, at least is generated as a site. So there are a lot of toposes of this sort. I still consider as toposes of pure variation. You do have a, in which the subcategory of separable objects is equivalent to those that have a site, have another site, all objects are epic.

35:00 Now when I, you have to be careful, of course the, is included in the topos. It doesn't mean that the maps in E are epic in the topos. But it means if you look at this small category in itself, epimorphicity is a property of quantifying well overall. So if you quantify only over the E itself, then you have cancellation. This is the notion of generalized space, which does include the fatigue etal. This is essentially the connected etal as the site. So between two connected etal, the maps. It means that if you have two maps between connected etals, which agree on some part at all, some not-zero part, then they're equal. That's exactly what epimorphicity means. We have a category of things over a base space, and we have two of these connected etal objects, and everything is a sum of connected, and so these do constitute a site. If there exists another etal object so that these two things agree when restricted to that one, then they are equal because these are connected. So just the existence of something, that's the epimorphicity property, and of course, in the Catilletal case, everything is a quotient of the sum of the adequate sum.

37:30 So again, the idea of Catilletal arose, as everybody knows, because of the lack of the implicit function theorem, but one still wanted to talk about variable discrete sets over some kind of domain, and it turned out to be. This topo is a good model for that. So we have two different ideas about variable sets, strictly variable sets as being a very special class, which I haven't yet said, but it should unite both of these. Now let me write down a condition here. We often want to, we can consider a pair FG as an equation. Is the equation true? If it's true, it would mean that F and G coincide. So, the cancellation law says that, well, yes, that the equation holds F equals G if something else. To say that A is epic is to say that if F equals G, that can cancel, but monomorphicity is also a cancellation, just the opposite of time. B is monic, says that if BF equals B, then the points themselves will equal F equals B. Well, you see, there's a common generalization of the notion of epic and monic about the condition of a category now. We've talked about the condition that all maps in a category are epic. All maps are monic, in the case of A times U.

40:00 So if we said, suppose we assume both, that's A equals N, and that's what that should imply. So that's a generalization. Again, that's a condition that all maps are generalized, because In the case, if it happens that all maps are epic, then one of these hypotheses is already enough. Or if it happens that all maps are epic, or harmonic, the other hypothesis. This is a reasonable, this is a, it's called a horn sentence, sorry about the logic, Hulz recently condemned all the recent work on topos theory as mere logic, but we just, we just found that we have to use a little bit of logic to understand what's going on geometrically. If you can think of this as a property of a category, O, O maps, A, B, F, G, in the category, you have this property, it's a horn, so it's very, you know, it's a very agreeable category if you take, if you take any category, C to the E again has the same property, limits, sums, it's a reflective subcategory, any category could be forced to be like that. And then forcing preserves finite products. That's what I just said. And so forth. So it's a nice... Well, the reason that I mentioned the reflection of path, that there is a left adjoint to the inclusion, and that left adjoint preserves products, is of course because, ultimately, I haven't told you anything yet about any kind of topos, but we would like to be able to mimic this idea that...

42:30 If a space, if a generalized space is really an object in a suitable topos, at least it should give rise to a corresponding petitopos. What would it mean to talk about variable discrete sets over an arbitrary space? So somehow the general topos ought to, the general topos x, if we localize it as an object, we should be able to collapse that to something called cheese on x. This is still a research project, although I've written several attempts. So the idea would be that if I have a site for x, and therefore a trivial x, I have one also for objects over x. So if I apply this forcing, I compress it down to pure variation. So the cohesion that's there sort of gets pressed down into the pure variation that it's capable of. I stopped. Again, for a long time I talked about Gros and Petit, which is very bad. It's completely misleading. It's not that we go from Petit to Gros, rather the opposite way. Last help of a column of clarity said, said that my observation is, well, if you have some definition of Gros, then Gros on one point is a branch of geometry. And then you can, then you can, then you can, maybe it's better to drop. And it's in any way qualitative distinction rather than large, rather than size. This is why I mentioned the nature of this condition. It's reflexive, which just applies not to the sites, to the site book. We might get a good notion of the relationship between some notion of cohesion and variation of discrete objects over any particular instance x.

45:00 To emphasize the logic here, this means if there exists an A and a B, we always put an existential quantifier on the left due to the left adjoining this existential. So it becomes a property of F and G and then we require that to be true for all F and G. So we get a class of sites now. Think of these as sites. You get a class of sites which includes both the etons U and the separably adequate whatever. He called these actually QD tokens, quotients. Every object is a quotient because if you take a sum of these separable things, it's clearly still separable, so every object is a quotient. In his book, he calls it locally. It's a little bit confusing because there are two quite different notions of local. There's actually a still broader class. So the game for the moment becomes, let's generalize this idea of this variable as much as possible without leaving that concept, without encroaching on the other end, which is going to be the cohesive thing. Well, there is a much simpler condition here. It's implied by this, not equivalent at all. It's implied by this. And that is, there are no hidden points. Precisely, the migration of points doesn't do anything. Vibration of points over such categories is the same as categories. Groups don't have any idempotence, neither do partially ordered sets.

47:30 Well, in fact, partially ordered sets don't even have any endomorphisms at all, so particularly no. But groups, we want to allow endomorphisms because of groups. In fact, monoids are consisting of monos or birth epis or even monoids in more general. There must be monoids that satisfy this without being... I think my first paper on this that was published in Bogota, Colombia, I talked about a very, very special class of Categories in which, how shall I put it? Yeah, okay, so I mean, sort of half the story about what a sheaf is, is that the stalks are discrete, so it consists of variable discrete sets. However, the sheaf condition is moreover something about how, paradoxically enough, these discrete fibers move continuously. Well, if we ignore the last one, it's hard to understand anyway, and just talk about discrete fibers. That's what's called unramified algebraic. It's more general than what's the slogan there. Etal means unramified, and for the moment, let's drop the smooth, let's just look at the unram. Anyway, and just think about pre-sheaves. So if you look at pre-sheaves on a certain category, and then objects over, you want to single out the things that have discrete fibers. Well, there's one of these unique diagonalization type properties that you can write down.

50:00 Which turns out to mean that you forced every math in a given category to be idempotent. So this idea of forcing every category to be idempotent, in fact, forcing every idempotent to be the identity. Sorry, the way it buzzes. Forcing every idempotent to be the identity. And interestingly enough, the context which I was talking about there, which has to do with generalizing graph theory, but in a very restrictive way, The site that you start with consists entirely of idempotence, well it's a monide with idempotence, all idempotence, a band, but now the thing is if you take a representation and slice by it you get a little category which is the site for that, which no longer has a property to all maps. You can kill off those idempotence again, so you start with things that are... A lot of things consist of nothing but idempotence, and then you kill off the idempotence and you have something left. That's because you localized in between. And this is a very good way of describing, in fact, things like bipartite graphs and bipartite graphs with various other graphs, B and so forth. So this is not devoid of interest, in any case, for combinatorics. I find myself getting more and more into some very elementary category theory that doesn't seem to have been. I think we all find that actually the simplest sort of algebraic calculations with composition have not yet been done, so we need to be very good at this, to do that, so it just keeps coming up.

52:30 So here's a very elementary thing. The vast generalization of this already. I'm going to go beyond then my idea of generalized space forever and forever. The category that satisfies that, if we use it as a site, Take any topos. If the topos happen to have a site like that, they'll certainly have the following property, that there are no central idempotens in the topos itself. Now, the center of any category is what? The center of any category is, if you look at the identity functor on it, and then the natural endomorphisms of the identity. It's automatically going to be a commutative monoid. And sometimes there is, often there is none. I think Bourbaki did this. The idea that to deny a statement is to say it implies a trivial statement. A trivial statement might be false, but more typically in ordinary discourse, it's a trivial case. So if I say there's no hidden potence, it means every hidden potence is one. There's a well-defined trivial case, so the center of a topos is the natural endomorphisms of the identity. So it means that... Really, it's a huge thing. To every object, there is assigned an endomorphism, and that commutes with every morphism, in the sense that the one I have assigned here, it's not surprising that you'd be...

55:00 I claim that if I have just a site, a site with that property, then the whole topos will have this property. There might be lots of, there will always be lots of, in a topo, in a topo, individual objects that have the graph of any map, for example. I have a map from A to B. It's graphed of the projection, and so there's a non-trivial impotent there, which is squashing the whole rectangle onto the graph. Unless every object is a sub-object of one, there are going to be non-trivial impotents on objects. The whole category is central impotent, so now I'm saying that if there's a site, if there's an adequate part, there's an adequate part that has no impotent, because you see if we... If you had a hidden potent that was central, then it would commute with every element you find over these generators. And it has a significance if there are enough elements or figures, like Volterra. They are confused by elements because of the set theory, so they call them figures. It's the same idea as Volterra's. Everybody understands geometrical figures. So if there are enough figures, whose shapes belong to... If shapes have no idempotence at all, then in the whole category, there's no central idempotence, because then they would have to commute with all the figures, and that would mean, well, just faithfulness, like they don't even need etiquette.

57:30 Figures of these types are faithful, even, much less than etiquette. Even that's enough, because you can compare the central idempotence with the identity by means of a suitable element, and the idempotence is equal to the identity. Now, the thing is that there are categories which do have central areas. In fact, toposes are very interesting. Again, the obvious example is the category of points of a given topos, the vibration of points of a given topos. For example, just the sets equipped with an endomorphism, but an idempotent one. If the whole structure is just one thing anyway, then of course that can be essential. And this could be any topos or any category of points. This is the function that takes either the fixed points or the orbit space as you like, it's the same thing. In fact, I want to give another shocking name to this. Two categories, for example toposes, and an inclusion function with a left and right adjoint which are however equal. This kind of a thing I call a quality. It's not even a quality. More exactly, I want to call this a quality type. Quality is a functor that goes into, it's a codomain for a quality.

1:00:00 More correct if you pick the quality. It's kind of like you if you paint red by red. Need more thought. So this is an example then of a quality type. Maybe I should give an example of why this is. We have quantities as well as qualities, mind you, but I just didn't get to it yet. Quality, quality types, quality types, or quality to a type of quantity. I tend to be calling this quantity. In inclusion, the closed basal punctures in the left and right adjoints are equal. Now, of course, that means that there's an infinite string of adjoints, all equal. All the ones going up are equal and all the ones going down are equal. So, this can be viewed as a special case of the longer string of adjoins in several ways, and that's, so again, a collection of graphs were a good example of an etan-do, and that's also true of the discrete dynamical systems. Operations of an idempotent are a good example of a quality type, but what's a good example of the kind of thing that I really am planning to talk about, cohesion?

1:02:30 There are actually simplicial sets. For the first example, this, of course, is a topos of a combinatorial nature, a very odd phenomenon that people who work with simplicial sets know about topos, but they never use the fact that this one is a topos. As qua topos, it has remarkable special properties, some of which they are implicitly using, but they never write down what these properties are. There are all sorts of ideological arguments. Cubical sets are better. No, superficial sets are better. Now, since nobody's ever pointed out the actual objective categorical properties to distinguish these, debates remain on the lower level. It's, official Dutch is an example of it, but an even similar example is the truncated, which I call also reflexive graphs. So this is totally opposite in nature to the anomaly. Over at each point you have a preferred loop. In that it's preserved by morphisms, the natural transformations, the appreciative category preserves the preferred, you see, if I, what I basically want to study is contrast between more cohesion and less cohesion, which was basically Cantor's strategy, which started, set theory was, he looked at all the cohesive mengen in mathematics and he said, I can take an extreme case.

1:05:00 You got the idea from Steiner, the geometer. I can take the extreme case where there's no cohesion at all. Of course, set theorists have ignored that. They've studied the cumulative hierarchy instead, but mathematics, practice of mathematics in the 20th century pretty fully internalized this idea of the totally cohesion-free sets, used them by double negation as models for, as background for all kinds of structures including Structures that are inclined to model particular kinds of cohesion, topological spaces, differentiable manifolds, and so forth, are always described as being over-structural sets. And of course, when I speak about the region of toposis, it's that sort of thing, except for... So really the thing is to look at...