Closed differential forms
Robalo, Marco
Derived de Rham complex, relations with negative cyclic homology. Examples: classifying stacks BG, smooth schemes.
 So, let's go for both parts. But first we have to talk about the Fridge of course. And the last one, we saw the Potential Complex. So very briefly, let me just say I will define the different two forms. You just take the complex, it takes hours and the element of the module will be back as it was before. Sorry, it's a complex. Okay, so P forms a degree M for basically n for cycles in this complex. But there's no rapid Fridge of here. That's just a complex complex module. P forms a degree M, let me explain. P forms x. The big Potential Complex takes hours and the fourth of the degree M, our penitention n cycle. So the way you mean when you P Lx is like the sheet of the Potential Complex, and you take local sections that are complex of the P Lx. Because what you want is a sheet, it's not complex. You take more sections of this. So after now, this way is a complex. Ceremonitation for P forms a degree M. Let's keep it like this for the moment, okay? I will read it. There is no question. So let me talk more about... As you start by saying that the explaining idea... These are P forms of degree M, and I will press this planning idea for what the closed form is. And then I will be more formal in explaining how we can put this P closed P forms a cycle of complex. So this idea of a top. This is assumed. Now I will talk about closed P forms. What is the idea? So if you go to X, or 2, or... E, you can construct these random formulas, okay? And... Okay, so you have... And you can do... You have the derivative for each row. Okay? You can take P forms, and you have the derivative for what? Okay, sorry. That's the other way. Over, sorry. No, we're in the time of omega 1A over something. Okay, that's it. So the idea in the last two settings is that you want to define P for a closed P form. So just take things that are closed and this is a differential. But if you go now to case where this is not moved, you want to have complexes here. So you have... When you find closed things, you have... These are going to fall in some homology. When we say these vanishes in homology. So you need something in degree here that's called... Okay? So the idea is that the definition would be, if you want, if these are now closed complexes, you're going to say that the P form has to be degree 0, like something like in here, the distance. You're going to say that this guy is closed if when you set it here, it's zero in homology, which means that there is omega 1 in the level above, such that it closes it, and you're going to impose the closure condition up to infinity. You're going to say that... So, hoping that this is closed, but now, for now, over some more, not talking about the P and scheme. I'm not... I'm not assuming that these are completely... These are complexes in general, okay? If you have complexes here, do you want to say that means to be closed? What I'm saying is that you have to say what causes it at each level, okay? So the notion isn't that... X is not fine, you're asking. Okay. So if you want this guy to be closed, this is the same X, same A, this... So that's it, only that it's a P form, okay? Of degree 1, of degree 0. It's going to be closed if there exists omega 1 in this guy, such that the differential here is the differential, it's the right differential of omega. Okay, and if you use this game of... you say now, also, so let's say, and... dR of omega 1 is d of omega 2. And it means... Okay, so this is the idea that I want to make formal during the course. Okay, so that's not the definition. It's not an definition, it's just to sketch out what was the... Okay, okay. People don't care that they have to deal with this as definition. No, no. The problem is that you explain this definition, okay? This definition. The idea behind the definition. Exactly, okay. Okay. So... So, in particular, so this is just a sketch. And what I want to explain is how to construct spaces. Let's say, px. And space, again, px is the right side. Space of p-forms of degree. Actually, I should express it in this little lecture, okay? Ha! I like the degree of p-forms here. So, space of p-forms of degree n. You can organize these guys, these notions here. This piece of... You can organize these guys into space, okay? Sorry, these guys first. You can organize them into space. And then you can also organize them into space for kind of space. It's a blaster space. How much of a space? I see, please. This would be a simple, shall I begin? Okay. So, okay, so this is space of p-forms, sort of, of closed p-forms. Okay, something we can do. Okay. So, you can organize these guys into space. And the idea here, so this is the first idea, so this is the first kind of definition. And if you think of an idea, then it's the idea that if you want to close a form, you need to specify these guys, omega 1, omega 2, omega 3, they're actually closed for you. Okay, so in particular, we can construct, I don't expect you to construct a map. From closed forms to forms, which is not an inclusion of the subspace. Okay. But how do you need this code? It's not in the... It's part of the data. I mean, I'll be cool. I just thought it was going to be a little bit... It's not going to be a commercial idea. No. The problem is going to be... Yeah, on fibers? Okay. So basically, for any point here, that's a form. You're going to have space here. It's called space of keys, not bandwidth. We see generally what's happening. These are the main keys. Okay. So, space of keys and ways to code them here. In the user, user and the words is... Sorry, you may have some comments about the space, but I'm sure you'll decide. Which generally is not the correct answer. Right, in the ending also. That's the answer. Okay. So, we'll work two examples. The first one is the rest of our examples. The x, for example, I'll leave more details. In the end, the x is good, for instance. And the y is good. This space here... Let's do zero. Form p forms a degree zero. Close forms, the pi zero of this guy is just classical. Close. Forms. Okay. So, this is Spanish. And pi is strictly degree zero. Sorry? Two things really strict. Yes. In this case, and if you want the pi zero up to the... Okay, that's great. So, pi zero is going to be h dot on p plus n, and then this goes up to pi n. So pi n is one. It's going to be h dot on p plus one. And pi n is going to be close to forms. And then it's finished. Okay? Just to give an example, we'll work out this example. Okay, let's try it. This is the closing introduction. Okay, I think this... And now, starting with something. So, we can say this is the call of the closer points. So, if you remove close, h dot on xn, that's the element of a clean space. So, pi n is just p forms, and then all those groups are simple. So, this is an example where the pi goes to the nth. So, h dot on xn. So, pi i is zero, except when i is for n, it is in p forms. Okay? x has to be a fine point. Okay? And smooth. And smooth. And easy. Sorry, this thing is... Wait, it is on smooth where the pi is the same, but pi i are non-zero after n. But the linear term will prove something. You have to change the height of the force of non-linear. Of a p xn. Not close. Okay, let me start now by explaining the... So, preliminary things. So, the first thing we need is the notion of a very complex system. So, what is a very complex? A very complex, just a complex. So, a complex. And on each hand, you have a very complex system. And the differential is compatible with the gradient. So, I will write, like in the paper, I will write dG gradient for the gradient of gradient. Okay? And the first thing, d. Now, m. And I will read that notion of v is the notion of a mixed complex. So, I mean, the most direct way to say what it is in this case is just it's a complex again. But where we put the differential between the other way? We give the structure, we give another differential that goes the other way. So, one way to say is to take k of epsilon modules with epsilon in degree minus one and epsilon square with zero dG modules. Okay? And we use the, let's say, right epsilon dG k for a mixed complex. The two differentials, commune and mixed. And epsilon has degree. Yes, epsilon has some degree with a degree minus one. I will talk to you, say yes, yes, yes. So, we use complexes. And then we also need, and then, we use minus one for the two differentials. I mean, there are two degrees. But you think of the polar coordinates or the graded units. It's not graded. It's not graded. It just means... Ah, sorry, yes, it means... So, I guess it's minus one. Okay. It's the... So, the key to shooting is to epsilon for both differential and element of grade. If I'm not mistaken. What is the action of epsilon? Sorry? You know, I can improve the notation if you... What takes after the one? I was just complaining that you're using epsilon in different ways. No, it's the same. This is the action of epsilon. If you take an n module for this guy, this one is exactly what we mean for the action of epsilon. It's not action of epsilon. It's not action of epsilon. It's action of epsilon. Why? Okay. Action of epsilon. So, when you write a description, k of x, k of x, multiplication by x is the definition of... Are you sure? Are you sure? I'm not sure. Okay. So, this is the thing of epsilon development in Algebra, but the same time epsilon is operated, we'll also take our countages. You can do dot epsilon. That's all right. It's fine. It's fine. Let's keep going. Okay. Now that we've worked out, we can just go on. Okay. If you want to work out... k of x of n of g of 1 is just something that we have. And then you can... If you want the action of this side, it doesn't work. It's the guy who keeps the map like this. This is the product of the sheet. This makes the sheet in degrees, right? This guy is in minus one. You're going to have just that on the map. From n... m of n to m of n minus one. Right? This is the map of the derivative. It's also... So you're saying that epsilon is a gradient, like in the ring? But is it in degree minus one? Okay. I think it's easier like this. Forget this. This is one more. You have a complex with the differential going in different ways. So then you have... braided... just a quick one. So, okay. Again, I will not specify this guy. I will just tell the extension of the idea of what the other thing is. Don't sketch. Tell us actually. Because it's confusing. Basically, I won't tell exactly what it is. It's a complex like this with a gradient on each level. And epsilon just shifts the gradient by one. So it is good if you have some element here. So it has a total degree minus one minus one? Yes. And the idea is that it's going to be one, two, one. You set yourself up with your notation really well in Europe. EID equals the strep sum of E and T. So I'm sure you can write what epsilon is. That's the problem. So explain this to me. I mean, you cannot just write it. He said... Yeah, yeah, yeah. Is this right? Okay, so... This is a complex like this. And this is a gradient. And he does it like this. So you can write the E and B and say how the potential will move the E and B. Yeah, yes. Yeah. So D is going from E and B to... E and B. Just what B? B? I don't even know what I do with the size. Can I just make it a bit like this on this one? Okay, so... Yeah. Is it okay? Yeah. It's okay. It's okay. It's okay. So there's always the B zero, which is the sum of these guys. And epsilon shifts to the weight by P by one. Is it clear? Okay. So they have something like this. So this one is nice, but it's possible. Yes, okay. So... Okay. So let's write epsilon Vgk graded for the category of these guys. Okay? So after these... Ah, some things. So these guys have natural notion of moving performances. So you can take the line in the category, so you can write the line in the category. So I'll just put... Can I not write the line in the category? I'll just put like this. Where is this? Yeah. I'll just write this for the line in the category. There is one important thing. Tell us what are the opposite because we have two different ones. But which one do you use? Yeah, that's true. I'm going to be second-line explaining that. You're wondering what are the two? Probably the first two. So the more people... The more people that just... Here in this case, I'm just a map of the reserve of rays. Here are the maps that are commuted with good inferentials. And there is the same thing in reserve of rays in intermediate inferentials. So we have two complexes and I'll say that some of these are not good. Yeah, just say that again. The map that we want to... I can pet both with both thresholds. Nothing with nothing thresholds. Okay? So... Do you want to be both ISO and both? Or... Okay, so you'll be both quite ISO on total? Okay. So you forget the position, you forget the gradient? Okay. And you ask that the end-use map on the other end... Okay. Always with D. It's always with D. So the map of graded complex needs to respect the splitting, right? Like you've written D as the direct sum of these other... Yeah, D is the direct sum but the conditional... These complexes, the D is the one that you have one in use on each... But I'm saying a map with graded complex, they don't further mix, because they have to respect the splitting. Yeah, the splitting. Yes, the splitting gets the weight. It's graded by this. It's graded by this. Okay. So what I wanted to explain enough is how to even this thing like this, construct one graded complex, the method of... So we have a weighted version of the ADC model. So how do we......lead these sides? Yes. Just a second. And the sides. So your chromology is still graded. There's still a chain complex in fact. Yes, but you take the sum and... So... D is the one that I'm saying. I wanted to get a... I didn't say yes, but I should say about this... this epsilon VGK graded the infinity category of these guys. And then afterwards, Antonio will explain his talk at some point, I think. He said, I mean, not this is the relation. So he's still showing that these guys here, these complexes here, are quite similar to the ADC category. Quite similar to VGK and VGN. For some of us, I'm sorry. This is what I'm saying. Graded complexes are the same as things... quite similar to VGN and VGN. Okay. And these, you can write Jesus... K, T minus 1, which is the chromology. And you can see the VGN is these guys. And if you want to see the VGN, these are the graded complexes. So this VGN is VGN... K, T minus 1, okay? So if you look at the data in this case, the chromology of this guy is just a... you have a map like this. And every element in M, you can write as if by that sum of guys, like this. You can actually see how it's graded. Yes, exactly. It's just... So these are the two ones. And you will see that the map... I'm not going to explain the auto-construction map like this. Okay. And afterwards, this map, we can identify this map that I will now explain to the same thing as... So we have... these are the things that are VGN. Okay. These will be quite a convenient sheet for the P. Now it has Gm, cross-product in VGA. Yeah, the map on this guy is just a... The fence of which nobody took you. So what's the number of stairs? You see what's going on in the background now? Okay. Yeah, this is where the problem is. And VGA or the... This looks really close to... I mean, if that was auto-VGA, and you grab it with these, and it's just like any one of these is auto-1. Yes. This is the actual... The full-faculty section. But you have to even auto-... You can auto-faculty this. So what do you say? But now you have to feel ordinary and have the perfect amount of auto-1. Here, in this... Yes. Push forward. Could you rewrite the ending overlapped forward? Here? Yeah. Why is it we're here to cheat? What's happening inside? Yeah, because we have Gm. Do we have an extent after... Yes, it does come from the floor. What I'm saying is that in relation to what I'm explaining now, and what I'm explaining later. Right now it seems like we have a product really, classified as Gm. Now you have to see this group, Gm, Sela, Ina, and Iqbal, and it's VGA. Take the automaton and auto-eculance group, and that's Gm, Sela, Ina, and Iqbal, and it's VGA. That's the auto-eculance of VGA. Okay, so you have to IGA the scale, and you can make VGA actually the same value. Yes. Do you need to send out to have an action from VGA? Another way to explain this, is the shoe or the......the shoe. I just need to automatonize it in any case, if I'm not mistaken. Gm. Of Gm. Of VGA. So sorry, is VGA acting on Gm, and the group's attack on... Gm is acting on Gm. Gm is acting on VGA. I hit the algorithm on VGA with two hands. Okay. Very good. Can I... I... So, what I want to explain is the map. And how this map already encodes is in the form of a X-ray, of course, for... Okay? If you... If the fundamental question is here, you take this map to keep it up for the next one. Sorry, I was explaining. Don't worry. So, let's start with the... Would you now take there? If you have not taken there, and you take the grid, then you can construct the following complex. You take......and see... Okay, let's... I will start with something not grid, and then I will modify those things so that it takes the gradient in this version. So... You can define a complex......by this formula. Okay? It's a product that has a 10-round space. No, I'm trying to... I just saw people... But now it looks like it's all over the place. What's X? It's five. There's five of you all sitting on the table. No. It's definitely not that. No, it's definitely not that. There we go. But... I'm happy. No. So this is what you want. You have a... No. No. You understood the idea. Why? They just want to make jokes. Two dollars a month. Yeah, it's a change. So this is not five. This is not fully built-in. And this is just a problem. Okay? So now you have this guy. And now you can do more or less... You can do the final construction so that you take the gradient in this direction. If you have something like this... With the gradient... With the gradient... You can define... Something. Which is a little bit right here. And it's the same thing you consider if you shift... Every time you make the gradient. So this is... I mean, this is the vertical guy. This is the... What's P? For each P. So this is gradient. So for each P... You can define this guy. I'm talking A, I, A. Ah, okay. Here's the gradient of... Sometimes C, W. I think one P. And I have this one. So this gives you a... A comp. And... Okay. So... Maybe you should use the same notation for the weight gradient. For the weight gradient, you need two parentheses on both sides. Because this... On the left side... Ah, because it's like making two power, you mean? No, no, no, no. You can put people side by side. There you go. I was using notation. Okay. So you didn't say what the difference was? Okay, so this is important. The difference shows... So let's just give an idea not to write in formula. So you have... This is just N. You have A and... minus 2. A and... minus 2. This is P. Sorry. Yeah. P equals 1. People's 2. And then you have... Here. Let me write. Say it again. So this is what... This is N. And it's 1P. But it was 1P. And... minus 1. P equals 1. Okay? So the idea now is that you would just want to use the action of... of epsilon. And your differential will be the sum of... of the two. The differential is... The differential is... the... on... Ah... So it's an element there, so it's a family. Okay? So this is going to be... on an order of J. So it's going to be... X1 of XJ minus... minus 1. This is... the... Okay? Okay? See? Here you can already see precisely how we call this differential form. What you wanted was exactly... So you have some... some form. Imagine that these are P-forms of A and... So what does it mean to be close? You're going to say that they're close if there's a differential on this. If there's a cycle with respect to this differential. What does it mean? It means exactly that its differential is close. And here if it is in the right differential, it's going to be equal to something and the same thing all the time. You see? So... I'll let you......... Okay, some other things you can do now. You can find... So we can find the differential. As you wait......................................................... The woman shows that Ncw...... is a... I didn't say that there are one structure here, this is right in front of it. The important thing if you want to, if this implies that when you pass the data in the line infinity category, this function deserves all the data. So, the infinity function has numbers, A corrector that help you to find some points, there areface noise, there is a fixed value I can't go unươnghoon in the airflow which a radial So you can also use some juice body transfer, you might be forgetting that one. Yeah, so I'll probably go away for a second. Yes, I'd invite everybody to do that. So this confused me for a little while. You should not think of epsilon as homological differential, it's some kind of extra symmetry. So the legal instances are really only with respect to T. So anyway, just think about it as having an extra action that you forget. I'm going to have a pre-action, so you can use that to change the motion. Okay, so another point that is important is that you have a natural map. From N to E, which is just even by the projection on the first coordinates, on the zero coordinates. They're not the same kind of coordinates. Okay, if you forget the action of T-mars. Okay, so I just get it. Okay, so this is more of an example, but it would be a map if you love it. This is the thing that the logic is striking about on close coordinates. Sorry, here on the formula, with this empty coordinates, this is just the projection of the first one. Okay, just keep E and... Keep the eye on the other one. Yeah, this guy, you see, you have 45 coordinates here. You get the copy of that. Oh, you have to grab them. And maybe I have taken the sum of all P, you're actually getting all P here. So these are the two positions there. Include these parts in the complexes. And I will explain how to do this, if you do this in the other random coordinates, you get the same sequence. You can continue. Yeah, this is the one. That is between the right and the left, or? Of course. Okay, so... How much time do I have? 40 minutes. Okay, 40? 40. You want to avoid your talk? How much time do you need, actually? Is it 45? I would say like 25 minutes. 20 minutes, yeah. We have 50 minutes. So let's go with faster. Okay, so what do you want to say? So what I want to say is that how do you find this guy? You can find the random complex that's written over this. A, a communicative EGA. And the random complex of A over K. And the random complex of A over K is written over C. Omega 1, should be right. Okay, so if you want to see... And super A. So explicitly this is the same as taking this sum. Omega... P1 over K. Now shifted by nothing is anything. Right? P1 over K is non-negative. What am I talking about? Oh, that's what I was saying. Can someone say what was actually said? That's okay. So yes, you have this. So if you want to sketch, more or less what is going to be? Is it going to be a random complex like here? Where the action of C is going to be even by the direction of my choice. If I have something like this. If I have something like this. Okay. Zero. One. Sorry, there's no way I want to say it. It seems to be wrong. I want to take it here. One over K. When is one? And here you have this one dimension. Which acts like this. Here. This is one of the best ways to write a choice. So, I'll be in A.K. So, I just wanted to explain this. Even though you're over for this question. I'm going to make it too much. Sorry? Yeah, it's much too much. One more question. You can have it. Everything below this diagonal line is just zero. So, it stops this. So, it stops this immediately. Now, zero. I just... No, just to that. It's not that deep. The A zero is also zero. I don't think so, but this is 100,000. So, it's something like this. So, okay. So, you see, now, this is the arm. You can already see it. You now apply this conversion to this guy. And let's say, A is for vibrance, and this is already for potential. But you want to get exactly the notion of importance. Of course, importance. So, let's take this guy. I like to apply Ncw of the arm, A k, on B. Okay? So, what you get will be zero. You get something like... Sorry, this is not what I meant. Okay, maybe a shift. So, the idea is for instance, you want to say that the form here is close. What does it mean? It means that the fragile is zero, and the round fragile is equal to something here. Okay? So, let's just write what it means. Your shift, we have two gradients, which is what you're trying to understand. Did you talk to Simon? Did he shift like this? Like what? The vertical. The vertical. Okay, I will... You get the idea now. I mean, this thing. So, basically, what do you want? Let's say you want to write down what are these close before. It's one close before, or deeper. You want to write this down. So, according to the definition I just sketched, in the beginning you would have to say, it's something that is here in the zero, because... Sorry, it's something that is zero here in the homology, because something is close. So, now if you go to the form that I introduced, this is what you get here. What do you find? You find it in the level zero. You want to have this guy, and something here, and something here, like this. So, now you shift to the basic. I mean, one, two, three, eight, minus one, two, and you see? So, now, what do you have in degree one? In degree one you have something like, starting in zero. That's key from zero? No, no, I have constructed, so this is a great complex, okay? But you key right now. You change this for... No, okay, so this is not a great complex. This is the key part of a great complex. You see, X and P is a great complex. I need you to... The P... He wants you to say the P is one. Yes. In your example, P is one. You know, okay? I'm specifying you what this is, okay? And how the common attributes of these guys get the cycles in this complex give you exactly the percentage of the need for closed forms. So, what do you do? You want to say what are one closed forms of degree zero? You start with these guys, and you construct this sequence. And now in degree one, what do you have? You make exactly the same game, but one level becomes below. So you have, in this case, zero. But then, as you have always to make it a sheet of two, you have exactly this guy. Okay? And now, again, a sheet of two, so you have... And this continues. You see? So basically, this complex here, I want to make this part, but basically, this complex, yeah, it's this one. What have I done? I just shifted. Once I want to consider two degrees zero, this shift, I start with this guy in degree zero, and then I'm giving you this peak of... just because I'm making a shift on the weight. So, you want to give that point of view to three, four, five? Three, four, five? Then, four, zero. Yes. That's the important one. Yes. Oh, first example. This is the complex. This is this complex here. Okay? If you want, this is the end. Yes. So, it's one. You see, basically what I'm doing is following. I want to... Can you just move on and give definitions, following definitions and just move on? We can do this size, for the sheet of paper. Okay, but I have... So, basically, the differential, what you get when you apply D equals zero, is that with the cycle, that's the notion of the... of course, but... So, let's continue. I have time. So... So, can you just give us the following definition of the complex A equals N? Yes. Just the complex B equals... Yeah. So, basically what you do is following. You find what is this place, this space. So, this is going to be... So, you take this complex, okay? And you truncate it, in degrees zero, and you can see the result kind of image as a signature circle. Zero is an addition to... usually everything... in particular, you see that the zero, the five zero of this guy, it will be exactly the zero of this guy. Okay? So, in particular, you also have a map from... So, remember, you have the map like this. Okay? So, you just leave everything above zero. So, you got a... Yeah, so then you have the number of times to be graded. So, remember that you have the map like this, it's just a projection that's sort of coordinate. And now you can apply this guy to the normal place. What you find is that the... Sorry, I forgot to make it P. P, I didn't say that... it's important. You find this, not closed, okay? These are closed forms, these are not closed forms, and you can see the book and... Okay. So, as you... Now, what you do is you... H, E, omega, and then you... And another statement. This is the thing, right? So, you have a natural map between the two, sorry, between the two, and this natural map will retain the dot cam realization of these guys. You get exactly the map. And these are closed forms. And these... are closed forms. Okay, so... I don't have much time. I guess... I have zero time? No, I have plenty of time. Okay, so basically I'll just... do an example. So, okay, first important thing is that if you use the nays of the fiber, all these workings you can take, and you can use the nays of the fiber, you have to derive this, compared to the convention complex, and you apply some of these nays of the fiber to the convention complex. So, this is the nays of the fiber, you just do it for a co-fibre resolution? Okay. So, to conclude, maybe I can sketch. There are two examples I want you to sketch. I will sketch. You didn't tell us already? This is going to be in the next talk. Watch one of our closed-fields of the half-style. No, I didn't. Are you supposed to do it? I don't know, I can tell. I was... I didn't know. Basically, the idea is the following. So, these guys here, you can prove that they define a stack for the topology on CDGA, okay? And this is essentially because, first, because the convention complex has a send for the topology, and secondly, because this NC guy here, the easy-to-find function, the easy-to-find function associated with the community-affiliating limits. So, if you combine these two themes, I guess you can prove that. So, let me sketch. So, what? A goes to AP closed A and A goes to AP A and R stacks with respect to get out the topology. So, just an ingredient extension we are... that I just said, that we use L-type to get out the topology send and NC to these words. So, okay? So, in particular, definitions if you have the right stack, you define AP closed X and there's a mapping space and... Yes. Okay. So, if you could find these as stack forms, yes, you can send a technique. Nobody can see here, you're saying that AP is here, like, space or mapping time? Space. Space, yes. It's fine if you could do it in 3D. Space is there. You could also take this one. So, and blue. Let's just catch the example when the X is moved. And X is moved by the scheme to recover the definition, the classical definition. So, all the summits, so, X is moved by A, X is moved by A. So, in this case, A is moved and you don't have to convert it anymore, but if you go to that complex that I had before with the digital type of complexes, you will see that there is always, only this guy here, A in degree 0, you have A1, A2K, and you have this, and then use the wrong term. And this is the speedos, right? Everything else can be zero space. So, in particular, I try to compute the space of the element for the element for element 1, the 1 and p, any p, just to give the idea basically what you want to do is call it, so, if you take this complex and see the amount of S2K, N minus 1, N minus, sorry, N minus 1, N minus p, p, so, this guy, what is this guy, can be zero, even though I have, so you have the p, p-forms of the, even though I have the p with zero, even though I come there, you have this guy zero, please. Okay, they have zero and two levels of all you have, please, that. In degree 1, you're going to have a short, and then you're going to run out. I think you already told us the formula, okay. So basically, I told you in the beginning the example, you can see that if you now this construction there to this guy, what you get is exactly, if you have p, N, N positive, you're going to have the multiple groups of this space, 5, 0, I want my 2, and you'll find N minus 1, I want to be the algebraic number more than 1, you'll find N having, for example, that we close to 0. Okay, we determined none of these is running. And it's actually only going to add lines of jargon or w Some. Interpraction too close You can, you know, there are spider complexes, and I mean, it's just cool to know the spider complexes. I don't know why we wrote these 80 clothes as I can't remember. I mean, sometimes it's just cool just to say you have a home to be in some space, and you know, you know, you can't really... Language, that's it. Obviously, the meaning of the word for it is it's something you don't have to forget completely. So if we want to think of these APs, it's not space, but actual, you know, a second or a complex. Yeah, a second. So we need some translation, right? So we'll be truncated the way so that we got something. Yeah, we turn off the zero translation. Now, the dole kind of is just the entire essential set of the spectrum, which is the infinity, truncated by definition. But then at least the end, so working with the families of 80 close to n for our xn is the same thing as working with the unknown context, because it's like a spectrum. So if you prefer, you can look at 80 close to n, essentially in groups together with condition n that makes it into a spectrum. So here I just have a comment about the definition ABXN that appears to be not produced, but I think we heard this morning, Dyssen-Pompon. So this is equivalent to global sections of x of UHP of x, which is like... Which is also going to be produced. Yeah, but feel that, because I think we're going to be these from the long-term and the definition, please. It doesn't work for close homes. For close homes, it doesn't work in the sense that I don't think LX as a quasi-quadruple worksheet has a... as a quasi-quadruple worksheet, so let's... You get the whole sheet after the key question. Can you see this then? I'm just saying that this last equivalent is way more intuitive. I mean, it's literally degree P or P-forums that are ancient.
 So, let's go for both parts.  But first we have to talk about the Fridge of course.  And the last one, we saw the Potential Complex.  So very briefly, let me just say I will define the different two forms.  You just take the complex, it takes hours and the element of the module will be back as it was before.  Sorry, it's a complex.  Okay, so P forms a degree M for basically n for cycles in this complex.  But there's no rapid Fridge of here.  That's just a complex complex module.  P forms a degree M, let me explain.  P forms x.  The big Potential Complex takes hours and the fourth of the degree M, our penitention n cycle.  So the way you mean when you P Lx is like the sheet of the Potential Complex,  and you take local sections that are complex of the P Lx.  Because what you want is a sheet, it's not complex.  You take more sections of this.  So after now, this way is a complex.  Ceremonitation for P forms a degree M.  Let's keep it like this for the moment, okay?  I will read it.  There is no question.  So let me talk more about...  As you start by saying that the explaining idea...  These are P forms of degree M, and I will press this planning idea for what the closed form is.  And then I will be more formal in explaining how we can put this P closed P forms a cycle of complex.  So this idea of a top.  This is assumed.  Now I will talk about closed P forms.  What is the idea?  So if you go to X, or 2, or...  E, you can construct these random formulas, okay?  And...  Okay, so you have...  And you can do...  You have the derivative for each row.  Okay?  You can take P forms, and you have the derivative for what?  Okay, sorry.  That's the other way.  Over, sorry.  No, we're in the time of omega 1A over something.  Okay, that's it.  So the idea in the last two settings is that you want to define P for a closed P form.  So just take things that are closed and this is a differential.  But if you go now to case where this is not moved, you want to have complexes here.  So you have...  When you find closed things, you have...  These are going to fall in some homology.  When we say these vanishes in homology.  So you need something in degree here that's called...  Okay?  So the idea is that the definition would be, if you want, if these are now closed complexes,  you're going to say that the P form has to be degree 0, like something like in here,  the distance.  You're going to say that this guy is closed if when you set it here, it's zero in homology,  which means that there is omega 1 in the level above, such that it closes it,  and you're going to impose the closure condition up to infinity.  You're going to say that...  So, hoping that this is closed, but now, for now, over some more, not talking about the P and scheme.  I'm not... I'm not assuming that these are completely...  These are complexes in general, okay?  If you have complexes here, do you want to say that means to be closed?  What I'm saying is that you have to say what causes it at each level, okay?  So the notion isn't that...  X is not fine, you're asking.  Okay.  So if you want this guy to be closed, this is the same X, same A, this...  So that's it, only that it's a P form, okay?  Of degree 1, of degree 0.  It's going to be closed if there exists omega 1 in this guy, such that the differential here  is the differential, it's the right differential of omega.  Okay, and if you use this game of... you say now, also, so let's say, and...  dR of omega 1 is d of omega 2.  And it means...  Okay, so this is the idea that I want to make formal during the course.  Okay, so that's not the definition.  It's not an definition, it's just to sketch out what was the...  Okay, okay.  People don't care that they have to deal with this as definition.  No, no.  The problem is that you explain this definition, okay?  This definition.  The idea behind the definition.  Exactly, okay.  Okay.  So...  So, in particular, so this is just a sketch.  And what I want to explain is how to construct spaces.  Let's say, px.  And space, again, px is the right side.  Space of p-forms of degree.  Actually, I should express it in this little lecture, okay?  Ha!  I like the degree of p-forms here.  So, space of p-forms of degree n.  You can organize these guys, these notions here.  This piece of...  You can organize these guys into space, okay?  Sorry, these guys first.  You can organize them into space.  And then you can also organize them into space for kind of space.  It's a blaster space.  How much of a space?  I see, please.  This would be a simple, shall I begin?  Okay.  So, okay, so this is space of p-forms, sort of, of closed p-forms.  Okay, something we can do.  Okay.  So, you can organize these guys into space.  And the idea here, so this is the first idea, so this is the first kind of definition.  And if you think of an idea, then it's the idea that if you want to close a form,  you need to specify these guys, omega 1, omega 2, omega 3,  they're actually closed for you.  Okay, so in particular, we can construct,  I don't expect you to construct a map.  From closed forms to forms, which is not an inclusion of the subspace.  Okay.  But how do you need this code?  It's not in the...  It's part of the data.  I mean, I'll be cool.  I just thought it was going to be a little bit...  It's not going to be a commercial idea.  No.  The problem is going to be...  Yeah, on fibers?  Okay.  So basically, for any point here, that's a form.  You're going to have space here.  It's called space of keys, not bandwidth.  We see generally what's happening.  These are the main keys.  Okay.  So, space of keys and ways to code them here.  In the user, user and the words is...  Sorry, you may have some comments about the space,  but I'm sure you'll decide.  Which generally is not the correct answer.  Right, in the ending also.  That's the answer.  Okay.  So, we'll work two examples.  The first one is the rest of our examples.  The x, for example, I'll leave more details.  In the end, the x is good, for instance.  And the y is good.  This space here...  Let's do zero.  Form p forms a degree zero.  Close forms, the pi zero of this guy is just classical.  Close.  Forms.  Okay.  So, this is Spanish.  And pi is strictly degree zero.  Sorry?  Two things really strict.  Yes.  In this case, and if you want the pi zero up to the...  Okay, that's great.  So, pi zero is going to be h dot on p plus n,  and then this goes up to pi n.  So pi n is one.  It's going to be h dot on p plus one.  And pi n is going to be close to forms.  And then it's finished.  Okay?  Just to give an example, we'll work out this example.  Okay, let's try it.  This is the closing introduction.  Okay, I think this...  And now, starting with something.  So, we can say this is the call of the closer points.  So, if you remove close, h dot on xn,  that's the element of a clean space.  So, pi n is just p forms, and then all those groups are simple.  So, this is an example where the pi goes to the nth.  So, h dot on xn.  So, pi i is zero, except when i is for n, it is in p forms.  Okay?  x has to be a fine point.  Okay?  And smooth.  And smooth.  And easy.  Sorry, this thing is...  Wait, it is on smooth where the pi is the same, but pi i are non-zero after n.  But the linear term will prove something.  You have to change the height of the force of non-linear.  Of a p xn.  Not close.  Okay, let me start now by explaining the...  So, preliminary things.  So, the first thing we need is the notion of a very complex system.  So, what is a very complex?  A very complex, just a complex.  So, a complex.  And on each hand, you have a very complex system.  And the differential is compatible with the gradient.  So, I will write, like in the paper, I will write dG gradient for the gradient of gradient.  Okay?  And the first thing, d.  Now, m.  And I will read that notion of v is the notion of a mixed complex.  So, I mean, the most direct way to say what it is in this case is just it's a complex again.  But where we put the differential between the other way?  We give the structure, we give another differential that goes the other way.  So, one way to say is to take k of epsilon modules with epsilon in degree minus one and epsilon square with zero dG modules.  Okay?  And we use the, let's say, right epsilon dG k for a mixed complex.  The two differentials, commune and mixed.  And epsilon has degree.  Yes, epsilon has some degree with a degree minus one.  I will talk to you, say yes, yes, yes.  So, we use complexes.  And then we also need, and then,  we use minus one for the two differentials.  I mean, there are two degrees.  But you think of the polar coordinates or the graded units.  It's not graded.  It's not graded.  It just means...  Ah, sorry, yes, it means...  So, I guess it's minus one.  Okay.  It's the...  So, the key to shooting is to epsilon for both differential and element of grade.  If I'm not mistaken.  What is the action of epsilon?  Sorry?  You know, I can improve the notation if you...  What takes after the one?  I was just complaining that you're using epsilon in different ways.  No, it's the same.  This is the action of epsilon.  If you take an n module for this guy,  this one is exactly what we mean for the action of epsilon.  It's not action of epsilon.  It's not action of epsilon.  It's action of epsilon.  Why?  Okay.  Action of epsilon.  So, when you write a description,  k of x, k of x,  multiplication by x is the definition of...  Are you sure?  Are you sure?  I'm not sure.  Okay.  So, this is the thing of epsilon development in Algebra,  but the same time epsilon is operated,  we'll also take our countages.  You can do dot epsilon.  That's all right.  It's fine.  It's fine.  Let's keep going.  Okay.  Now that we've worked out, we can just go on.  Okay.  If you want to work out...  k of x of n of g of 1 is just something that we have.  And then you can...  If you want the action of this side,  it doesn't work.  It's the guy who keeps the map like this.  This is the product of the sheet.  This makes the sheet in degrees, right?  This guy is in minus one.  You're going to have just that on the map.  From n...  m of n to m of n minus one.  Right?  This is the map of the derivative.  It's also...  So you're saying that epsilon is a gradient,  like in the ring?  But is it in degree minus one?  Okay.  I think it's easier like this.  Forget this.  This is one more.  You have a complex with the differential going in different ways.  So then you have...  braided...  just a quick one.  So, okay.  Again, I will not specify this guy.  I will just tell the extension of the idea of what the other thing is.  Don't sketch.  Tell us actually.  Because it's confusing.  Basically, I won't tell exactly what it is.  It's a complex like this with a gradient on each level.  And epsilon just shifts the gradient by one.  So it is good if you have some element here.  So it has a total degree minus one minus one?  Yes.  And the idea is that it's going to be one, two, one.  You set yourself up with your notation really well in Europe.  EID equals the strep sum of E and T. So I'm sure you can write what epsilon is.  That's the problem.  So explain this to me.  I mean, you cannot just write it.  He said...  Yeah, yeah, yeah.  Is this right?  Okay, so...  This is a complex like this.  And this is a gradient.  And he does it like this.  So you can write the E and B and say how the potential will move the E and B.  Yeah, yes.  Yeah.  So D is going from E and B to...  E and B.  Just what B?  B?  I don't even know what I do with the size.  Can I just make it a bit like this on this one?  Okay, so...  Yeah.  Is it okay?  Yeah.  It's okay.  It's okay.  It's okay.  So there's always the B zero, which is the sum of these guys.  And epsilon shifts to the weight by P by one.  Is it clear?  Okay.  So they have something like this.  So this one is nice, but it's possible.  Yes, okay.  So...  Okay.  So let's write epsilon Vgk graded for the category of these guys.  Okay?  So after these...  Ah, some things.  So these guys have natural notion of moving performances.  So you can take the line in the category, so you can write the line in the category.  So I'll just put...  Can I not write the line in the category?  I'll just put like this.  Where is this?  Yeah.  I'll just write this for the line in the category.  There is one important thing.  Tell us what are the opposite because we have two different ones.  But which one do you use?  Yeah, that's true.  I'm going to be second-line explaining that.  You're wondering what are the two?  Probably the first two.  So the more people...  The more people that just...  Here in this case, I'm just a map of the reserve of rays.  Here are the maps that are commuted with good inferentials.  And there is the same thing in reserve of rays in intermediate inferentials.  So we have two complexes and I'll say that some of these are not good.  Yeah, just say that again.  The map that we want to...  I can pet both with both thresholds.  Nothing with nothing thresholds.  Okay?  So...  Do you want to be both ISO and both?  Or...  Okay, so you'll be both quite ISO on total?  Okay.  So you forget the position, you forget the gradient?  Okay.  And you ask that the end-use map on the other end...  Okay.  Always with D.  It's always with D.  So the map of graded complex needs to respect the splitting, right?  Like you've written D as the direct sum of these other...  Yeah, D is the direct sum but the conditional...  These complexes, the D is the one that you have one in use on each...  But I'm saying a map with graded complex, they don't further mix,  because they have to respect the splitting.  Yeah, the splitting.  Yes, the splitting gets the weight.  It's graded by this.  It's graded by this.  Okay.  So what I wanted to explain enough is how to even this thing like this,  construct one graded complex,  the method of...  So we have a weighted version of the ADC model.  So how do we... ...lead these sides?  Yes.  Just a second.  And the sides.  So your chromology is still graded.  There's still a chain complex in fact.  Yes, but you take the sum and...  So...  D is the one that I'm saying.  I wanted to get a...  I didn't say yes, but I should say about this...  this epsilon VGK graded the infinity category of these guys.  And then afterwards, Antonio will explain his talk at some point, I think.  He said, I mean, not this is the relation.  So he's still showing that these guys here, these complexes here,  are quite similar to the ADC category.  Quite similar to VGK and VGN.  For some of us, I'm sorry.  This is what I'm saying.  Graded complexes are the same as things...  quite similar to VGN and VGN.  Okay.  And these, you can write Jesus...  K, T minus 1, which is the chromology.  And you can see the VGN is these guys.  And if you want to see the VGN,  these are the graded complexes.  So this VGN is VGN...  K, T minus 1, okay?  So if you look at the data in this case,  the chromology of this guy  is just a...  you have a map like this.  And every element in M,  you can write as if by that sum of guys, like this.  You can actually see how it's graded.  Yes, exactly.  It's just...  So these are the two ones.  And you will see that the map...  I'm not going to explain the auto-construction map like this.  Okay.  And afterwards, this map,  we can identify this map that I will now explain  to the same thing as...  So we have...  these are the things that are VGN.  Okay.  These will be quite a convenient sheet  for the P.  Now it has Gm,  cross-product in VGA.  Yeah, the map on this guy is just a...  The fence of which nobody took you.  So what's the number of stairs?  You see what's going on in the background now?  Okay.  Yeah, this is where the problem is.  And VGA or the...  This looks really close to...  I mean, if that was auto-VGA,  and you grab it with these,  and it's just like any one of these is auto-1.  Yes.  This is the actual...  The full-faculty section.  But you have to even auto-...  You can auto-faculty this.  So what do you say?  But now you have to feel ordinary  and have the perfect amount of auto-1.  Here, in this...  Yes.  Push forward.  Could you rewrite the ending overlapped forward?  Here?  Yeah.  Why is it we're here to cheat?  What's happening inside?  Yeah, because we have Gm.  Do we have an extent after...  Yes, it does come from the floor.  What I'm saying is that  in relation to what I'm explaining now,  and what I'm explaining later.  Right now it seems like we have a product really,  classified as Gm.  Now you have to see this group,  Gm, Sela, Ina, and Iqbal,  and it's VGA.  Take the automaton and auto-eculance group,  and that's Gm, Sela, Ina, and Iqbal, and it's VGA.  That's the auto-eculance of VGA.  Okay, so you have to IGA the scale,  and you can make VGA actually the same value.  Yes.  Do you need to send out to have an action from VGA?  Another way to explain this,  is the shoe or the... ...the shoe.  I just need to automatonize it in any case,  if I'm not mistaken.  Gm.  Of Gm.  Of VGA.  So sorry, is VGA acting on Gm,  and the group's attack on...  Gm is acting on Gm.  Gm is acting on VGA.  I hit the algorithm on VGA with two hands.  Okay.  Very good.  Can I...  I...  So, what I want to explain is the map.  And how this map already encodes  is in the form of a X-ray, of course, for...  Okay?  If you...  If the fundamental question is here,  you take this map to keep it up  for the next one.  Sorry, I was explaining.  Don't worry.  So, let's start with the...  Would you now take there?  If you have not taken there,  and you take the grid,  then you can construct the following complex.  You take... ...and see...  Okay, let's...  I will start with something not grid,  and then I will modify those things  so that it takes the gradient in this version.  So...  You can define a complex... ...by this formula.  Okay?  It's a product that has a 10-round space.  No, I'm trying to...  I just saw people...  But now it looks like it's all over the place.  What's X?  It's five.  There's five of you all sitting on the table.  No.  It's definitely not that.  No, it's definitely not that.  There we go.  But...  I'm happy.  No.  So this is what you want.  You have a...  No.  No.  You understood the idea.  Why?  They just want to make jokes.  Two dollars a month.  Yeah, it's a change.  So this is not five.  This is not fully built-in.  And this is just a problem.  Okay?  So now you have this guy.  And now you can do more or less...  You can do the final construction  so that you take the gradient in this direction.  If you have something like this...  With the gradient...  With the gradient...  You can define...  Something.  Which is a little bit right here.  And it's the same thing  you consider if you shift...  Every time you make the gradient.  So this is...  I mean, this is the vertical guy.  This is the...  What's P?  For each P.  So this is gradient.  So for each P...  You can define this guy.  I'm talking A, I, A.  Ah, okay. Here's the gradient of...  Sometimes C, W.  I think one P.  And I have this one.  So this gives you a...  A comp.  And...  Okay.  So...  Maybe you should use the same notation  for the weight gradient.  For the weight gradient,  you need two parentheses on both sides.  Because this...  On the left side...  Ah, because it's like making two power, you mean?  No, no, no, no.  You can put people side by side.  There you go.  I was using notation.  Okay.  So you didn't say what the difference was?  Okay, so this is important.  The difference shows...  So let's just give an idea  not to write in formula.  So you have...  This is just N.  You have A and...  minus 2.  A and...  minus 2.  This is P.  Sorry.  Yeah.  P equals 1.  People's 2.  And then you have...  Here.  Let me write.  Say it again.  So this is what...  This is N.  And it's 1P.  But it was 1P.  And...  minus 1.  P equals 1.  Okay?  So the idea now is that you would just want to  use the action of...  of epsilon.  And your differential will be the sum of...  of the two.  The differential is...  The differential is...  the...  on...  Ah...  So it's an element there, so it's a family.  Okay?  So this is going to be...  on an order of J.  So it's going to be...  X1  of XJ minus...  minus 1.  This is...  the...  Okay?  Okay?  See?  Here you can already see  precisely how we call this  differential form.  What you wanted was exactly...  So you have some...  some form.  Imagine that these are P-forms of A and...  So what does it mean to be close?  You're going to say that they're close if there's a differential on this.  If there's a cycle with respect to this differential.  What does it mean?  It means exactly that its differential is close.  And here if it is in the right differential,  it's going to be equal to something  and the same thing all the time.  You see?  So...  I'll let you... ... ...  Okay, some other things you can do now.  You can find...  So we can find the differential.  As you wait... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...  The woman  shows that  Ncw... ...  is a...  I didn't say that there are one structure here,  this is right in front of it.  The important thing if you want to,  if this implies that when you pass the data in the line  infinity category, this function deserves all the data.  So, the infinity function has numbers,  A corrector that help you to find some points,  there areface noise,  there is a fixed value  I can't go unươnghoon  in the airflow  which a radial  So you can also use some juice body transfer,  you might be forgetting that one.  Yeah, so I'll probably go away for a second.  Yes, I'd invite everybody to do that.  So this confused me for a little while.  You should not think of epsilon as homological differential,  it's some kind of extra symmetry.  So the legal instances are really only with respect to T.  So anyway, just think about it as having an extra action that you forget.  I'm going to have a pre-action, so you can use that to change the motion.  Okay, so another point that is important is that you have a natural map.  From N to E,  which is just even by the projection on the first coordinates,  on the zero coordinates.  They're not the same kind of coordinates.  Okay, if you forget the action of T-mars.  Okay, so I just get it.  Okay, so this is more of an example,  but it would be a map if you love it.  This is the thing that the logic is striking about on close coordinates.  Sorry, here on the formula, with this empty coordinates,  this is just the projection of the first one.  Okay, just keep E and...  Keep the eye on the other one.  Yeah, this guy, you see, you have 45 coordinates here.  You get the copy of that.  Oh, you have to grab them.  And maybe I have taken the sum of all P, you're actually getting all P here.  So these are the two positions there.  Include these parts in the complexes.  And I will explain how to do this,  if you do this in the other random coordinates,  you get the same sequence.  You can continue.  Yeah, this is the one.  That is between the right and the left, or?  Of course.  Okay, so...  How much time do I have?  40 minutes.  Okay, 40?  40.  You want to avoid your talk?  How much time do you need, actually?  Is it 45?  I would say like 25 minutes.  20 minutes, yeah.  We have 50 minutes.  So let's go with faster.  Okay, so what do you want to say?  So what I want to say is that how do you find this guy?  You can find the random complex that's written over this.  A, a communicative EGA.  And the random complex of A over K.  And the random complex of A over K is written over C.  Omega 1, should be right.  Okay, so if you want to see...  And super A.  So explicitly this is the same as taking this sum.  Omega...  P1 over K.  Now shifted by nothing is anything.  Right?  P1 over K is non-negative.  What am I talking about?  Oh, that's what I was saying.  Can someone say what was actually said?  That's okay.  So yes, you have this.  So if you want to sketch, more or less what is going to be?  Is it going to be a random complex like here?  Where the action of C is going to be even by the direction of my choice.  If I have something like this.  If I have something like this.  Okay.  Zero.  One.  Sorry, there's no way I want to say it.  It seems to be wrong.  I want to take it here.  One over K.  When is one?  And here you have this one dimension.  Which acts like this.  Here.  This is one of the best ways to write a choice.  So, I'll be in A.K.  So, I just wanted to explain this.  Even though you're over for this question.  I'm going to make it too much.  Sorry?  Yeah, it's much too much.  One more question.  You can have it.  Everything below this diagonal line is just zero.  So, it stops this.  So, it stops this immediately.  Now, zero.  I just...  No, just to that.  It's not that deep.  The A zero is also zero.  I don't think so, but this is 100,000.  So, it's something like this.  So, okay.  So, you see, now, this is the arm.  You can already see it.  You now apply this conversion to this guy.  And let's say, A is for vibrance,  and this is already for potential.  But you want to get exactly the notion of importance.  Of course, importance.  So, let's take this guy.  I like to apply Ncw of the arm, A k, on B.  Okay?  So, what you get will be zero.  You get something like...  Sorry, this is not what I meant.  Okay, maybe a shift.  So, the idea is for instance,  you want to say that the form here is close.  What does it mean?  It means that the fragile is zero,  and the round fragile is equal to something here.  Okay?  So, let's just write what it means.  Your shift, we have two gradients,  which is what you're trying to understand.  Did you talk to Simon?  Did he shift like this?  Like what?  The vertical.  The vertical.  Okay, I will...  You get the idea now.  I mean, this thing.  So, basically, what do you want?  Let's say you want to write down what are these close before.  It's one close before, or deeper.  You want to write this down.  So, according to the definition I just sketched,  in the beginning you would have to say,  it's something that is here in the zero,  because...  Sorry, it's something that is zero here in the homology,  because something is close.  So, now if you go to the form that I introduced,  this is what you get here.  What do you find?  You find it in the level zero.  You want to have this guy,  and something here,  and something here, like this.  So, now you shift to the basic.  I mean, one, two, three, eight, minus one, two, and you see?  So, now, what do you have in degree one?  In degree one you have something like,  starting in zero.  That's key from zero?  No, no, I have constructed,  so this is a great complex, okay?  But you key right now.  You change this for...  No, okay, so this is not a great complex.  This is the key part of a great complex.  You see, X and P is a great complex.  I need you to...  The P...  He wants you to say the P is one.  Yes.  In your example, P is one.  You know, okay?  I'm specifying you what this is, okay?  And how the common attributes of these guys  get the cycles in this complex  give you exactly the percentage of the need for closed forms.  So, what do you do?  You want to say what are one closed forms of degree zero?  You start with these guys,  and you construct this sequence.  And now in degree one, what do you have?  You make exactly the same game,  but one level becomes below.  So you have, in this case, zero.  But then, as you have always to make it  a sheet of two, you have exactly this guy.  Okay?  And now, again, a sheet of two,  so you have...  And this continues.  You see?  So basically, this complex here,  I want to make this part,  but basically, this complex,  yeah, it's this one.  What have I done?  I just shifted.  Once I want to consider two degrees zero,  this shift,  I start with this guy in degree zero,  and then I'm giving you this peak of...  just because I'm making a shift on the weight.  So, you want to give that point of view  to three, four, five?  Three, four, five?  Then, four, zero.  Yes.  That's the important one.  Yes.  Oh, first example.  This is the complex.  This is this complex here.  Okay?  If you want, this is the end.  Yes.  So, it's one.  You see, basically what I'm doing is following.  I want to...  Can you just move on and give definitions,  following definitions and just move on?  We can do this size,  for the sheet of paper.  Okay, but I have...  So, basically, the differential,  what you get when you apply D equals zero,  is that with the cycle,  that's the notion of the...  of course, but...  So, let's continue.  I have time.  So...  So, can you just give us the following definition  of the complex A equals N?  Yes.  Just the complex B equals...  Yeah.  So, basically what you do is following.  You find what is this place, this space.  So, this is going to be...  So, you take this complex, okay?  And you truncate it,  in degrees zero,  and you can see the result kind of image  as a signature circle.  Zero is an addition to...  usually everything...  in particular, you see that the zero,  the five zero of this guy,  it will be exactly the zero of this guy.  Okay?  So, in particular,  you also have a map from...  So, remember, you have the map like this.  Okay?  So, you just leave everything above zero.  So, you got a...  Yeah, so then you have the number of times to be graded.  So, remember that you have the map like this,  it's just a projection that's sort of coordinate.  And now you can apply this guy  to the normal place.  What you find is that the...  Sorry, I forgot to make it P.  P, I didn't say that...  it's important.  You find this, not closed, okay?  These are closed forms,  these are not closed forms,  and you can see the book and...  Okay.  So,  as you...  Now, what you do is you...  H, E,  omega,  and then you...  And another statement.  This is the thing, right?  So, you have a natural map between the two,  sorry, between the two,  and this natural map will retain  the dot cam realization of these guys.  You get exactly the map.  And these are closed forms.  And these...  are closed forms.  Okay, so...  I don't have much time.  I guess...  I have zero time?  No, I have plenty of time.  Okay, so basically I'll just...  do an example.  So, okay, first important thing is that  if you use the nays of the fiber,  all these workings you can take,  and you can use the nays of the fiber,  you have to derive this,  compared to the convention complex,  and you apply some of these nays of the fiber  to the convention complex.  So,  this is the nays of the fiber,  you just do it for a co-fibre resolution?  Okay.  So,  to conclude,  maybe I can sketch.  There are two examples I want you to sketch.  I will sketch.  You didn't tell us already?  This is going to be in the next talk.  Watch one of our closed-fields of the half-style.  No, I didn't.  Are you supposed to do it?  I don't know, I can tell.  I was...  I didn't know.  Basically, the idea is the following.  So, these guys here,  you can prove that they define  a stack for the topology  on CDGA, okay?  And this is essentially because,  first, because the convention  complex has a send  for the topology,  and secondly, because this  NC guy here,  the easy-to-find  function,  the easy-to-find function associated  with the community-affiliating limits.  So, if you combine these two themes,  I guess you can prove that.  So, let me sketch.  So,  what?  A goes to  AP  closed  A and  A goes to  AP  A and  R stacks  with respect  to get out  the topology.  So, just  an ingredient extension  we are...  that I just said,  that  we use  L-type  to get out  the topology  send  and NC  to  these words.  So, okay?  So, in particular,  definitions  if you have the right stack,  you define  AP  closed X  and  there's a mapping space  and...  Yes.  Okay.  So,  if you could find these as  stack forms,  yes, you can send a technique.  Nobody can see here, you're saying that  AP is here, like,  space or mapping time?  Space.  Space, yes.  It's fine if you could  do it in 3D.  Space is there.  You could also take this one.  So,  and blue.  Let's just catch the example  when the X is moved.  And X is moved by the scheme  to recover the definition, the classical definition.  So, all the summits,  so,  X is moved by A,  X is moved by A.  So,  in this case, A is moved  and you don't have to convert it anymore,  but if you go to that complex that I had before  with the  digital type of complexes,  you will see that there is always,  only this guy here,  A  in degree 0,  you have A1,  A2K,  and you have this,  and then  use the wrong term.  And this is the  speedos,  right? Everything else can be zero space.  So, in particular,  I try to compute  the space of  the element for  the element for element 1,  the 1 and p, any p,  just to give the idea  basically what you want to do is call it,  so, if you take this complex  and see  the amount of  S2K,  N minus 1,  N minus,  sorry,  N minus 1,  N minus p,  p, so,  this guy, what is this guy,  can be zero,  even though I have, so you have the p,  p-forms  of the,  even though I have the p with zero,  even though I come there,  you have this guy  zero,  please.  Okay, they have zero  and two levels of all you have, please,  that.  In degree 1,  you're going to have  a short,  and then you're going to run out.  I think you already told us  the formula, okay.  So basically,  I told you in the beginning the example,  you can see that if you now  this construction there to this guy,  what you get is exactly, if you have p, N,  N positive, you're going to have  the multiple groups of this space,  5, 0, I want my 2,  and you'll find N minus 1, I want to be  the algebraic number more than 1,  you'll find N having,  for example, that we close to 0.  Okay,  we determined none of these is  running.  And it's actually  only going to add  lines of jargon or  w Some.  Interpraction too close  You can, you know, there are spider complexes, and I mean, it's just cool to know the spider complexes.  I don't know why we wrote these 80 clothes as I can't remember.  I mean, sometimes it's just cool just to say you have a home to be in some space, and you know, you know, you can't really...  Language, that's it.  Obviously, the meaning of the word for it is it's something you don't have to forget completely.  So if we want to think of these APs, it's not space, but actual, you know, a second or a complex.  Yeah, a second.  So we need some translation, right? So we'll be truncated the way so that we got something.  Yeah, we turn off the zero translation.  Now, the dole kind of is just the entire essential set of the spectrum, which is the infinity, truncated by definition.  But then at least the end, so working with the families of 80 close to n for our xn is the same thing as working with the unknown context, because it's like a spectrum.  So if you prefer, you can look at 80 close to n, essentially in groups together with condition n that makes it into a spectrum.  So here I just have a comment about the definition ABXN that appears to be not produced, but I think we heard this morning, Dyssen-Pompon.  So this is equivalent to global sections of x of UHP of x, which is like...  Which is also going to be produced.  Yeah, but feel that, because I think we're going to be these from the long-term and the definition, please.  It doesn't work for close homes. For close homes, it doesn't work in the sense that I don't think LX as a quasi-quadruple worksheet has a...  as a quasi-quadruple worksheet, so let's...  You get the whole sheet after the key question.  Can you see this then?  I'm just saying that this last equivalent is way more intuitive. I mean, it's literally degree P or P-forums that are ancient.
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So, let's go for both parts. But first we have to talk about the Fridge of course. And the last one, we saw the Potential Complex. So very briefly, let me just say I will define the different two forms. You just take the complex, it takes hours and the element of the module will be back as it was before. Sorry, it's a complex. Okay, so P forms a degree M for basically n for cycles in this complex. But there's no rapid Fridge of here. That's just a complex complex module. P forms a degree M, let me explain. P forms x. The big Potential Complex takes hours and the fourth of the degree M, our penitention n cycle. So the way you mean when you P Lx is like the sheet of the Potential Complex, and you take local sections that are complex of the P Lx. Because what you want is a sheet, it's not complex. You take more sections of this. So after now, this way is a complex. Ceremonitation for P forms a degree M. Let's keep it like this for the moment, okay? I will read it. There is no question. So let me talk more about... As you start by saying that the explaining idea... These are P forms of degree M, and I will press this planning idea for what the closed form is. And then I will be more formal in explaining how we can put this P closed P forms a cycle of complex. So this idea of a top. This is assumed. Now I will talk about closed P forms. What is the idea? So if you go to X, or 2, or... E, you can construct these random formulas, okay? And... Okay, so you have... And you can do... You have the derivative for each row. Okay? You can take P forms, and you have the derivative for what? Okay, sorry. That's the other way. Over, sorry. No, we're in the time of omega 1A over something. Okay, that's it. So the idea in the last two settings is that you want to define P for a closed P form. So just take things that are closed and this is a differential. But if you go now to case where this is not moved, you want to have complexes here. So you have... When you find closed things, you have... These are going to fall in some homology. When we say these vanishes in homology. So you need something in degree here that's called... Okay? So the idea is that the definition would be, if you want, if these are now closed complexes, you're going to say that the P form has to be degree 0, like something like in here, the distance. You're going to say that this guy is closed if when you set it here, it's zero in homology, which means that there is omega 1 in the level above, such that it closes it, and you're going to impose the closure condition up to infinity. You're going to say that... So, hoping that this is closed, but now, for now, over some more, not talking about the P and scheme. I'm not... I'm not assuming that these are completely... These are complexes in general, okay? If you have complexes here, do you want to say that means to be closed? What I'm saying is that you have to say what causes it at each level, okay? So the notion isn't that... X is not fine, you're asking. Okay. So if you want this guy to be closed, this is the same X, same A, this... So that's it, only that it's a P form, okay? Of degree 1, of degree 0. It's going to be closed if there exists omega 1 in this guy, such that the differential here is the differential, it's the right differential of omega. Okay, and if you use this game of... you say now, also, so let's say, and... dR of omega 1 is d of omega 2. And it means... Okay, so this is the idea that I want to make formal during the course. Okay, so that's not the definition. It's not an definition, it's just to sketch out what was the... Okay, okay. People don't care that they have to deal with this as definition. No, no. The problem is that you explain this definition, okay? This definition. The idea behind the definition. Exactly, okay. Okay. So... So, in particular, so this is just a sketch. And what I want to explain is how to construct spaces. Let's say, px. And space, again, px is the right side. Space of p-forms of degree. Actually, I should express it in this little lecture, okay? Ha! I like the degree of p-forms here. So, space of p-forms of degree n. You can organize these guys, these notions here. This piece of... You can organize these guys into space, okay? Sorry, these guys first. You can organize them into space. And then you can also organize them into space for kind of space. It's a blaster space. How much of a space? I see, please. This would be a simple, shall I begin? Okay. So, okay, so this is space of p-forms, sort of, of closed p-forms. Okay, something we can do. Okay. So, you can organize these guys into space. And the idea here, so this is the first idea, so this is the first kind of definition. And if you think of an idea, then it's the idea that if you want to close a form, you need to specify these guys, omega 1, omega 2, omega 3, they're actually closed for you. Okay, so in particular, we can construct, I don't expect you to construct a map. From closed forms to forms, which is not an inclusion of the subspace. Okay. But how do you need this code? It's not in the... It's part of the data. I mean, I'll be cool. I just thought it was going to be a little bit... It's not going to be a commercial idea. No. The problem is going to be... Yeah, on fibers? Okay. So basically, for any point here, that's a form. You're going to have space here. It's called space of keys, not bandwidth. We see generally what's happening. These are the main keys. Okay. So, space of keys and ways to code them here. In the user, user and the words is... Sorry, you may have some comments about the space, but I'm sure you'll decide. Which generally is not the correct answer. Right, in the ending also. That's the answer. Okay. So, we'll work two examples. The first one is the rest of our examples. The x, for example, I'll leave more details. In the end, the x is good, for instance. And the y is good. This space here... Let's do zero. Form p forms a degree zero. Close forms, the pi zero of this guy is just classical. Close. Forms. Okay. So, this is Spanish. And pi is strictly degree zero. Sorry? Two things really strict. Yes. In this case, and if you want the pi zero up to the... Okay, that's great. So, pi zero is going to be h dot on p plus n, and then this goes up to pi n. So pi n is one. It's going to be h dot on p plus one. And pi n is going to be close to forms. And then it's finished. Okay? Just to give an example, we'll work out this example. Okay, let's try it. This is the closing introduction. Okay, I think this... And now, starting with something. So, we can say this is the call of the closer points. So, if you remove close, h dot on xn, that's the element of a clean space. So, pi n is just p forms, and then all those groups are simple. So, this is an example where the pi goes to the nth. So, h dot on xn. So, pi i is zero, except when i is for n, it is in p forms. Okay? x has to be a fine point. Okay? And smooth. And smooth. And easy. Sorry, this thing is... Wait, it is on smooth where the pi is the same, but pi i are non-zero after n. But the linear term will prove something. You have to change the height of the force of non-linear. Of a p xn. Not close. Okay, let me start now by explaining the... So, preliminary things. So, the first thing we need is the notion of a very complex system. So, what is a very complex? A very complex, just a complex. So, a complex. And on each hand, you have a very complex system. And the differential is compatible with the gradient. So, I will write, like in the paper, I will write dG gradient for the gradient of gradient. Okay? And the first thing, d. Now, m. And I will read that notion of v is the notion of a mixed complex. So, I mean, the most direct way to say what it is in this case is just it's a complex again. But where we put the differential between the other way? We give the structure, we give another differential that goes the other way. So, one way to say is to take k of epsilon modules with epsilon in degree minus one and epsilon square with zero dG modules. Okay? And we use the, let's say, right epsilon dG k for a mixed complex. The two differentials, commune and mixed. And epsilon has degree. Yes, epsilon has some degree with a degree minus one. I will talk to you, say yes, yes, yes. So, we use complexes. And then we also need, and then, we use minus one for the two differentials. I mean, there are two degrees. But you think of the polar coordinates or the graded units. It's not graded. It's not graded. It just means... Ah, sorry, yes, it means... So, I guess it's minus one. Okay. It's the... So, the key to shooting is to epsilon for both differential and element of grade. If I'm not mistaken. What is the action of epsilon? Sorry? You know, I can improve the notation if you... What takes after the one? I was just complaining that you're using epsilon in different ways. No, it's the same. This is the action of epsilon. If you take an n module for this guy, this one is exactly what we mean for the action of epsilon. It's not action of epsilon. It's not action of epsilon. It's action of epsilon. Why? Okay. Action of epsilon. So, when you write a description, k of x, k of x, multiplication by x is the definition of... Are you sure? Are you sure? I'm not sure. Okay. So, this is the thing of epsilon development in Algebra, but the same time epsilon is operated, we'll also take our countages. You can do dot epsilon. That's all right. It's fine. It's fine. Let's keep going. Okay. Now that we've worked out, we can just go on. Okay. If you want to work out... k of x of n of g of 1 is just something that we have. And then you can... If you want the action of this side, it doesn't work. It's the guy who keeps the map like this. This is the product of the sheet. This makes the sheet in degrees, right? This guy is in minus one. You're going to have just that on the map. From n... m of n to m of n minus one. Right? This is the map of the derivative. It's also... So you're saying that epsilon is a gradient, like in the ring? But is it in degree minus one? Okay. I think it's easier like this. Forget this. This is one more. You have a complex with the differential going in different ways. So then you have... braided... just a quick one. So, okay. Again, I will not specify this guy. I will just tell the extension of the idea of what the other thing is. Don't sketch. Tell us actually. Because it's confusing. Basically, I won't tell exactly what it is. It's a complex like this with a gradient on each level. And epsilon just shifts the gradient by one. So it is good if you have some element here. So it has a total degree minus one minus one? Yes. And the idea is that it's going to be one, two, one. You set yourself up with your notation really well in Europe. EID equals the strep sum of E and T. So I'm sure you can write what epsilon is. That's the problem. So explain this to me. I mean, you cannot just write it. He said... Yeah, yeah, yeah. Is this right? Okay, so... This is a complex like this. And this is a gradient. And he does it like this. So you can write the E and B and say how the potential will move the E and B. Yeah, yes. Yeah. So D is going from E and B to... E and B. Just what B? B? I don't even know what I do with the size. Can I just make it a bit like this on this one? Okay, so... Yeah. Is it okay? Yeah. It's okay. It's okay. It's okay. So there's always the B zero, which is the sum of these guys. And epsilon shifts to the weight by P by one. Is it clear? Okay. So they have something like this. So this one is nice, but it's possible. Yes, okay. So... Okay. So let's write epsilon Vgk graded for the category of these guys. Okay? So after these... Ah, some things. So these guys have natural notion of moving performances. So you can take the line in the category, so you can write the line in the category. So I'll just put... Can I not write the line in the category? I'll just put like this. Where is this? Yeah. I'll just write this for the line in the category. There is one important thing. Tell us what are the opposite because we have two different ones. But which one do you use? Yeah, that's true. I'm going to be second-line explaining that. You're wondering what are the two? Probably the first two. So the more people... The more people that just... Here in this case, I'm just a map of the reserve of rays. Here are the maps that are commuted with good inferentials. And there is the same thing in reserve of rays in intermediate inferentials. So we have two complexes and I'll say that some of these are not good. Yeah, just say that again. The map that we want to... I can pet both with both thresholds. Nothing with nothing thresholds. Okay? So... Do you want to be both ISO and both? Or... Okay, so you'll be both quite ISO on total? Okay. So you forget the position, you forget the gradient? Okay. And you ask that the end-use map on the other end... Okay. Always with D. It's always with D. So the map of graded complex needs to respect the splitting, right? Like you've written D as the direct sum of these other... Yeah, D is the direct sum but the conditional... These complexes, the D is the one that you have one in use on each... But I'm saying a map with graded complex, they don't further mix, because they have to respect the splitting. Yeah, the splitting. Yes, the splitting gets the weight. It's graded by this. It's graded by this. Okay. So what I wanted to explain enough is how to even this thing like this, construct one graded complex, the method of... So we have a weighted version of the ADC model. So how do we......lead these sides? Yes. Just a second. And the sides. So your chromology is still graded. There's still a chain complex in fact. Yes, but you take the sum and... So... D is the one that I'm saying. I wanted to get a... I didn't say yes, but I should say about this... this epsilon VGK graded the infinity category of these guys. And then afterwards, Antonio will explain his talk at some point, I think. He said, I mean, not this is the relation. So he's still showing that these guys here, these complexes here, are quite similar to the ADC category. Quite similar to VGK and VGN. For some of us, I'm sorry. This is what I'm saying. Graded complexes are the same as things... quite similar to VGN and VGN. Okay. And these, you can write Jesus... K, T minus 1, which is the chromology. And you can see the VGN is these guys. And if you want to see the VGN, these are the graded complexes. So this VGN is VGN... K, T minus 1, okay? So if you look at the data in this case, the chromology of this guy is just a... you have a map like this. And every element in M, you can write as if by that sum of guys, like this. You can actually see how it's graded. Yes, exactly. It's just... So these are the two ones. And you will see that the map... I'm not going to explain the auto-construction map like this. Okay. And afterwards, this map, we can identify this map that I will now explain to the same thing as... So we have... these are the things that are VGN. Okay. These will be quite a convenient sheet for the P. Now it has Gm, cross-product in VGA. Yeah, the map on this guy is just a... The fence of which nobody took you. So what's the number of stairs? You see what's going on in the background now? Okay. Yeah, this is where the problem is. And VGA or the... This looks really close to... I mean, if that was auto-VGA, and you grab it with these, and it's just like any one of these is auto-1. Yes. This is the actual... The full-faculty section. But you have to even auto-... You can auto-faculty this. So what do you say? But now you have to feel ordinary and have the perfect amount of auto-1. Here, in this... Yes. Push forward. Could you rewrite the ending overlapped forward? Here? Yeah. Why is it we're here to cheat? What's happening inside? Yeah, because we have Gm. Do we have an extent after... Yes, it does come from the floor. What I'm saying is that in relation to what I'm explaining now, and what I'm explaining later. Right now it seems like we have a product really, classified as Gm. Now you have to see this group, Gm, Sela, Ina, and Iqbal, and it's VGA. Take the automaton and auto-eculance group, and that's Gm, Sela, Ina, and Iqbal, and it's VGA. That's the auto-eculance of VGA. Okay, so you have to IGA the scale, and you can make VGA actually the same value. Yes. Do you need to send out to have an action from VGA? Another way to explain this, is the shoe or the......the shoe. I just need to automatonize it in any case, if I'm not mistaken. Gm. Of Gm. Of VGA. So sorry, is VGA acting on Gm, and the group's attack on... Gm is acting on Gm. Gm is acting on VGA. I hit the algorithm on VGA with two hands. Okay. Very good. Can I... I... So, what I want to explain is the map. And how this map already encodes is in the form of a X-ray, of course, for... Okay? If you... If the fundamental question is here, you take this map to keep it up for the next one. Sorry, I was explaining. Don't worry. So, let's start with the... Would you now take there? If you have not taken there, and you take the grid, then you can construct the following complex. You take......and see... Okay, let's... I will start with something not grid, and then I will modify those things so that it takes the gradient in this version. So... You can define a complex......by this formula. Okay? It's a product that has a 10-round space. No, I'm trying to... I just saw people... But now it looks like it's all over the place. What's X? It's five. There's five of you all sitting on the table. No. It's definitely not that. No, it's definitely not that. There we go. But... I'm happy. No. So this is what you want. You have a... No. No. You understood the idea. Why? They just want to make jokes. Two dollars a month. Yeah, it's a change. So this is not five. This is not fully built-in. And this is just a problem. Okay? So now you have this guy. And now you can do more or less... You can do the final construction so that you take the gradient in this direction. If you have something like this... With the gradient... With the gradient... You can define... Something. Which is a little bit right here. And it's the same thing you consider if you shift... Every time you make the gradient. So this is... I mean, this is the vertical guy. This is the... What's P? For each P. So this is gradient. So for each P... You can define this guy. I'm talking A, I, A. Ah, okay. Here's the gradient of... Sometimes C, W. I think one P. And I have this one. So this gives you a... A comp. And... Okay. So... Maybe you should use the same notation for the weight gradient. For the weight gradient, you need two parentheses on both sides. Because this... On the left side... Ah, because it's like making two power, you mean? No, no, no, no. You can put people side by side. There you go. I was using notation. Okay. So you didn't say what the difference was? Okay, so this is important. The difference shows... So let's just give an idea not to write in formula. So you have... This is just N. You have A and... minus 2. A and... minus 2. This is P. Sorry. Yeah. P equals 1. People's 2. And then you have... Here. Let me write. Say it again. So this is what... This is N. And it's 1P. But it was 1P. And... minus 1. P equals 1. Okay? So the idea now is that you would just want to use the action of... of epsilon. And your differential will be the sum of... of the two. The differential is... The differential is... the... on... Ah... So it's an element there, so it's a family. Okay? So this is going to be... on an order of J. So it's going to be... X1 of XJ minus... minus 1. This is... the... Okay? Okay? See? Here you can already see precisely how we call this differential form. What you wanted was exactly... So you have some... some form. Imagine that these are P-forms of A and... So what does it mean to be close? You're going to say that they're close if there's a differential on this. If there's a cycle with respect to this differential. What does it mean? It means exactly that its differential is close. And here if it is in the right differential, it's going to be equal to something and the same thing all the time. You see? So... I'll let you......... Okay, some other things you can do now. You can find... So we can find the differential. As you wait............................................. ............ The woman shows that Ncw...... is a... I didn't say that there are one structure here, this is right in front of it. The important thing if you want to, if this implies that when you pass the data in the line infinity category, this function deserves all the data. So, the infinity function has numbers, A corrector that help you to find some points, there areface noise, there is a fixed value I can't go unươnghoon in the airflow which a radial So you can also use some juice body transfer, you might be forgetting that one. Yeah, so I'll probably go away for a second. Yes, I'd invite everybody to do that. So this confused me for a little while. You should not think of epsilon as homological differential, it's some kind of extra symmetry. So the legal instances are really only with respect to T. So anyway, just think about it as having an extra action that you forget. I'm going to have a pre-action, so you can use that to change the motion. Okay, so another point that is important is that you have a natural map. From N to E, which is just even by the projection on the first coordinates, on the zero coordinates. They're not the same kind of coordinates. Okay, if you forget the action of T-mars. Okay, so I just get it. Okay, so this is more of an example, but it would be a map if you love it. This is the thing that the logic is striking about on close coordinates. Sorry, here on the formula, with this empty coordinates, this is just the projection of the first one. Okay, just keep E and... Keep the eye on the other one. Yeah, this guy, you see, you have 45 coordinates here. You get the copy of that. Oh, you have to grab them. And maybe I have taken the sum of all P, you're actually getting all P here. So these are the two positions there. Include these parts in the complexes. And I will explain how to do this, if you do this in the other random coordinates, you get the same sequence. You can continue. Yeah, this is the one. That is between the right and the left, or? Of course. Okay, so... How much time do I have? 40 minutes. Okay, 40? 40. You want to avoid your talk? How much time do you need, actually? Is it 45? I would say like 25 minutes. 20 minutes, yeah. We have 50 minutes. So let's go with faster. Okay, so what do you want to say? So what I want to say is that how do you find this guy? You can find the random complex that's written over this. A, a communicative EGA. And the random complex of A over K. And the random complex of A over K is written over C. Omega 1, should be right. Okay, so if you want to see... And super A. So explicitly this is the same as taking this sum. Omega... P1 over K. Now shifted by nothing is anything. Right? P1 over K is non-negative. What am I talking about? Oh, that's what I was saying. Can someone say what was actually said? That's okay. So yes, you have this. So if you want to sketch, more or less what is going to be? Is it going to be a random complex like here? Where the action of C is going to be even by the direction of my choice. If I have something like this. If I have something like this. Okay. Zero. One. Sorry, there's no way I want to say it. It seems to be wrong. I want to take it here. One over K. When is one? And here you have this one dimension. Which acts like this. Here. This is one of the best ways to write a choice. So, I'll be in A.K. So, I just wanted to explain this. Even though you're over for this question. I'm going to make it too much. Sorry? Yeah, it's much too much. One more question. You can have it. Everything below this diagonal line is just zero. So, it stops this. So, it stops this immediately. Now, zero. I just... No, just to that. It's not that deep. The A zero is also zero. I don't think so, but this is 100,000. So, it's something like this. So, okay. So, you see, now, this is the arm. You can already see it. You now apply this conversion to this guy. And let's say, A is for vibrance, and this is already for potential. But you want to get exactly the notion of importance. Of course, importance. So, let's take this guy. I like to apply Ncw of the arm, A k, on B. Okay? So, what you get will be zero. You get something like... Sorry, this is not what I meant. Okay, maybe a shift. So, the idea is for instance, you want to say that the form here is close. What does it mean? It means that the fragile is zero, and the round fragile is equal to something here. Okay? So, let's just write what it means. Your shift, we have two gradients, which is what you're trying to understand. Did you talk to Simon? Did he shift like this? Like what? The vertical. The vertical. Okay, I will... You get the idea now. I mean, this thing. So, basically, what do you want? Let's say you want to write down what are these close before. It's one close before, or deeper. You want to write this down. So, according to the definition I just sketched, in the beginning you would have to say, it's something that is here in the zero, because... Sorry, it's something that is zero here in the homology, because something is close. So, now if you go to the form that I introduced, this is what you get here. What do you find? You find it in the level zero. You want to have this guy, and something here, and something here, like this. So, now you shift to the basic. I mean, one, two, three, eight, minus one, two, and you see? So, now, what do you have in degree one? In degree one you have something like, starting in zero. That's key from zero? No, no, I have constructed, so this is a great complex, okay? But you key right now. You change this for... No, okay, so this is not a great complex. This is the key part of a great complex. You see, X and P is a great complex. I need you to... The P... He wants you to say the P is one. Yes. In your example, P is one. You know, okay? I'm specifying you what this is, okay? And how the common attributes of these guys get the cycles in this complex give you exactly the percentage of the need for closed forms. So, what do you do? You want to say what are one closed forms of degree zero? You start with these guys, and you construct this sequence. And now in degree one, what do you have? You make exactly the same game, but one level becomes below. So you have, in this case, zero. But then, as you have always to make it a sheet of two, you have exactly this guy. Okay? And now, again, a sheet of two, so you have... And this continues. You see? So basically, this complex here, I want to make this part, but basically, this complex, yeah, it's this one. What have I done? I just shifted. Once I want to consider two degrees zero, this shift, I start with this guy in degree zero, and then I'm giving you this peak of... just because I'm making a shift on the weight. So, you want to give that point of view to three, four, five? Three, four, five? Then, four, zero. Yes. That's the important one. Yes. Oh, first example. This is the complex. This is this complex here. Okay? If you want, this is the end. Yes. So, it's one. You see, basically what I'm doing is following. I want to... Can you just move on and give definitions, following definitions and just move on? We can do this size, for the sheet of paper. Okay, but I have... So, basically, the differential, what you get when you apply D equals zero, is that with the cycle, that's the notion of the... of course, but... So, let's continue. I have time. So... So, can you just give us the following definition of the complex A equals N? Yes. Just the complex B equals... Yeah. So, basically what you do is following. You find what is this place, this space. So, this is going to be... So, you take this complex, okay? And you truncate it, in degrees zero, and you can see the result kind of image as a signature circle. Zero is an addition to... usually everything... in particular, you see that the zero, the five zero of this guy, it will be exactly the zero of this guy. Okay? So, in particular, you also have a map from... So, remember, you have the map like this. Okay? So, you just leave everything above zero. So, you got a... Yeah, so then you have the number of times to be graded. So, remember that you have the map like this, it's just a projection that's sort of coordinate. And now you can apply this guy to the normal place. What you find is that the... Sorry, I forgot to make it P. P, I didn't say that... it's important. You find this, not closed, okay? These are closed forms, these are not closed forms, and you can see the book and... Okay. So, as you... Now, what you do is you... H, E, omega, and then you... And another statement. This is the thing, right? So, you have a natural map between the two, sorry, between the two, and this natural map will retain the dot cam realization of these guys. You get exactly the map. And these are closed forms. And these... are closed forms. Okay, so... I don't have much time. I guess... I have zero time? No, I have plenty of time. Okay, so basically I'll just... do an example. So, okay, first important thing is that if you use the nays of the fiber, all these workings you can take, and you can use the nays of the fiber, you have to derive this, compared to the convention complex, and you apply some of these nays of the fiber to the convention complex. So, this is the nays of the fiber, you just do it for a co-fibre resolution? Okay. So, to conclude, maybe I can sketch. There are two examples I want you to sketch. I will sketch. You didn't tell us already? This is going to be in the next talk. Watch one of our closed-fields of the half-style. No, I didn't. Are you supposed to do it? I don't know, I can tell. I was... I didn't know. Basically, the idea is the following. So, these guys here, you can prove that they define a stack for the topology on CDGA, okay? And this is essentially because, first, because the convention complex has a send for the topology, and secondly, because this NC guy here, the easy-to-find function, the easy-to-find function associated with the community-affiliating limits. So, if you combine these two themes, I guess you can prove that. So, let me sketch. So, what? A goes to AP closed A and A goes to AP A and R stacks with respect to get out the topology. So, just an ingredient extension we are... that I just said, that we use L-type to get out the topology send and NC to these words. So, okay? So, in particular, definitions if you have the right stack, you define AP closed X and there's a mapping space and... Yes. Okay. So, if you could find these as stack forms, yes, you can send a technique. Nobody can see here, you're saying that AP is here, like, space or mapping time? Space. Space, yes. It's fine if you could do it in 3D. Space is there. You could also take this one. So, and blue. Let's just catch the example when the X is moved. And X is moved by the scheme to recover the definition, the classical definition. So, all the summits, so, X is moved by A, X is moved by A. So, in this case, A is moved and you don't have to convert it anymore, but if you go to that complex that I had before with the digital type of complexes, you will see that there is always, only this guy here, A in degree 0, you have A1, A2K, and you have this, and then use the wrong term. And this is the speedos, right? Everything else can be zero space. So, in particular, I try to compute the space of the element for the element for element 1, the 1 and p, any p, just to give the idea basically what you want to do is call it, so, if you take this complex and see the amount of S2K, N minus 1, N minus, sorry, N minus 1, N minus p, p, so, this guy, what is this guy, can be zero, even though I have, so you have the p, p-forms of the, even though I have the p with zero, even though I come there, you have this guy zero, please. Okay, they have zero and two levels of all you have, please, that. In degree 1, you're going to have a short, and then you're going to run out. I think you already told us the formula, okay. So basically, I told you in the beginning the example, you can see that if you now this construction there to this guy, what you get is exactly, if you have p, N, N positive, you're going to have the multiple groups of this space, 5, 0, I want my 2, and you'll find N minus 1, I want to be the algebraic number more than 1, you'll find N having, for example, that we close to 0. Okay, we determined none of these is running. And it's actually only going to add lines of jargon or w Some. Interpraction too close You can, you know, there are spider complexes, and I mean, it's just cool to know the spider complexes. I don't know why we wrote these 80 clothes as I can't remember. I mean, sometimes it's just cool just to say you have a home to be in some space, and you know, you know, you can't really... Language, that's it. Obviously, the meaning of the word for it is it's something you don't have to forget completely. So if we want to think of these APs, it's not space, but actual, you know, a second or a complex. Yeah, a second. So we need some translation, right? So we'll be truncated the way so that we got something. Yeah, we turn off the zero translation. Now, the dole kind of is just the entire essential set of the spectrum, which is the infinity, truncated by definition. But then at least the end, so working with the families of 80 close to n for our xn is the same thing as working with the unknown context, because it's like a spectrum. So if you prefer, you can look at 80 close to n, essentially in groups together with condition n that makes it into a spectrum. So here I just have a comment about the definition ABXN that appears to be not produced, but I think we heard this morning, Dyssen-Pompon. So this is equivalent to global sections of x of UHP of x, which is like... Which is also going to be produced. Yeah, but feel that, because I think we're going to be these from the long-term and the definition, please. It doesn't work for close homes. For close homes, it doesn't work in the sense that I don't think LX as a quasi-quadruple worksheet has a... as a quasi-quadruple worksheet, so let's... You get the whole sheet after the key question. Can you see this then? I'm just saying that this last equivalent is way more intuitive. I mean, it's literally degree P or P-forums that are ancient.
fine point. Okay? And smooth. And smooth. And example. This is the complex. This is this stacks with respect to get out the topology. So,