Strong Nullstellensatz for Euler continua in a cohesive topos / Jonsson-Tarski toposes

0:00 Again, I want to speak about a continuation of two years. Namely, we have to describe in a common way many of the categories applicable in mathematical analysis. And I've chosen the name cohesive for this theory. Just because words like continuity and connectedness have already been assigned well-established meanings that you would be better not to. But I believe that his presence is an intuitive idea of what space is all about. It is a definite rejection of the idea that a determining feature of space is open sets. Already, in 1949, with his invention of J-spaces, the principle that if a category was unsatisfied with what we now call artesian closure, that we should change the category.

2:30 Since the earlier stage of analysis, the fact that the one space is based on another, or in a well-defined space, he featured in all his studies. We try to exploit that by assuming that the archaic figures do have connotations and other features that they have to be extensive in the journal. Now, this, of course, the locale expression is not the determining feature, does not mean, of course, not using it. Rather, like the principle laid down, put forth by James Clark Maxwell, In the time when people were really debating about the affiliation of the ether, it's commonly said that the ether has been overgrown, but I think the practice more accurately say that ether has properties quite unlike what was imagined in 1875, but the principle that Maxwell initiated in his borrowing his experience as an experimenter. The level of precision used to be screwed up or screwed down according to the needs. You have the factor, you have the combinatorial spaces, there's topological spaces, analytic spaces, and so on. It is an indication that probably the end of space is an object in one of these categories, but more exactly it's a thread that goes through a whole system of these categories. There is no practice among algebraic topologists to use that category variable in that way, expressing again this, always the practice, whether we recognize it here or not, the practice of screwing over down the precision as it's needed.

5:00 As I intimate, although I'm going to work down the details, even the numerical computations If you go into predicting the weather, it's one of those levels of precision that's a problem. I've arrived at the basic context that I study, and it has to do with category E, which is And the basic idea is that this is more cohesive, this is less cohesive, as Herndon emphasized, everything needs to be perfect. And so even what we call non-cohesive relative to a certain discussion might turn out to be cohesive with respect to another, or a person, basically. This may or may not coincide in some way with the previous notions of connectedness based on others preserving coordinates and all that, which is a possibility. So this voucher determines then three others by adjointness, non-street as a special case of street and cohesive as a special case of street.

7:30 As I say, I think of this pair from Cantor's, what he called the Magnum, imagine that his fundamental move was to deny that and say, what else is relative to that? This is thought. As a contrast, in particular, the idea that there's a special case. And finally, we have the code streets to ensure, in a computational way, that the point structure... That's the main use, except, of course, in order to, of course, there are many such well, sacred dust powders, conditions, the most fundamental are the pieces of preserved powder.

10:00 This seems to be quite fundamental to the whole thing. I might even say, in the spirit of the radio, that it's not true you should change the categories to make it. Now, in this case, that would probably mean changing the base, conjecture, for those who... Now, congrats for that. Even if your original topology is defined over group steps, it should be possible to raise it up, to factor in the rest of human work in such a good way, such that the first one is backing up from Cantor to Galbraith. We find that for many reasons, the algebraic geometry over the non-algebraic field, the proper progression of the interlocking set is not assessed, but rather an operative of the topos and atomic sheaths on the category of finite field extensions. Because that absolutely has much better properties relative to the set as it still is. I don't know if these are properties or not. I'm going all the way to Kantorian deviation. But this, so this, this plays a crucial role in many ways. For example, the fact that these three adjuvantnesses are all enriched adjuvantnesses. They're all enriched relative to the base K. That depends on this, these K enriched adjuvantnesses.

12:30 Some of them may even be enriched because of this. We can define the labels of category just by explaining the degree of k of each category. Defining this branch in my paper, that should be, this defines a new category, it depends on the type of defunct or T-shape, or at least the closed function. Now, ideally what we're trying to achieve is that this, that is, this is a new, or I think it's a closed... It's easy to calculate. This is, again, Cartesian closed and extensive distribution and from this property. And it has this puncture. It's a, it has a very strong cross. It should have a very strong cross. It's not the right edge of the thing. Fine in this, in any case, but this is a very desirable property. The contradictions which are both in the work of Gabriel Wiesmann. It is basically that while category is the official objects in K, or cubical objects in K, and so forth, this kind of category is an example of all the other categories that it doesn't quite achieve.

15:00 This is where the whole story of fractions, the basic construction of Eremius, which works for continuous categories, but doesn't quite achieve everything that Plunger wants with a category. Something as, this statement, this statement, that's the intuitive idea of what are you doing with your past and the components of your science, you're declaring that what were previously components are now also the point. The problems are represented by one, clearly, you put x equal to one, now with one y, it's just a component of y. So that, that's why it's easy to, it's easy to see, which says that the The thesis from here not only preserves finite products, but infinite products as well, where it can be expressed in finite terms in this way, that we take a constant object, so-called constant k, and raise any space to that power, then the components of that are reduced to the power k in the lower category.

17:30 It seems to me often a good idea to think about what you might call a finite tool, this type of thing will put a dramatic picture there of the pulmonary types of these objects will satisfy this property. Again, as I say, in the infinite combinatorial case, this is not true, and so this led to the condition of fractions. Specific problem, not this is probably only a fragment. I mean, their actual goal was to retrieve the process of homology, and you can perfectly hand it back in extensions, but obviously, again, this actually was almost too obvious to mention, so air quality is one of those terms that I've introduced to many concepts that are sort of automatically generated, almost like objective wonder, under every statement, and we are generating statements, but as a byproduct of constructing... The concepts that flow out of this sort of situation, so making precise the use of the word equality that I made in, for example, the theory of Florence, this hunter who has two adults that are the same in general, I call it equality type, in the sense it's the kind of recipient for some specific equality, and equality is itself a functor from a given situation into... The quality types, rather, turned out to be a very special case, namely where these bunkers collapsed in two.

20:00 It seems to be both a reflective and counter-reflective subcategory, or subcategory. I'll say at this point that very important strong air conditions, which is not satisfied by quality types, not satisfied by, I don't know, sufficient. To attempt to capture really the idea of a category of spaces we've really got, this actually means that if at all there exists something, it could be a children's piece. I'm asking for the existence of a contractible object. When a contractible object has to be empty, it's terminal in the array of its category. When a particular is connected, it can move on, but every function space is also. I'm using the word connected as an example to teach me people's minds. It turns out that this is true. In the case of the topos, if only an omega itself is connected. In that case, when I point it, when I connect it, all of its exponents are automatically connected.

22:30 It can serve sort of as an input to connect things, and that lifts up to the function space. For example, to show that this condition implies... We could take the power set, or we could take the partial math classifier, which is smaller, and I think of it like this, which has these fibers tuning the force out of existence. The fibers are connected, so you can see everything out of existence in some way. Visualizing stacks, what I've gotten from it is that we assume that we assume one thing, namely that 2 and k is an injective condition, as it seems to be the natural thing here, an injective object, where the k is 2 and the 2 is... Also, in the context being discussed here, contractible and injective turned out to be, say, another way of phrasing this idea is that injective objects are connected, their dimensions are connected, but as is often, it's useful to think of injective in a loose way, you know, there exists an extension, but on the other hand, in algebra, there is a total of better context to the economical extent.

25:00 It comes up so often that we're aware of it. On the other hand, it's useful just because it doesn't exist in any sense. Anyway, so, Rowe's next point was that there exists, there should exist, some object I, which would remind you of an integral, with the property that there are two points whose equalizer is empty, and by itself. So that implies it's equivalent to this side hypothesis in some cases. And basically the proof of this that he used was you can actually find the characteristic map of true into omega, this condition will mean that it's not true. One of the points, the characteristic map of one of the points are true, but then the other point would be false, and so therefore this I can be used to construct a homotopy. The truth value object is isomorphic to one, to the other. I wanted to say that the is what I call extensive, extensive quality. The central point being that the book was sort of collapsing the situation with 400 on the situation with only two in terms of quality.

27:30 But it should be, it should agree with the components. So in other words, if you take the cumulative rate of H, the cumulative rate... The ages of extensive quality, if it preserves the leftmost, by Q lower shape, is the same as Q lower star, so by symmetry, intensive quality is the opposite. Intensive quality means that the entire L should agree in points and steps. But also a few lower stars, or a few lower degrees, which I don't know if you'd like. As opposed to the S, let's say S lower star, it's actually a few lower stars. But the points of the intensive quality are the same. There is an idea of the value of the construction of such a thing. The economic is the least thing you would think of, the intensive type. It is to be hoped that this is actually an adjuvant, but it is too adjuvant, just as the extensive part of the work.

30:00 So this I call form. It seems to be an actual quality of form and a quality of substance. And as I said, somehow you had an idea embedded in the conjecture. Motivation for calling this a substance comes from chemistry, say that water is H2O, a sample of water is H2O, but we're not caring whether it's ice or steam or whatever, we're just saying that the individual molecules have that sort of combinatorial configuration and that's the kind of configuration which is retained when we pass to the canonical intensive product. Okay, what's the definition? Well, here we are defining a full sub-category of all those objects for which, in fact, these are isolated. You have the two hunkers going down there, and then you have the tangent mission between them. Oh, sorry, I should have said, in general, we have the canonical map because of... Points are, points of the space are contained in the space and the components is equivalent, and so there is a chronical map. I assume this is a basic axiom of the physics of memory work. This I call the Nolstein thoughts.

32:30 I call it the Nolstein thoughts because it's basically a statement about the existence of points. So the classical Nolstein thoughts of Hilbert was about the existence of points. The existence in an extension field. I'm sorry, Seth would say that you have irrational points, typically points, but there's something almost as much as that. Among the X's, there are those for which this epimorphism is actually nice enough. And if I consider that subcategory, it turns out that it has, by construction, P and two steps. There's an S lower star, which is... I think I hope there could be an explicit formula for the right. My construction is intensive quality. And the more exactly, I think that we need to know that we're...

35:00 There will then be another kind of mathematical map between these adjoins, which I'm calling the cooling map. Thus, intuitively, the idea is that we extract the sense of quality of an object, we're just taking the, it's as if we had a sample of material and heated it up so much that all the particles are so far apart that there are no longer any interactions. But the internal interactions at each point are the main. So that's somehow the substance, as I said, the technical form of the absolute thing will be still contained here. On the other hand, if we imagine taking the same sample and supercooling it, then it will become a bunch of superaddicts, where the points that were in the same component before all become the same point, and all the interactions between them become self-interactions. So a specific example, we could have directed graphs, reflexive graphs, being reflexive, they could be reversible or not reversible, unless the same thing goes through. If they are reversible, then of course they're more numerical invariants than they can be points. So anyway, so this all becomes quite obvious. In other words, this is extracting the points from the graph, and this is taking the connected components as you would expect. All things that help on the graphs in which every component has exactly one point. That's nothing but a graph that consists only of loops, so it can have arbitrary loop structures. There's no midterms. A boolean map, sometimes in Spanish they call this the Maddenwell's map, explains what happens when you take the superheat, which you're trying to add. Some of the information will be retained, will be contained in this map,

37:30 There's a lot of information that's not in these funders as such. The essential map, and one that I speak to, is that as low as the start of this course is up, it does not contain the other properties. There's no attributes at the top, and there is no address publisher. The information of objects are moving at the interference. The general idea is that using this sort of structure, One should be able to define all sorts of geometrically meaningful constructions and check out what they mean in various cases, like homotopy, like intensive substance. One of the, I shouldn't have said that this is sufficient, but it is true in synthetic differential geometry and also for sufficient sets that really look like categories of space. I remember when I was listening, I watched a Plains lecture in Chicago that was instructing there's functors in all these areas and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on

40:00 That's the way of neglecting it. And this is very fun to work with. Now, this is, of course, a common construction in all the great topologies, that one has a subspace and collapses it to a point. Later on, I thought, well, actually, whole basic construction should be adjoints. This one is not quite an adjoint, from spherical spaces to pointed spaces. It's much better to be considered, for example, a subspace. The space, too, is, in some sense, the correct definition of S. The day smashed, I think, involves pi-zero. So, of course, the day is connected. Oh, pi-zero, of course. So that one can work in a cyclonic category. And then, duly, by the way, sometimes you use absolute value for this. You speak from discrete space. That's why the S starts in space. The mathematical invention of both the stars and the trees is a very harsh, very harsh literature in that way. So anyway, so this, this construction, of course, this is, look at the, you've got two different focal discs, and this is essentially the articulation for that. Basic calculational rules should also be, I like to call this the exterior of a, because the,

42:30 The problem is, it's sort of like an alphabet, except it's really a code, but it's still very ungrouped there, which is the basic problem. But also, it still is the exterior of something. It's just that the A was like this, then, upon splashing, you get the exterior of A, which is still the exterior of something. So actually, there are several pieces that I'm going to work with, where ordinary language is captured by formal logic and used to think of columns as non-substantial rather than more accurate models. So there's a third construction. The exterior of the substance actually has one letter also on the composite. The substance of the space x. But I mean, the fact that this is a modern work doesn't mean all of it. But remember, this is an object whose components and points are set, so equally about the same, but that's really the same components.

45:00 So we're pushing out here, we've got another quotient, x modulo, anyone have a suggestion for a single letter in there? x modulo, now what does this look like? And in the case of graphs, reflexive graphs, well, one thing is just the loops of x, we're getting something which is internal. But here you're getting a graph which has no loops at all, it has connections between points, which points themselves at those holes, those are external. It still has the same components at the same points, as you can see. You see to what extent a general object can be determined as sort of an extension. Given two objects of this kind of develop, whatever the object is, you have these particle invariants, which you have to use, and especially if you have two sort of contrasting ones, it's a natural question to what extent is that going to determine the outcome. Okay, I'm talking about the strong and also having thoughts.

47:30 I've said that the continuum consists of ratios. This turns out to be true in synthetic and conventional parameters, homothetic, so this should relate to a kind of versus, we should really consider this fact of generating topos, not just by taking co-limits, but by taking exponentials, because that way, if the decimal is only, we can arrive and consider the famous project of Dean, and we have the combination of zero. Then the typical line of synthetic differential geometry is the sub-border like this on the monoid multiplication of real numbers which is automatically determined in terms of the composition of the categories of the next description. In this way we obtained a general of finite dimensional varieties by taking these finite limits. In that sense, the reals are the ratios of these different tests. The action is called multiplication. By the way, there is a commutative reflection. It cannot be commutative. No exponential algorithm can be done to be one. But the fact that this can come into practice is composite. The conclusion followed by this is nice and emphasizes that this multiplication is commutative

50:00 But in general, as I say, all the quantum conventional varieties can be obtained by taking finite limits of explanations of these really infinitizing objects. And lots of seasoning, I claim that this was Wiley's idea. It's an idea that we've liked, an idea that's been thought of for many years as well, but it's not really good mathematics. On the contrary, it is a very, very good idea. Instead of building up the categories, what are the endosomes? And this general said, well, you don't know exactly how to capture, you're going to get two generals to capture the, I mean, it could always possibly happen, objects with atoms, in the sense of this thesis, and that, of course, means that these constructions are simplified. In particular, I should say, this is one of the two different patterns that you start off with in a smooth world. If you apply the Antonian abstraction and negation, there's a special way that we take those objects x to the d. If d is an atom, as is the idea of conjecture in Cambridge thesis, assuming there's an object for which x is x to the d, or d is an object which is x to the right-hand joint, you get a good theory of the speed. Okay, so, however, it's just to be the things that tell itself, whether it's the objects, the components, which are related to these objects, or are related to any, the strong or the most dominant sides, for some object R. I'm sorry, I didn't necessarily mean this R. But for example, a specific example of R would be the generic speed of your brain.

52:30 A standard example of art to connect this most common sense. I say the mere cost spirit. What I'm saying may or may not have any relation at all with sub-directly irreducible objects, which is of course another argument, but the spirit in the sense of the strength of the object, since you have not just mere existence of points, of various sort of points, but you have enough so-called points that it's infinitesimal paths, or whatever. But to actually distinguish between functions is a very, very convenient fact in commuter algebra in particular, not just in universal algebra, but in commuter algebra, in order to verify equations in the commuter algebra that we construct by general simulations. It's very helpful. You can always assume that the algebra is quite a dimensional aspect of space, Which happen to be, more or less, a finite dimension of the spectrum. It's very useful. So in this case, we're going for what I'm calling the Strauss-Rostel massage for any given object.

55:00 For any object X, we add two R-value functions which agree on the substance. The engine has a core. They should be provided. Again, it's helpful to think of these composites as maps from the... Substantial core, as formal functions, so in other words, at every point you're getting a formal power series, something like that, so to say that such a factor can be extended is to say that this formal power series can be integrated, this infinitesimal can be integrated to an actual function on all of X. And the Straum-Molstein analysis is just a statement that this convergence is unique if it exists. Of course, it usually doesn't exist, but if it exists, it's unique, i.e. extendability. Formal functions. It's an easy calculation to see, talking about classes of R, it's easy to see that if the Straum-Molstein is not due for some object B, And so finally, we have the result that if the extended most common thoughts, the strong most common thoughts, again by definition means the finite limits of, what's the extended most common thoughts?

57:30 Well, this is just the fact that likely it was the simple most common thoughts would be. For maths from NESTAR, the cooling map is epic. Of course, the cooling map is not epic for graphs, but the expectation is that it's true in synthetic differential econometrics and this is what it is. One example of a situation where you have a string of four adjoints like this where the leftmost one preserves finite products is categories over sets. Strict 10 categories over n-1 categories. That seems to be a very different kind of example, or is it? I mean, how far does it fit into your thought patterns here?

1:00:00 Yes, that's an example, yes, yes. And that's a comprehensive visualization of that. Is that what you're saying? Okay. And that seems to be a good one. And is that a question or a statement? That's the word right there. Intensive. Because it does this to... The components go to the point. Ah, yeah, okay. And in the application, the first part is going to specify the point. So I see a situation where the whole, well, I don't know what the point is, but it would look like stuff. It would look like stuff and see if the whole thing goes through. I don't know. So not only the gap, but also the time. It's useful to look at. Yes, about the distribution, I should have mentioned that math, what I call substance, is not a, it's not a, not some kind of distribution. It's a parametrized method. So the method of math doesn't matter. Any other questions? Well, then that's my question. We have five minutes. Any other questions? Yes, um... I also made a discovery about the organisers, which is that at least one of them has a mischievous sense of humour. The first time that I had any sort of theory was about nine months ago, so you can imagine how I reacted when I looked at the programming source. I'd been scheduled to follow Professor LeBair. I feel like someone who's just learned to sing Happy Birthday.

1:02:30 He was told that he's got to go on stage after a capparotti and do a good bit. Nevertheless, here it is. The sole aim of this semester seems not to have been studied before. What induces here could be changed to is, in the sense that all recent categories are geopolitical.

1:05:00 What we'll be considering today is where value didn't stop rather than set from there.

1:07:30 This way we can use contrary purposes and on the other hand, see that you will notice. I'll just give you some of these facts, it's easy enough. You just have to know that you can resolve xi into its two components, r left and r right, with this site as being the counter-construction.

1:12:30 It's first of all clear that you have two sides of a, y, t, or m. A mimicking definition of the class M-algebra is that the...

1:17:30 The next thing is that the reality I'm only wishing for is the same as the reality on. It's exactly the same pair of triangles as in the classical case. And then the final thing is that this thing is, in the classical case, I gave you that cycle. You take A, one new arrow for each example so that one of those arrows points out to me and points out to this. I think I'll just swap these slides in.

1:20:00 Okay, so we're taking a finite discrete category whose objects are called 1 up to n, and a finite modulo, a finite legence, and it takes values in finite steps. So, what is this module? Well, it's just a square matrix that's now measuring probability. What is the Youngstown Task Force, then? Its pre-sheets are directed graphs.

1:22:30 And there's a certain two-sided aim property that, coming out of it, gives you the quotable pairs. So this is called that in the graph x. In a two-sided module, there are two ways of thinking. Topos are instant tasks. In other words, which topos are of the form J, T, M?

1:25:00 You can show that every Grecian topos is also a topos, but only x is equal to x, plus x into x.

1:27:30 This is the type of distributions from the Tutskys, a very rare situation. Anyway, so, problems in Tutskys do not have problems. There are many examples, but the specific problem of Tuposkys is not Tuposkys. Let me move on, but we, I feel that we have to, you know, anybody else, I feel it's probably, you all know how M. Kennison Argyra, by the, those X, which is X's, which M stands for X, and he's nice, and I'm sure that that is so true, and it's a good thing that in America, there's a whole lot of examples of sources for which you can have a distribution of our problems. Well, I've just been given a copy of your book. It's in our book. Okay, any questions? Thank you for your attention. Thank you.