Open Problems in Cartesian Closed Categories

0:00 In the morning session of Sunday morning, a particular pleasure for me to introduce the first speaker. He's a particularly appropriate person to have the presence of. Connections between John Lapierre and David Scott go back a long time and I've ever gained the knowledge to say that. In fact, one of those people who has well-estated pride in many things, and perhaps states pride in his work. I didn't recognize that there was something going on from time to time. Okay, Bill. Can you all hear me? Okay. I just thought I would say a few words, first of all, about the origins of the, what we call nowadays, the Cartesian closing categories. I think the fundamental motion of thought which is involved was clearly quite an important and definite role 300 years ago in connection with the calculus of variations because a variation in the calculus of variations means that a path in a conscious space And these variations, a variation of a point is like a variation of a name of a map, perhaps, so the way that one's saying the depth is optimal with respect to some functional. So, well, it was necessary to parameterize these paths and to recognize that they are exactly the maps when x cross i is applied.

2:30 That is nothing else except the very definition of right adjuvant to the product, and that was already, as I say, used to study what are variations and solve all sorts of problems, long before there was any notion of topology. Maybe earlier, I don't know, but certainly in the work of Molteira in 1987, in which she explicitly, well, it's very interesting because the first lines of this paper, the very first lines, start off by saying that a space has other kinds of elements than just points. Described by saying that a space is determined by figures and incidence relations, he essentially had this idea that there are several kinds of figures that interact, so points but also curves, also parameterized surfaces and so forth in a space. These he called elements. He recognized that among the elements were points but also there were others. These are not just locked together in some sort of union or untyped situation. His point was that if you have figures or elements of a certain type A, well then you have functions, functions of elements of shape A, figures of shape A, generalizing or extending the idea of function on X itself.

5:00 There being one such kind for each relevant sort of figure. So he developed this idea. Later, of course, Voltaire's friend, Audemars, in France, called these kind of functions, functionals. The word functional was not coined by Voltaire, although the subject of functional analysis was. His definition of homotopy categories is that the objects are the same as the spaces you start with, but the set of maps between three defined to be set components of the function space doesn't exist, but it affects itself for another function space, as I pointed out. It never lives in the category of topological spaces. So Horavitz, he was also interested in functional analysis where similar problems arise. So for both the purposes of algebraic quality as well as functional analysis and their relationships, Horavitz sought something better than the earlier construal as described not by its figures but by its Sierpinski valued functions.

7:30 So Huravitz put the problem, put this problem in a letter to Fox, who in 1945 published a paper referring again explicitly to this letter to Huravitz. The basic point about this paper was that, at least one basic point, was that the category of sequential, sequentially determined topological spaces has exponentials. This is something that was implicitly known before to complex analysts where they talk about sequential convergence, a sequence of functions, f sub n converges to a certain limit, if and only for every convergent sequence of points, x sub n. The randomization f sub n of x sub n converges to the limit f of the limit point. Figures mentioned in relation to the figures of a space or the elements of the space in that sense being points on the one hand, for example, and the main thing is convergent sequences, the set of convergent sequences in a space is taken from that point of view to be the determining factor, the main type of element of the convergent sequence, and then everything, including the existence of exponentials, is quite easy. Essentially, when you have a structure which is defined in a manner by means of figures and incidence relation, then essentially never, when it's described primarily by functions, whether they be Sapinski functions, or real-value smooth functions, real-value continuous functions, whatever, all those notions are parts of interest, but should be regarded as derived from... As being the maps into those special co-domains which just happen to preserve the nuclear incident structure and the achievement of exponential basis.

10:00 Ker-Avitz continued to work on this. He never apparently published about it, but in 1950, the Proceedings of the American Mathes Society, again, he says that Ker-Avitz has He defined a category called category of K-spaces. As a student, I learned about K-spaces from a book published five years later, namely one by J.L. Kelly, and many people did learn topology, and that's a very nice book. He talks about K-spaces, but he doesn't actually ever mention Karevitz, and I think many people assumed that K stood for Kelly. Gabriel and Ziesemann in their book on fractions and homeostasis. They actually call these things Kelly spaces. The basic idea, I mean there are variants, compactly generated spaces, K-spaces, Kelly spaces, and so on. They are all very similar, but the point is that one takes a very broad notion of a figure, namely a figure is in a continuous map whose domain is compact. The compact basis taken as given presents a finer role of the figure type and one, again, one that in a sense trivially achieves the desired Cartesian closure of the existence of exponentials, of course, leading to further problems to figure out exactly what that new category is like. Gale's paper is entitled, or the main subject is the Ascoli theorem. The new and better form of the Hathaway theorem, which even preceded Voltaire's, in 1897, the first international congress, the discussion of topology, where the French were saying, gee, maybe it would be a good idea to work out something called topology, and the Italians were saying, well, look, we've been doing this for years. But basically the issue being, again, the same thing, really.

12:30 How to describe the fact that a function space of two cohesive spaces, in some sense, ought to again have its own notion of cohesion. Since 1950, also in 1950 by the way, another fundamental category was introduced by Eilenberg and Zilber, the simplicial sets. This is again a category with free exponentiation. It's a category, again, which has several figure types. One for each natural number. There's a figure called n-simplets. So the figure type called n-simplets. So the simplicial sets were very important. Again, the first line of the paper often reveals what the author thinks is really important, even though, of course, the theorem may be more involved. So they say, very important to realize that the simplicial sets are not determined by their points. They have points, but another example, which is already current, which had arisen because of cohomology and cohomology of groups, was the standard complex, whatever it's called, of a group, which has only one point, no matter what the group is. Now, since 1950, I think algebraic topologists who need to use these function spaces all the time They have, in effect, completely abandoned the idea that the open set definition is the fundamental one, but in another way, they haven't really pushed this point so much because usually they're only interested in things up to homotopy, and it turned out that the simplicial sets or the topological spaces and so forth, the Taylor-Kelley spaces, they all have the same homotopy category. So from the point of view of algebraic topology, all these different categories are equivalent.

15:00 It's important that they should be Cartesian closed, but didn't care particularly which one. In contrast, these ideas are obviously, from the beginning, fundamental for functional analysis, but quite unfortunately, functional analysts haven't really picked up on this. We haven't heard very much yet about the measure theory and so on, higher order functional analysis, which lurks there in any topos, actually. Indications are that it's a very realistic sort of functional analysis, not some kind of science fiction. And yet, it has not been investigated by very many people. Anders Strauch has a couple of papers about it and had some ramblings about it, but it really should be pursued. These were recognized as being functors. We know it's a contivariant functor of the hexagon and a covariant functor of the base. And in the original paper on categories, it was given as examples of this new concept of functors. Now in 1958 or 57, they were recognized further that they are in fact adjoint, again as an example of adjoint. In 1962, on that occasion, 40 years ago, when Dana allowed me to speak in the Tarski seminar, there was a further wrinkle there, namely, it's just a sort of idiot's conclusion from Kahn's work. Kahn showed that these fundamental constructions are adjoints. He also showed that adjoints in general are unique. Therefore, if we want to study In a particular category in which these things exist, we should take as an axiom the adjoints itself. You know, this was pretty successful in logic in the sense that quantifiers were defined by rules of inference, which are nothing but adjoints,

17:30 and so the whole development of the rules of inference flows out of the assumption that basic operations are adjoints to each other, still in the same spirit. To axiomatically describe the category of categories, I took as the first axiom, and still do, the fact that it is a category, and that it has a little finite code category that helps to reveal about the objects, and it has an object called the category of finite sets, but I'm sorry, I've got that too, small sets. But the most fundamental thing is that it has exponentiation associated to any two categories, A and B, right? Yeah, and B. There is another one, B, G, and A. Now, later the philosophical significance of this continues to reveal itself, realize that actually, that this is actually the form in which A vast number of mathematical constructions appear. One can take A as a kind of abstract theory, B as interpreting that theory, and C then as a category of all concrete interpretations of that theory in that you might want to impose more properties at a smaller c as it were, but the starting point is this and in fact In some sense, always can be taken this way through your sketches. So, that's the sort of background in which I wanted to introduce some open questions. So to speak, what about Cartesian closed categories? Everybody knows why they're called Cartesian closed.

20:00 That is, I've called them, I've called these categories with exclamations. In 1965, when Eilenberg and Kelly published their great work on closed categories, They needed an adjective to describe how these ones are particular, so the point is that the tensor product is actually the so-called Cartesian product. Later we resolved the problem with that and thought, well gee, maybe actually Galileo is one of them. Anyway, fortunately CCC could also be interpreted as... Categorically closed categories, closed categories in which the tenses of the categorical problem So the point is, let's take this as a subject in its own right. That is, what are Cartesian closed categories in general like? What are typical examples of how they were made and so forth and so on? Well, it's often said that there is a developed formalism for that, a machinery for Like in the theory of groups, there's the theory of presentations of groups, word problems, reductions of word problems, and a whole technology of symbolism which goes into presenting groups for anyone to calculate the presentation. It's often said that there is such a system also for Cartesian closed categories. I wasn't present, but Eilenberg is said to have remarked when he heard Someone pointed out to him that a lambda calculus is really nothing but a Cartesian closed category. So obviously his joyous response was, good, that means we can forget about lambda calculus. But in fact, I have the impression, I'm not really an expert on the whole literature of lambda calculus, but I have the impression that by and large, One is rechurning again and again features of the free Cartesian closed category, whereas, as in group theory, we'd like to be able to present examples which are not free. Most of the actual concrete examples that you come up with, like pre-sheet categories, like directed graphs and so on and so forth, are not free, probably.

22:30 Tell us anything about them. In particular, will it be of any help in solving the following problems? So, as we know, the first achievement of domain theory was probably the fact that there is... Basically, there are just lots of Cartesian closed categories, which there are lots of objects which are their own exponential, self-exponential. Maths, you see, could be thought of as an equation in the sort of Cantorian abstraction of the category, right, in other words, or actually it was a Burnside rig construction because of associated to any Cartesian closed category with sums, let's say. There is what you might call an exponential rig. This is an abstract algebraic structure essentially Satisfying Tarski's high school identities, because 11 equations for addition, multiplication, and exponential natural numbers. Rig is the word that Shanwell and I devised. Rigs have the property that by adjoining n, you get rings. That's the negative. And the adjoint is the left adjoint to the obvious conclusion. What rigs are, indirectly. Coming commuter to rings, that's not a rig. We're finding that it's actually quite useful to look at algebraic geometry over rigs, not just over rings, so for example, over rig geometry. That's another story. So anyway, basically we have addition, multiplication, zero and one, distributive law, commutativity and associativity of both operations, all of which are obvious properties of addition and multiplication in a Cartesian closed category.

25:00 So then if we add to that the binary operation of exponentiation and the well-known laws of exponents, which are also theorems about isomorphisms, Cartesian post-categories, then we see that in general, if you start with the Cartesian post-categories and sums, You make the Cantorian abstraction, that is, you consider each object in itself, where in itself means up to isomorphism, and you forget about the other morphisms, as crucial as they were to defining these operations in the first place, and you get this abstract algebraic structure. So, already, we call this whole thing objective number theory. So the objective number theory involving just addition and multiplication, the Burnside rigs associated to categories with addition and multiplication, these are already proven to be quite complex and some of them interesting, some of them in particular also everything for computer science, but now I'm looking further ahead and say, well, what about adjoining exponentiation to that? What kind of exponential rigs can actually arise in this objective way? By objective number theory, we mean the study of rigs, exponential rigs, and so forth, from the point of view that they are abstractions of actual objects, that is, the elements and quantities are abstractions of actual objects. His definition is because there are some kind of stupid things about zero that he wanted better to avoid, but at first, you know, X, sort of an infinite number of identities that are really not very interesting, just about, but, so, you know, from his point of view, looking at the purely equational theory, that was perhaps a wise move, but obviously, from our point of view, we can't believe zero, for example, perhaps we can change things by working on it. So anyway, so the situation with respect to that is that a few years ago, remarkable discoveries were made to the effect that Tarski's conjecture was wrong, namely that there are identities for the exponential rig of actual natural numbers, which of course is the objective number theory of the category of finite sets. That's really the best way to think of natural numbers.

27:30 All of this is necessarily true in all cases, which don't follow from the obvious axioms. So, in other words, the natural numbers have more identities than follow from the obvious axioms. Now, the examples for that, as far as I know, that have been constructed, are not objective. It may be that the exponential rigs that arise from Cartesian bonus categories are almost very general, almost typical, or maybe they're very, very, very special. So information on that is very much needed in order to progress toward the question of whether the sort of objective analog of Cartes' high school problem, and of course all sorts of other, not just the equational theory, This equation is not going to be true for all objects, probably the most special property, you know, more than just an equation constructed with, well, since I've got a million more of them, which is the next problem. I'll point out that Horavitch's homotopy category, in order that we don't have to worry too much about it, let's say these are simplicial sets, and some other equations. Horavitch's definition of morphism gives rise to, so this is a functor which preserves

30:00 Exponentiation is strictly reserved up to isomorphism, as is multiplication. This category has, that probably was never dreamed of in lambda calculus, maybe I'm wrong, maybe the following. See, well, first of all, I should say that there is some sort of discrete space, notion of spaces, just think of them as abstract sets, as they often are, and the discrete And this inclusion actually has both a left-adjoint and a right-adjoint. I can just take the points from this. This is the Cantorian move. You see, it threw away all the equation and the number of points. So this vector pi-zero is what Cantor called cardinalzau. He did not make the further move of Cantorian abstraction. To arrive at the name solid, because carbon non-solid was already the abstract set, and we made it further abstract, of course. We also studied what are now called carbon numbers, but unfortunately the word solid has been changed there. In any case, the points. So when we talk about the set of discrete space, the maps between two arbitrary spaces, That is realized as being the points of the function space. That's how the abstract category underlying x is derived from the Cartesian flow structure. And of course that means in particular the points of y, we can always think of as om of 1, om of y.

32:30 But on the other hand, the left adjoint, added to a crucial special property, There are millions of examples where you have these adjoins, but very few where pi-zero preserves finite products. Pi-zero, in fact, the idea that this really gives a category at all depends on the fact that pi-zero preserves finite products, because we have to be able to define the composition, homotopy classes from x to y, homotopy classes from y to z. If we rise to the moment from x to z, but if you remember that this is pi zero of a function space, and this is pi zero of another function space, the point is that since pi zero preserves products, this is pi zero of y to the x cross z to the y, and of course in any Cartesian closed category, the composition is lowly triple of points expressed by a single map. So you take phi z, apply the number phi zero to that composition map, and you get that into phi zero of z of x, which is by definition that other thing, so the definition of composition in the Eurevich category seems to depend on at least having a map in one direction, but phi zero of preserving products, let's say. But now the distracting thing is that actually Again, it contains the discrete spaces, it didn't change them in the past. But now you see, by bracketing, the one comma y, pi zero y to the one, pi zero y. So the points, quote unquote, from the point of view of this criterion closed category,

35:00 have become the components in the previous sense. But on the other hand, what happens is that actually these two counters are equal. The left and right adjoints are the same. So the pi zero has become both left and right adjoints to the discrete object. So we have a Cartesian closed category in which the inclusion of discrete objects exists because, well, it's another story, but you can see how to do that. So this is really an inherent property of a Cartesian closed category. Now by the way, that depends on a bit more. It depends on the pleasure of preserving discrete powers. And it's related to, if you look at it concretely, it's related to the fact that one of the realizations of the... Involves the continuous intervals, which can be subdivided as much as you like. The simplificial picture doesn't, you know, you have to generate the equivalence relation of the homotopy. The generation involves proofs of a certain length, and so if you have an infinite product, that is to say, discrete power, you may have proofs of different lengths, and so there's no single proof. But by contrast, with the continuous... The continuous interval seen as the object which is affecting the components, that is, things are in the same component if and only if they can be joined by a continuous path or some kind of continuous interval, then that's the further equation you need to verify that indeed this pi-zero-like function really is also the right adjoint to the inclusion.

37:30 So the conjecture now, I think, is obvious in my picture. We have Cartesian closed categories in general. We have a Cartesian closed category with this striking property that the right and left adjoins at that level are the same. Can we find a universal ? If you can't find it in that context, then you see that the hope would be, one conjecture would be, that you don't really need these two conditions that I mentioned. You can simply force everything and get the Cartesian closed category with this property, which will happen to be Ruriewicz's one in case those two properties are true. Homotopy theory raises its lovely face. This property of having left and right adjoints the same also comes up in another way, namely to do with infinitesimals. This is a sort of intuitive picture of the infinitesimal, that is to say the figures, the infinitesimal, figures in an infinitesimal space, not necessarily that the space itself is small, it might be big, but it looks like it has points, and then these points are surrounded by little infinitesimal clouds, imaginary, since it has emotions away from those points, the reason they're imaginary is they never get to any other points. The purpose of these things is to condition actual motions by means of differential equations and the like, as I may be able to mention at home.

40:00 Anyway, so this is the kind of picture I have of an infinitesimal space, and you see, again, it is somewhat discreet. It does have this feature that the left and right adjoins are the same. This space has exactly as many points. All of these are categorized as it has got components. Even though the components individually are bigger than the bare points, they're in one-to-one correspondence. Now this happens, for example, in algebraic geometry or algebraic number theory. The speed is not the category of sets. I often find out that actually it was Galois who showed that the category of sets is not a good base of those in geometry. Because, you see, these might actually be points defined over different fields, because they'll be the bare points or the rational points defined over one, but, so essentially, these discrete spaces could be, in a certain context, could be the spectra of products of fields, extensions, basically, not all the same, whereas, by contrast, this X here would consist of Similar spectra of by like products of just arbitrary finite dimensional algebras to the base. So that essentially these, the one way that you detect these infinitesimal clouds is precisely from the function point of view. Not with the Sierpinski phase, but with the Algebra of Mind, where you find that there are, in the function algebra, such as things, there are Milko developments. There are some very simple examples like that. If you take a single infotainment rating on sets, and that's a category which keeps coming up implicitly in all sorts of discussions even here, I think, well then this is obviously true, you see, because the cloud now is all the things that have that canonical form or that particular champion or hero of the village or whatever it might be.

42:30 If you just have an idempotent structure on the universe, there's the fixed points, which are sort of the chosen leaders or whatever, champions or whatever of each group, and so, but this object has again this property that the components, the number of components, the set of components, is exactly the same size as the set of points, even though these clouds themselves can be arbitrary size. So they're very simple examples. Now, so again, there's a question of, is there, what are you, the point, part of the point of this is to say that if one succeeded in finding a universal way of mapping any Cartesian closed category to one in which this was true, and thought of that as the sort of qualitative essence of each of objects. That is, what remains after all the motion that the collision makes possible has actually been performed, so that all things last. Well, there might also be something I'm talking universal on the other side, because again, at least in algebraic geometry, we can—these infinitesimal spaces on which these two things agree are a tiny subcategory of all algebraic spaces. Including the functional analysis over that and so on. Where these adjoints exist, they of course are different. So really, if you have these adjoints, you can just extract the objects that satisfy the cell radius equation. So that seems to be something that's universal on the other side. Another Cartesian closed theory, the topo exact.

45:00 It's probably universal on that side. But now having mentioned infinitesimals, Another, even more striking property that infinitesimal spaces tend to have, and now I've changed, I mean the simple components, the connected infinitesimal spaces, as it were, connected infinitesimal spaces, seems to have a strong tendency to be ATOs. By ATO, this is called amazingly tiny objects. When I introduced this word amazingly 20 years ago, I think people thought It was just a matter of enthusiasm. Of course, there was a lot of enthusiasm, but actually, it's supposed to be the translation into English of Grosvenor's notation, because Grosvenor got this notation, you see this, it has an F lower star, a left adjoint of that called F upper star, and these ingredients will always have any geometric morphism of toposes, but then sometimes you might be lucky and have something further. Which is left adjoint to that. So that is determined by a lower exclamation point and of course that implies in the offertory the exactness of F-upper stars so any triple adjoints like that does determine the subtle kind of morphism of toposes. But then sometimes one had instead an extra adjoint on the other side. This is called F-upper street. So the idea of the yoga was that the streets were there in exceptional, surprising cases.

47:30 And of course the upper and lower has to do with whether you're going forward geometrically or backward algebraically. So anyway, so this, the existence of these kind of spaces is connected with the, directly connected with the exterior right-hand joint. So that's what the word amazing means. So that means these are objects A with the striking property that the exponentiation with exponent of A has a further right adjoint, which I've been denoting, I believe, as a fractional exponent. So this means basically that functionals of arity A are actually just functions. Remember, ordinary functions. On y, you see, from the point of view of y, they're just ordinary functions, except that they take their values in a much bigger ring or whatever, rig or whatever, you see, I mean, being a right adjoint, any algebraic structure is preserved, so whatever structures he has, like the real line or the ring or something, that again will be carried over to this larger object, perhaps indicating Carnot, as I sometimes wondered at. I've been criticized for thinking that heat is a function and it should be a differential form, but no, actually, now we know that differential forms, which are a special case of these kind of functionals, infinitesimal erratics, are actually, you see, functions of space, just as they take their values into place. So the tininess, well, many aspects of the tininess, you could say, but one point is certainly that being a, this function being a, left a joint must preserve sums, so we say that A is strongly connected, internally connected, because if you try to map one of these ATOs into a disjoint sum, it has to go into one piece, it can't grab it.

50:00 But then, of course, it also means that Latin QDA preserves epimorphism, preserves quotients, so it's even very special among the connected objects. Now, I've adopted this terminology because another terminology which was used by Thomas Koch in an excellent book a number of years ago was atoms. Fortunately, these things often participate in something that starts with M, namely matter in motion. So, these objects equip with a certain further structure, forming these little riches, sort of, called ATOMs, amazing tiny objects. Realized here, as a fundamental example, tangent bundle function. They've been with us implicitly or explicitly for 300 years. The key ingredient of differential calculus is the tangent bundle puncture. On the other hand, there's this general fact of Yoneda and Kahn and so forth, which I started to think might be significant 40 years ago, that adjuvants are unique. Representing objects are unique, in particular. So, if a functor is representable, there's only one object that can do it. Tangent bundle functor is representable in several categories. So there is a fundamental kind of object, either the basic or differential calculus, that needs to be studied. D turns out to have all these properties, and it has a little object which represents the tangent. Proof, or at least complicated from the point of view of functional analysis based on topology, based on contemporary information, for example, if you take an arbitrary function space and that's what its tangent bundle is, well that's obviously by the laws of x-quantum another function space where the co-domain has been replaced by the tangent bundle of y.

52:30 It should be, and is, a totally trivial elementary calculation of fact once you recognize that the functor is representable by an amazingly tiny object or by any object, a cheek, but as I say, although the fact is needed, I think probably most functional analysts think it isn't always true, even, because it's so hard to prove in the cases where they prove it. I think the recognition of the representability of the tangent-thumb of the concrete is already implicit in Leibniz, actually. I think I'm misreading him. He said there should be an extension of the real numbers, which, for education, I think misinterpreted. There should be some kind of elementary extension in standard analysis. But in fact, Leibniz said something like the complex numbers, which is not an elementary extension. Now, in fact, there's something much simpler, the function algebra of the representogram, the dual numbers of Studi, because Studi was someone who 100 years ago recognized the key role of this algebra and its role. It probably is what Leibniz had in mind because the rule of multiplication in it was nothing else but Leibniz's famous product rule.

55:00 This was all made, again, much, much more explicit by Koehler, and so it became overwhelmingly incorporated as part of the standard machinery of the theory of algebraic groups or something that had to do with spaces as such, whether they had group structure or not, because of the particular efficiency with which this point of view leads to the practice of re-algebra of the group and so on. It has received its most explicit attention from people who work on the new structure. Anyway, there's again the obvious problem here. Can formalism like the Lambic dogmatists be extended so as to account for particular objects which have these extra bright edges? There's a further way of generating types, a further way of generating terms and so on of importance. We certainly want also to be able to present Cartesian closed categories with some specified ATOs, atoms, as it were. So, finally, the last problem I wanted to mention was this, something probably, now there probably aren't as many about this, I just don't know it. A topos, why is a topos a closed category? The original axioms said so, but the laws did not go to these original axioms. There was a theorem. The French knew about it, but they didn't take it as an axiom. Anyway, a more minimal presentation of the notion of topos is simply, basically a category with power set. What does power set mean? It means something that represents relations, etc. And the truth value object, in particular, implies, as a special case, Cartesian closure, just as we always believed it did, namely, it's possible among many things, if we have relations, we can think of maps as special relations.

57:30 You see, there's a question, I guess you'd call it proof-theoretic strength. Is having the power set really a lot stronger than having exponentiation? Of course, that has to be conditioned by various sites to probably get a reasonable answer, but, intuitively, it seems that it does because of the basic construction of the internet intersection. Once you can represent sub-objects in both ways, then you can collect classes of sub-objects, and you can take the intersection of all the sub-objects, which, for example, contain a given thing, of all the sub-whatevers, They contain a given thing. So this is the definition of the subgroup generated by a set of elements and so forth and so forth. It's an infinite intersection, which is meaningful in the strong, you know, internal sense, or whatever, in the category where you have a power set machining. And so in particular, you can create bad infinity that way. An object which is not dedicated and finite, an object which is not dedicated, which is dedicated and infinite, means an object that has, let's say, a monomorphic endo-map, but not a, for example, if you take a half line, you can shift over it by one, that's monomorphic, but it's not surjective, so, in other words, there are lots and lots of very good geometrical reasons for things that should be objects that are dedicated and infinite, but then, which in itself is relatively innocent, you see. It's complicated, but it's called theoretically doable, computable. If you have the power types, then you can take a point, let's say in a half-time, take the smallest, intersect everything that contains that point, and it's closed under shift.

1:00:00 And you get an object, which you can then prove, oddly enough. All of this is recursively generated. It actually satisfies the recursion. You get this left adjoint kind of mysteriously out of this right adjoint, this infinite intersection. We know from bitter experience that the sense in which this natural number object is really generated is slightly illusory because we move to another category. The same thing is no longer that generated. There could easily be things that are missed by... Which are just not yet by this intersection. So we have to construct this subjective infinity, this left-side joint, within the given category, if it has... So, the question is then, in what sense are the exponential types really tamer than the power types? So there's a whole subject in logic, which goes under the strange name of O-minimal. Theories, hominimal models, and so on. McIntyre. Ever since I realized that this field exists, I thought it was probably of extreme importance for geometry foundations of physics and analysis and so forth. But, of course, most of that work is, you know, not in terms of higher types. Now, we know basic methods, you know, native embeddings. We're embedding any category into something with higher types. But if we just take the things that are generated by exponentiation from some objects which are, for example, I will not lie, the question is, how bad is that from the point of view of recursion?

1:02:30 Is it really just some more or less soft extension, or does it really introduce, does it, for example, you see this whole business about girdle scare is effectively bypassed by these... The key point there, subjectively you can put it within the decidability of various theories and so forth, but there's an objective aspect which is at least almost equivalent, namely that every object has only a finite number of topological components, so even though you have quite complicated functions in higher dimensions, you can take equalizers and so forth, you can produce things with many components but never with an infinite number, with a space having an infinite number of components. In another way, it sort of automatically would lead to a number object and hence to undecidability. So the question is, if we take the exponential category generated by, for example, a homonymous model, does it have objects with potentially many components? Or even if it does, can these somehow be mastered in a way that is not so undecidable? A couple of key spaces as a subcategory of topological spaces is that products are no longer products, and that confuses you if you don't take this, which is only another reason why topological space is kind of good. In other words, it's precisely, you see, that the following thing arises in analysis. Of course, you have a group. The group is acting on space. Now that there are functions on that space or measures on that space, the group should still act on that. Well, this is all quite trivial if you have a Cartesian closed category. But with topology, it's not true, because you find lots and lots of papers and analysis are revolving around the problem of separately continuous action, group action on the functions base that is merely separately continuous to the verb, but it's not jointly continuous as it should be by ordinary classifiable mathematical prejudice.

1:05:00 What's good is the products are different. A little while ago, several people joining in this room are studying ecological spaces, opological spaces with an equivalent selection. The sand becomes a category. Have you thought about uses for that? Do you think it's a fruitful thing to look at? No, I haven't thought much about it. I think, yeah, I think is in that category actually a direct limit of tokos. If you take larger and larger spaces as a site, I think it's actually, it has all the exactness properties, but the only thing wrong with it is that it has a small set of generators, so it can be viewed just as a direct limit. But there are a lot of subcategories there that are more or less the same. Yeah, I mean, that's certainly an attractive, large environment where a lot of things happen. So, for some reasons that have to do with computer science, I got involved in studying this. That's the guys who led to the program. Yeah. So we'll be talking about that in a couple of days, but the amazing thing is that the wiki counter-samples have objective number theory meaning. Ah, great, yes, thank you. And this has consequence for the, yes, for the equation and theory of bison-morphism. Thank you very much. If I confess to some embarrassment, I'm not quite sure which of the...

1:07:30 Two advertised speakers is the next speaker, and I'm also in the embarrassing situation of not having met the next speaker, which means that I don't get around enough, so can I call upon either Benjamin or Roberto to come and give their talk? Or are they not here? That's an exciting moment. So, I feel that if you organize those things, you know, you never mind emailing them. There's some new information here that they didn't get money to count this. Sorry. Yeah, right. Probably lying to you, because they're resilient. I had heard that today in the morning and I was coming with another Brazilian girl and she said that they, well, we're having some economical problems in Brazil and they didn't give them anything. I thought that was Argentina. Well, we're closing up, right? Yes, I guess. Okay. Well, I'm not sure that I, I mean, much as I'd like to entertain you for some long, long speech, I'm not sure that it's really as profitable as all that. Perhaps we could all have the opportunity to have a few quiet discussions, which is what, after all, workshops are supposed to be about. I would propose that we quietly retire. I also propose that at the 11 o'clock session, the speakers have a little more time for their talk sessions. What if the 11 o'clock is, as it were, identical? Then the other two speakers can have more time. Well, they can have more time if they want it, but they shouldn't have more time if they don't. I'll see how I have it. Very good. I think I'll close this session. We'll start later. So thank you all very much.

1:10:00 And so on and so on and so on.

1:12:30 I'm sure by using this computer, I'm going to be able to learn a lot of things.

1:15:00 I've read a lot of these works on mathematics, but I don't know how many of these works I've read, and I don't know how many of these works I've read, and I don't know how many of these works I've read, and I don't know how many of these works I've read, and I don't know how many of these works I've read, and I don't know how many of these works I've read, and I don't know how And more generally, there is an ability. So most times when we think about quantum theory, it has to be personalized. It has to be independent. It's not that it's made out of a finite number of weird things and it doesn't have any good meaning. It should again count as good. This is the qualitative difference between good and bad. Precise interpretation of all of them. This way we don't have a limit.

1:17:30 So, taking the topos generally upon that should give an example of the integral being totally already known, such that we never notice a difference between the specific goal of the problem with the pathological functions of the physics. Thank you for your attention. Thank you for your attention. Thank you for your attention. And there is still something that I don't know exactly what, but I think it's also interesting to look at that. Thank you for your attention and see you in the next lecture. Thank you for watching.

1:20:00 I think I can tell you something about little books, little books. I don't know whether they would have that to see if those are children's books. I found a desperate precinct across the other side of the, I forget what the square is called, at the end of the street where my hotel is. And I was in a little Greek restaurant the first night there. It's just across from the shop. Fantastic. You go past the work of music on the way. Oh, third race of time. We can walk around the place. It's absolutely amazing. I've had a lovely talk and I've got very, very good satisfaction from all of the lectures this time. But the one thing I was quite surprised was that you didn't mention, of course, making the point about the, you know, the specialness of the case, where the components from the first product was, but of course, the initials of the first product is capital.

1:22:30 I'm still trying to get my head around it in this way. I'm just wondering how, you know, it's quite like, if it's something very possible that you're a petite, but not really a, not somebody who's a proper, I mean, it's a kind of a two-way space. Anyway, so let me... Thank you for your attention. Thank you for watching. Thank you for your attention. Thank you for your attention.

1:25:00 I have permission to buy some pixie books for our new friends at the bottom of the list of best children's books. Well, some of them are. Some of them are. I know. I was looking at them in the catalog and I saw some of them. No, no, they're quite different than what is common. They seem to be formative. It's like that. It's like a pumpkin is going to come and eat all of us. Thank you. Thank you for your attention. Thank you for watching.

1:27:30 Anyway, that should be coming out in a few minutes, but now I'm going to check on the process of the time for the education program, because this is difficult, and therefore I'm going to have to wait until the end. We were planning on the last moment to handle Gonzalo's retirement party, but we couldn't make it. I wrote to him and said, is it Marie's 15th birthday or question mark? Because he wrote about the big particle, he didn't mention what it was. So we thought it would be a good time to go there, and we all got very bad sick, so I wrote to them and decided to make it a very fun lecture. No, because they went into the mountains, so they have a little hot water, so Gonzalo wrote to me that it would not be easy to reach the next. So are you working on further aspects of mathematical equations and all that? Well, I think we are not really able to finish well, also because I have not been quite a mother for the last... Five, six months, which is one about the heat equation in distributional terms, because there's some sort of functional analysis that is airy, whereas the wave equation, that is essentially finished and easy and works synthetically, whereas the fluid heat equation would have to go into a particular tool to make it work, namely the Caillou tool. So, this is incomplete of what we have posted on the Rosalabos website, but it's incorrect to find the heat equations concerned.

1:30:00 Thank you. I was invited to a French-language conference in New York and talked about distributions. Thank you. Thank you for your attention. This is the thing you're looking for. That's right. The introduction starts off by saying, Thank you for your attention.

1:32:30 Thank you for your attention. Thank you for watching. Thank you for watching. When he started translating conceptual math, he also learned x-y-x. He became a more expert in the community of physics. But that's easy to understand. They sort of can't insert that. Bottom has been fighting in my absence. Well, I'm sure she's ever stood in the fellowship. She has. I hope you enjoyed this video. If you did, please like, share, and subscribe.

1:35:00 Several jobs have been made available to the faculty and staff of the University of New York University of New York University of New York University of New York University of New York University of New York University Thank you for watching. But it's two, three natural numbers, and it has to be followed if you're going to come up with the opposite answer. What's the name of the opposite answer? Forty, apparently. Tell us number three. Forty. I see that. So he found this example. I mean, the equation was three. Thank you for watching. There's a lot of people playing with topology, but there's a lot of people playing with mathematics, and there's a lot of people playing with mathematics, and there's a lot of people playing with mathematics, and there's a lot of people playing with mathematics, and there's a lot of people playing with mathematics, and there's a lot of people Speakers include mathematics, geometry, algebra, mathematics, physics, quantum mechanics, physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics,

1:37:30 And then he put me in a place where I couldn't do anything either. The West Coast, yes. There's a lot of north-western callings in the United States for mathematics. North-western people in the U.S. are really proud of it, so they're really proud of it. I believe this will go to the end, that's for sure. Thank you very much. Thank you. Thank you. Thank you. Thank you. Thank you. I'm not sure what the path weight is, but I've tried to find it. I've tried to find the path weight, but I've tried to find the path weight. Thank you for your attention. Thank you for your attention. Thank you for watching.

1:40:00 I don't remember if I've actually been together in Paris, but many of us grew up in Paris. Thank you for your attention. I think that there should be some kind of a random section, a random section of the people, and subtraction of the local functionalities. So, basically, the process of distribution. So, there's a distribution of code. So, of course, how you do it may not mean you have to be a scientist. Thank you very much.

1:42:30 All of this is part of a series of lectures on mathematics, geometry, algebra, mathematics, physics, quantum mechanics, algebra, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, Thank you for watching. And it's, it's, it's, it's, it's, it's, it's, it's, it's, it's. Thank you for your attention. Thank you for watching. Thank you for your attention. Symmetric? Symmetric? Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah.

1:45:00 Yeah. Yeah. Thank you for watching. Thank you for your attention. Thank you for watching. It's quite specific in the sense that it does not take care of all the elements. Speakers include mathematics, geometry, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, mathematics, physics, algebra, The inclusion may not preserve quotients, but preserves new ideas, new products, and the adjuncts present subs of quotients, and so on, helping me to learn how to use them all, and so the quotients depend on exactly which way you apply them, and so in particular, it's allowed me to express them both on the side and on the other side, and I suppose it's quite easy.

1:47:30 The names of the atoms in the quantum parameter are different from each other, because each of the atoms is characterized by a new conjunction that one can think of. So the linear spreading can be based on, as I said, pairs of first-order amounts, and so on and so forth. Thank you for your attention. Thank you for watching. Thank you for watching. Thank you for watching.

1:50:00 The category of success, it's like science, by the way. It's like, you know, you've got to do the best you can to get the most out of yourself. And that's the two, you know, I've heard before. You've got to get the best you can to get the most out of yourself. And if you're going to do the best you can to get the most out of yourself, you've got to do the best you can to get the most out of yourself. And that's the two. You've got to do the best you can to get the most out of yourself. Thank you for watching. Thank you for your attention. Thank you for watching. I was confronted with this really in my encounter with the United States of America, where it was not about mining, but it was generally about the relevance of the elements, rather than the functions of the elements. And on their end, I systematically learned that there is a labor force, which is generic. Their labor force is not precisely the same as the force of the economy. That's the sort of generic their labor force is. Well, of course, it's not an element, in that sense. An element, of course. Well, it's not an element. An element, of course. No, it's not even an element. It's not an element, anyway. Right? Because it's an element, I would say. An element, a figure, a shape of being, is a derivative.

1:52:30 Thank you for your attention. Thank you for your attention. For example, we talk about elements, where, of course, it has a label, but the extra part of the D, or the generic extra part of the D, requires some verbal explanation, and the explanation of the exact figures. Thank you for your attention. Thank you for your attention. Thank you for your attention.

1:55:00 Another point, which is generic enough, is that you cannot express it in terms of figures, so the standard took us in the same direction. But I'm sure it's the original distribution of math, mainly because it's based on the real answer. Based on the real answer, you do a lot of explanation. Thank you for your attention. Thank you for your attention. You are referring to some of the things in the talk about the topocentric functional analysis, except for the things in the online talk about the topocentric functional analysis, right? Yeah. Ah, Friday night. That's it. So, the general point you write about... Yes, sure. How much external does it work? I don't expect so. Implicitly. I mean, there's nothing in the field. I mean, there's no size of the basic general structure of the space. Now, does it work? You're only ready to be structured up right now. Thank you. Realize. The functional analysis is a classic illustration. There's been a steady exchange of tools, but really low conceptual understanding.

1:57:30 The idea that part of it must simplify is not very agreeable in some ways. It seems actually really remarkable that the function algebras and distributional spaces There are, in fact, that complete technique. Take any distribution, take any math or mathematics, take any distribution on x and y, take any distribution on x and y, and if there's any interpretation as a form, a bunch of answers, it sounds like it costs a lot of money to do that, but tautologically, by the way, of course, it kind of costs a lot of money to do that, but it comes automatically. In the monadic category, you have three algebras, you know, so it's nice to have that. Esoteric space is the most concrete one, which actually has that. To what extent is the infinity of elements where they then have this kind of infinity? Well, that was Walbrock's paper in Volume 1 in Functional Analysis, specifically about the infinity of elements. Let's see, what is it then that... He showed there that... I mean, the difficulty is that what I'm saying about this completeness is true for... Function algebras on finite dimensional manifolds, but of course we have function algebras on arbitrarily big spaces, and it's not clear exactly how they... Well, maybe I'm putting it wrongly, but the problem with C-infinity...

2:00:00 It's precisely the fact that the central part, i.e. the core part of billions, does not have an interpretation in purely linear terms. Oh, yeah, yeah, yeah, yeah. So that's right. There is this nice monadic category, but whether it has a tensor or not is not so important. Thank you for your attention. Then the person comes back to watch and says, hey, what did I tell you? It doesn't even exist. Well, I think nobody, because of the non-appearance of a zillion speakers, they're now starting to look... Thank you very much for your attention and I hope to see you again soon. So, what is the object of the black cloud of the Milky Way? So, somebody explain, right? So, the thing is, you know, you remember or know Wilkie's identity. I can tell you. If you want, I can tell you. I don't want to impose myself.

2:02:30 The questions are true. In fact, for the Witten equations, you need... No, even one generator is in there. Oh, yeah, it's also for anything. Okay, so that's like the torsion tree. Thank you for your attention. So the surprising thing for me was that the film, you see Miki's identity and his justification for the fact that it's true for numbers is essentially because you factorize some polynomials by a polynomial with integer coefficients and you place some algebraic tricks as a manipulation. So there is this proof, there is another proof. The fact that you have the cancel function over there. Exactly. So I set myself to prove that this thing could not be an isomorphism in the three things and after understanding the three categories in quite some detail what it came out is that in fact you can write Lambda terms that realize the identity, which gives you a pure combinatorial proof of the identity.

2:05:00 Well, I mean, if I understand what you're saying, it strikes me as this is quite unlike these seven trees in one phenomenon. In other words, Tarski's identities, the mere equations, of course, don't capture the fact that these things are adjoints. You have actual maps. So in this case, that adjointness as such is implying more equations. But it could be that in fact they are equivalent. So I'm starting to suspect that my current state of conjecture is that the isomorphisms that are true in the three categories are exactly The ones that are equations for the natural numbers. I'm starting to believe this is true. Without a constant whole series. So if you put series, it's not true anymore. Yeah, as I was saying, somehow the zero messes up the essence of the problem. Yes, but it could be too much. I don't really know how one would go up to fail with such a result. Yes, I mean, you see in the written situation, the big hurdle we had to get over with, to get started at all, was that if you just consider additional multiplication, then at least there's a general tendency if you assume that objects satisfy some polynomial equation as an isomorph, then you take this equation and you... ...do purely rig-theoretic consequences from it, you get exactly the same things that are true in all the categories or evolutions, which is, you know, something to be proved because at first glance you might say, well, the category is much more structured, these things are adjoint, there might be more ways of constructing maps, there might be more ways of constructing isomorphisms other than those that come from just substitution, but the fact is, no, at least...

2:07:30 For most equations, Robbie Gates showed that you don't get more isomorphisms and you get really abstract things. But if I understand what you're saying, it seems to be the opposite situation. And somehow, things that could be proved just from the exponential grid, axioms, may not exhaust any of them. Because that's more or less Tarski's identity, more or less Tarski's identity. I don't think it's the zero which makes a difference. So my conjecture now is that, in fact, they do exhaust, but they may not, as far as I know. Do you mean what is zero? So take Karski's side. So the conjecture would be that the isomorphisms that are true in the free, well, CCC with sums, well, binary sums, are exactly the ones that are true in the, for with the ring, the ring axioms. And as far as I know it could be true. I know what you're saying. The Wiltke identity is true. The category is true. It's not a consequence of the identity. Yes. Yes, yes. I'm not sorry. That's very confusing. No, no. I said something wrong just now. This thing, so the weak identity is true in the categories as a consequence of the identities, but so the conjecture is that what is true in the category of finite sets is exactly what is true in the three categories, so in all categories. Right, right, and it was really totally the opposite sort of thing, the objective example of finite sets, not the deduction systems, what determines. Exactly, yes. This could be quite true.

2:10:00 Yes, because you can write the natural numbers, so lambda f, lambda x, f to the n, applied to x. There are infinite number of isomorphisms. It may not be true. I mean, technically, it should not be true, but I haven't thought about that. The x to the x to the x to the x is sort of like the x to the x to the x is sort of like the x to the x to the x. That was relatively simple of the natural numbers, right? They were also defined by cases. The ones I'm thinking of were not defined by cases. That's possible. Maybe you don't have enough definability to define these parameters. That's right. Interchange 3 and 4, interchange 6 and plus 1 are very simple permutations, but they're precisely defined in 3-4 cases in this church numeral thing that you may not have the possibility of. ...dividing things. Yeah. Yeah. Is that root for getting into community communication? Probably doesn't work. Probably doesn't work. Maybe some other kind of definition would work. One thing... No polynomial. ...you would have... I came across a little fact relating, well that seemed to me relating distributive categories portioned by some equation, polynomial equation, and something in algebraic number theory, which I don't know much about, but so take the three distributive categories, so this would be Shanwell's view of...

2:12:30 And now take the distributive category. Now what I'm going to do is to add an i, the imaginary one. So it's t equals 1 plus t plus t squared. So from t equals 1 plus t plus t squared, you can prove that t equals t5. The universal property of the three things from the first thing, the first distributive form, category, Adi-1 to the one Adi-9. Also, you can show in the second one that i to the 3 has the same property as i. So you have the convolution. The conjugation, so this is like, oh yes, it's the one that, oh sorry, so if you take the rig for the first one and then ask for cancellation on top of this, you get the integers. For the second one, if you take the rig and ask for cancellation, you get what is called the Gaussian rate, yeah, integers. Dimension rig has only three elements. For all equations of minor order and non-linear equations, you get sort of minus infinity to zero infinity. So the rig itself, even without that, would be a ring. It's basically the Gaussian interviews, sort of the three levels. Yes, I see. And so, the remark is that... Okay, so now you have conjugation at the level of distributive carbon. And this allows you to define norm and trace. The norm is sense x to x times x bar. Yeah.

2:15:00 And I trace the same to the class. Realization, the categorical realization of trace and norm, in my conjecture, I have not worked on that, but in fact, I mean, they wouldn't be using that. This inclusion has two adjunctions, and these are the normal traces. You see what I mean? Yes, yes, yes, yes, yes, yes, yes. That's possible. Yes, but this, okay, it has a center on the second thing, but in fact it can be cut down to the first three, and that they are left and right adjoined to the inclusion of the... The conclusion is probably not full, so the number of times in the usual is just numbers. The analog would be if the inclusion were full, which would probably define more maps, I think, than the secret pathway. So what the typical thing is, if you get the classic equations in terms of a junction, you get that what is always an equation is usually a map. It's the adjunction there, but when you reduce something, like the fractional exponents, it's not true that y to the a times 1 to the a is the y, it reduces the a canonically. So I think you'd probably do something like that, or something like that, and you'd trace it more. They may be hard to learn, but they won't be retried, because it's more complex, I think. Yeah, no, but it probably sounds like a very beautiful line of development. That's what I was wondering. No, I don't know the reason to pursue this remark. Yes, I think so. I always wanted to understand the presentations better, but I never thought about you writing down the traces of your work. I just thought because... You're right, I just thought about how you sent through it. No, no, no. It just came to me because I don't know number few, so I opened the book and I saw that it didn't match what I was thinking about. Yes, yes, yes. And that's a lie.

2:17:30 The thing is not that these are the computers, these three things you can't do. So you can define the functions but not prove that these are joints, it's going to be a piece of work. Yeah, it would be good. Thank you for your attention. We can talk about the features and tactics of construction, but it's always nice if you can physically embed them into some double-function space or something and see more concrete stuff. Maybe that's the thing that I think has been part of all these different diagrams, but it doesn't mean it couldn't be. Which would be a big help. I find it, as you say, very difficult to understand. So, I mean, we can always take precepts on these two categories and then we will get an assumption out there. Oh yes. And then it will restrict how you understand if it does. That's the, I can't think. Yeah. I think that would be great. Yeah. Well, I think Gates is a big step forward. It's easy to describe the type theory for this, but this doesn't help that much. Let's have a look at Gates' paper to see if he's kept. It's also just inconceptual though. It's not an inconceptual furniture. It's something that was reading systematically and I thought it was better than publishing it. It's the idea of an infinite zero. If you take the generic distributive negative generated by some objects and by fixed-point equations to a higher degree, it could also be a vector equation.

2:20:00 It should be hard to read. The linear phrase will not apply here, so the original negative sets are hard to read. But all these are hard ones. Okay. So then you always have just two dimensions. So an infinite object, what's up with it? Because, you know, intuitively those are the syntactical, those are the things that genuinely contain some variables. That's right. Thank you for your attention. I mean an object which when added to any infinite object doesn't change. It acts like a zero in the world of infinite objects only, forgetting about these high line ones. Now, the amazing thing is, such a thing is unique. Thank you for your attention. Well, you know, in other words, it behaves like minus one. Minus one, you know, the infinite negative one, multiplied by any infinite element, added to that infinite element, will be the infinite zero. That's also unique. This is just rigged theoretic. T-square would be minus one. The infinity. The infinity would be minus one. Yeah. That's right. It's the same. That's right, because you see, in other words, you're going to have this picture of the Gaussian integers,

2:22:30 and you have these other layers. So, yeah, I mean, that's... If you go around the circle, you get the thing that you thought was minus one from the point of view here. Right. Well, it lives over minus one. Just so it doesn't play out. So some calculations that I have done start to make more sense. If you use this point of view, a lot of them make amazing sense because you see the minus one infinite is unique and so is the infinite zero. It's just an incredible thing. So now, so even more amazing. The infinite elements form a ring. With the addition and multiplication agreeing, but not the zero and one agreeing. With the infinite zero, infinite minus one. It is a ring, so you have... You take the rig, there's always a universal ring, which is surjective if there was a virtual minus one. But anyway... So, what we're saying is that in these places where you have higher degree equations, this has a section which is preserving nearly everything except the reference points. It's almost like a ring. So, the earlier ring is actually a substructure of the ring and its consistentality is infinite elements. That's quite sedentary. So once we realize a lot of those calculations, and we've been doing this, these re-theoretic calculations, we always think that there's some kind of miracle to start off with these three equations, you get these longer and longer and longer expressions, pretending if you're clever, suddenly the x to the 7th equals x. That's just like some mystery. But this is all very much more understandable now. If you think systematically, in other words, given any such equation or presentation, you should first figure out which element is minus one, which element is zero. And this tells you how to do most of the calculations. Excellent. So I've been doing these calculations a lot, and I came up with the families of equations that are true. So, for instance, t equals one plus t plus t to the n. It implies t equals t to the 2m plus 1, something like that, and this has the exact calculation of this mean. You go up, you get a bit bigger, but then you can also get smaller and smaller.

2:25:00 So inside my calculation, there is your conceptual view, I guess. Yeah, because as soon as you make it fun enough, you introduce some things with genuine X's, and so you've got this infinite zero in there. Infinite zero has the effect of just dropping out, and you've got other infinite elements in the equation. And yet it's not perfectly rigid, you see. That is the amazing thing. With sets and recursive sets, we're accustomed to all sorts of things being isomorphic, but here it's all perfectly rigid. There's only one infinite zero. That is incredible. Does this help to prove that I want to work with a ring of polynomials of a natural number portioned by an equation t equals p of t? Am I interested in deciding the world problem? Thank you for your attention. You can't prove that the world rules are multistyle. As long as you do the multi-equation in the end, you know that every polynomial has at most degree n-1. And then, by careful analysis of the examples, you can understand exactly how your polynomial should look like. I think that what I'm saying is it seems like if you're given a polynomial like that, that you want to use them. The first task should be to determine whether this is algorithmic or not, I have really no idea. But at least maybe for a class of polynomials, you can guess it algorithmically, or you write the polynomial so that there's always a certain power of the variable, things like this. This could be true. So is your book accessible secretly? Not quite. It's just spotty. I've written parts and not written other parts, so I've got to be comprehensive. That wouldn't put me off. It's just that if you don't like to... Well, I'll ask General. I'll tell him that you're making this progress, and I'm sure he'll be happy.

2:27:30 I'll be in the States around October in Pittsburgh. It's not that close to you, maybe, but I'm happy I would be. It's not far. No, I'm happy. I would be happy to go if you have the time. Well, Shaniwell should be there. I should be in Nancy. Oh, yes. We're not going to Nancy for the whole of October. Not for the whole of October. It's just the first days of October. It sounds to me like you have a very, very fruitful conversation. We're at the end of our time. I should be back. Shenuel will be there in any case. He would be very interested in this. I have no idea how long this project is going to take. I'm ready to come. I was going to ask a much, much more naive question about the general conceptual motivation behind your remark about... I'm thinking of these things that are actually generated by real objects, and how far it connects up with this trying to obtain the particularly intensive ones that seem to have a concrete mode, and also the functions of analysis in a very concrete way. It's just that I'm an hour slow because I haven't actually figured out how to change this watch. Should I want to see that or do you want to leave this? I just want to stand there. Of course they've changed this kind of way now. Ah, yes. These things are not now coming. Curriculation of Scott Crowe's set lattices. Yes, they're by the, uh, this, uh, Chinese... Bo Wenjing. Well, I'm working with Toyster and, uh... I want to get a hold of Steve Howdy anyway, because he was one of the people on your wish list for next year, actually. I wanted to tackle him. Steve Howdy? He was just standing here a moment ago. Well, actually, about half an hour ago now. I'll get to him while he's around. Anyway, we'll talk to him.

2:30:00 Thanks for having this fascinating tutorial. So, yes, let me know how you've reached this. And the aim of my talk is to go one step further and describe other theories, namely those which are generated by an arbitrary end of number of sets, which is finite. As a typical example, Helmuth's paper gives us a description of three iterative theories and one binary operation, let's say. One of the examples that I often speak about is free iterative theory about one commutative binary operation, because that can be expressed by the final breaking part. But before I go into this, because it's rather technical, and to remind you of what free iterative theories are involves a lot of technical details. I will tell you how I achieved it. So I will tell you something quite clear, and that is a description of the final or terminal cohomology graph for such functions.

2:32:30 Description of terminal cohomology graphs for the finite array and the functions of sets. When I told you how this went out, I will tell you the consequence, how the iterative theories work. Before I do, I'll explain all, or shortly remind you of all the things that are in this slogan. So, first of all, coagula. Given an element of any category by an algebra, I mean an object, so in this case a set, And given to such, then, a homomorphism function from A to B, such that these square functions. Now, this gives you a category of algebras and homomorphisms, and an initial object is a category, shortly initial algebra. Each algebra can be more specific. It's something that plays a central role in algebraic semantics. As a concrete example, observe that if the functor is the homonymy of the functor mentioned above, then the category of algebras is the classical category of sigma algebras and sigma homomorphisms, and the initial algebra has various descriptions because it's, of course, only unique after isomorphism, the one that I prefer for my talk is that it's the algebra of all finite sigma trees. Where a sigma tree is simply a tree, where every node is labeled, the labels are operation symbols, and every day of the operation determines how many children there are.

2:35:00 So that was not my promise. I promised I'd tell you what is a co-algebra and what is a dual. So a co-algebra is then a set. Together with a structure morphism, which now goes in the opposite direction, and given to such, you define a homomorphism by the commutativity of this square. And again, we get a new category, the category of homo-algebra and homomorphisms. I am not trying to give you the motivation for, but the fact is that this is not a formal dualization, there is rather a motivation-from-system theory. Anyway, the object which is terminal or final in this category is simply called a terminal co-object. And for those who may wonder at this point, what is my talk related to? This workshop is Dana Scott's birthday. Well, it is, because one important fact in this famous model of unbiased language calculus was that in that environment, terminal algebra and initial algebra coincide with some important things including the school assignment that was coordinated. Okay, that's how I see the connection. Not in the main scheme of the structure, but the quantity is connected. That's what I wanted to say. Okay, example. What is the paradigm of quantum mechanics?