Final lecture (partial) / Meaning of category theory for C21st philiosophy

0:00 So today I'm going to discuss a specific wave of mathematics, in a way motivated by Cantor's The abstractions process is quite different in that we really presume that the world is some kind of cohesive category, and within that we try to isolate the so-called discrete. I say so-called because on the one hand, due to... We see that in some sense we can never really hope to reach the absolute absolute in the sense of Cantor's vacuum. But on the other hand, and on the other hand in practical situations, We see that, in fact, it's very useful to think of that contrast in a relative sense, so that the lower thing, the relatively less cohesive, is even in some sense qualitatively less cohesive, is actually still clearly got a little bit of stuff going on in it, which is very useful. The study of algebraic varieties of all dimensions and degrees of complication was clearly guided and influenced and helped by studying the case of varieties with only one point. But varieties with only one point, or destroying some of those, do form in another topos, but actually it's a Mugan, even a Mugan topos, There are interesting spaces in that way, but it's still non-trivial because it essentially is just the tokos incarnation of Dao Galois.

2:30 So, the kind of revolves around, if you like, a Galois connection of the following types, S and X, from S into, and you can ask for that in an isomorphism. And as such, it gives rise to the Galois connection, and in many cases, this Galois connection is not a mere poset, a poset thing, but a geometric work as in some kind of adjunct or a functor itself. And this is some of the contradictions between sort of relatively contingent and relatively discrete. The idea is that s is so discrete relative to x that there's no x-parameterized motion possible in s, or the other way around, that x is so connected relative to s that the same thing is true. So this is the sort of connected versus discrete as a contradiction between two objects. All of these terms are related to the y-axis of the x-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the y-axis of the

5:00 The list is the idea of a prolongation operator. We might instead require that C, following the upper star, is the identity order.

7:30 And then in that case, I would say that both of these kind of operators come up. The averaging operator is this. You consider all the matter in the solar system. Then you partition it into planets and the sun and so on. So the mass gets averaged. Planets, for example, in the prediction case, often relate to the mass in such a way that there is disposed prolongation and the averages in fact are equal and it's divergible in the brackets. It doesn't go with the generality of the same thing. In the latter situations, you don't think of it as, say, you think of S being the street, quite the contrary. This is kind of the basic algebra. Oh, and I should say also that this, yeah, this goes back to the original one.

10:00 This makes sense even, of course, in a hiding algebra. So in a hiding algebra, two elements might meet in this relation, and that again generates the Gallagher connection. In fact, one of the ultimate ways of expressing the process of generating a group decypology for which given pre-sheets happen to be sheaths, which is due to André Joyal, makes essential use of that particular, that same galvanography I'm describing. Okay, so now I want to, we have a, usually we don't say that. Do we have a given object key for the talk, for a given object? We, for one thing, we will by the way consider the category of all S for which S is a key. Now this, in fact, should have said that already in the discussion, that there are some different kinds of objects that won't be in such keys. They fail to be ethical from R2 and use ethics and monics or whatever it is. Of course, it won't work because the existence of either kind of sectional refraction, prolongation, or averaging will imply that you use math as you perhaps want, but that's not sure that it will work, so somehow this won't be good. So in the particular case of the math from Key to 1, there was a lot of, 20 years ago, by Peter Fry and others, to the effect that this construction doesn't really...

12:30 In general, it is an extra structure, but the category of key-to-speed objects, supposedly, so consider this as a full subcategory, and I guess we'll call it S, but of course one should always realize that it might not be. The species that can't go laundered might be more like a gallop launderer, or something quite different. We'll see when it comes out on the 7-pound-feet basis. And so you can think of these as a fatigue history, or we call it for terminology, the object which perceives as being a fatigue thing. Now, this category is different. If we had infinite products, Equalizers and so forth. This would be closed under Newton. So, it should be the case that there is no left-add joint. If you have pi zero, you should call it pi sub t, because it depends on t, and it may not be as extreme as pi zero, as we know and love. Now, of course, the construction of this left-add joint can be complicated, just because it seems as though it should exist. We don't have anything to do with each other.

15:00 So unfortunately, underneath what we want to make about t, there is an easy construction. If we take an arbitrary object x, raise it to the power of t, cross that with t, either evaluate that general value, that that's pi zero of x. This is a reflexive. In this factor I have the consonants, of course, and in the other one I have the two of them. So I have a reflexive structure there. And so if this is the left-hand line, then we will have an important equation that applies to all of these products. All of this has the universal property that is, if you map x into any object which is t-district, there will be a unique extension, just by the property of differentiation.

17:30 So, the reason that this simple construction actually gives me, without, you know, for example, iterating this, I'm going to transform that into something like that. All you have to show really is that this pi zero of x is to your street, that this co-equalizer is to your street, and then it can preserve the name of pi nearly. So that will follow from the, in Florence I came up with a sort of semi-joke and tell you I want to exploit the fact that in Lots of languages. Points come in two genders. There's Punto and La Punta, and the, I mean, I don't claim to have full grasp of the linguistic subtleties, but it seems many of the everyday examples of the contrasting uses of these two types of points to think that the maximum point is really like a bare point. Mathematical point is unusual. I want to suggest the intuitive idea that this map here is really mellifluous You see, what we're driving at is the idea that the cohesiveness of space is intimately connected with the possibility of motion.

20:00 And that, in some sense, if you have like pure cohesion, it's providing at least a basis and the possibility of motion. By contrast, the discreteness, in the discrete space, you can't move. Of course, the reason you can't move is because you think of time as being connected or cohesive somehow. It has to be cohesive. And some other kind of general space, where the general space somehow in itself really just describes variation, maybe some sort of sophisticated variation, not necessarily one opposed to it, but in some qualitative sense pure variation as opposed to pure gravity. But still, you see, if you consider all the possible criss-crossings of patterns that have taken place in this room over the centuries, then you will have a notion of cohesion. And then some motion that moved to the other. History of what has happened is a qualitative belief that the locale of tokos could become one in which phi-zero reserves products and their reality can be embedded into or connected out, which the locale of tokos should have, which are sort of two earmarked properties of tokos as a purest cohesion, or at least relative dominance of cohesion over variations.

22:30 So, also in my studies of the kinship relation, I noticed that the first model of ancestors, in some sense, pure motion, if you truncated that topos, you would get one of that, which really comes first, I don't know, but there was a slogan of Hegel, which was, All of these terms are related to the whole past, what it was, what it went through before, what it gave asin, that's just the asin part, so that somehow gave rise to, so that is the vision of pure variation giving rise to, if you think even the category of reflexive graphs, which is one satisfying at least those two axioms, it's supposedly a quotient of a locality of those, so what is the nature of that locality of those, it's somehow that. There is a minimal amount of zooming around that has to be considered and then identified in order to recover the idea of the universe. The technique here, mainly the sort of opposite view that I want to construct, this I think for 400 years, has been the main scientific attitude that you construct this vision of space, when we say space, you model time as a particular kind of space.

25:00 You model bodies, too, as a particular kind of space, so that the whole dialectic of bodies and time giving rise to space or time giving rise to placements of bodies, bodies giving points in past space, that whole issue is literally regarded as taking place in a category of faces, the body itself and the time in which the laws of motion sit in the context of the also Cantor's point of view means that we have had a fourth accident informally. It's just kind of a general guide on the thoughts of the three. So, another point is that if you ever tried to be capable, and I tried before I succeeded, but on the first page he says it's absolutely necessary that we begin with being. That we can't start with becoming. We have to start with the being. And he argues this in a great light that he obviously considers to be very important. So I think again, you see, the idea is, well, you have to recognize that you have a being. But then what you've formally developed is trying to climb back up to what this being is in some kind of more explicit way. You don't know anything about it, but you do have to recognize that it's there. And the third example of that stance, as I was saying, is Cantor, because he says, well, we start with the main one, and from those we extract by this subtraction process that I'm trying to model here.

27:30 The abstract sense of the cardinal. So it was essential that we had those magnets to start with, but to make it there, it doesn't really do much about analyzing it. A centuries-long process to sort of climb back up to an analysis of what many of you are, after all, because the first step was to make a massive abstraction. Recognizing the top, but not very much about it, went way down to the bottom by a massive abstraction and then voraciously climbing back up, this is at least part of Cantor. In fact, I want you to assume that this T is what is called an acronym, like the Boonis word, for example, and the, I haven't stated the axiom for that yet, but... This could be thought of as an acronym, for example, as an amazingly tiny object making motion possible. This expresses, in some sense, the idea. And also, so there are philosophical slogans. The experts will tell you it's probably a parasol, but in any case,

30:00 Philosophy and mathematics, since Bolzano hated the Gagin reason, great French geometers occasionally made a mistake as proof that the whole geometrical approach to mathematics was the philosophy of mathematics, and the students are told these philosophical formulations are nonsense, but what I propose is that, well, they're not at all nonsense, they have a very nice way of mathematically modeling them. So one of these philosophical formulations is that motion means being in one place yet in another at the same time. So this dialectic formulation of the meaning of motion is usually said to be contradicting, but not necessarily inconsistent. And the usual mathematical way of dealing with dialectics, the usual scientific way, is to sort of pull the aspects apart, neglect the 19th iteration of the dialectical relation, just cut off there, and be consistent, that subjective systems don't really matter too much, give us a reason.

32:30 Is it one? Or in other words, this is the basic contradiction, the value of f1, f2, is the place where we are, and yet, to make that completely impossible, if little t is a map from 1 to t, then that implies that there are no other global points, so this is clearly the place where we are, yet, we're not just there because g is not. This map is not an isomorphism, and that also, namely, again and again, two obvious points, but there's also the name of the constant, which certainly has these two canonical points, and so now we take the equalizer, and we require that that's the empty object, and we take the equalizer, and we require that that is an isomorphism to the object, and so now this can only be possible if t is not 1. This says in a very definite way, which we do not want, this says that T and F are very different. There's only one global part. So this is the fundamental contradiction of all motion possible.

35:00 Now, of course, the further critique that's going to complete this is that T to the T is a T-connected operator. This is a weak point in my presentation. I don't know how to prove that in a more basic way. And so in some sense that is one thing. So we have this actually. Sorry about the dispersal, the space, these segments. And the other one I just said is that, oh yes, sorry, the equation. If s to the t is equal to s, in other words, if s is t to the t, then that should imply that s to the t to the t. In other words, if T to the T is connected, the Galois connection, the Galois connection is that one side is called discrete, then the other side is connected relative to, here we start with T, T is the starting point, or if you'd like, one point is a retraction of T, obviously a point so, so you want this, for all, for all spaces S, if S is T to the T, then it's also T to the T to the T, or in other words, T to the T is connected. So now having put all of these together, we have, I haven't yet totally made the timing of that. I'm sorry I didn't really work this word right here. I'll come back to that. Well maybe I should come back to it here, because among other things it will imply that this co-equalizer really is a simple computation of the left-hand one, which, as I said before, ought to exist more generally, but usually is more topical than it is huge.

37:30 So yes, F.T. is an actual, whatever that is. I'm used to having more. Sorry about that. Well, okay. I was starting on the question that the allegedly nonsensical philosophical characterizations are, after all, incredibly reasonable. So, another important slogan is that curves in the decimal or in the small become straight, become equal to straight curves, become equal, divided, okay, in the small, insufficiently. The idea is that RT is precisely that, sufficiently extreme way, or, arguing from another angle, since the last 300 years, one of the most important contours in continuum mechanics and geometry and so on and so forth is the tangent bundle of punctures.

40:00 All of these terms can be used to define the function of the function of the object. There is a fundamental lemma that says that a function is representable if only one is an object. Then there is a single particular space there that obviously deserves an incredibly detailed analysis, because it's somehow the germ of everything. And so I want to say that, well, that is just like the tea. This business of being one place and yet becoming something else is exactly like that. I'm going to call this the tangent of a puncture and we'll see that there are more axes. So if we have any map, we have the curve, the curve path, the curve surface, the x and the t, the contradiction, and likewise here, the t, so we have this. Now, moreover, if we use any point in x, and so we have the, what do I call it, piece of x. The tangent space at, and likewise we have g sub f of x, which is the subspace of the function space y of t, pulled back and there to there, so we have curves.

42:30 And of course, all the little x's which have a given x is zero, we'll map into yf, total little y, because then y is equal to f. So we have this infusion of f sub x. In the small. Now, of course, t to the t algebraic structure, you know how to solve it in that way. Now, of course, to this point, there's a part of t to the t consisting of parameters equal to zero. Now, this, in the category, even though t has just one point, and therefore any globally defined enum of t, of course, takes that point and gives it a point. Nonetheless, when you look at the function of space, it actually consists of endomorphisms defined over some arbitrary parameter of space, and there you might have things that didn't always take place, but this is a really proper part of it, and I'll talk a little bit about how similar and how different it is, but for the moment, we've noticed that we have this subsystem of endomorphisms, I'll call it typical. There's no lambda, but it's a sort of homophage, or a dilation, speed up, or something, those paths, just by applying, since this is the part that preserves zero, the r, the r itself doesn't act on a piece of little y, or little points, faces, and of course it operates naturally, and so we have that f...

45:00 It's the part of t to the t that actually takes zero to zero. There's the evaluation map, an evaluation of zero, of t to the t to t, and it also is a consequence of a map, an equalizer. Clearly, r is also a sub-monoid, and it's a monoid as well. But now, you see, so I say that this is certainly a manifestation of three things. And in a probability context, this is in some sense the only thing we can really say about straightness, that f sub x is straight even though f was a curve, and all of this even though we don't yet know what g and r really mean, but later. So, curves in this form become straight, at least at a very precise reflection, at least a reflection of this thing. You could say, if you'd like to, throw around philosophical conclusions. This is a radically synthetic development of geometry, in the sense that I'm not postulating any structure on T at all, it's just an object, in some category or space, it's just an object, so it has it, and whatever it means to be in that category, it has it.

47:30 Beyond that, you know, it doesn't have a ring structure or something of that size. It does have this property that it has only one point, where you could say, well, this zero is a structure, but this is one of those cases where it's a property-like structure or a structure-like property, because, say, you give the zero, it's the same mistake as when you want to give it. But even that, if you want to distill it, that's an object that you want to make mention also in terms of a structure that you prefer. In fact, that is a good idea because there are many, many examples of children who may not have any. But the point is, now, we've got this multiplicative monoid. My claim is, well, that is the real numbers. In other words, the model of the continuum of group-sure quantities I propose is just R. And I think there's something to find things in Euler. Again, we may have Euler scholars here who know the place and the exact quotation. But Euler says somewhere that real numbers are just ratios of infinitesimal. And as I said before, that doesn't mean that there's some process that has divisions. A ratio is a homomorphism which might have some input and might have some output. The real number is determined as by which infinitesimals it maps into which infinitesimals it is multiplied by. So multiplication is the composition of speedups, speedup wheels, and becomes equipped with the multiplication just because it's...

50:00 And then the space of endomaps in the given category. This multiplication is in some sense not an accostumated structure. It comes from the fact that we have a category. You took any work in the space and you took a part of that. Now, so you may say, well, where does the addition of meals come from? Well, I have brought up three different stories about that, all of which would be important in analysis. I don't know if they are equivalent, but at least for example, Maclean's 1950 paper about characterizing additions in a linear category uniquely can be sort of applied to it, which is crucial, of course, because if a certain thing applies to a category, not a single one of it, but there is a nice little category, even though the powers of R, so the homogenous maps between the powers of R are having unique additions. The important feature of experience, which is that, well, I remember, for example, some of Walter Newell's papers on material elements and products and all that, he talks about maps being homogeneous, and he means linear, and from the point of view of abstract algebra, these are different, but in fact, in the smooth world, homogeneous implies linear, that is, homogeneous across zero. In other words, the Euler's theory of homogeneous functions is sort of non-trivial for functions that are not even defined at zero, but for those that are defined at zero, homogeneity at degree zero, of course, that implies that the earth, because of Taylor's work, we're taking that experience and using it as a guide for saying that if you look at the powers of our It seems multiplicatively, and look at homomorphisms between them, that that should be a linear category in the sense of McLean. In other words, that the obvious product within that category should also function as the co-product within that category.

52:30 So that is one approach to, you know, realizing that addition is also inherent just in the nonlinear category we started with. We would like that the space of homogeneous maps from r to the x multiplicative thing, so according to the power of the Leibniz way, we would like that this function is called m of x, the natural idea of distributions, on an arbitrary object, even on an infinite dimension of an object, even on the true value of an object, I mean every object x has its name, but, you see, so we would like... The axiom is an extensive and linear sense of quantum. So m of x plus y should be m of x times m of y with exponential law, which has a sense because of the zero map, we can define a map in one direction. That's the work of it. So the axiom that this factor should be a linear extensive quantum type implies in particular that all these values are additive by the length of the calculation. You will pretty much get out from that condition map. Another way, a third way of seeing that there is a condition is a little more interesting than going into the actual structure of the object T. If you look at the tangent bundle of A, it has a evaluation map to A. This is actually going to be homomorphism.

55:00 If I say Lie, because it's really the Lie algebra of A, then A is a good group object. I'm not talking about the Lie bracket here. That does seem to depend on where they've been expressed in manifolds. On the other hand, by the way, the higher Lie brackets, I think, could be talked about for the arbitrary quantum in A by consistent speaking about A to the power of T squared and A to the power of T cubed looking at the system. Anyway, I'm just here just talking about the fact that this is still a monoid. So, we can compute, for example, the Lie monoid of t to the t. That's one thing, but also Lie of r. If you look at this carefully, this is contained in t to the t to the power of t. So these are certain functions of two variables on t with values in t. In the traditional example, these are the functions of the form of, say, 1 plus, using these formulas that I have mentioned before, lambda times small s, where s changes over the final exponent, all multiplied by small t, where t ranges over the inner exponent. So this is, for an arbitrary real number of lambda, this is a certain type of function of two infinitesimal variables, and it gives an infinitesimal value, obviously, because it's multiplying. Now, so it turns out that, again, in the traditional example, that this thing is exactly those functions of Peter, who are various lambdas. But how do these things multiply? So this is really the additive structure. And if you look at, now that you have the additive structure, you can look at the endomorphism of that additive structure.

57:30 The endomorphism of an additive structure would always form a ring structure. So it turns out that there's a map from R into the mom of respect, or let's say respect of plus, but it really is just the media of R into the media of R. So this thing is intrinsically got both addition and multiplication. This is a map that preserves multiplication, but if it's a bijection, Then you will get that R is a ring. Everything in the way. So this is the third. This is the more contextual and getting ready as to why R should have an addition as well as an addition. I already said that from the very nature of accident delay beyond this is the fact that T can power T. It could also be T-connected. T itself is topologically T-connected. And then the fact that T is a property of T, no left at the top is with our peer group, and I don't really know how to prove it, but it's true in all of these examples, and I do assure you that in the ultimate algebraic geometry, the truth of geometry, the analytic geometry, and many other things in between, it is indeed the case that R is arranged in this way, and it is indeed the case that T for T is connected, and hence, O, R itself is connected. Another thing is that the inclusion of r into t to the t has a unique retraction and it should be called dot.

1:00:00 This is really just taking the derivative at zero. It may not deserve zero, but it has a derivative, and it does deserve zero. And that's the one of them. If you remember the chain rule, this g dot o g dot times Where the times is also a composition, so that doesn't quite look right, but you see the point is that these, oh, this is, sorry, evaluating at zero, evaluate the chain loop at zero, and then you notice that because these daffs are necessarily linear, the f's are necessarily affine linear, therefore a g-dot composes a daff, and I think it's the same as the g-dot, that g-dot is constant. So you get the literature going on and on and on. And again, all the examples, this retraction is unique. Now here's another. I have actually, this is, I'm in favor of the axiomatic method. I've reasonably not written it down yet. I will conceal it. I'm not writing down the dual numbers and so forth. It's actually really much more general. On the other hand, I think that one should be able to write down a sufficient list Strong properties of the team, which would imply all the others, but the present situation is that I've got a list of twelve properties, and I don't know that any of them imply any of the others.

1:02:30 The way that I get these properties, of course, is peeking at the numbers. I regard that as not in any way a flaw in the program, except for the importance of statements and results. So here's another very extremely mysterious fact. The multiplication of R is commuted. Now, you might say, well, what about T to the T? Well, T to the T can never be, never be commuted. No object in no jugglers has the property that T to the T is, unless T equals 1, or something like that. Because, so to speak, I mean, here's an intuitive, it's just a calculation, an intuitive explanation that the... T to the T includes all the constants. I mean, that's what my original axiom was, that that inclusion actually showcased it as an isomorph. And if two constants commute, they have to be equal by the very nature of constants. So, if this could be commuted to the T, it's essentially wrong, and then you set up the number T. So, here again, in some way, there's a huge contradiction, a real contrast in this very small... Somehow by preserving this one point, this necessarily non-communicative becomes commutative. Now, this is of course essential for many calculations, but I'm sure all those people who believe in the magic of non-communicative will say, well, gee, maybe even infinitesimal times non-communicative, and run off and do all sorts of fancy things. Then some of those things will be correct. I don't know, but I have no idea myself. All I know is that it seems to be quite essential that this time you're doing well, you have commutativity, and whether or not it's non-commutativity is why not way, way above that, why don't you like epsilon or something like that? But in particular, I just wanted to remark that the fact that there's a retraction at all, monoid or not,

1:05:00 Impliance of R, 2 of R also. Of course, this is a personal guide to that, but also the pure quantities. I say pure, by the way, because in general, endomorphisms, if you want to attach physical dimensions, endomorphisms must be thought of as pure, because they get back the same tiny thing as you feed in. And at the same time, those quantities that have dimensions of that, in that sense, will be represented by maps that are not endomorph. So, one little theorem that we can prove, and it may seem surprising, is that maps that have zero derivatives are constant. I'm surprised by this because it didn't seem to add quite enough. But, of course, we're using a, you want to say, locally constant. Locally constant maps are simply, this is the two-sided ideal. There are a number of maps that factor through the T-district, remember that, it's not so surprising maybe that's the true way. So when I say that the derivative of the map is zero, of course I mean that it actually factors through the subtraction.

1:07:30 For example, if they factor through one, they're really constant. What's the word? Bi-ideal. Two-sided ideal. If you compose on either side... And even potent bi-ideal, that's important, because that... the meta-level gets rise to substantial subtopology, which factors through some... through S. Now, so I want to say that if derivative of F is zero, in the sense that I have an eigenman here, then that should imply Now, the proof of this depends on the same axiom of being abnormal, which is also based on what that is. Explanationally, the use of that, of course, is that taking the tangent bundle function that is basically the topology preserves co-numbers. So, not that preserves explanation and products as explanation of anything would. But preserves also sounds and quotients and so on. So we have this diagram and we apply a key to it. Oh, sorry. We look at the image. We're assuming that we're in a total image factorization. So if we take the, this method here is that the image will itself be, and that, of course, that map is supposed to be induced by the map in the middle.

1:10:00 And, but you need to use the fact that the first rule remains, you need to use the fact that, you need to use the fact that one engine can prevent this problem. So any map that has a derivative of zero is locally constant, it doesn't factor through its image, but its image would be the street result in a concrete example of an ideal. Now, again, we can really use this, well, we can really use this if we have, you know, the contradiction between, the contrast between the speech and the negative. So, this there really means what it's supposed to mean, because R is negative. From R, anything, if they had to do a derivative. Another, I'll talk about it later. It's surprisingly true this year in the holidays. There's lots of time between sessions if you could make us a short break. Yeah, okay, I'll take a break. Hi, do you read? Yes, I do. Are you kidding me? I'm trying to. There's one prayer out here that's hard, and it's right. It's been an hour and a half. Come on, let's have a chat.

1:12:30 Thank you for your attention. Thank you for your attention. Now, if you're going to be smart, you might want to take a look at some of the questions that we've asked today, and I'm going to go through them one by one, and then I'm going to go through them one by one, and then I'm going to go through them one by one, and then I'm going to go through them one by one, and then I'm going to go through them one by one. Thank you for your attention. There are a number of key terms, but I'm not going to go into all of them. Thank you very much for your attention and I hope to see you again in the next lecture. Thank you for your attention.

1:15:00 But then we have some different kind of academic lectures also to give you an idea of what's going on. Yeah, there's a lot of stuff, but we want to give you a sense of what's going on, especially if you have some questions. Thank you for your attention. So, on your angle of R, this is your sun, T, this is your sun, this is your sun, this is your sun, this is your sun. Thank you for your attention. Thank you for your attention. There's a lot of these, you know, a few of these examples, and then I don't know why I don't think there's a bunch of these. Thank you for your attention. In this field, we're going to set up a very broad sheet. I mean, it's actually a very simple sheet. You know, I'm trying to sample things directly from this project. In terms of developments, in terms of programs,

1:17:30 we're going to have to do all sorts of things. I mean, monetization. All right. The thing corresponds to a hundred and a half million. In other words, we're going to have to take them all out. Thank you for your attention and I hope to see you again soon. Thank you for your attention. Thank you for your attention. Thank you for your attention. Thank you very much for your attention. Thank you very much for your attention and I hope to see you again soon. Thank you for your attention.

1:20:00 Thank you for your attention. Thank you very much for your attention and I hope to see you again in the next lecture. Thank you very much for your attention and I hope to see you in the next lecture. Thank you for your attention. Thank you very much. Thank you for your attention. Thank you. Thank you.

1:22:30 Thank you for your attention. So, uh, you must be. Thank you for your attention. Thank you for your attention. Because although many of the complex numbers are not in elementary, the analogy breaks down if you insist on elementary.

1:25:00 On the other hand, the basic being an atom of T is that the plank of the T has a right-hand point. The plank is an example of an essential form. And so the plank of the T is the inverse image part. And so that means that the language of the V preserves all positive logic until it passes over to fixed base. If you have any other type of thought of as a classifying, since you have, as a mirror, V, well, it will always be a classifying morphism under the structure of the V. Geometric morphism is a classifying matter under the T. It is an elementary extension of R, in a sense, which is both stronger and weaker than the tailwind. This is the example of Leibniz, I think, because I think Leibniz must have meant the dual numbers. It's very simple.

1:27:30 Leibniz was the discoverer of what? Leibniz's rule. And the multiplication rule for the dual numbers is exactly Leibniz's rule. The basic example of this, which is the following, I want to give an example, an example of algebraic geometry, let's consider the category of A, and what we look at is set value punctures on A, and then we are E, there's a certain category of sheaves, With respect to a suitable group of people there, and T is just the function that assigns to everything completely. One sees that T is actually, again, a suitable quality. This is a representative function, which is represented by the dual numbers for K, that object. So this is a representative of a representative.

1:30:00 Here's an additional example, and now we've known for 35 years, 40 years, that much the same kind of construction can be made in the realm of C-infinity algebraic spaces. Analytic, we've now articulated the line of the open disc. And there are lots of other related examples that they've fully investigated. I don't know if you've got as best of two, but you might say, well, now, what is this category S of key to speed objects? It turns out that, so that is a characterization of the kind of exactors of topology we took, but for many choices of topology, we get what Barr called the atomic topology. We shift to the separable algebra and the algebra that they cite.

1:32:30 So, in this example, again, the appropriate choice is a probability on E, in fact, probability for several choices, S thus reduces to the Boolean, in other words, of bar atomic, bar atomic, completely different from atomic, bar streams on the algorithm, well, sorry, on the category of finite field extensions. If S, the originals, does satisfy the classical logic and all that, it could be reinterpreted as simply the continuous representations of the Galois group with a separable closure gate, but in fact, of course, that site tends to have a lot of properties, of course, it's not really a group, it turns out to be Boolean, but over different sorts of bases, that reduction to the Galois group, obviously, every map is a cover. That would be more of a perspective of human attention. There is a particular topology, a particular condition on topology that is especially convenient when it comes to mathematics, and that is that we should take this as an example of maps that have sections, split up, because of course they're closed under pullbacks, or atributes. On the other hand, if we... every space has its own... And so it has a space called, when it all downgrades, a space of distributions, non-actives, as a linear object. So we can ask that a given map not have a section as it is, but after we've mastered the spaces of distributions,

1:35:00 we'll then have a linear section, a probability distribution. When you push that back down, you get the, in algebraic terms, a computer in terms of the tensor product, the precise function of both these local numbers. In fact, the problem is that it's sort of extendable without changing it too logically. Now, the reason that I like that topology is because of the expression that we have G cross T, and because we have that unique point zero, there are two maps there. There's also the diagonal map, but I'm not looking at that. I'm looking at the axes, as it were. Now, there's a different construction. I could take the switching map and the identity, and take the co-equalizer there, and that would deserve to be called T squared mod 2 factorial, the symmetric power of T.

1:37:30 N factorial means the group of automorphisms of N. So, T squared mod 1 T factorial. Again, because G is so tiny, since these two are really the same, this seems to be identified much less. We're just identifying this with this and reading what we get, whereas here we are doing the whole flip. So obviously one condition is contained in it, but there's a map between these two. Now, so, what I'm getting at is that I know a second order, and what this represents is the usual. It's not usually put this way, but when one tries to find the notion of the second order differential equation, one is led to consider, well, t to the t, well, that's the second tangent bundle, but we don't want to look at arbitrary sections there, we want to look at the sections that are symmetric. So the second order differential equation is some kind of map like that, basically because... We first look at the regression equation which is a section of a section of a tangent, a second order one in and of itself, but one that has this centric position.

1:40:00 So now what this C is, is either the prolongation, prolongation just means a section of an induced map. So we have this fixed map alpha, T, all the way to this constructed second order. Then we get a new category, basis x, equipped with the equation of x. The fixed alpha, for example, we get a simple category of algorithms involving tuples of t and tuples of t. That's how we get one particular axion equation in the fact that this, now, so one of the remarkable facts about this is that because of t being That one, having this extra right-hand joint, or, if you like, the Leibniz-Chan's principle fragment, a cognitive fragment. This is again a totals. And in fact, there are, the forgetful counter, in a different space of the prolongation law, forget the prolongation law, has both left and right-hand joints, so that we have a geometric morphism, an essential geometric morphism, that we need to recognize. So this is essentially the free and the chaotic and the laws generated by a given phase.

1:42:30 Now, this adjuvant is really only over s. It's not an e-morph. And this was really the original motivation for the definition of this. Because if you look at the fact that length and t has a right adjuvant, some kind of right adjuvant, Is S strong, is E strong? Because R acts on the speed of lambda, of course, is on T in such a way as zero. It also operates on the second term. It does so in a way that's compatible with the alpha. The dynamic laws which are homogeneous with respect to this action. Those are essentially exactly the classic characterizations. Now, the idea, so, this is essentially, Newton says that the body continues the straight line unless the force acts on it and all that.

1:45:00 This is the straight line motion, this is the notion of straight line motion that has to be given. And I look forward to the slogan that curved lines become straight and small, that's why we call it the standard case. I think it's probably true that the very, very short paths for any half-line connection with the known as the notion of friction is in the large, so it only depends on the half-line connection, and the very small one depends on the true-line connection. So, right, and this, again, if you take the right topology, as I said, the only random sections of the maps that should be used, that is, you find that, in fact, The symmetric power of that case, of that kind of a case, the symmetric power is also itself the spectrum on the dimension of it, namely.

1:47:30 I want you to study this function of a second order differential equation. So, E-alpha is a category of these objects and the laws, and it's worth it. It should be somehow just a little bit more between laws. If you have the general law going on there somehow, then a motion, a particular motion in the narrow sense, is some kind of map of the one-dimensional space of time. But the fact that this arrow satisfies the differential equation is really just saying that it's compatible with a standard law of evolution of time. Now this is literally true if we have an affine and if we structure the line, you've got the obvious idea that x double dot is zero there. x double dot is zero is an example of an affine. It's the law of capital F. So the morphisms which preserve it in the sense that if u is some time interval, then the x following the law is given there is equal to the given law on x. But that is the differential equation. It's saying, you know, that's just saying that it's commuting with the actions of morphism. So, I think this idea is correct. But now, what is exactly this time? Well, as Dr. May has emphasized, this x double dot equals zero is not general enough. Because, if you look at second order morphism, this second order is moving into the affine part of the other one.

1:50:00 If you had an affine, definitely, but at least the law at least has given that this does not have that homogeneity properties. Now there's a theorem of Jacobi, which even Arnold says he doesn't understand, but I think that the concept of it is in fact it can change the homogeneity definition in order to make a typical law into an affine. The very idea of the states in E, with the second order of laws, where you call the state space the tangent number for itself, and then the motion is, well, they're really just paths in the tangent number, which have to satisfy the same law, but not the same law as thought of as the special first order law, instead of the second order law. It's the first word of all on the tangent. Now, we're still in that simple set, so adding it over to square zero acts on u as an example of an electric field and all the objects here are electric fields.

1:52:30 These are just alpha-prolongations where the domain is one now instead of t, the co-domain is t. So, if you at least simply hand a log of a precise choice, you would probably be able to do it. You see, if we really believe that's a typical physical law, it's a second-order law, then time is not a mere one-dimensional base. It's exactly this. So, the sort of, okay, go back to the philosophical idea of the concute general and the abstract general. I think if a concute general is a category of dynamical systems, then time is the abstract general corresponding to that. So the abstract general that we can extract from this situation is the output. It's certainly not just a line. It's a richer thing. It's generated by a line, but as a general, as a generic second order result, instead of this trivial generic second order result going purely to the time. Now the other point is that inertia is essentially an affine connection. The actual dynamic law of the system is that Another one, so we have two. We have this notion of a richer category where the objects carry two second-order laws, one of which is affine.

1:55:00 Now, what is force? Well, force is a measure of the discrepancy between the two. Neither one in themselves is what we call force. Force is the discrepancy between the two. So one might ask, well, could we extract the notion of inertia just from a single given? And, of course, the configuration states, spaces, that we are most interested in are those where we take ordinary space, B, and raise it to the power of B, where B is the space that's standing for the body. And all the possible placements that are sent to the body of the space, B should be the configurations. The basic placements of the body into ordinary space. And suppose that we have a second order dimension itself. So we move this left side section up there. Now, can we extract from this situation the composites of the given c along mu?

1:57:30 So mu of y to the power of g. This is the law on E0 itself. Now, we can get the law on the full axis by taking that to the power V again. So, that is a proposal for the straight line, the sort of straight line . So now, to complete this story, we should somehow know that C0 is actually affine. But topically, that's what Erickson's theory of mechanics seems to be really basic with this idea that somehow it's a miracle that by averaging the nonlinear law, but listening between the lines, I wonder, can you say what sort of technical developments you need in order to carry out a program?

2:00:00 The right topos are not so, especially because we have very, I guess, infinitive topos, and they will satisfy all the basic requirements, which is a matter of development, if you want, partly of developing this theory that was written back in 1970. Parallel to that, there is the axiomatic program, which remains explicit as much as possible in the actual properties. Of that purpose, and many like it, it will suffice for a U.S. formal development which could be computerized without necessarily composing any particular sort of foundation for those two general approaches. The investigation of that particular kind of topology that I mentioned in how to break geometry, I don't think there's any problem with topology. Which is sketched in my recent paper on JP and A is the fact that there are many situations in mechanics, across a lot of surfaces, the material elements and so forth, which really seem to be, although they have been stated in terms of the visual analysis, really seem to be a matter for the bodies to be themselves are infinitesimal objects in one way or another. And so the actual development of that particular fragment that we will need. And what happens if T is not a point, if it just is a sub-object or has a sub-object?

2:02:30 Because coming to the beginning seems it also can be interpreted as a motion in a more general sense, not like being at the same time in different places, but like being in a different time in different places. Yeah, well, yeah, I mean, okay. The question is how general is this action? You see this, am I amazing? The extra right adjoint. Well, of course, in any pre-sheet trokos, on a trokos where the site has products, and every representable object, or exactly a representable object, you have this extra right adjoint, so there are lots of them. Moreover, and by a small development from that idea, you can see that given any rotating trokos, given any object in it, it's included in a larger trokos, but that object has become amazing. And so, it becomes, this conceivably would be an interesting, useful technique, because you see these prolongations, I've been just talking about prolongations that could make sense for any two objects, that they form a tokos, that depends on g being half, but such a prolongation, see, if g is half, I can also pick up the math from X into the fraction of A over T. And so we have this completely different way of dealing with it. We can deal with, you know, various operations that are defined there, so we can apply it to the things that make direct sense for this. So, the idea of the problem is that we're calling it the boundary value problem. So, any sort of boundary, it might be useful to think about the boundary, per se, You make it tidy by enlarging it until you can still do it, whether it's profitable or not.

2:05:00 Can I ask, how is this construction of the real that you have here today, how is it related to the one we know from the standard textbooks of the reals, as the sieve of real value continues? Well, I didn't talk about the ordering, but in that real situation. Some of these objects, in some way or other, partake of orderings. I mean, the discussion about Frantano's splitting the line and finding that the point that's left over has a direction, or whatever you said, I think that, I think that, no doubt, is literally true and an appropriate topos where one takes account of the directions more than the models that I was speaking about. So if you have any orders out there at all, so there are many orders out there. For example, a really local rig, meaning one in which a sum is invertible, used after being invertible, if and only if one or the other rig has a copy of the positive rational. In this sense, the word objects are automatically used by the Omega idea, Omega to the power of Q, in order to determine. So that is an instruction. You can see that quantities which may happen in the 10-month term, they can cut. Put it up to more abstract way a little bit. Well, that is, that's the idea that's going to get killed by doing that.

2:07:30 The definition of an R is that mathematics is exactly that. Yes, of course, my definition of an R is that it is related to mathematics. But then the question was, how does that relate to medical literature? It would seem that somehow these are almost two different aspects of the continuum. But you kill that and the other one doesn't see that. Good question. I think we should thank you again. How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? How long ago? The title actually, I know, sounds a little bit pretentious. What I intend to suggest today is a series of little issues, ethically, for which use of category theory can turn out.

2:10:00 Problems, hence the most important, dealt with, quintessential laws to provide response based on quantum mechanics, which is the first example is related to modern domain and the second example is

2:12:30 Concerned the notion of theory. The third example is the categories for the philosophy of business and milking the mechanisms that the pathos for the working condition. And I refute working philosophy. And so that is the closing idea of the session. I will skip the third example of the previous legacy which was ascribed to

2:20:00 And moreover, because the existential generator or the separator is likely to a constraint on the indication and the extension, it was mainly through the use of different tools and extensions, models, by set of and basically a relation of intuition and speculative logic.

2:25:00 So, as I already said, extensionality has various forms, different from the point of this speech, which I wrote a quite good version of later, to set the radar.

2:27:30 And each of the various forms of extensionality acts as a measure instinct of attention, in relation to which different categories are used. What the idea is, it's not just that, you know, intentions are so defined, structured, entities, that no category of objects can grasp and grasp what the idea is like. You have to find what processes you assign to, as a meaning of...

2:30:00 And the idea is even more important than these two of logic is not here, but it is the relationship between the spaces, next slide, concerns the idea of the second idea, that's nice, and there is also the class of structures, where the sum of these functions provides you with a good idea of what is on the field, rather than...

2:32:30 It is a much more strict notion of theory. From the categorical perspective, theories are models, and the basic idea goes back to models, thus, are suitable factors from theories to categories of possibly different concepts. So, these, and in large, of course, the practical categories of theory, we consider.

2:35:00 In this way, you see, I approach the notion of meaning and try to precise the notion of semantics or produce. There is a method, an idea, or architecture, which was developed by the Burbati school.

2:37:30 The notion, however, that the Burbati school had, overall, is the idea of providing the right notions to unify them in the respect of which, and so on.

2:40:00 The various learnings and features that make up this way, the greatness of this, we have incorporated.