Towards a Dialectical Logic of Concepts

0:00 I will not hide the desire to open this session on the theory of categories, as I said earlier, I will do it simply for the people who do not really know the function of the seminar, I think that the categories are present here, perhaps. This seminar was founded in 1995 in the idea of making possible, something that we consider more possible here in this school than elsewhere, making possible meetings and confrontations between scientists, disciplines and philosophers. Not only philosophers, but also sociologists, historians, etc. The activities of the laboratory are archived according to thematic numbers in the O8 synthesis framework of the Sintet center. So I'm not going to expand here, I'm not going to make the presentation last too long. I'm not going to dwell on the protocols, the mode of functioning, the directives of the seminar and the laboratory. There are also philosophical generatology on which we propose, but I refer you to the number 1 of the synthesis book of 1999 entitled The Thoughts of the Sciences, where there is a clear description of the mode of operation found in the text of Jean-Claude Saint-Exupéry and René Thaume.

2:30 Today I will simply start with a question that should seem to be devoted to the category theory, a complete session in a philosophy seminar. Thinking of philosophers who could relate to it, I believe that the best explanation is to come together from the simple description of motivation, Fundamentally, a technical treatise of this theory can present the potential of mathematicians. Why study categories? Several reasons for this. Point 1. Abundance. Categories abound in mathematics and in fields that are linked to them, such as computer science. Entities such as ensembles, vectorial spaces, groups, topological spaces, banar spaces, varieties, orderly ensembles, automata, languages, etc. include all categories without exception and in a natural way. 2. Penetration in similar constructions. Similar constructions intervene in completely different fields of activities. The theory of cataclysm offers the means to invest such contributions simultaneously. 3. Usage as a language. The theory of cataclysm offers a language that allows to describe precisely the dangers of similar phenomena that occur in a different mathematical domain.

5:00 For example, the space of the equation has a finite dimension, is uniform in its duality, and the same in its non-duality. The second one, which is considered natural, is the first of the three. The category theory allows us to determine the interaction between the two natural isomorphisms. Second example, topological spaces can be defined in different ways, namely by means of open and closed ensembles, bobbins, convergent filters, and fermion operations. Why are these definitions essentially describing the same objects? The theory of categories must have a response to the notion of concrete isomorphism. For example, initial structures, final structures and structural structures intervene in many different situations. The theory of categories allows us to formulate and invest in such concepts with appropriate degree of generality. Fourth point, adequate symbolism. These are particularly complex effects by means of diagrams. Seventh point, problem transport. The theory of categories offers a vehicle that allows the transport of problems from a mathematical domain, which has adequate functions, to another domain where the solutions are sometimes easier. For example, algebraic topology can be described as an analytical topology analysis, which has adequate functions, by means of algebraic methods. Last point, duality. The concept of category is well-known because it offers an economic and useful duality. Thus, in the theory of categories, the principle of two horizons functions. Each concept is two concepts and each result is two results. The reasons for this progress show that this familiarity with the theory of categories will help those who are faced with a new field to detect, analyze and connect With the familiar domains, to organize the new domain appropriately and to separate the concepts, problems and general results from their specific occurrence, justifying specific research.

7:30 Categoric knowledge is therefore to direct and organize one's thought. If I add this declaration of MacLean, it seems impossible to draw it all the same, and according to which, in the theory of categories, the notation of the arrow leads to a concept, any additional commentary aimed at justifying the importance of the concept may therefore be practically useless. The use of notation leads to a concept. This is for us, in the name of legitimacy, a philosophical questioning. It is practically enough for oneself. What we will try to show this year through the confrontation to the category discipline is how to adapt to the level, from an epistemological point of view, by the new concepts introduced in the field of mathematical constitutions. We are simply at the opening of the notion of transformation that has been introduced in the hierarchy of fundamental concepts. We could also think of the concept of adjunction, which we consider to be one of the most fascinating mathematical phenomena and has come to that. This notion seems to contain in power both the combinatory of Algerian theories and the essential procedures of transition between theories, for example the passages of topology to physics, certain principles of symmetry, and even more so the logic of quantifiers. It is for these very reasons that William Noweer considers the concept of adjunction as one of the main principles of knowledge. The language of the categories is particularly good at transporting and transferring intuitions from the field of mathematics to the field of fundamental philosophical concepts. I quote, I am sure that in the next decade and the next century the technical progress due to category theorists will serve the galactic philosophy by giving a precise form.

10:00 Through mathematical models with ancient philosophical distinctions, such as the general and the particular, the location, the being and the becoming, space and quantity, equality and difference, quantitative and qualitative, we can see categories of space and quantity, in contrast to Echevarria, the editor, philosophical, epistemological and historical explorations. I would like to add that with the theory of categories, even sociology could find its way around it. Thus, we can say that the theory of categories realizes a sociological approach to the mathematical object, in the sense that it is no longer considered as an individual and as a member of the community of its assemblies. Here, I quote Manin In an article by the Encyclopédie Einarwee, a remarkable encyclopedia, we unfortunately do not have a translation in French, to the application article. Last but not least, William Le Weir gave a title to his book that should not be left indifferent to any philosopher in the world. It is undoubtedly one of the best titles in mathematical literature in the world, Conceptual Mathematics. The pursuit of this exact knowledge that we call mathematics seems to imply fundamentally two dual aspects that we can qualify respectively as formal and conceptual. For example, we algebraically manipulate a polynomial equation and we visualize the corresponding curve geometrically. At first, we focus on the deduction of theorems from the actions of group theory.

12:30 Then, in a second time, we consider the classes of the effective groups to which these theorems refer. From then on, the conceptual appears, in a certain sense, as the very content of the format. We can read it in The Jointness in Foundations, published in the journal founded by Ferdinand Ronset, Dialectica. Vol. 23, p. 2, 1969. Finally, an additional emphasis on the particular interest of the present here, in these walls, in this school, an attempt to present the historical and theoretical relations that this theory was able to maintain between them and the Roman Empire. One of these two founders, category theory, proposed in his time a hypothesis, this time of institutional political philosophy. It may well be that an effort of collective and anonymous organization, such as Bourbaki, is only possible in a highly centralized country, like Paris, and where schoolchildren are exposed very early to a wide discussion of philosophy. This is a concept, a category, a perspective published by Peter Duren, the 5-year-old mathematician Amélie Ligard, in 1998. I will not go into the details of the speech any longer, but simply, in this seminar, which must, directly or indirectly, and more specifically at the opening of this specific session, I would just like to remind you of the decisive importance that the one we affectionately call the cat had to have, because he called himself the spirit. These are the categories' theories. Let's go to Jean Toussaint de Santy. I had drawn a list of categories.

15:00 On the 4th, the cat, I refer you to Jean Toussaint de Santy's Dominique, in La liberté nous aime encore, which was published in 2001, and in particular in the 5th chapter, with a cat and a god. A cat and a god. This is what Jean Toussaint de Santy said in 2001. A concept needs to be welcomed as soon as it becomes productive, that is, as soon as it announces and engages a form of work of thought that we could call, if we like such a language, the spirit that animates the concept, and which can be profitably inspired in another field than that of the theory of origin. We are not here to look at how to apply the theory of categories to our object, but to strive to make us, as some of the specialists of the theory, categorically minded, faithful to the spirit of the concept. This is what McLean says, that we must strive to be first attentive to morphisms and not to objects. Those of us who have been asked in this context about the theory of relationship, about the reality of relationship in relativity, in relativity. Strangely, this kind of motor seems to be able to echo a phrase by Wittgenstein, which ends paragraph 18 of his philosophical grammar. A quote from Wittgenstein, which says to the reader to take advantage, if he can, of this philosophical confidence according to his own interests.

17:30 On the other hand, the concern of our interests leads us to keep together, in echo, the motard of categories and this confidence of Wittenstein. And indeed, when after nearly 30 years of distance, I wonder who, in the theory of categories, My philosophical complexion of that time, to the point of encouraging me to learn something, was that I hoped for it as an open promise, as a version of what mathematics had already produced. A reciprocal mobilization of available structures, in a way. Due to this mobilization, I imagined obtaining the production of an even richer field of creativity. It was taken from the philosophy of a dream by Flamboyant, Variation Philosophical II, published by Grasset in 1999. I would like to point out that Jean-François Nessenti mobilizes, if there is any particular category, in particular to realize a commentary on Dante, which is a comedy in verse 1, verse 4. Since the two sessions of the year will repeat the same order, I wrote a letter to Jean-Pierre de Saint-Exupéry on May 25, 2001 about the Gilles Châtelet Colloquium that I was organizing and about his inability to participate in it. I could write a few pages that, if necessary, would appear in the bags. I don't have a title yet, but the theme could be the rationality in the mobile in-between of living bodies, isn't it this gap between the two that shows the root and the enigma of diagrams? This question torments me, because you may know it, me too.

20:00 I make an effort to be categorically minded, which is not a bad thing. You are connected, as we have done this year, with the latest trends. So, I repeat again the pleasure and honor that Gerdoil gives you by opening this seminar. I am very bad placed to tell you in two words what you represent as one of the major representatives of the theory of categories, founder of what we may call the second generation. As well as that of Eilenberg and MacLean, the categorists would be better placed than me. I would simply say that it is in the 1960s, under the influence of William Lowin, that the theory of categories could be used as a foundation for mathematics to begin to be considered and developed seriously. This approach leads directly to the development of the theory of mathematics, which had to take root in the characterization of abstract categories through which various branches of mathematics were formed. In the development of our idea, the notion of topos is present in the 1960s, although it is certainly the work of Louis Ruthiernet, of the elementary axiomatization of this concept, published at the beginning of the 1960s, which gave rise to this notion, both in its fundamental status and its impetus. Enough rich to develop the essence of ordinary mathematics, that is to say the essence of what is taught in mathematics courses for a long time, and it is also in a generalized ecological space that allows us to connect directly logic and geometry. This may be enough to intuit the fundamental importance of the fields of mathematics invested both by category theory and by the world of the world.

22:30 Thank you for the opportunity to speak. You have heard some attempts to summarize category theory and develop by people who have not worked in category theory and also by some other people who have worked in category theory, such as Adam McHarrick, McClain, and others, and of course to know whether these summaries and judgments To be far accurate, one needs first to know what category theory is and what it has become. I presume that's why you came tonight, so I will try to give some indication. The title speaks about logic. Logic, according to Hegel, is the movement of thought in its necessity. And so the science of logic endeavors to determine what are the actual laws of the motion of thought and its necessity. Of course with the aim, why should we want to know these laws, with the aim to do mathematics better. However, there is a much more narrow concept of logic which is in common use, the logic of properties.

25:00 Or you could say sub-objects, the algebra of sub-object lattices and how these transform when the universe of discourse is changed, transforming by a substitution or a pullback or the adjoining of an economic modification. So this algebra of sub-objects is the objective content of the subjective presentations known as Now, I think, however, that a great disservice was done to science by Frege, who termed his system the Griff shift so as to suggest that it had something to do with concepts. In fact, he presented an algebra of properties only. He says, I found a passage in Frege where he says, it seems reasonable to identify properties and concepts, doesn't give any further justification for that. So I will try to indicate, as I said, the title would be a more adequate logic, which includes explicit care for concepts. The Mourglaki chapter on structures, disliked as it is by nearly everyone, nonetheless does contain a certain advance in this sense, which I don't know any previous attempt to address. And that's what one could call the logic of the given, the given concept.

27:30 Of course the existential quantifier is a formal symbol and one often hears even experienced logicians begin the discussion by saying let G be a group on which there exists, a set on which there exists a group structure. This existence of the structure of course is not at all what's meant. One means that there should be a given group structure but that did not have a formal place. And that logic in a narrow sense. However, word by case, specifically the discussion about deducing one structure from another, which as I say, I don't know what the previous treatment of, this it seems to me is a beginning for a lot of these people to include, so I'd love to be given. Which should be a part of a dialectical logic of concepts. Now there is one, at least one, mathematician in history who used a direct connection between philosophy and mathematics, and that of course was Grassmann. Grassmann began to show how to find the subjective in the object. He called the subject of formal science to indicate some knowledge of the motion of thought as such, whereas the objective he called real science, the aspect of thinking which attempts to form a picture of reality.

30:00 This is inevitably general in character, whereas reality is made up of particulars. Now, perhaps one could imagine a mathematical vision of the way in which these generals, which are forever different from particulars, nonetheless manage to approximate. In the case of tangent planes, we see that it is quite general, but the generality is more and more useless as we get further away, but then of course the same body can be approached by other tangent planes and other tangent planes, so the appropriate system of generals makes a better and better approximation, or you could also imagine approximation by... Cubic curves or so, always the same principle, never mind. Okay, so in particular, in a particular case, we want to, as Rosman did, we find algebra in geometry. And the algebra we find in geometry will lead then, dialectically, to... In particular to algebraic topology and functional analysis as higher ramifications of this relationship. So as concerning the relation between structures and properties, I would like to offer the following paradigm of structure plus property, not a concept but a presentation of a concept in the algebraic sense.

32:30 A certain abstract general A, objectified into a category, represented in a background B, may constitute an adequate modeling of a certain concept C. The properties are imposed on the structures of shape A and background B in order to arrive at concept C. But of course, it is indeed a presentation because you might have different A1 and A2, which are genuinely different. But which, together with the properties that were imposed, nonetheless gives rise to the same C. So, in that sense, we refuse structuralism because mathematical objects are not equal to structures. They can be represented as structures, but not uniquely. Because the underlying background, the underlying set function, the underlying space function, if you will, Can be chosen in genuinely different ways in some circumstances, so to say that a mathematical object is a structure is just as inaccurate as to say that a group is a presentation of it. Now what is this background? What is this background into which we interpret? I don't know, perhaps I didn't say it in the product. This contrast between the abstract general and the concrete general It is totally necessary for the advancement of thought within the abstract general via substitution and higher amplifications of substitutions such as quantification, etc.

35:00 One can determine certain amount of the properties of inside the structures, but on the other hand, We also can survey and investigate the totality of structures with properties that are so described. We can do formal calculations in the theory of groups, but we can also investigate the category of groups. This role of the concrete general is necessary in particular for making plans. The exploration of the possibilities, mental exploration of the possibilities through, as it were, simulation, is possibly the answer to the question of the, how is it stated, the unreasonable effectiveness of mathematics. People thought it was unreasonable, perhaps didn't take into account this particular move. At any rate, what is the background B into which we interpret the abstract general with additional properties in order to arrive at the concrete general? Another point I should say, or it says that the morphisms in the concrete general are the natural transformations. Generalizations, I think, are all the ad hoc definitions of morphisms. They can all be construed as natural transformations, natural respect for the movement of A. On the other hand, but also from a pre-mathematical point of view, the concrete general should be something like, even outside mathematics, it should be something like a mathematical category. Why?

37:30 That is to say, it's not just a class where it's a necessary way of comparing the things, because they are all determined by the abstract, not individually determined, but they have a common feature which has a common name, which are the objects and maps in A, and so there is a suggested way of comparing the totality of realizations of an abstract general. This is inevitably admitting a notion of comparison between instances going beyond the mere equality and so on, just because of the fact of participating in the given abstract general A. So that said, now, finally, coming back to the question, what should be the background? Or, the category of sets. Now, of course, the category of sets is an extreme case of the category of space. There is a specific notion that should be mentioned here, which is the idea of Cantor that the spaces should be as devoid of structure as possible. Less structured background, etc. McCantor thought, for many reasons, but one was the invariant on this card analogy that she's looking for. But there's another reason, which is a logical reason. You see, the calculations that we do inside A, how do we know that they're precisely and nothing else relevant to the C? It seems to depend on the background. So, if the background is as abstract as possible, if it consists of what Cantor called cardinal solemn, and I call abstract sets, then one would hope that there is a minimum of interference of the properties of the background with the properties of the structures, so that the logical...

40:00 The content of the logic should be distorted as real as possible. By the way, in reading Cantor, one has to realize that he used the term cardinal solve not for the cardinal numbers, in the sense that we'll remind you later, but in the sense precisely of the abstract set which has a cardinal, but on the other hand which is a little bit less abstract, because abstract sets can support mappings which cardinal numbers can't. So it's in this connection, by the way, that I have the opinion that, in fact, Gödel did prove Cantor's continuum hypothesis because the continuum hypothesis, Cantor's conjecture, which immediately followed his discovery of this idea of the total abstraction, or near total abstraction, and his... And so on and so forth. Then he says that it seems reasonable that there's no sets between a set and its power set in size, and so this depends very much on the idea that the sets are totally abstract and constant, free of cohesion, et cetera, et cetera. And if they are sufficiently so, I believe Yertle actually showed that this is the case. On the other hand, the models constructed by set theorists showing that things like the continual hypothesis are independent of other axioms and so forth. These are, of course, constructions of categories which are not totally abstract. They have some vestige of collision and motion in them. That's why it's possible for the power sets to be figured. But at the opposite extreme, you see, for certain purposes, it's well to think that the background is as simple, abstract as possible, which, of course, entails in order to discuss the same...

42:30 Category C, we have to use very vague and complicated A. For example, if we want to discuss the notion of Lie group, thinking of it as a theory A which is being interpreted in abstract sets, it has to have many components. On the other hand, the same notion of Lie group can be viewed as interpretation of theory of groups, which is something much simpler, into the category of smooth spaces. And this is a general methodology, which you could say was started by Lee but continued by Ekman and many other people, the notion that actually the background could be quite rich, quite a rich category of spaces, and that both kinds of extremes are desirable for different purposes. It seems that one thing, one property which should be in common to all these is the explanation, which I have symbolized like this. The category B has categorical products, or Cartesian products, and that function in turn has a right adjoint, which, as you can see in particular, has the property that it points. All of this space are corresponding to the maps from A to Y, and therefore it's called a function space, but it has lots more besides points, as you see by choosing various nexus, so that it actually has cohesion, uniquely determined cohesion of the same kind as the other objects in its category. Notice the fundamental nature of this construction was, I think, Volterra, who began an explicit consideration of such matters, although they had been used, of course, for a couple of hundred years before that.

45:00 Notice that certain facts about such function spaces or whatever, map spaces, It can be discussed by circumlocution without actually introducing those spaces, but that does not permit a feature which is possible if this notion is representable, namely simply the notion of a map whose domain is such a map space. First functional was named by Adhemar, a concept Adhemar recognized as due to Volterra. Now, the Volterra School of Functional Analysis was somewhat marginalized by the theory of topological vector spaces, It's well worth reviving, in particular in the analytic category. This was studied a lot by Bulgera, by Zorn, by Pellegrino, by a whole list of people, including Ademar, actually. General homotopy theory, to which I will return, who pointed out the crucial nature of this operation, for homotopy theory in particular, but also for functional analysis, as he worked in that area.

47:30 So he asked Fox to investigate whether this is true in the category of topological spaces, who found out no, but it is true in the category of sequentially determined spaces. Now, by 1950, in the first volume of the Proceedings of the American Maths Society, there was an article by David Gale about Ascoli's theorem, where Gale states that Horowitz, in the meantime, had invented the notion of K-space, and the notion of K-space is due to Horowitz, not to Kelly, as some authors have... But it is a modification of the traditional category of cosmological spaces, dealing with compact spaces as basic figures, which does have this operation in a restrictive sense. In a sense, I think one could say that since 1945-1950, the traditional notion of topological space is actually obsolete because this was one of the fundamental purposes for which it was invented to be able to deal with continuity of functionals. The basic reason why it didn't work, I think, can be seen because really it is... The category of topological spaces is in its essential conception a form of algebra, not of geometry. Now, the algebra of functions is valued in the Sierpinski space. This algebra of functions is the basic concept by which the structure of topological spaces is concerned.

50:00 This is the fundamental notion of morphism in this category. So it starts with algebra, not with geometry. Of course, it approximates rather well geometry, and in many branches we see there is a dialectical relation, whereby some simple notion, either algebraic or geometric, is transformed into a richer one on the other side and into a richer one on the other side. I mentioned that there is a very important observation of Hadamard in 1923, which makes possible, beginning with a geometric point of view and arriving in five steps at reasonable category spaces, Essentially, this paper of Adhemar is about a pedagogical problem, how to teach calculus. However, Adhemar was a great analyst and a characterist. He proved what he was saying was correct. He doesn't give any proof in this case. But the crucial point is, and if you haven't... A manifold, a smooth manifold, well in general if you have two smooth manifolds, then an amorphism between the two is determined by curves and functions. Well, let's say that that's just a function on the manifold to start. So if we're all smooth curves, B U is smooth, then that implies that's true if and only if B itself is smooth. So that the notion of derivative in higher dimensions can be determined just by means of paths.

52:30 Now of course these paths, this test, P-U, is ending up in the monoid of maps from R to R. In other words, in the space of maps from one dimensional space to itself. If we think that C is something like the monoid of continuous deformations, let's say, of an open interval or a certain one-dimensional space, then starting just with this geometrical idea, we can construct a big category of spaces. We look inside the pre-sheaths on C, the unit embedding, and we take the finite product closure of the representable, the whole finite product representable, the whole subcategory. So, in other words, the basic algebraic content of the geometrical construction is that somehow the natural transformations between the finite products Which are just natural with respect to, in this case, the one-dimensional transformations, but these are some of the correct transformations. A non-trivial theorem, indeed. It was proved later, in 1967, by Beaumont, that I feel quite confident that Heidelberg must have had the proof as well. We can proceed. Oh, yes, by the way, I was thinking of the smooth category, but exactly a similar result holds for the algebraic category, that is, for commutative algebra or algebraic geometry over a field which is not countably infinite, so that could be the fields, the complex, or any finite field. Again, the same result is true, but the function of invariables...

55:00 All of this, which is compatible with every polynomial in one variable, is in fact a polynomial in a variable. And there are several other instances of this phenomenon. I'm mentioning it because I don't understand it. That is to say, why it should be in so many cases that the way the expression proceeds isn't easy. So the finite product-preserving punctures from this category of finite products intersects, this is a category of universal algebras, and then we can isolate, we can find a classifying topos, E sub c, it's a topos that can be constructed without Explicit reference to Groten-Dietz apologies, it is just, again, a full puncture category on the finitely presented algebras of that type. And well, and then finally, you may be, you may find some reasonable Groten-Dietz apologies you can depose on that, The full glory of the analysis and geometry that was inherent in the original transformations of the one-dimensional space, conceived of it as a one-dimensional construction, the content of this one-dimensional, including all of the functional analysis and all of the higher-order infinite-dimensional geometry and so forth, is basically to be found in this topos.

57:30 Focus on the dialectical logic which relates space and quantity. In fact, in a way, every category permits an analysis which you might call pre-geometrical as well as pre-algebraic. Let me draw a picture. This is the category. Take any object X, and then you have all these objects of A. Look at maps from the A's into the X. We think of those as figures, geometrical figures in X, of shape or form. I'm hoping that some linguistic expert can explain to me which comes first, shape or form, but whichever it is, that's what it is. So a shape A, right? So for the various objects of A, we have the figures of that shape. But then we have also incidence relations between these figures. Anyway, if we have a map A, little a, it might be followed by one figure to equal another figure. For example, a point might lie on a circle, a line might lie on a plane, and so on and so forth. So in this way, we have a kind of attempt to represent, once again, these spaces are kind of geometrical concepts, this is a way of choosing a notion of structure whereby to represent them. One might choose a different A, represent it differently, but my pre-sheaves on A, we can thus represent the object by having chosen the A. And if this subcategory A is what's called adequate in the terminology of Isbell, then this will be a fool.

1:00:00 It's a fortunate fact of life that a great many very big categories have rather small adequate subcategories. For example, all the modules... Or, on the other hand, we could instead imagine another subcategory, let's say, of values, and represent an arbitrary object by functions with such values, and by algebraic operations, theta, on these functions. If the value category is chosen and closed under products, then notice that these operations include binary operations, binary operations, and all kinds of other things. So in this case, we get a representation in the opposite category as covariant factors on the value category. And again, this might or might not be . So when I'm saying that the usual notion of topology is algebraic and not geometric, what I mean is it's not defined in terms of figures and incidence relations, but rather in terms of functions and operations they're on. It's a big category, B. It's all the powers of the Minsky space, but still that algebra. Functions, there is an induced definition of figures, namely, just by naturality always, naturality given natural, and conversely, given analysis in terms of figures and incidence relations, there is an induced notion of algebra, which is given by natural transformations once again. So, this, of course, these two types of analyses... We'll be more successful in some cases than in others, and as we know, for example, for toposis, that this one tends to be quite successful.

1:02:30 So coming back then to this idea that the best categories for the geometrical sort of analysis, we could call them what we call base categories, are those which at least have exponentiation. Or perhaps even our TOCO. TOCO is just a category of exponentiation in which the notion of sub-object lattice is representable. The logic in the narrow sense is representable. I think I'm emphasizing here the exponentiation. So again, this contrast between cohesive space and... If I say cohesive, Erasmus said continuous, but you see this word continuous has been purloined by the specific definition of topological space, which we are accustomed to, so it can no longer be used as a variable, but cohesion has not been taken. Physicists used to say continuous when they meant differentiable or analytic or depending on the situation. There are many situations. Okay, so how is this direct contrast between the category of cohesive space, the space category, and the extreme case of the street to be understood or analyzed? Well, the typical situation is that they will be linked by adjoint counters. Similarly, the set of components of a space, connected components, this is left adjoint to the inclusion of the discrete spaces, the special spaces that are called discrete, that's left adjoint to the functor that was discovered by Cantor, namely the points, where you can take...

1:05:00 Continuous space, extract from that the abstract set of points, throwing away everything else. So, although we've come to think of points as a rather banal concept, I think really it was Cantor who said we should perhaps take those things seriously in themselves. And often there's still further adjoint, another inclusion. This situation, again, is quite an intuitive content here, but it brings into immediate play the notion of unity and identity of outgoing opposites. Because, you see, I'm getting a 5-0 for the moment. These are two subcategories, the street and the industry. They're completely opposite. Opposite sides of the category, in a very real sense, as the gadget on this expresses, because to map the points of any space x into a set S, a discrete space S, is the same as mapping continuously, or smoothly, or out of range, or whatever, x into the indiscrete of S. Because it's the same as mapping just the kind of points, whereas by contrast the QMAP is set into the points of X is equivalent to mapping the discrete space with S points into X. These composites are the identity, these are full inclusions, so the points underlying an indiscreet space are the same as the points underlying the corresponding discrete space, namely the abstract set we started with. So these two adjointnesses clearly put these two extreme kinds of spaces on opposite sides of the category, they're very opposite. They tend to intersect only in the truth values, only in the sub-objects of one. And yet they're identical at the same time, because they're both isomorphic to S.

1:07:30 You take it as a category in itself, it's just S. You take it as subcategories, they are opposite. And the points functor unites these two opposites, in fact in a very precise way, that they are respective adjoints. If you follow one of the fundamental theorems about adjoints is, adjoints are unique. So all four of these factors are uniquely determined by any one of them, and you can start with any one you'd like to discover in your particular category, what these things are, if they all exist. So now, how can we arrive at this situation? Well, there are many ways, but one example is... In connection with smooth geometry and algebraic geometry, I'll come back to that in a moment, sometimes we have a preferred object, T, in E, which is what's called sometimes an atom, where A stands for amazing and T stands for tiny, and O stands for object, and M stands for moving model. All of this comes from Rotendieck's notation. If you have an extra right edge line, you do it like this, with an exclamation point, and that's amazing. Now what does that mean? That means that we have exclamation, but in fact, exclamation by t has a further right edge line. All of this can be equivalently expressed as functions of x itself into something that depends only on y and t, which I like to call y to the power 1 over t, a fractional exponent. Now, this is indeed amazing also because people who work in lambda calculus have surely never heard of it.

1:10:00 But, in other words, there are well-developed formalisms about exponentiation by itself, but the fact that some particular exponentials could, in fact, themselves have furthered right-hand lines is something that has not been at all exploited in much degree. I like to also mention the figure of Mozart, Cournot, who was very famous, of course, for many reasons, This theory of thermodynamics is sometimes fascinated by saying that he had the wrong idea, that heat is not a function, but actually heat is a differential form. Well, with suitable choice of T, these functionals are exactly the differential forms, and so you see that they are functions, it's just that the values are lying in a more complicated ring. So perhaps you'll be vindicated. In any case, I mention also that this particular construction is leading to a very quick construction of the Eilenberg of plane spaces, because if you think of T as being an n-dimensional cube or something, and Y, say, as a building group or a ring, then these are the co-chains of dimension T on X with values in Y. And this says, well, okay, don't change the dimension t. There are actually functions on x. It's just that they have their values in this other thing. Now, between these things, there are suitable boundary operators so that one can extract the cohomology. And the point then is that the two of these maps are homotopic, if only they are actually homologous. In other words, there are about three steps in the construction of Hilbert for plane spaces. The biggest conceptual leap, I think, is this one.

1:12:30 Suddenly these co-chains of higher dimension turned out to be functions on x by itself. I will see in a moment why I'm especially interested in the tiny objects. I didn't mention that yet. One construction which one can make, one can define, given a category with exponentiation, given such an object, or perhaps several such objects, let's just think about one, so then we can define the subcategory of all those objects x and v for which there are no paths of shape t. So these are the objects which are intuitively discrete from the point of view of T. If moving is what T allows you to do, then in these spaces you can't move, they're discrete. Well, some properties follow from this automatically. But if E is a topos, so is S sub T. And between E and S, there are all these connections. Now, you may say, well, of course, that S is not as discrete as Cantor wanted it. It's only discrete insofar as this T is considerable. So in particular, in certain cases, it turns out that this is not totally abstract, not totally abstract sets, but actually the tokos of sets in which the Galois group acts. There was a certain sense in which Galois' discovery was that the base tokos were algebraic geometry over a non-algebraic tokos field is not sets.

1:15:00 It's another Boolean topos. Why is that correct? Why is that different from going all the way down to sets? It's because there's a very special property that I didn't mention yet, which tends to be true for many Category B which deserve the name of space category, and that is that pi 0 preserves finite products. By the way, the pi zero, I can say, in this case, the definition of pi zero is by a co-equalizer. Key might not have any points, and you can consider any pair of points as possible endpoints, quote unquote. And then you have two evaluation maps, at one point or the other, going into x. So you simply take the co-equalizer, and that's pi zero x. So you see, in other words, it's quite natural, using this definition. Just by combining the various adjoins in the right way, you can construct the form. This gives the definition of pi zero, and as such, this condition, of course, in the empty case, is that the one point in space should have one component. Now, you see, this property may fail if you go all the way to the more abstract sets. If you go only partly down, you may go way down from a non-Boolean to a Boolean situation, but still retain this good property. Why is this a good property? By the way, of course, you would not expect a size unit to be left exact. It preserves products, but not equalizers. Most of the interesting spaces, which have more than one connected component, are constructed by equalizers. Take a curve like this and another curve like that and intersect them and you have several components, even though you started with spaces that have, that are connected. So, well, I mean, of course there are situations where pi zero is left exact, but it's completely different from the usual, that's what we imagine about geometry.

1:17:30 Derevitch's definition of homotopy. Derevitch said that the homotopy classes are matched between two spaces is exactly pi zero of the function space. So this is one reason why he needed the map space, in order to define this. This connected component, a concept uniquely determined by the setup. There's one composition, one composition of these abstract, more abstract mappings, and that's possible because in any case where you have exponentiation, there's always an internal composition, y to the x cross z to the y, close to z to the x, that's a special case of the gradualness, but then we take pi zero. So, it's the fact that Pi zero preserves that product that enables us to define this map just by applying the functor Pi zero to the composition map that you have anyway. So, this gives a new category. I already used H for a hundred marks, so I have to say that it is more letters. That's the usual homotopy type. The category is homotopy. This is essentially the identity of an object, but replaces y to the x maps, not points of y to the x, mapped to the class zero of y to the x.

1:20:00 So this is a functor which actually preserves on the nodes, which is unusual for addition, multiplication, and even exponentiation. So this is again a category with exponentiation. By the way, I should have said that in the literature, categories of exponentiation are also called Cartesian codes, and counter-Cartesian codes. So this again has exponentiation, as you just verified by seeing that the original function space plays a direct role in respect to that. Well, actually, in general, much more generally, if you could have any category C, which is enriched in X, what does that mean? Kind of internal, not internal, but have a confunctory value to x so that the points of it are the usual set of maps between two objects. Now this is the enrichment. So in other words, C is a category, yes, but better than that, the maps between the two objects form a continuous and smooth algebraic space. So any category that's enriched in X will turn out to be enriched in a different way of C, is the one whose maps are the pi zero instead of the points, pi zero of, in other words, both pi zero, which preserves products, and points, which of course preserves products, these are two different functors that preserve the closed monomodal structure of E and S.

1:22:30 So that really anything that's enriched in E has, in the sense of points of an enriching column, are the usual maps, as another enrichment which is Y of Y0. So, of course, there are all kinds of categories enriched in X, pointed objects, group objects, map objects, etc., etc., etc. Okay, now, another thing that I can do is I can form a subcategory of X consisting of those objects which are infinitesimal in a certain sense, or at least three different senses of infinitesimal, which I could be using here, and it's still an open problem to see what are the implications between them, but here I mean infinitesimal just in the following way. I want to take only those objects for which that isn't relevant. So you see, at least I picture that sort of thing like this. It's a very special kind of space with the property that it has these consistently built infinitesimal clouds with different degrees of infinitesimality. And each cloud has exactly one point in it. Again, the points are, you know, objects in S, which might be tested with the alloy action or something, but never mind. Let's take it one point. So you see that the components of this space are these clouds, but there's exactly as many of them as there are points. Subcategory. Now, what I'd like to suggest is that I skipped the part about functional analysis, about extensive and intensive quantities in the sense of Grossman, for lack of time. It's described in that paper that Charles cited from the Science of Grossman.

1:25:00 This is the qualitative rather than the quantitative version of X. Extensive quality and intensive quality. Homotopy theory with extensive quality. You see why, I mean, you imagine that the space itself permits all sorts of motions, quantitatively. But now if you imagine that all those motions have been done, what's left? So that way, it's the quality of the space, but in an extensive sense, I'm emphasizing. On the other hand, I would like to develop the idea that the infinitesimal space underlying, of course, this thing has an animal too, I should say. The infinitesimal skeleton of the general space. This is the infinitesimal skeleton of the general space. This is basically where the real algebras live and so on. This should be thought of as the intensive quality as it can be described by partial differential equations and so on. Now, these two are radically different things and they are radically different. Nonetheless, they do have a special property in common. There is another category that no other category with exponentiation that I know about does have, and that is that, well here by construction, you see, pi zero equals the points, but that's also true in Pomotovian theory, that you see that if you look at the other definition, if you put x equals one, you see that the...

1:27:30 Now, of course, that formal remark wouldn't be so significant in itself, except for the fact that, indeed, you still have the discrete inclusion into the homotopy categories, left over from here, of course, and the pi zero, the old pi zero, is left adjoint to that. But the same old pie zero has become the writer's one. So a striking feature of the homotopy category is that it is Cartesian closed, it has exponentiation and so forth, but it has this property which again I think is not well known in the lambda calculus and so on, that the relationship with the extreme spaces is collapsed. The two opposites have become... Equal celestial in both kinds of qualitative categories. And the object is the same as the one that was said, that is to say, to describe...

1:30:00 Protonix has said several times that the definition of topology is not the right one. It was said in its own way, with force. And we would indeed have tried to develop this idea. And the idea was the same, to describe a space by the figures, by the shape of the figures. Should I repeat that? I just heard today that Michael Maynard, Walter de Gretemanski, written approximately in 86, about a theory of shapes, and basically they are, as far as I can guess, the same as you express, that space should be described as a... Yes, yes, of course there are several other reasons for the... There are several reasons for being dissatisfied with the traditional definition of topological space, another one of which was addressed by Brondy under the slogan of tamed topology. Yes, exactly. One should be able to avoid the so-called pathologies, which I think really are pathologies. And he agreed. Actually, this program has been carried on by logicians with the slogan of O Minimum Models, a very strange slogan, but some of the most interesting work in logic, I think, is growing from that idea of building a change of quality. Would you consider the lack of epithelial closure for the category epilogia as being one of the pathologies? Itself a pathology? Well, yes, in a way. So, filter species, for example, Kurodowski's filter species would be better? Yes. Convergent spaces. Convergent spaces and similar constructions. There is a recent one of Dana Scott called Ecological Spaces which...

1:32:30 In this category of all topological spaces into an even bigger one, which becomes Cartesian closed, so that construction shares with convergent spaces and several others, the fact that one needs a proper class of figure shapes, it's familiar from functional analysis, you might need sequences that are arbitrarily long in order to describe convergence. So for that reason, you have a proper class of features that are necessary, which I think is also really not very relevant to everyday mathematics, geometry, and so on and so forth. Do you see that there are peaks in a topos? Yeah. There exist bigger topos. This first step, basically the Heidelberg step, replaces appreciation of the bigger one, where you have products. Let's say you can go to these topos, which are the sub-topos and the pre-sheet topos in a certain way, then what you can do is you can embed C in a larger category of D, for example, you can have a finite product, and then you will get, yeah, that's right, you will get a... So now you have this inclusion, and, well, you could...

1:35:00 You could imagine taking something smaller as appropriate. But the basic point is that appreciating the category D, where D has finite products, has the feature that the atoms are exactly the objects of D. That is a representable function, and it's computed this way. A-shaped figures in it, where A is also in D, and then by Morita and so on, you have that this is the maps into Y from this functions-based object. Now, these are two representable functions. Of course, the exponential itself in general is not representable. It might be, but one can simply define it. In fact, amusingly enough, you can always make this definition. Define y to the 1 over d as value of a to b, the natural transformations from this function space, a to the b, that we know exists in any type of space, into y. So now we have the set value, we have another pre-sheaf as a of errors. You know, it's like elementary algebra. If there were a solution, it would have to be this, and you have to verify that this works. So this definition actually works. It is adjoint to the power d. You can just verify that. Now, in fact, you see, even if we don't have this condition, you need the condition about products in order to make that verification. But if you didn't have the property of products, you could define a structure this way, and it's clearly right adjoint to something, but that something that it's right adjoint to is sort of, I don't know exactly, but I think of it roughly as being the maps of compact support from D into X.

1:37:30 All of these are adjoints of this function, in any case. So you can say, well, you can define these two things in general, they're adjoints, but if d has finite products, then in fact this is equal to x to the d. So we have the verification. So any surface can be embedded in one in which this is true because h of c has products. And then, of course, you could worry about trying to approximate E as well as possible and obtaining that by taking the strongest topology on H-C for which not only is it representable in E, but all the results of applying the fractional argumentation are also sheaths, which is in some sense an approximation to E, but it has... This sort of thing you see is, as you say, if you're asked about this, if we have a map, alpha, for example, in first order and in second order, and on any object t, we could consider the possibility of a prolongation operator, c, something which, given a map on t, in fact, it extends it in an indefinite way. This is precisely a second-order differential equation in the case whether this is a first-order or a second-order. So what we have is the result that this category of second-order differential equations is itself also a topos, not just the spaces.

1:40:00 The laws of motion themselves are topos. And this is possible because, you see, this notion of prolongation, which would make sense in any case, can be transformed into something like this, the real fraction of T2 divided by T, so the same information can be represented like this and so you eventually arrive at a Adjoint cobalt net. And it's known that adjoint cobalt net gives rise to further toposes when you look at the reactions. So it's quite remarkable and still largely unexplored, although I have a paper coming out shortly in the JPAA. It's still largely unexplored the potentially tremendous ramifications of this fact that the second order differential equations themselves form a topos. But now we see, going back to this other remark, this looser remark about, well, we can always achieve this tininess if we are willing to enlarge it better. You see, there are several other situations where prolongation operators would be interesting. For example, suppose we have a body B, and we look at the boundary of B as included in B. So essentially, such a prolongation operator, it might be, for example, a solution of a boundary value problem. It's a definite smooth way of extending quantity variable over the boundary to a quantity variable over b. So it's not a differential equation, it could be the solution of a differential equation. We usually think of this as described by differential equations, but a solution has to have a solution. Well, you see, if we embed our category... If there's a larger one in which the boundary of B becomes an atom, then we can say that in that realm, indeed, the boundary values for all these solutions are also perfect for whatever purpose that might serve.

1:42:30 It may turn out that this enlargement destroys too much of the information. It seems to me this is a pure projection. It seems to me it's worth investigating. I'm not a mathematician, so it's a very nice question, but can you explain about the infinitesimal oscillator? Is it linked with non-standard analysis? No? Okay. It's not jacked in one second. Okay. We're going for actual space, and we're not going for formula that we can say is true. I heard Robertson launch his theory in Jerusalem a long time ago. The title of his talk was Formalism, 1964. The point was that going along with this mathematical construction, which of course is very interesting, the formalist philosophy that accounts as what formulas we can claim to be true, not what they're actually about, in conceptual time, that was, I think, attenuated. It wouldn't even exist without that.