Categorical structures & cohomology (lecture 2) (contd.)

0:00 This presentation of delta-m cross delta-m allows me to show the following questions that I have already mentioned and that have expressed topological spaces and at the same time by geometric realization of x, as I said, in the sense that I have applied. I will give an indication of the demonstration, as it is based on this, but I will not do it. The indication is that in the presentation, we will know nothing, if you will, x equals delta m, y equals delta m, according to the previous arguments, that is to say, x equals... Thank you. So, we go back to x and y, delta n and delta m, by writing x as before, as the co-limit on the delta n and delta n, and also, we go back to this situation there, and so the presentation, so it's really just indications, the presentation of delta n cross delta m,

2:30 In fact, I have not completely written here, but it is a meeting of delta, i.e. a meeting here of delta m plus n plus 1, which can be seen in theta n, 3 theta n, this presentation where I have said alpha and theta like that. And so, from that, we can move on to the realization of this presentation. We move on to the corresponding realization like that, to the corresponding diagram. With the corresponding degrees here, and this is sent, in any case, to delta n cross delta n, the threshold that we looked at, the threshold that is also determined at this level. And so, to demonstrate that we have denounced it here, it is necessary to know that this is also a presentation. They are aware of this because we have the realization of delta-n in relation to delta-n in a family of arrows like this. So to see that this arrow is equivalent to our foot, or even our foot, you just have to look at what's going on in the presentations. Well, I'm not going to say anything, I'm not going to say anything, but let's say... We refer to this case and we use the presentation. This is what we need to remember from this. The presentation of the delta L. And so, in this way, we obtain a corresponding presentation for those. And so, since at this level, geometric realization gives exactly that. So, in our case, we can... Well, it was just an indication. But I wanted to point out that this statement, there is an idea in this case. Well, then, let's go back to the question of...

5:00 The notion is a relationship, therefore the problem, the relationship-problem. Well, the H-relation is reflexive and transitive, but not symmetrical. You see, the intuitions are very rigid and therefore we often do not have enough terms, it is not like in topology, and therefore it is reflexive. So, there are two solutions. The first is to take the equivalence relation. In the case of homotopies in topological spaces, we just have to invert x, y and y, we invert the interval unit, we put t on 1 minus t, and in this way we obviously have the symmetry, just by looking at the homotopy in this direction, we don't have this alternative, but this is going to be something very big, and so the second solution... The first one is to restrict the spaces of the simplicial ensembles, to add conditions to the notion of the simplicial ensemble.

7:30 This is the second one. This is what we are going to do here. We are going to look at simplicial spaces, at simplicial ensembles that are better than others, that are better than others. So, the following definition plays a fundamental role in the Simplicia theory that we do not see in the usual topological theory, because in this one, this kind of problem is always resolved. And so, the definition is that a Simplicia set satisfies the condition of extension. So, it is even more common to use the name of the inventor according to the inventor of this theory, K.A.N. Modern terminology is fibrant, which is what we will talk about in the category of models. We will take this notion, we say that it is a fibrant simple set. When any diagram follows, we will be sure that there are enough elements in a simple set that is not too thin. And this coordinate was, of course, on any diagram like this, in a commutative diagram, with a y here.

10:00 Any application of a coordinate in x, except for an n-coordinate in x, extends to an application of the n-simplex entirely in x. This is the definition. So, a really combinatorial translation. We know that an application of delta-m over x is the same as an n-simplex of y, we have 2x, we saw it earlier, and so we can see that it corresponds to a cornet lambda-kn. Well, let's say that the data of x, given the content of the presentation, which I think I gave the last time, the data of x equals n and x0, so we list them like this. So, from x0 to xk-1, from xk-1 to xn, so from one element, so this is an element of this one, di of xj is equal to dij-1 of xi, which is strictly inferior to j, and i, j, both are different cases. The presentation, we saw it, there was a presentation with lambda nk, here there was delta n-1. And then there are the relations, so to give an application in an x is the same as a certain number of applications of delta n-1 by x, i.e. elements of xn.

12:30 And so, the condition of Kahn can be rewritten, and we will do that in our end. In xn-1, satisfied by the star condition, we can extend it, so I do not erase this, there is y in xn this time, such that dy of y is equal to y in question, obviously different from k, since for k we do not have xk, the condition does not make sense for the term xk since it is not there. This is the y, the data of the lambda m, the data of this value. So this is a kind of very partial subjectivity, well, partial, reinforced, in order to have operators facing each other. We'll see that these are spaces. These are the good spaces for the theory of homotopy here, but the point is that later on there will be theorems that will tell us, or at least I don't know if we will see them, when a space is satisfied with these conditions, and this is often the case.

15:00 So, a short example for you to understand. N is not fibrous, you see, from the beginning. For example, let's take the case of the identity application I1 in delta 1 and 1. And let's take xx2 equal to the image by gestation of the element 0, again it's in delta 1, 1. So the drawing is here I have 0, 1, and here I have x1, I think. This is x2, which is like this, which goes from 0 to 1 here, 0 to 0 here with an equal. The elements 0, 1, 0, you see, the elements that must be ordered in the right direction, do not exist in delta 1, 2, since they are increasing sequences, such as d1, 2, x2, equals x2, and d1, 2, x1, equals x1. What we should do is take the sequence x0, x1, x2 with this example. The following are the details of x and y, for instance, a y. I would like it to be this, an element like this. If you look at the order, you will see right away that... Well, there are different versions where some coordinates can be filled, but not others. And, well, these coordinates intervene. And in a more modern way now, we can talk about situations like this, in the theory of categories and of n categories, in which we know the names of n categories, of n cohomologies. I see Yann has introduced the notion of inner horns, that is, situations where this condition is satisfied each time that K is not the zero or n, but only those inside.

17:30 And so in this case, we come up with an interesting notion, but we draw from classical homotopy, it is all the corneas, it is for all K, and not only for this notion of inner cornea. The news is not as depressing as that, so that was the consequence. Now, some examples. Topological space Y, S of Y, in a centric sense, is vibrating. Indeed, you can see that if I have a continuous application like this, All phases, except the fourth delta n, can be extended continuously to the interior of the delta n, and therefore, by restriction, can also be extended to the last phase. Each time I draw like this... An application in a space Y is a continuous application, so I can fill it up in a continuous way here, something like that. The point is that this simple set S-Y of all the continuous applications of delta N per Y is a very large space. There is a lot of things in it, a lot of continuous applications. With a given simple set, it is something that can have very few elements. On the contrary, it can be a very small finite set. It can be very, very big if you want. This is what we are used to by topologists in art, to look at things like that, but it is very, very big and therefore it is not surprising that it satisfies the condition of extension. So another example that will be important in the course later, in the things on which we will specialize later in the course, is example 2 where an object is an official in the category of GM, each GM is a group.

20:00 DI and SI are homomorphisms, satisfactorily in natural conditions. Everything can be a set of satisfactorily homomorphic relations. What has changed is that it is a group and that the arrows are homomorphic, so it is a simplifiable group. Well, I say then that any group G is a group. Satisfy this condition of extension. Note that I did not say Abelian group, I said any group. It would be important for us not to restrict ourselves to the Abelian group, Simplicio, in the Veraclis, in the Parra, but in all the other groups. So I will give a very brief course, but very... Very effective, I think it really fits into the heart of things. Most of the ancient manuals, there are three pages, there are a lot of calculations here. Here is a paper that I think you can look at. We gave ourselves, as we know, from x0 to xn, with an object in gm-1, star.

22:30 To show, by recurrence, that we can extend the whole complex L-1 to x0, y2, plus k-1, k-1, I'll stop there, and we'll do something weaker. I'll start now with L-1 plus kxn, with L bigger than k-2. So if n is 4-2, it's 4-3, it's not good, so we're going to put an integral on top of that. We're going to take some intervals like that. So we're going to extend this together. We're going to put a y in gn, such as dy dy equals xi, to say everything when this makes sense. And by inducing this in the case of L-1, it will be K plus 1, we will have finally demonstrated the thing, you see. The more the distance between K and L is large, the more it is weak. For example, it could be the one that is all over the place, there, etc.

25:00 So that's the idea. And so, that is to say, that we consider the partial coordinates more and more full. In this recurrence, suppose that there is a y equal to di of y is equal to xi for i less than or equal to k minus 1, And this time, only i greater or equal to L, we want it for L-1 here, and I suppose that I know it only here by L, by recurrence hypothesis, that we know that, and considering the family of N-1 simplex, following, I take 1 everywhere here, 1 is the neutral element of the group, and then in L-1 position, I take xl-1, so it's a calculation, so I don't explain, times dl-1, 2y-1, and then neutral elements everywhere else, so all that is a single term, the protocol, here, here, there, so by all neutral elements. Except exactly in these terms, xl-1, dl-1, y-1, there must be something happening, it's the new place, and so I say in this case, this is a compatible family, in the sense that the relations we have, the DI, are trivially satisfied, the DI of that corresponds to the DI of the latter, of the way we live, and so we correct.

27:30 So, Y, the Y that we have obtained here, in a new Y tilde, and it will be essentially produced by these things, so the definition of Y tilde, it will be S L-2 of X L-1. In other words, I have multiplied my y, which I have, terms by terms at the levels of A. So this was, I added, I increased, for example, the degree by SL-2, and nothing has changed in the y, except in this degree, L-1, where I corrected by that, and we check. This is the end of the demonstration. This time, dy is equal to xi, and this time for i-1 is equal to k-1, and for i is greater than or equal to L, not L anymore, but L-1. From L, nothing has changed, since it's just things at this level, L-1 and L-2, that play a role. Otherwise, it's really the y of earlier that hasn't changed. So there is a modification, but here we started from a Y that was satisfied with its reference, we modified it, and so the proof is finished, but of course we must add details, and that's the idea, we knew how to skillfully, and we do this progressively, until we fill the whole thing. Well, so let's continue today with the Simplicia, one more moment.

30:00 Wait, I'll come back in a moment. There is an interim that we will try to demonstrate in part, which is quite technical, the reason to demonstrate it is that Professor Kessler is going to give a lecture in the following month and he told me that he is using the statement that I am going to give there now, and so he would be happy if I demonstrate it, I think, so there is a reason for this that can interest you for the future, and so what is it? It is generalizations of that. And so I will not demonstrate anything of what I say here today, I will just emphasize the following thing. So, in fact, the extension condition of the Kahn group is much stronger. When we add Cn simplex, well, it gives us much more than that. So we will see, there is a large family of A-contributed fissures A-contributed B. And not only the lambda nk contained in delta n, such as when x is vibrating in this set of conditions, then we have the property of the relief.

32:30 So this is the consequence, not a reinforcement of the definition, we will demonstrate, although I do not tell you today what this family is, but just some given terms for X and B, this inclusion, so there, we had given it a name, maybe F here, so there is a G, a simplificial application, that we call the inclusion diagram. At the beginning of the theory, we specified in a precise manner the cases we could extend, the generalizations of corneas, cornea products with simple X, things like that. Finally, we decided to discover that there is a whole great class of inclusions that works, and so the notion is more pleasant than the opposite. We do not need each time to demonstrate a statement like that, there is a general statement that I will not cite today, but I will also immediately look to introduce the definition more generally, the following definition. It is important to note that in 2009, more generally, we may call LP a simple morphism, you will see the generalization of what we have said, we say that LP is a vibration in the case when in any communicative diagram, yes it is, there is a renewal that I have here.

35:00 Zeta n times x equals the two triangles, and if the application of x times a star is a vibration, the star is the final set, the final object of the category of inflexible sets, that is to say, in each degree n there is only one element. This is the single tone of each degree, and the DI and SI applications are obviously identities, always, and all in a simple way, they are sent in a single way to this one here, and so it's immediate, it's this case here, if I have this, you see that delta N, delta NK, every time I have this, I draw like this, this, this, this always comes together because it's the most obvious object. The search condition for a G here, the commutativity of the lower and automatic square, and the only remaining condition is the condition of the expansion of the distance. The theorem that I hope to be able to show you a little later... I was not going to say anything because it was precisely given in the introduction.

37:30 This is the following, I will repeat it later. We want to look at a new level, a new section, we want to look at, it's quite short, we want to look at the spaces, simple and functional, simplicio. So either x and y, they are a set of simplicio, we are looking to define the set of simplicia that we are going to call Hohm, this time Hohm-Souligné. Remembering means that this is a simplicial set, not a set of applications of x and y, such as the exponential value of x in this simplicial space, in the simplicial set of a, b, y, and z, or an isomorphism in a double injection with a, b, x, 3, y, which we have already defined, I have only mentioned this one at this stage. So, we open up the set of synclistic applications, and we follow the synclism of x and y. In the exponential formula, the definition is reinforced

40:00 by putting x equal to delta n in the exponential formula. The definition of Hom-YZ is possible. We want the applications of ΔN in Hom-YZ, i.e. the n-simplexes of YZ, to be nothing but the set of applications of X cross ΔN of Y cross ΔN. We know what is Y cross SN, we know what are the syntetial applications, so it can only be the definition. And of course, for it to be syntetial, for any application, let's say sigma, injective or surjective, it corresponds to DI, DSI, co-degenerescence, etc. Remember y, z, n as you remember y, z, m by composition by sending u in the obvious way, y cross delta m. If I gave myself an element there, z, u in home like this, n, of course, I send it on the composite, a new sigma which goes from y cross delta m. In y3, delta n. In z, sin1, sin3, sigma, and so on. The arrow is induced by sigma. Remember that an application like this implies that it gives an application of delta m in delta n, since delta n and delta m are respectively representable as contravariant factors by m and n objects. So when I have a morphism of one in the other, it gives an arrow like that.

42:30 I'm doing the obvious thing here, and because of that, it's automatic that the Simplificial Relations here give us these Simplificial Relations, and these Simplificial Relations give us these Simplificial Relations. We've defined the Simplificial Ensemble well. So, afterwards, it's easy to define the exponential arrow. I'm going to leave the exercise, I think, because it's an immediate exercise. So, now, on the other hand, there is another, more concrete way to see this. Here, we notice the same thing. By returning to the case where y is equal to delta n, we can define an element of the form y, z, m this time. By using the representation of delta m cross delta m, I can find n-simplexes again. We write a formula for y equal to delta n, since I changed the notation between delta m and delta m, and then it's universal, it gives me a formula everywhere.

45:00 An element y, z, n corresponds to a family of applications yi to zi plus n, this time with i greater or equal to zero, which is the family of applications, like that, agi, satisfying, has identities, maybe I should have noted them there, for analogues to those, we will say like that, it is cheated since I have not noted them. To prove, in the case of n equal to 1, to define, we have to note that rho of y to the n in degree 1 is exactly the elements of rho of y3i to the n, and we really studied them correctly, they were applications of yi to the ni plus 1, that is, we need identities, and here it is the same thing, since we actually have after that... We found all the identities that were needed here. I didn't say what it was. I said that there was a presentation, didn't I? So maybe for the definition, it's a bit crazy how it's marked. But there you will find this presentation by Gabriel Zisman in the book that I quoted. Apart from that, I could not say anything else.

47:30 So, we have that. Next time, I will give you this list of what it means to have it written formally. The way to understand it is by this, where we have a presentation of the delta n cross of n. So, I told you that later I would show you a more general theoretical approach than the one we saw earlier for the vibrations. We will demonstrate the following statements. So what we will demonstrate is all I wanted to say at the moment. It is the following thing. We will see later that if x is a larger object than an integral, and that y is an integral of any kind. So, the artificial set, man and fibrous, is something very useful to extend the number of functional spaces whose goal is fibrous, especially fibrous, especially in the expansion transition.

50:00 Well, then, I will do enough, without a doubt, of purely technical things. In the last half hour, I started to talk about formalism and model categories. We will come back to this because we will have to compare them. There is a formalism that allows to replace the family of corneal inclusions in the Lenz-Sinplex with more general things and to understand what is really there. We will now look at a completely different subject from what I have done so far today, saying that it is a new chapter, since we will come back to this chapter later. So, this is what we call in English the theory of Krillen in 1968, maybe in 1967, something like that, and since then it has been imposed as the good case for the theory. It allows us not to avoid boring calculations like the ones we started to do with the presentations of products, corneas, and all that, but to put them at the beginning somewhere, in a series of axioms that we have to check, and then, if we have checked these axioms in a particular case, in a particular context, then it works if we want. So before we start, I need the following definition. We say that an arrow

52:30 F in a category is a retract, an arrow, G, X, Y. If we have a commutative diagram, as we said, we talk about a retract of an object, we say that an object is a retract of another. If we have direct factors, so if we have sections, so if we have a situation like that. So here, if we have a situation like that with the proposed identity, We would say that A is a retraction, traditionally, of X, but here we would say that it is a retraction if we find arrows, if we have a diagram like that, so F is the arrow G, and if again, it is the diagram, like that, is commutative, so it is a retraction at the level of arrows, not objects, in particular objects, but also arrows. We have to refine it a little bit, but Cullin has a lecture notes in mathematics 47, well, less than 50, I think, in which, and you'll see if he still looks at that, the theory hasn't really changed much since, but there's very little demonstration in it, and it's worth looking at. This is not the best reference now, but let's say that in 35 years it has not been completely cleaned up by others who have improved a lot. It is said that a category is a category if it has three categories of arrows, called the weak equivalences, the geocodantes, the coflux, the fibrations.

55:00 And cofibrations, which satisfy the following conditions. There are five families of directions. C and M, I think, is for cross-border. So, one. Before that, I have to say something. Before that, we say that a coefficient, there is a terminology. We say that an element of a vibration and also a weak equivalence is called a vibration. There is also a weak equivalence, vibration.

57:30 With these terminologies, we can enunciate the functions of Kuhlman. In Kuhlman, the category has all the limits and co-limits. In the case of Kuhlman, recently, there were the finite limits, but finally, we now prefer the terminologies of limits and co-limits. And so this implies in particular that there is an initial object which, as in the case of the social set, is noted in an empty set, as in the cases of people, and a final object which is the limit on the empty set. Then, FH, a commutative diagram, of composable and non-composable arrows, in the form of a simple equation. So, I say that in this case, if two of these arrows are of a weak equivalence, maybe I will be able to describe, as we traditionally do, WE, weak equivalent, while the weak equivalence are of a weak equivalence. This is the kind of property satisfied by bijections in the category of ensembles and isomorphisms in this category. And up to this point, what plays the role of isomorphisms is equivalences. We call this the action of 2 over 3.

1:00:00 So, if f is a retraction of g, as in the previous lecture, it is an arrow, G is a weak equivalence, respectively, a vibration remains a co-vibration, F is also a co-vibration. Now there are two properties that are really serious and characteristic. The first one we have already seen a little in a particular case. We have not talked about vibration in a simple sense, we have talked about its properties. Either by doing a commutative diagram of C with J as a cofibration, Q as a fibration. And, in addition, one of the two is acyclic, so it is a magnetic sphere, and the J is magnificently symmetric as it is called, where Q is acyclic, where there is a line phi, where B is X, such as the two triangles. We recognize here, in the case where the Q was a vibration of the simplicial ensemble, and J was the inclusion of a cornet in the N-simplex.

1:02:30 In fact, this is the case where it was J that was in fact the cyclic in this case, we have not shown it yet. In this case, there is a phi, which was the definition of a vibration in the simplicial ensembles as we defined it. And finally, the last property. These are the two conditions, because there are two, either we assume that J is a low equivalence, or we assume that Q is a low equivalence, and it tells us in a sense that, for example, assuming that Q is a low equivalence, then by equivalence there is the idea of the inverse, so there is a kind of property that says, then, I can take the inverse of Q, that is to say, of phi, which is the application followed by Q-1 in a sense. This is a common diagram, but it can also be relative to a sub-assembly, like this one here, which is marked by these positions. So that's it. The idea of this theorem is to invert the arrows, but in very particular contexts, where they are acute and in an inverted sense, in quotation marks. This is what equivalence means. In equivalence, there is the idea of inverse. Either it is counteracted, which is invertible. So we can say that it is counteracted by inverting this arrow on this subject. So in reality, it's not quite like that that we do it, but... This is what explains, when I say relief, these reliefs of this one, I was thinking of the case of Q and fibrations, there we would not say relief, we would say extension. I have a arrow from A to B, so they have relief or extension depending on the case. In the case of the condition of Kahn, this is what we call the condition of extension. We extend a morphism from A to X, a morphism from B to X. I'm going to write them separately so that we can mix them up together.

1:05:00 So we have a factorization like this, I and Q, with a cofibre and a Q with a cyclic fibration. And also, when we try to put them together, I'm going to distinguish the letters, a factorization. So with another space, with again, except that the letters change, with again the first J, a cofibration, and P, this time acyclical, and P, a fibration. In both cases, cofibration is followed by fibrations, an injective followed by a surjective. In the first case, the second is the equivalent, and in the second case, the first is the equivalent. These are two different constructions. So we have an obvious definition, we say that x in C is fibrous, remains cofibrous, the star arrow is in fact an acyclic cofibration, an acyclic fibration, so it is acyclic, we can say, the arrow is the initial energy of x.

1:07:30 Thank you for your attention. So, what we will show next time, which I wanted to do in detail, we will come back to this theory just after, there are some basic things here that we have not seen yet that we will be interested in, but the thing that I wanted to show is, obviously, one of the goals here is to show that the simple category of SimpliScio ensembles is a modern category. So, before doing that, it's still complicated objects, they seem simple, so I'll give you a simpler example, and so the definition here is quite standard, but we call it, I'm going to fix the chain, but it's complex, in a homological degree, it's going to be quite zero. You all know what I'm talking about. I'm going to talk about the complex of chain, which is a very famous terminology. I'm open to discussion. The chain complex is not ideal. It's complex. It's less oriented, etc. And so, I just want to tell you what we have to demonstrate.

1:10:00 So, if you want, the theory of the model... The homotopic algebra is the algebra of space, intuition, and everything is homotopically close. But a much more familiar notion of everyone is the homological algebra, which studies just complexes like this. And these are complex abelian groups, so this is the abelian theory, if you will, and the other is simplicius. These are either simplicius groups, this is the theory of non-abelian groups, or even simplicius sets. This is something even more cowardly. And before I show you that we have a model category for the category in simplicius sets, I'm going to demonstrate the following sentence. We have a category of models corresponding to complex chains. And so, in a sense, it's totally useless. Because it means that in algebra we know how to do it without introducing these notions. The most important thing is projective and injective resolution and things like that. And we can show that it fits into this context. And so, in this context, it's not that easy to demonstrate. It's a different way of seeing this standard theory of algebra. This is a good learning for general notions. So this will be the first example. And I will end by denouncing it. I did a lot of denouncing without demonstration at the end. So, the first example. We will take for C the category of modules. We will take for A the category of modules on a commutative line called R. And so C, the category of theorems. The chain category of R-modules is a category of models. To conclude, I will tell you what the three classes are for which morphism 1.1, a weak equivalence.

1:12:30 This is the most standard notion in logical management. It is a quasi-isomorphism, that is to say, for those who do not know this question, that the arrow indicates in homology h i of f, which goes from h i of n point to r h i of n point, is an isomorphism. So it is a notion of equivalence, you see, but with a weak sense and more or less of style, for those who know, categoricality. So I have to tell you what the other two notions are. Very simple, with FM applications, degrees per second, FM in the same, and on objective sources. So there is 1, be careful, not 0, we just need from the first step, the first step, and finally 3. So this is the idea, this vibration is the subjective idea, and indeed there it is really subjective, but a little more general, we do not write it in the lowest degree. Finally, 3 co-vibrations are injective ideas, if you will, MN in FN is and in OUTS, but in a very strong way, and the quotient MN modulo the image of MN of the injectors is a projective module. So if you want, in fact, it's a familiar situation that we know well in algebraic math, there is a situation like that with a P here.

1:15:00 And we have 120 here. It's a question of projectives, since projectives are a standard. And so it means that it's really written as a direct sum of m and n with a projective factor. So it's not that easy to demonstrate that. Let's say that it will be the last word. If we look at this factoring, if I have the application f of m in n, I think this is the scale, I don't remember the scale anymore, it's the first two or the second. So, cofibration here. Now it's cofibration. And so, in a sense, I said that when it cycles, it's as if we had an arrow in the other direction. And so we saw that in the case of cofibration, at least the factor on the other side is projective, so it's a wave. I don't know if it's a geological thing, but it's something like 54 to look at all the derivatives of Rome on a projectile here and to take Rome 9. I didn't say it was the same thing, but that's what I suppose. Sorry? I wanted an abelian category, so I wasn't sure. I preferred to do a commutative. I didn't say there was a problem. It's still abelian, yes. Listen, what is it? Yes, we should talk about bimodules, etc. I don't know, I don't know.

1:17:30 Since everyone is still here, I will continue informally my survey among the people who are here today. Who are the people who are doctoral students or students who can speak a thesis? Not necessarily here in Paris. Can they raise their hand just so I know how many of you are here? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 51, 52, 53, 56, 57, 58, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 56, 57, 59, 51, 52, 53, 57