Noncommutative topology in geometry & physics, String Theory (final lecture)

0:00 So if I have two sets, so one set is finished and we have to represent it, represent a set, and represent it in a naive way, with this way, in a totally honest way. In other words, this is a set, there is this common set, it is the elements zero, In this case, we have a set of elements called zero to n, which is the set of n plus an element, which is noted at n, and these elements are noted at zero to n. And this arrow is called a chain. So what is this arrow? Well, this is where the order comes in. This arrow, the arrow F, is an application of non-increasing. So that's not really that obvious. Well, what does that mean? Equals increasing non-strictly. For example, constant, where we see all the elements of M, which is a single element, and the growing ones, but in a plastic way, since there is a case in which there is a case in which there is a case in which there is a case in which there is a case in which

2:30 In other words, why is everything ambiguous, that is to say, we will write, it is less ambiguous, if i, interior h, implies that f of i is inferior to f of u. Well, strictly speaking, that's it. It's not strictly constant, but that's the definition. So, definition now. We call, together, a counter-variant, x, we call it x. Delta is the opposite category of Delta, the ones whose objects are the same and the arrows are reversed. In the category of Hans is the category of the sets. And Delta is the category opposite to Delta. Or arrows, or the direction of arrows is reversed. Concretely, for example, there are arrows in the state that play a fundamental role. While we're at it, it's going to be part 1, so definition 1.

5:00 Then we take step 2, which we call, together, co-sympathetic. A counter, this time, co-variant. And so, it's just to say, so we don't lose the arrows in the category of textiles, and so it plays a role, a little less in theory, but recently, people have been doing that, and it's still there, so it's worth mentioning. So, examples of arrows, to better understand these definitions, examples of arrows. In the delta category, there are two series of arrows that play an important role. If you don't have enough paper when it circulates, there are more people who have other sheets here. After that, you have to fill in your form. So, first of all, if I start from n-1, the category of the set of n elements, and I go to the set of n plus one elements, there is a family of di. This is for zero, an integral equal to i, an integral equal to n. This is the arrow. This is the arrow, 300 of course, 300. The application, for example, what do we have here? This is the application, 300, omit i. So the image, in other words, d of j, is simply to write c equal to j for j less than i. It's equal to i plus 1 for j equals i and to, yes, it's equal to j plus 1, I didn't get it wrong, and then the capital is equal to i. So it's a simple drawing, of course, it's this, the h and the i. So this one, i plus 1, it goes to i plus 1, i minus 1 goes to this, it's like this, and then it's like this.

7:30 So, TI arrows are injections, that is to say, it is the only way to detect whether a part of the arrow is treated as mercury or not in the image of TI. It is a cross-reference indication, which is the characteristic of X entirely. And in the other direction, we also have S-injections in the same number, which have this value of n plus 1 times n. And this is the application, I call it the 300th application, obviously, because that's what we're talking about, which repeats, in other words, which is to say that SI of I is equal to SI of I plus 1. So two elements there, and I'm going to do that on this side, obviously, there is an AGI plus 1, and there the two are going to be the same. So, of course, DI are injections, you have to do this in detail, you have to do things in detail, but this is absolutely the basis of all these machines, DI are injections, DSI are surjections, and so the hardest thing to remember immediately is that these arrows correspond to the identity, to the simplicity, So, what are these finances? Well, to look at what happens when we compose PSI and DSI together, or with the DI, so the first one is the following. Note here, be careful, that the Ds, which are entangled by these things, these contravariants, the values that are at the top, will have the DI, the DSI with the DI at the bottom. So, what are these? They are the following. So, DJ of DI is the same as DI of DI-1.

10:00 On the one hand, you can verify that with the characterization that I said, with the elements that are repeated, and on the other hand, S j of S i equals S i S j plus 1, this is the second iteration in this case, this is when, ah no, it's different, so this is for... I is less than I and SJSI equals SI SU plus 1 because I is less than or equal to SI SU plus 1, so we first have these rules here, they say how to compose between the relations when we compose between two D and two S and now the other thing we have is if we take a D followed by an S and so there are three A's in fact if I do SU of DI. There are three cases. First, it's equal to d i of s j minus 1 when i is less than j. On the other hand, it's equal to, we're going to write it, we're going to write it correctly, it's that s j is easier to say like that, s j of d j. If I look at it directly, when d equals j, it's equal to the identity and it's the same thing when I take s j of d j plus 1. We are now defining the two cases where there is an acceleration and an deceleration. And finally, the last case is Lj Di is equal to Di-1. So these three series of rules always tell us when I take a random d, I get an s. How to write it differently? There are three situations. d is smaller than j, and j is bigger than j plus 1. And in case where j is equal to j plus 1, it's the easiest one. And how do we compose ourselves? We have three simple symbols, right?

12:30 Yes, that's right. So in a drawing like this... It's a situation where we have m, for example, n, for n, if you want, for example, for this one, in this one, we have n, n plus 1, the gndi, and then I would do n plus 2, the gndi, a situation like that, with the s, it's in the other direction, but especially the case, the third case, is when I have s. N to the power of n plus 1 is equal to n to the power of n plus 1 to the power of n plus 1 to the power of n plus 1 to the power of n plus 1 to the power of n plus 1 to the power of n plus 1 to the power of n plus 1 to the power of n plus 1. This diagram is a diagram of this type and other different things for other types of physics. So, these arrows are not hard to see. In fact, none of these things are hard to see. But it may be a bit complicated now that we are using the arrows of physics. And S-J-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G-E-N-G.

15:00 The vector, which is written in a magnetic way, again, is equal to d i s d i 1 s j t plus k s j 1 i 1, strictly inside i s, and j 1, strictly inside j t. And so it's worth seeing here that... Since it is an application that must go from N to L, and so, in fact, it is a bit complicated to say that, since we have to go back here, P is the same thing as M-S. By definition, it is the number M-G which is equal to the number M-S. The goal of the IBS is to be the main source, and if we put something like this, we have this idea of appearing the same, so it's like that. And so the proof there, I'm not going to go into the details of the proofs, but a little thing that I forgot, well, no repetition, there is an element that lives in the source, and there is a repetition in some. R2j is equal to Rj plus 1. I remind you that each of these j corresponds to the repetition of two successive elements. And so here are the frequent successive elements. So look at what could be done with this application. It would be like this. There are the j's, or the pastries application, where there is repetition. It makes the list of elements that are given.

17:30 So it's great. And likewise... The I1s are inferior to the I s. The I in N, this time. So here, the I in N, which is an appartment of images for the application S. And we saw that the applications of the I1s avoided, precisely, the entire ones. And so these entire ones that are avoided, they have the same collection, and they are between the I1 and the I s. And so, with this lexicon, you can see that this application has this property, and on the other hand, it is in a public way, and so we have something stronger than the following identity, in particular, we have found the authorization of F, the F is an application of N, and in fact, it is the application that goes there in this definition, in this subjective, and there is N. This is an objective of first of all of the equations and then of the spaces that are, I don't know, the question of the pen here or the arrow. So this is the category of data. Do you have any questions? Excuse me. In which direction are you rotating the composition from left to right? Oh, is that composition of arrows or stuff? I have to read. So first I'm applying the S's. I should call this maybe, so here in our case it's one S, and this one P, where it's the composite of the previous one.

20:00 In the set of x index n, by definition, it is the vector x applied to the object n, with n at the top, and for all m, a pair of positive numbers, and for e, we will say, for all e, f, we go to m. I'll show you an example of an application that would be a Vx of f, which is always like our little technology, we'll talk about that later, which, as I said, was contravariant in the other direction of xn or rxn, such that a member of the component g, since it's not the indicator, but in the air, f, g, g of f, well, I'll say that x... All of these terms are used to describe the application of g to f, which is an application of Cxg, which means that the Xs are in the same direction as the Xs. This is just the definition. There is a better description. Here is an auto-description of the best or worst frequencies that we can do, the initials.

22:30 So, what we are going to do is that, for e to the power of e to the power of n, let's say that i, by definition, is going to be, we are not going to use it, d i equal to x. What did we call DI earlier? So DI, did you remember that DI had xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn-1, xn And here, it's always an indication of the excess, xn, towards x, n plus 1. Basically, that's for all i between 0 and n. So, the way to remember is that from xn, there's always n plus 1 arrow, but the same number of arrows here as the number of arrows there, only one arrow in the other direction. So, now, if that's rule 1, We deduce, so this is an interparameter of the definition, so it's an implication of the same axiom as the other one, we deduce false-inficiable identities, in this case, and the following simple-inficiable identities on the tables. The second definition is the traditional definition of an academic lecture.

25:00 It's a family. I'll come back to it later if you want. So we have DI, DJ. It's just the previous rules, but the order is reversed this time because of this contrapartition. And SI, SJ equals SJ plus 1, S limit. There is always a multiplication here in this sense, so it's still the same answers as before. This is for the interior of J, and this is for the interior of J. And finally, we have the three rules corresponding to the precedence, S, J, D, I, S, D, I, S, J. I'm going to put them on the table so that you can see them in contrast to each other. I'm going to add everything here. DIJ equals DI-1 DI-1. SILJ equals SJ plus 1 SJ plus 1. And finally, the third series of rules. DISJ equals 3K SJ-1. And finally, it's equal to the identity when i equals j for u equals j plus 1. And finally, it's equal to lj times bi minus 1 when i is greater than j for u equals j plus 1. I would like to say for those who see the meters for the first time, one of the actions is also to mix the identities, the identities that I use here, the application, if there are no arrows in this table, there are arrows in the ensemble, you see the identical application on the identical application, otherwise you will not find this arrow in this table. So these are the applications. So this is an artificial set. So, as I said before, an artificial set is made up of n plus 1, slash, xn, xn plus 1, n minus 1, which I call di, and n plus 1, slash, xn, xn plus 1, which I call sd.

27:30 Speakers include star identities or stars of the relationship of physics. There are names for these operators and I will explain why. The application DI refers to the operators AS. The application SI refers to the degenerations. This pronunciation is just a consequence of what we learned in this way. Why not DI for degenerative sense? Sorry, DI for? Why not D for degenerative sense? For DG? Why not D for degenerative sense? Exactly. But D is for differential, so it's good too. Okay, so what is S for then? Thank you for your question. Yes, it's a good question, but we are so used to it that I never thought about this possibility. For me, it seems quite natural. That's not a suspension. That's not a suspension. Yes, it's not a suspension. That's not a suspension. Yes, it's not a suspension. Yes, it's not a suspension. Yes, it's not a suspension. Yes, it's not a suspension. Yes, it's not a suspension.

30:00 Yes, it's not a suspension. But no, that's not exactly it. So, in the same way, a data set, a co-synthetic set this time, is equivalent to the data set in a co-synthetic set in the form of y, and the members of the set y are equal to zero. So here we write at the top, because it's contravariant, Key terms may include D-indices, Y-indices, Y-indices, Y-indices, Y-indices, Y-indices, Y-indices, Y-indices, Y-indices, Y-indices, I am very happy to be here today to talk with you about my research on quantum mechanics and its applications in the field of quantum mechanics.

32:30 Equals t, u, s, u, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, Next, if we want to be very specific, we will say that, more generally, C is a category, therefore, we call the object of C, the counter of delta 0 in C, the countervariant Y. It may be the opposite of the value in C, in other words, we have a counter like this, in other terms, corresponding this time to the elements in Xn, which are not all of them, but both objects of the category C, but for all of them, and the application of Xf, Xf is an arrow, and Xn is Xn, an arrow in the category C.

35:00 We will talk about these terms in the course of the course of the course of my life. We will talk about these terms in the course of my life. We will talk about these terms in the course of my life. We will talk about these terms in the course of my life. We will talk about these terms in the course of my life. In all these cases, you may have noticed that in particular, the arrows xn to xn-1 di and xn to xn-1 si are now morphisms in category C. This is just an example of the group of adenians. Here is a group of adenians in a special case. This is an example of a group of adenians in a special case. This is the data of the family of the group of adenians Xn.

37:30 Family BU per xn per xn-1 plus SI per xn per xn-1. So, family of what? Of homomorphisms. This time, we will not talk about them together. Homo-morphisms. Correlation of the quintessential identities. And what? We could have said inventors, etc., but we saw that it was the same thing because of the way in the delta category the features are generated, and we can multiply these infinite examples. This one is the most important for us. We could say that we can dominate anos, simplicio. In this case, there are anos, commutative, for example, and di, anomomorphism of anos, etc., etc. We could talk about simplicial spaces. C is equal to C is equal to C is equal to C is equal to C is equal to C is equal to C is equal to We have not seen that there is a threshold in this category. We must now, in any case, define it. By definition, we call, a ridiculously abstract definition, but since we have started as a center, as we say, we call morphism I between two, we will call morphism U, between two axiomatic ensembles.

40:00 And I would say that both of the meters of the category and of the category as a whole are the data, since they are the meters, the data of natural transformation, or APU itself, which goes from the meter x to the meter y. So, I want to detail that, it's not like that that we want to see, for those who know the notion of natural transformation of the category, everything is said, let's say more precisely, more concretely. I don't know how to translate this, but I'm going to do it with a pen and paper. Here's what it's going to look like. For all n, an application um of xn is true to yn, compatible with the first letter, with the last letter, and with the last letter of the essence. Thank you for watching this video.

42:30 C to D, a pointer for the great being between the two categories. So, for all, together, we have an official object, C, which we proposed f for x. This is an official object. I don't know what that is. It's an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, The group is not linked together, it is generated by them, generated by them, together by them. So, Pocahontas tells us that if A. is a small group, it is linked by a structure, a non-sensational structure, which is the Euclidean group. At the moment, since the group is a unification between them, conversely, if...

45:00 X, X, M, together, the initials, then Z, M, X, defined by Z, M, X, X, M, which is equal to Z, which is represented by X, M, which is represented by X, M, which is represented by M, which is represented by M, which is represented by X, M. Well, that's all there is to it, but now we have to look at the examples of scientific concepts. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. I think we're almost there. Yes, okay, I'm going to show you the categories, the forms of the categories. Yes, we're going to look at the examples. We're going to take a top category, that is, topological spaces,

47:30 So, there is a category of infectious spaces, of infectious spaces. A component in the real world is an XM space. The DI and the DSI, phase operators and generations, are continuous applications of XM2, XM1, XM2, XM1. And so we have these infectious spaces. Are there any questions about this part? So, you probably have examples. This is a small one. And what we will see is really a small one. That is to say, it is implicit in what I said before, that the objects of these portal assemblies are larger. So, the following example is quite fundamental because it is going to associate us with a category. All of this is the point of passage between the theory of categories and the theory of algebraic topology, which is studied without simplification. So the construction is as follows. We define the binary matrix of C and the binary of the category C. So you see that it is a bit inconvenient for one thing.

50:00 It's just these objects here. That's why we needed an integral set. We needed the place of all the objects in the integral set. That's why we needed these objects. For degree 1, the component of degree 1 becomes C. That's going to be the integral set of all the arrows of C. And more generally, the component of degree 1 of this is going to be the families of n possible arrows. Thank you for your attention and see you in the next lecture. So this is also an ensemble, an ensemble of all the arrows are an ensemble, an ensemble of all the arrows. So the arrows compose a composite, which is obviously an ensemble. And so I have to tell you that these are the components, I have to tell you what the operators are, the degenerations. So if I have u0 to n, I would like to fix that. All of this I will send to an operator S.I. who will send me a reposable arrow that has one more element. This is the one in which we start to identify. U0, U0, 1, U0. And then alpha 1, here alpha n. It's the same in this way. So, I started with n composable arrows and I associated a set of n plus 1 composable arrows in the only possible way without making a choice.

52:30 The first arrow was the identity arrow. And on the other hand, if I have u0, it becomes f0. And more generally, if I have the SI, it's the same thing. That was the last one, and now it will be u23, u0, u1. And even the DI, so the DI arrows, two arrows, at 74 degrees, which are a little less than the top, and so it's just a little bit of an extension, which is DI0, to U-1, and that's obviously alpha-1-1, and here I see U-1-1. And so this arrow here is going to be the arrow n plus 1 plus i. So the usual convention would be to compose two arrows. We would have wanted it to be in the other direction, n plus 1, but that's obviously how it works. And so that's the T arrow. And we can see that it's a composable arrow. So it's n minus 1 arrow, as you can see. So it's going to be an alpha arrow. Alpha is minus 1, alpha is minus 1. So we have a minus 1 in here. So I forgot to say that there is still one arrow missing here, I could have asked which one is missing, I don't remember, so I could have specified in the study that the arrows, well, there is one, you can't be sure, for the arrow, if I start from C, in degree 1, I have two arrows, D0 and D1, which go to the objects of C, and so the point is that D0,

55:00 The answer is P1 of alpha, which is U0 times U1 times alpha, and 0 of alpha is equal to P1 of alpha, which is P1 of alpha, which is P1 of alpha, which is P1 of alpha, which is P1 of alpha, which is P1 of alpha. The diagram you see here, by Alfa, is pure convention, you can take the dimension you want, this is the one I would take, which is zero, so it's not in the order of zero to one as we want, it's here we named zero and here we named the top, it's the top that we remove from the count in the indexation, so it's one thing and it's the other. This is the category of a single element, a single object, I called it an star, and therefore the set of the arrows of the star, of the star, the arrow with the note g, this time it means a single object of all the arrows on this plane. We have g equals the set of the arrows of the star, of the star, the composition of the arrows. I'm going to take a group. I'm going to take a composition law of applications. So the composition of G and H is H times G. So I'm going to take this group. And so with Schoenberg's law it worked well.

57:30 So it's a monolid. We assume now that the category is a group, what we call a copoid, that is to say that any arrow in C is invertible. In this case, EG is a group, which is less than 7. And so, the subject, the whole subject is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, is, So, let's sum up what it is. bg is an object, so it's just a single element, the star of earlier, g1 is g itself, and more generally bgn is g to the power of n, isn't it? It's a star, a star, a star, g1, gn here. The operators that we want, if we don't have them, we just have to change the group of their proposed group, if necessary.

1:00:00 I think that's it. We define the DI of G1, Gm, G1 to G0. That's it, it's G2 to Gm. The DI of G1 to G0, where there is a phase operation. G1, Gi, 3Gi plus 1, Gn, 0, i, i, n, that's where it's in this class, as if in a linear deficit, it would be composed, it would be in the other direction, it doesn't work at all, and finally, G1 to Gn minus 1, This is just an example of the two syndicals. I will give you two other examples now. I'm going to show you one more. It's space. It's called Euclid. I'm going to jump a little. I say space. It's not traditional. It's rather traditional. Now, we're going to do a little topology here, and we're going to write another ensemble of these tables. So, consider one of the three.

1:02:30 It's an example of a simple equation. For this reason, there's a lot of delta, but there's a bar around the delta n. So, delta n is the convex envelope Rn plus 1. The basic vectors are dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, dn plus 1, From zero to Tn in Rn plus 1, we have the sum of Tn and Tn, as we know it, the coordinates, the definitions of the coordinates, for example, in relation to the affine base from zero to n, in the space of Tn. I hope that everyone agrees that the drawing of Bethlehem, I think that everyone agrees that if we have E0, E1, we have two basic vectors, so if we take these two points, we have E1, now if I have three, E0, E1, and E2, so I thought that I had this question here, so it's a segment, an interesting area on the other side.

1:05:00 Thank you for your attention. Okay, so I want to point out some things that we have seen and that we will see again, but that we will discuss in the future. An application f of eta corresponds to the continuous application. So f of eta m corresponds to eta m in this direction, such as f of ei. r is applied to the point ei. The sum of E, I, C, E of F of I is extended by the originality. We can also use the formula of specificity. It's a method to solve a problem. We can use the formula of specificity. The application T, I, which we have just sent, M-1, N-1. But there is the formula of general specificity.

1:07:30 That's the thing. It is called the Neocrystalline N-Syntax, in particular because, as I said, if I put n-1 in n, theta n, it is the inclusion of theta n of what we can call the i-n-phase, that is to say, the one which omits the top and the same SI. The sum of n plus 1 plus n is the projection that defines the sum of i, or I can call it EI. It is the projection that identifies the sums EI and EI plus 1. And so, the point of all this is that... The collection of delta-m for m-variable and n-variable forms a co-sympathetic space.

1:10:00 Each component here is a space where the arrows are in the same direction as there. This is a co-variant, not a contravariant structure. It is therefore a co-sympathetic space. The topological space is called the quintessential space. So, the notion of the inverse of chance follows that for all topological spaces, we associate that Atiyah is associated with the whole. So, here, Saint-Juliet is the end. All of this is equal to the sum of the applications of delta-m and r. So, I will end with two remarks. The first, and we will see later, I will develop, is what we call singular homology associated with space. That is, homology h-n-5. In fact, the way of classifying these homologies includes the applications of space-synthetic OCEAN, but this is a complex design.

1:12:30 All of this is a complex design. So, the context is very close. And secondly, these synoptic techniques tell us that this is similar to homology, for example, We have cells, as it seems to me, but where the cells, it would be the same things, the cells are images, the delta m is an image, it is an entire element, an image of the delta m, and so it can be very simple, it is not at all the delta m, it is not beautiful delta m or balls that we put in, but their images, in any continuous application, are parts. Eventually, it is very ambiguous since the details in topology space are not the same. I'm going to ask where is the sheet I made circulate, because already at the beginning there are people who are going to say that everything is there. It's a language that everyone has to repeat. It's a language that no one has to put 120 seconds of the absolute. No one has done it. Another thing maybe, the question is how many people here do not understand French. Naturally, everyone has stayed. How many people really don't understand the question? In any case, let's say at some point in the course...

1:15:00 I don't like French, but do you know who was trying to write for you? Well, that's a problem in any language, but I will try. Yes, so don't hesitate also. This is what I should tell those people for the time being, and we'll see how it goes. But they should not hesitate. So now a arrow, we have to say that it is not an ensemble, a arrow, an indication that we have a very beautiful structural arrow. We are talking about structural here, not about generations or compositions that enter the structure. So we go from delta n, m, to delta n, m prime. When I have m, m prime is completely greater than zero. And finally, here, I'm going to send this here, so this is Mesh times delta of M over N. I have to send this to Mesh times delta of M prime over N. This is the result in a representative counter. And so I gave myself here an alpha of the category. And so, this application is defined by the value of n over n, because there is an object in the set of these arrows, an element of this component, and of course, I send it on the object n prime to n, so maybe the composite, alpha, so alpha n prime, combined with phi, phi over phi over alpha, in other words,

1:17:30 In other words, an element of delta n is an application of n-crochet n, so it is given by the image of the elements in n-crochet n, n plus one element in it, and it is an orderly family of elements here, so delta nn, another way of saying it, is to say that it is an entire V0 to n. These terms are defined like this, so that Vi0 is equal to Vm is equal to M. So, M simplex in delta M is the data with K Vi, by definition, is equal to image of I. This is an application of the image of each of the elements, and since it's 300 months that everyone has this, it belongs to the next generation, so it's very interesting. This is another very important thing with punctors, like this one, or like this one. I'll try to explain it later. Besides, to print in delta induces an autism.

1:20:00 In a simple sense, we will define this, which goes in the covariant sense this time, delta n to delta n prime, and there, which sends an element of delta n, you can remember that it is an arrow, that is to say, in the future, the application of delta n in the mth component here, in the mth component here, which sends an application m. In this way, we conclude that what remains is the antithesis of this symbol, the arrows U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, Star up, star down, since it goes in the same direction as the moon, so it goes in the same direction as the system. The morphisms... So it obviously converges with the operators of phase and generation. So this is the application of the artificial sense. And the morphisms of the impulsive sense, u star for all, u in the arrow for delta,

1:22:30 In the simple category of the quintessential ensembles, each element in a crochet element is associated with a quintessential ensemble, and it is covariant in the objects when u, r, y are in the same direction, i.e. it is a u star that is in the same direction as delta n. So it is in the non-inverted direction of the quintessential ensembles. And so a notation is sometimes, we can call it delta, we can consider delta m n in two if we feel like it's too much, in two indices, m n, and by definition it will be delta m m, it's a little graduated, here I have parentheses that I have to read, and if you want it's a little complicated object because it's n. A space, a co-sympathetic, co-sympathetic set, a co-sympathetic object, and so it is syntactic in M and co-sympathetic in M. So these things are interesting, we will not talk too much about them. The most interesting elements, the ones we will not talk about at the moment, are the bi-sympathetic sets, so bi-graduated and bi-synthetic. Simplicials in each of the two directions, in the first variable and in the second variable. Here, I have to point out that this is an object of the concept of simple.

1:25:00 Why is this delta n with the red index playing an important role in our theory? Well, I say that for all together, it is a fissile, and all whole. One reason is because it looks in detail with the bars and the n-subtractive types that we saw earlier and we will specify the relationship. The other reason is purely inside the quintessential theory of a dissection. There is a natural dissection between the whole of the arrows in the quintessential sense. The delta n value in x is in conjunction with the xn set, the n-degree component of the integral set x. It gave itself an element in there and gave itself an application like this. And so, first of all, the proof is that I said that the integral sets were the factors of the delta n value in the sets. It is said that delta-n is representable by a crochet object, so the proof is very elementary anyway, and what we call the Yoneta theorem, which is the fact that delta-n is representable, and therefore an object is representable, an image of an object on the project, on the object representing the object, and so now another proof, more explicitly, if we are familiar with the scientific language more extensively, which exists ...

1:27:30 An object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, an object, And so, this allows us to ask the application of this graph, which is defined by applying an alpha element, if I take the definite, so for alpha, the arrow of alpha n to the value of x, So, if alpha equals alpha applied to the alpha index n, alpha means in particular that the alpha n, delta n, n, in xn, is a component of the alpha arrow, in the transverse inclination, and I can apply this alpha n to the identity. Identity plays a universal role, in this way, we have a flash like that, and so that's enough, I don't know, for the very elementary verification, and conversely, everything is in there, and it comes from something in a certain way, it's really an example, so the word NEM and NEMETA include, it's a big NEM, in fact it's a very small NEM, but it's not a bad thing. So, we introduce the NEM.

1:30:00 We say that an element, an element M of a quantum artificial ensemble is a non-degenerated element, which is not an image of a degenerate operator, which is not of the form of an element X. In Xn there is an insensitivity, an element of EGN, and if there is no force, X equals S squared, with S being an operator of EGN. And an example, a wonderful example here, there is a non-degenerate element of Eta n index n. So, it is important to factorize, not to factorize with even just a little bit, not to factorize in the image, because that's what I mean by the operators of the universe, and of course... In the past, it was always used as a single component. These were elements not in the image, they were elements that did not have the same element of identity.

1:32:30 They were both adjectives and surjectives, so one of them was one of the others. So, to go a little further and also to understand a little more the relationship with geometry. So there, they still have, in the rest of the synoptic, definitions. So, I'm going to start with a little bit of geometric language, which we call, in a different way than what we used to call it, we call it P-simplex, sometimes I call it that. It's a set of a little A, a big X, all-emitting, of the component xp of x. And as I said earlier, it's not P-simplex in the sense of topology, it must be very degenerate because the application is so obscure. Thank you very much for your attention. Ah yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, The sub-assembly, in the sense of the word, the sub-assembly of the synoptical consists of the P-simplex of the X.

1:35:00 For example, we have here all the P-simplexes with P-impeccable to the K, which is the K-skeleton, and the two types of degenerations. We have S1, S2, S3, S1, S3. There are a lot of different types of images, but the most common ones are the ones that are less than or equal to k, k-skeletons. So we have the skeleton of x-k, x-p, and nothing else than x-p when p is less than or equal to k. And now everything can be said if we take a material case, x, a reunion of all the states of the great hierarchical order, of the four states, x. So there is precisely a state which is very important, and we put, we call the edge of the n-th integral of the state n. I'm sorry, in which sense is this a simplicial set?

1:37:30 We're saying it's a simplicial subset existing in all of this. So I didn't define a simplicial subset, but I did define a morphism between simplicial sets, so an ejected morphism. But in which sense is this decayed skeleton a simplicial set? I don't quite see... It blows all the degrees less than there is no problem. And if I have something which has degeneracy, then by using the simplicial identities, if I have a dj of si1, si2, then the simplicial identities will allow me to write it as si of something else, of lower, of length, one less or the same length. So in that sense. Yes, I would like to ask you an important question, which is an important one, which is an important one, which is an important one, which is an important one, which is an important one, which is an important one, which is an important one, which is an important one, A sub-assembly can be used to define a mass greater than or equal to k dm because it corresponds and it's pretty much the same as xd itself. Now, for an element like this, i.e. the degeneracy of a given object, it must be seen that for degeneracy operators, it's something like that, it's obviously something degenerate as well because I'm going to add an S in front of it. On the other hand, for low-level operators, such as D-indices, S-indices, and J-indices, it is there that simplicial identities allow us to describe S-I-D-J-I as S-I-I-I-I-I, or something else, and so the image of a degenerate will also be degenerated in simplicial identities. All of this is a subsamplex of x, o, middle, and identity.

1:40:00 I think that's enough to say that in the form of an education, so that's enough to say. So, we have a problem. Is the formula for f, k, x, or p, 1, 2, 3, 4, 1, 2, 3? I think so. I think that we can write an explicit formula, since we saw that every element has to be 20. Something with Disney and then DS. So we can write in this case in a unique way, routinely, under this form, I think, with a more rare rule, with an order on the links that we have to state. We have seen that this is a unique composition. Now there are common... well, this is not the best answer, but at least we saw at the very beginning that everything could be written as something followed by b and then by s and then by b to the decomposition. So in particular if something is this way and with these orders, not just arbitrators, then it should be written this way. So instead of these, that would be something towards the formula of g. Okay, so, now we have the form of the simple expression. So, keep in mind that delta n, I'm going to give you two examples, but it's things like that, it's... We saw that Theta 2 was the tritonium, Theta 3 was the tetra R, and Theta 3 was Theta M, and so we started in the case where Theta 2 was with the interest of centricity. So here, the edge of the simplex is Theta M, the geometric drawing is the edge, and so the definition here is this, and the edge is the quintessential ensemble. For the delta of delta n, or I should have said s of delta n, because it's an image, but we can't say delta of delta n. Or there are other names, if there is this name there. It's not very good because we think about the interior. It's not very good because we think about the interior. So they are torsion of delta n like that, defined by, well, it's the n minus s of delta n.

1:42:30 So, for verification, we verify that, in fact, Ansi has an important point, which is that if we look at the degree n-1, and that, it has exactly n plus 1, obviously, which are exactly the e, i, 2, i, n. So, if we have n, it is in delta. N, N, that I have N plus 1 in the operator phase, which goes from i to n to n minus 1, and so these elements are in there, and they form the elements that are exactly there, and we have these elements there, and delta, delta N, in this dimension, is the greater or equal to N. This is an example of an image of these n plus 1 elements by the operators of the fluorescence. For example, for delta 2, for the drawing we took, it's 0, 1, 2. That doesn't mean anything because I didn't put any gavels around these images that we must have. All these terms are constituted by three

1:45:00 These are the three elements that are the three elements that are only the same now. Yes, but the degenerates... Every time I generate an element, I have all these degenerates, so... No, but there, if we were to connect the edges of f to n, n-1, n-1, n-1... Yes, it would be enough here to say, I gave you the image in which there are things that are degenerates, but everything degenerates is the same as everything that is an image of something in n-1, even if the degenerates are lower. There is a series that goes back to the first minus one and then it is generated and then it is enough. It is the same thing to do that. It is generated by the line, it is generated by the minus one. And so we have three, three, three, three, three, three, three, three, three, A delta N would be a component of the degree N of delta N which is not in the edge, because the edge is no longer the interior, it is the imaginary edge. So here is the first sort of good result, even if it is a bit hard, but in fact we have a presentation of the delta N edge in terms of reunion.

1:47:30 This is the next thing, so when I have a presentation, I will write in the category of the 5-physio ensembles, the object delta of delta n as a co-equalizer of a copy of delta n, so in the category of the 5-physio ensembles, the edge of delta n is the co-equalizer If I put delta n index, I have the hooks because it's not the i's from earlier, so this is just a copy of delta, actually it's the n-1 index that I have. So it's the same thing as delta n-1 for all i's. It's just that I have several factors, I want to index them by the i's, and also delta n-2 indexed by 2s and j equals delta n-2, so this time i and j. So, what is the diagram that we can co-evaluate? I will take, so by this I take, it means disjoint union, in other words, the co-product in the category of implicit ensembles.

1:50:00 It's just because in the category of ensembles itself, the co-product is disjoint union, because it transfers from component to component. And then there is the co-equalizer, the co-limit of the arrows, so two arrows here, and there I have at the level of delta n-2, the co-equalizer of a value of, once again, And so the interest of this index is to tell you what the arrows are, so this will be sent to what interests us, the edge of delta n. The element here, n-1, will go to di of n, here. We said that this is the element in the body, this degree, n-1, so delta i n-1, and here delta n n-1, it will be di of n, of the body, and here, the arrow here, so it will be like an arrow. So this is induced by the arrows that I called DI, it's better to say it like that actually, excuse me, where DI is the arrow from ZAM-1 to ZI-SAM that we had defined previously, and so they fall into something like that, and so there are arrows DJ-1 and DJ-1 and DI. Key terms are dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1, dj-1,

1:52:30 All of these are co-equalizers of this law. So, a co-equalizer can be an example. So, this is something in the category of this law. So, the interest of this is that if now I see, I want to look at... So it allows us to say something very pleasant, and if I want to look at an application of this with a value of x, the translation of this statement, since in some books it's like this that we see things, an arrow with the property of being universal of the commutative calculator, which is the collimite of the diagram.

1:55:00 Here we have a very simple map with two arrows like this. In fact, in fact, the community of this is exactly the cohort of this. We have given an arrow, r, with a delta n, rt, in simple. So that we can, in a traditional sense, start with the coordinate. Equivalent to this. It would be the same thing if it was given an arrow like this, as the images are made by the two arrows here, and this arrow here, as a symbol. And that's exactly what the data is composed of as a symbol when I compose with one or the other arrow here. So that's an F arrow, let's say, and after that, I'm going to add an alpha and a beta. Acell, the arrow, the reunion on i of delta n-1, i to rt, such as f' for alpha equals theta to rt, co-equalizer, co-equalizer, it equals, it becomes equal, the two images are equal, it becomes f. So this is a very good point, but we have seen these copies of the deltas, and we have seen that a morpheus of a n-simplex in an ensemble by the length of a delta is an element of the corresponding degree of the simplicial ensemble. So we have the following translation of this dimension. Just to be clear, we have the data of a morphism on the edge of delta n in x, in the line, which is equivalent to that of a nuclear n plus 1 in a working way.

1:57:30 X0 to Xn for each of the Xi is in Xn-1. This is exactly what Ftils gives, because Ftils earlier is a family of n plus 1 arrow of delta n-1 in X. So it's a family of n plus 1 elements Xi. But as there is a presentation, so as we have the following developments. DI equals DI-1 for all, zero, less than or equal to I, less than or equal to J, less than or equal to L. So, we see, maybe we don't understand so easily with this, this kind of drawing, but a morphism of delta A minus X. This is a combination that is not as pleasant as usual, and that is much more common than that, but this gives an explanation. The simple algebra is full of objects like this, where we talk about different relations. First of all, the vision of these relations is important, and here the vision is the drawing from earlier. We have the goals here, this is x0 with something like x1. And then you have to take the relations that happen at the top. So when we take the operators from the top, we have the relations given by the scientific identities. And so the proof of this presentation, I just want to say the following thing,

2:00:00 the proof that the diagram of a co-evaluator in a presentation like this These are just some of the possible identities for Simplicia.