 First of all, thank you very much for the opportunity to speak in such a beautiful conference and such an event. And of course, somehow, the subject of talk is very closely connected to what Arthur did over the years. First of all, perverse chiefs now even admits that he is part of it. And also, somehow, I will talk about infinite dimensional generalizations. So, of course. OK, so let us, I will write down a plan. One of the reasons why I want to write, because I'm not sure how far I will go. But first, this will be motivation. Namely, it will be finite dimensional situation. Second, I will formulate my goal. Third, I will talk about characters, categories. Categories of eladic chiefs on prosthetics, or maybe infinity prosthetics. Next, I will talk about, somehow, there will be some categories. Categories will be infinity drive categories. Or stable infinity categories. And next, I will say about perverse structures, perverse chiefs on, let's say, nice prosthetics. Namely, categories will exist always in very big generality. And in order to define perverse structure, we will need certain smoothness kind of restrictions, or whatever, or some finiteness restriction. Finally, I will formulate main result. And after that, hopefully, I will get in order to this, because I will need, somehow, my result. We'll use some stratification. And stratification we will need is Garevsky-Kotviksman first on stratification. And the last ingredient I will want to, you know, I'm optimistic, you know, otherwise. You know, it's impossible to, you see, it's impossible to live in Israel without being optimistic. Smallness, or smallness of Grotendik Springer, a fine resolution. But it is actually just dimension count, so it's not. Okay, let's see. Okay, first of all, let us start. Start, first of all, motivation. So motivation, of course, comes from, as you probably all have known, I'm coming from a representation theory. Namely, I want to apply some algebraic geometry and category theory to get concrete results in representation theory, and specifically something connected to Langlund's program. So motivation is false. So let G will be the usual connected reductive group over some algebraically closed fields. Good example, algebraic closure over finite fields if you really want to apply to representation theory. Then I want G with G is a joint function. Okay, and then you can consider, yes. No, because it's actually a very nice notation because there are three actions. One is the left, right, and a joint. Actually, so at least this self-explanatory notation. It's a quotient stack, okay. If you consider on the level of points, it will be conjugate classes, but it's also very good. Okay, so far it's, you know, art and stack of finite type, it will disappear. Okay, and then we will consider D, is the category that usual familiar category, which is, you know, D, B, C, okay, L, or L bar doesn't matter, where L is prime different from K, it's a usual drive category. And then it, it's a- This will be finite since I want to define it there. Yes, yes, but actually, actually, actually I will soon give you a way that in some sense if you work with infinity categories, that to define it actually can do without no work, essentially. We'll talk about it in a moment. You're right, of course. It's, it's still, but somehow the, okay. And then it contains very nice sub-object, which is called character shifts, which I'm not going to discuss now, but I'm going to introduce one important example, which I'm going to generalize, or at least try to generalize. Example, which is called Grotten-Dexpringer-Schiff. Consider the following, consider B cross B, which is the same as a joint quotient where B is barrel. And what is Grotten-Dexpringer-Schiff, which I will call S, this is just pi of the, maybe I want to say that somehow this is very nice tech, which is smooth of dimension zero. In particular, there is a very nice shift here, which is dualizing. The constant shift with dualizing shift are just the same. No, even shifts and twists. So it's D of B. And Springer-Schiff, of course, is of whatever, QL. And all the factors will be derived. I will not write R in any point, okay? And this morphism, of course, is proper. So I can put I as the Shiliko star, but at the moment, at least, but I will put this one. Now, what is known about this shift? Okay, known is then. What is known as is perverse. Second, that S is minimal extension, where J is the embedding into G, regular semi-simple. Regular semi-simple is the second property. Third property is that double, double affine viral group. So let us, let us, I will say even stronger assertion. The set of automorphism of S is just equal to the W, which is viral group of G. No, automorphism, not endomorphism. Group of automorphisms. No, but if you're, if you're not set, then if you're not automorphism, then the morphism algebra, which is a QL algebra, automorphisms form the units in this QL algebra, which is bigger than that. Ooh, yes, what I want. That's sexy, say, you are, you are. Yes, okay. This was this, of course, of course, what I actually need, but what of course, yes. And the morphism is a fine viral group, and moreover, for if you have a irreducible representation of W, then you can consider the, is a typical component. Let us say, in this case, it's finite group and the characteristic is zero, so it does not matter. So let us, I will write down the reason I'm writing down, because I have in mind something for the fine-setting that most likely I will not have time to talk about this. But okay, so this is irreducible, okay? And this, of course, example of the character shifts and more general character shifts, you do similar thing, except you replace here B over B by P over P by parabolic subgroup, and you will replace the constant shift by the shift, which is pullback of the hospital local system, whatever it is, which George defined and studied for many. Okay, intensively, and of course, some of this is more general, okay. Now the question is what is, so first part was okay. Now we are going to our goal. Go to extend everything to the fine-setting. Okay, so I want to, so what does it mean I have to extend? I want to extend this, then I want to extend this, then I want to extend this, okay. Springer shift and this result. And the answer is that it's possible to do it, there is some small modification. Okay, first of all G, you replace by what? But the loop group of G. What is the loop group of G? Which is the loop group of G, and what it is by the definition? By definition, loop G is a factor from algebras over K to sets, right. And what is lg of a is by the definition is G of a. So as a factor is defined, now it's known by, I am not good in history. Probably Gert was the first one who quoted carefully, but I'm not sure, so I don't want to. So that's somehow it's in, it's in scheme, but in scheme in a really in scheme, namely that's somehow noted like in the previous talk, in the previous talk there were in scheme, but it was inductive limit or finite dimension of varieties. Here it's really, okay. Now you can do, now you can, let us try to generalize. What you want G over G? You want to try the same thing. Okay, now we can ask what it is. And the answer it is you want a factor from affine K to, to what you just want to group weights. Namely, you just take, you know, every A goes to, so lg of a is a group object. So this is just the group point action of the group by the group, or if you wish it's the category whose objects are here. Okay, and actually I will talk, and later I want to say that of course later I will use the remark that it's group points will be part of the infinity group points, or what is the same spaces. Now you can ask the following questions. Can one define, can one define? First question, what is the D of lg of lg? Second question, can you define this element S? Grotten-Dexpringer-Schiff. Third question, can you define the, okay, can you define perverse destruction? And for is S is perverse. And question five, let me put the same thing. Just, okay, perverse and maybe some others. Question five says that let a, let, okay, yes, what I want. Okay, maybe you know what I will say even more here. The question is whether I want here, whether I want here. This is defined by a fine while group. And question five, let V will be representation of let us say irreducible representation. Irreducible representation of a fine while group. And you can ask whether you can ask SV by using similar format, whatever it means. First of all, can you define it? Whether is, is, okay, defined, perverse, and irreducible. Okay, so far I formulated some questions and now let us say trying to give answers. So first I will give answers, excuse me. When you form the LG mod LG, there's no notification that you're doing, right? No, no notification. There's no need for notification and I will comment on this later because later I will talk about chiefs and then yes, you're right. What is the main idea I learned from paper of, I guess, Dennis, that somehow that's, you don't need to, sometimes it's better not to chiefify things. Okay, so at least for certain questions, it's easier. Then you can ask what happens whether something changed after you chiefify. And fortunately in our case, the answer would be no. You mean the group would be changed after? No, no, no, the group says yes, but the category of chiefs would not change. Category of chiefs would not change, would not somehow, before chiefification, after chiefification would be canonically equivalent. Yeah, it is more like chiefification, or is it? Yes, okay, yes, chiefification means chiefification. Yes, you're right, you're right, you're right. Okay, now let us say answers. This is our joint work on partial and mean. It's not written, so it's almost done, but I don't know. I will write in progress to be on the safe side with Alexis Boutier and David Cajdan. Okay, first of all answers. A1, A2, so the answer to this question is then says yes. Okay, A3, A4, then says yes after some modification. And in a moment I will say what kind of modification you will need, and what is the reason? Okay, and answer five. Okay, first of all, it's okay. First of all, this is SV is defined. Most likely meaning that somehow it is, so we are sure that it is, let us say, conjecture, or let us say almost theorem, but not in the sense of almost mathematical. SV is perverse, okay? Namely, that there are some, okay, we have a reason, but we don't, there are some technical details to check, but what is not true, SV is not irreducible. Not irreducible in almost all cases. For example, take G equal SL2, and take V equal sine representation. If it takes sine representation, it would not be irreducible, it would contain two irreducible summands, and actually I'm not sure what is semi-simple, most likely no. Okay, anyway, this is something that we have to carry out. For example, on the other hand, we don't know the answers for this trivial, it's very easy to answer, the answer is yes, it would be irreducible. It could, can it also be zero? For some, can it also be zero? The answer is no, because we know over the regular semi-simple, over the locus where the reduction is regular semi-simple, it's a regular representation. Now, like in the case, like in the finite case, over the regular semi-simple, it's just a regular representation. Okay, so there you go at what you should get, yes? Yes. No particular, not zero. Yes, in particular, it's not zero. You're working with representation on that fine, very good. Yes, I see. No, as I say, in the fine case, it's also true. In the fine case, it's also true. Over the nice locus, it's precisely the regular representation, and you can ask whether it's extension. Okay, I will, in a moment, I will say you. I think we need a fine value of the big translation part, what do you mean, the regular representation? Regular representation? Even infinite group has a regular representation. On the dice? It will be, it will be infinite dimensional. Like Dennis constructed, it will be not constructible. Okay, it will be, you know, okay. This D would be not constructible, it will be inconstructible. Yeah. Well, in a moment, I will define the category. No, it's definitely not constructible, you can ask. Okay, good. Oh, okay, yes, yes, maybe I want to comment on this. What you do is the following. Let N, which is not the same as N, which is actually, maybe N is not good notation, but it's, it's what is called an important school, but important to mean anthropologically important. But actually now, okay, so it's not even anthropologically important, but the set of elements with, okay, let us say with LG, will be the set of locally closed, whatever, sub, subint scheme. Size it, let us, I will write what is, it's a point. Let us do the notation, let F will be K over T. So, LG of K is of course the same of G of F. And you want that I will say what is N of F, okay? What is F valid point, not F valid point, K valid point. It is the same as G into G of F, such that the G is regular semisimple, semisimple, and also I want that the characteristic polynomial, let us say G is bounded, let us say G is bounded. But it is the same as characteristic polynomial has coefficients in OF. And I hope to be, get to a point that I will define it properly. Okay, again? And determine it is invertible or something. Yes, yes, yes, yes, yes, yes. Determined, yes, is invertible. Okay, maybe I want before, so, yes. And now what is, let me write down precisely what is, what I mean almost, and answer, okay. And answer three says the following that there exists perverse structure on, what? Why is it called N? Because I don't know what is a better term. No, it's bad terminology, any suggestions I will, I will say better after the lecture. I don't know why it's, okay, I say somehow N is, because it's like topological and important, but actually it's even not topological and important. It's bounded, I don't know what's bounded. No, but v is overused, I don't know. How do you go there? Do you go there? I don't know. Okay, anything, okay, you know. You know, let us go with this. Okay, so we will perverse structure on d of g. You can ask why is this, okay, why is this condition and the answer is as follows. We'll see later when I explain the perverse shifts, somehow our spaces are very, very infinite dimensional because they are not only infinite dimensional, there is the int part and propot, okay? And it turns out that the propot is not a problem, but the int part is a problem. So let us consider the quotient, lg divided by lg. Then both lg's has int part. And in some sense, this is both of them cause the problem and you want somehow to kill these int parts. And now what I claim is that if you consider n model lg, that we essentially killed both int parts because here this end with this characteristic polynomial is bounded. And here is that all the stabilizers are not bounded but they bounded up to some letters. But it's all topology and topology is, if you have discrete stabilizers, it's not a problem. Because the stabilizer is what? The stabilizer of each point, it's what? It's the loop group of the centralizer, which is the torus, right? So it has a propot and it have a latest part. Latest part is not a problem. So this is the reason, so our restriction has actually, there is a good reason why we need it. Of course, it would not explain why we, so somehow why it gives a structure because we need more somehow. Formally speaking, we're using more than this. But if you consider a torus, these values in the north side with a two string, the loop group, I mean, you can have in the loop group some... But it's a topology. It's a topology, it's not important, we can disregard. You're right, there are plenty of, you're right, there are plenty of importance which has to be taken with care. But fortunately, the not affect the etallic carmology. Okay, now let's start, okay, the next. Now we want the part number three. I want to give a categories of the, is it maximum? It's maximum, is it? Okay, yes, and here I am very fortunate that Dennis gave talk before. And before he did a part of the hard work, except we need actually more general setting. But not much more general setting. It will be categories, okay, categories of elliptic shifts on infinity prosthetics. Okay, first of all, what is infinity prosthetic? Oh, prosthetic. Yes, maybe I want to say that because, so despite of the fact that all the objects are infinite dimensional, but I personally, very ignorant person, I only know what is the elliptic shift on the variety of finite type of a field. Even more over here, I even need only variety of a fine variety of finite type of a field. And the idea is that actually what we want to do, we want to start from this finite dimensional station and then apply category theory and hope it will solve all our problems. It does not help, unfortunately, but in some sense it does. Okay, so what is infinity prosthetic? Infinity prosthetic is a factor from a fine, oh, just a minute, what I did, I want, what was it before? I was of algebras, I know what I say, a fine opposite to infinity group weights, which is the same as spaces. Okay, and actually at least all the prosthetic, okay, the prosthetics that will actually appear will be take values in the usual group weights, but if you want to develop the whole theory, you would better to use this set. Now you want to say that now for every, so go for every this prosthetic, ah, yes, yes. Okay. No, it's not, so the funder, okay, the funder should understand in the following way. You can ask what are the infinity group weights or spaces form? It's form, infinity category. This is a category, in particular infinity category. It is a factor of infinity categories, whatever it means. This is a big thing because many people work and work in infinity category. But nowadays it's, you know, what? No, no, but let's ignore it because there is no, no, no, I want to talk to some, I'm not, okay. I want to convince you that if you use the language of infinity categories, then everything becomes much easier, but of course, if I cannot do all the details. Well, let us assume that infinity categories are like categories, which is the correct way of thinking. And there are some thousands of pages that people, Ludi et cetera wrote. So in some sense, nowadays the theory is rather settled down. Okay, one has to, of course, one has to be careful. You're right. And maybe I will, I even point out the points when one has to be careful. But still, Prostek is a certain factor. For example, the Prostek we will be interested in is this one, LG divided by LG, at least you kind of understand what it is. Okay, let's do it. Okay, for every Prostek, our goal will be to define certain category which I will write it, namely, somehow there's actually several choices. There are at least, actually will be two categories. This one and this one. This means constructible, will be small category and this will be the big category. And big category, for a big category, there are four choices, kind of, because you have to choose its category with a stack to shrink or start and you want int or pro. And for some reason, I will, I mean, I will use the same terminology as Dennis. No, actually, by the same reason, pro-based Dennis. But I might want to say just the remark that if you want something like she functions correspondence, then it's better to consider verdiadual version. In a sense, verdiadual version will be, it will be actually star and pro. So the choices that you mentioned, the pro one give different categories? Yes, it will be different categories. Two of them will be verdiadual, but two of them are not. We will be verdiaduality, but it's not verdiaduality within one category? Between the pair. Yes. But the pair is not. It's also, it's also, so, so, so, so, but some, some, some, some causes, it's something less useful, somehow. There are reasons why, why, why is, why this choice is better. And of course, one of the reason is Dennis explain it, for example, that this is, that shriek has a, has a left adjoined. Okay, and Dennis explains there was a general, you know, the category of sets, it's not self-doll. Set is not equivalent to set opposite. And it's therefore left adjoined and right adjoined, there are some, some, some, some different features. What? Okay. Okay, let's start. Okay, so what you do, we do the same, the procedure is, is as Dennis, indicated case A, let x, x will be, let's say, a fine. So the fine does, does not need to find a type of k. Then we define in dc of x is dcb xql, but I want to consider the corresponding enhanced version. So it's the infinity drive category, whatever it means. Okay, I consider this one and also it's the other one, it will be just the int. Okay, then what is, what, what is important that all the six operations, or at least four, four functors can be lifted to, upgraded to this setting. In particular for every x, y, there exists a functor which I say, as I told you, the main functor for me is this one, the shriek pullback. Okay, so fun, so good. Okay, except of course, this is almost the same category, except as I said, that it's not, it's not a triangulated category, but it's called stable infinity category. Now the next step is you want to say let x be a fine. But for fine schemes, you know x is a limit over xi, where xi is a fine or finite type, am I right? And then what you do, you define d, d of c of x, is by definition is co-limit, actually I was careful enough that it has co-limit in the category of categories, except, okay, one digression, all the co-limit will be rheumatopical limit, and I don't want to, so everything would be rheumatopical limit, or if you wish, in other words, somehow the infinity categories call, so there is infinity category of infinity categories, and in the infinity categories, in every infinity category, there is a notion of the limit. And so this is the limit inside of this infinity category. What? Yes, yes, yes, but so it is something rheumatopical limit. Despite that, of course, in the applications that we had in mind, actually what happens that x, this factor would be fully faithful, and then you can really define everything on the level of triangle categories. Then the rheumatopical limit is actually the same as co-limit one used to. Yes, but now what's important is co-limit with respect to, yes, and now claim is, it is independent, what? Yes, it is just a pullback, yes? Star pullback, as I said, our main factor is a star pullback, you have a projective system here, you have an injective trick, trick, trick, trick. Oh, pullback, sorry, sorry, pullback, sorry. So on the operation would be, okay, so it's independent on the presentation, okay? It's independent on the presentation. Now, okay, this is, okay, now functionality for any morphism from x to y of affine schemes, there exists a, okay, sorry, I forgot, okay, I will say in a moment, there exists a natural factor from d of y to d of x. But since you're doing it over a filtering category, probably the rheumatopical limit, because the higher derived direct limits are zero. No, why? Because it's probably in groups when you have a filtering category, so I'm not sure, this is the case that that's the level in the int triangle. No, no, no, it's not fully, fully faithful map. It's not fully faithful, it's very important. If you pass to homotopy categories there, it's also still a homotopy column, right? No, no, what's that say, no. I'm pretty sure, okay, if this is fully faithful, no, if this is fully faithful, the answer is yes. In general, I'm pretty sure that the answer is no, I cannot give you a counter-example on the stop. No, the transition maps. If all the transition maps would be fully faithful, then the answer, of course, would be yes. Otherwise, I'm pretty sure that the answer is no. But let me not... Because it does not... No, you know, I'm not good in finding a counter-example on the... No, but I know the definition of the calling. Okay, no, there is a notion, okay, no, as I said, send the following, the correct definition is as follows, there, okay, you want... What? Compatible system of some kind... Yes, some. Okay, this is a... Okay, maybe, actually, you know what, maybe what Peter has is correct, but I'm not sure in this... You are working some world of replacing primary categories by what you call stable and... Yes. You didn't say what it is, so we cannot... I mean, I don't know it already, I cannot know what... No, no, but... It's the calling within this sense. Okay, okay, suppose I have a category, usual category, then there is a notion of limits inside this category. There is really certain diagram, you can... Now, what I claim that suppose your category is infinity category. Then there is also essentially the same definition which says that inside of infinity categories there is a notion of co-limit. Now, what I say that each this of guy is a stable infinity category, but stable infinity categories form a infinity category, and you form co-limit inside this infinity category. Whatever it means. I'm really sure that on the level of the categories. It might be, okay, but maybe it's correct, you see. It just means part of the usual... You know, I... Yes. No, no, no, it's definitely false. So, it's definitely for, okay. Each object comes from here. What is the mapping space? The mapping space, take two objects here and take the mapping space. And you want co-limit of mapping space that doesn't come used with taken by zero. It does. It's filtered. It's filtered. Filtered co-limits. Sorry. Then this. You're right. Yes. Okay, you're right, you're right, you're right. Yes, you're right, right. Here it is. But anyway, there is this one. Okay, now is the question. Okay, now is the third. So, let X be any pre-stack. And then what you do is the following. I define, yes, maybe it's the only thing. It's better to do it here. And I want to say what is D of X is in D C of X. Or it's the same as co-limit in the same sense that Dennis explained in the previous talk. It's co-limit of D of X size, but in the sense of co-limits or inside of categories which has all co-complete. Yes. No, okay. Okay, C is, okay. Then some post C, some call, let now X be any pre-stack. Then we define what is D of X is either I something and then I will explain what is going on. The question, take any morphism, take D of A, where A is a fine. That is to give a element object in the category, it is the same thing as is for every, you want for every morphism from the affine scheme to X to give it's D of A, which is functorial in A. Is it like some I? Not just functorial, but compatible with pullbacks. Yes, yes, no, compatible with pullbacks, yes. And also some I of coherence. Yes, of course. Yes, no, yes, as I said, limits here is everything. No, it's limits of infinity category, but somehow, but here, but limits somehow, it's actually rather usual operation in the sense that it limits you essentially can calculate naively. What? No, not at all. Like here's a really a lot of coherences in there. Yes and no. No, no, no, no, no, no, no, no, no. The indexes sense is complicated. What is the object? No, no, you cannot spell out, it's true. Like to call them at these events, the limit is extremely complicated. Okay, it depends what you call, okay. Okay, okay, okay, let us give example, which is, let us give example, which is compatible. I mean, if you want to spell out, you need infinitely many data, it's true. But then you even don't try. Compatible with pullbacks. Let us say, say, say example. What is suppose you have some, well, let us say even those more general situation, it will be even easier. Where's g, x, x, everything is finite dimension or affine. From these forms, we want one, then, of course, lg goes to lx. Then you can consider what is the drive category on lg over lx, model lg, okay. And then you can ask how to calculate it. And the answer is that actually, it's sort of, it's easy to calculate and you write the same bar complex that people would assume. Somehow d of lx, lg, limit of what d of lx, d of lg. And it does not matter whether you put errors in this direction or not. But now, what's important, somehow, it's told that somehow, usually, you can see that the drive category on some complicated object of the simplicial schemes. And usually, it's very, very complicated object. Here, you in some sense do the same, but you package the difficulties differently. You just write the same diagram as before. This is the usual bar complex. And you just take, take limit, but limit has to be taken in the appropriate sets. If you try to spell out it, it's impossible. But in some sense, it's an advantage you even don't try because the only thing what you have to do, you have to use some certain properties. And why, okay, if you wish why, and the reason is that lx, model lg is just called limit of this diagram. And this, after you spell out the definition, it's essentially, it's almost a tautology. If you write the down, down, down, what is the definition of collimits and you use that collimits in the collimits commutes with base changes, then some, some, some, some. It's just, remember that we are in the category of functions, right? So you should check any, any, your test affine scheme and you calculate its value here and value here. And then, then, then it's just some general nonsense, so, speculation. So, so, so it just needs some factorial properties. Yes, and this, from this is essentially a tautology. So this kind of definition, what it is. The only, somehow, the advantage of infinity category is that if you try to do in the derived categories, in triangulated categories, there's no limits. You cannot, you cannot even write this thing, you can write, but usually it doesn't exist. There's no limits inside triangulated categories so the limit is often good categories. Okay, now let us, somehow, now we can ask what about perverse shifts? Okay, as I said, somehow, on perverse shifts there is essential difficulty because, you see, perverse shifts, it's something about, in the definition of perverse shifts there is a dimension involved. And here everything is infinite dimension. You can ask what you can do in order to overcome it. Okay, but let's start. Perverse shifts on some rest x. What you do, of course, the same thing that we did, did before, but then you have to be a little bit careful. Okay, first, you won't let x be usual finite dimension of what I say, a finite, which is not necessary, a finite type of k. Then, then you can ask the following, okay, then your d of x has a perverse t-structure. The problem is that now you want that this t-structure nicely behaves in a functorial way. Problem is that if even it takes the nicest possible morphism, which is smooth, no, nicely. Smooth then is somehow t-exact only up to a homological shift, all right? Because there is some homological shift, and therefore, if you take limits, it will be infinite homological shift. It's not good because we don't want to get something. But there is, of course, the very easy solution, how to overcome this problem solution. We consider I shifted perverse t-structure. For example, let us say x will be equidimensional. So what you do is you just apply a homological shift in order to make it compatible. So you define, you say that d is db, is, let us say, I shifted perverse, if and only if, if and only if the f, I hope I do it correctly, is perverse. For example, that you have to make in the line, the dualizing shift is perverse in this new shifted t-structure. Sorry, if x is smooth, then you want the dualizing shift is perverse. This is what, x to the, sorry, it's ql, twice dimension, right? And you want to check when this perverse, the answer is when you minus this, it will be perverse. So just shift it in the correct direction. But now what is nice about this is that somehow the shift only depends on the x. So for every x, you know how to shift. Of course, there is some small technicalities what to do in the noticable dimensional situation, but I don't want to discuss it. It's possible to do it, and unfortunately there is a more than one solution, which I don't like. I mean, there is one solution which is true in general, but there are the other simple solutions which would have worse for functorality properties. Which is enough for many applications. But nevertheless, what we do, we just take the usual perverse shift, perverse structure, and then you shift it by dimension. Then, okay, this is what we did for the, let's say, for the finite type. Now b, okay, definition, let x be, okay, say x, say definition, let x be, an affine scheme, x is called placid. If x can be limit of xi, xi or finite type at all transition maps are smooth. For example, there is a, all the, all the, some from example, if you take, let's say, L plus of g, or let's say any R groups of this type. So it's just projective limit, projective limit of something which is all the transition maps are smooth. But in this case, we defined, okay, in this case there is a claim which is actually easy exercise to see that, that's what, what is our category? Our category that we know that dx, dx has a unique t-structure such that all transition maps, such that all maps for all t-exact. Why? Because I forgot whether this exists, oh, it's already, yes, because what is this guy? This guy, it's co-limit over dxi. So, let's see. No, but if you shift all of them, as in this, by the same integer. No, no, it's not the same integer. In, it depends on x. On every x, I change it by the dimension of x. Yeah, but in the claim, you say all maps, no, no, no, no, no, but t-exact with respect to shifted t-structure, in shifted t-structure, the shift depends on the xi. Okay, okay, okay, you are doing, yes. Yes, so this is that with dimension, okay, yes, yes, okay, I say it's possible to do it not because it's actually locally equidimensional case, but actually you can do it even in general, but let me ignore this issue, because anyway, I have to finish very, very soon. Okay, and now let us put c. And now, what is the functoriality? Functoriality says the following, okay, is a claim if x, y is prosmooth, then is t-exact. No, it's essentially general nonsense, because. And now we want to do the general case, okay. Now you want to do something which is not a finance case, but let us consider, let us do an example, which is essentially what we do. Let x will be a quotient, e x by r, no, mean quotient x by r, where we have the r is, where you want that x and r, sorry, r is what its prosmooth groupoid in placid affine skips. Okay, let's not discuss what is prosmooth, but no, but you can say it's the following. I guess I'm going for this purpose, it's enough to show this is the inverse limit of finite dimensional, this is infinite limit of finite dimensional, and you can assume that there exists a presentation such that it's the inverse limit of the smooth maps. Let's say this is not good enough. Okay, this is, and this is enough for this purpose, because you remember some how all the categories are coming from finite dimensional level. In particular, everything actually comes from finite dimensional level. Okay, let us do of placid affine skips, then what you want to say is that d of x, r, which is remember what it was. It was a limit over dx, d whatever, it was r, d of whatever, what was r, r, x, etc. All right, the request has unique t structure such that what you want that it's pullback is t, exactly. And the idea is very, very simple. What you want? You want, for example, you take any object here. You want to write it as exact sequence of less than equal and greater than equal. Take it here, consider this diagram. Take it pullback here, pullback here, and pullback here. Here, here, and here, you have already t structure. So you know how to decompose this as the exact sequence. And now the fact that all the maps are approximately means that they're compatible. So you define, for example, that the corresponding exact sequence here as the limit of the corresponding exact sequences here. Of course, one has to write it, but it's essentially formally follows. Okay, and now I have roughly zero time, but okay, yes, and now maybe I will say d the last time, unfortunately our space n is not like this. But now assume that our space x has a stratification, x i sides z, all x i has t structure. Then what I claim is that there is a general thing which is called co-gluing, that you can glue t structures on each of the x i's, and then you get the t structure here. Aren't you basically in the set up with g is acting on x, five dimensional affine, and aren't all these maps from lx times lg to lx, et cetera? Okay, okay, no. Right, but even on lg mod lg, why is it not pro smooth there? Which one? If you want to advance to lg mod lg, are you not in the situation where this is pro smooth before it, in this sense? No, no, no, because it's int. If it's l plus, if it would be in the case of l plus, everything would be perfect. No, no, if it would be in the case of l plus, everything would be perfect. Our problem is that lg is not smooth, because affine gas mania has an inductively limit of projective schemes which are not smooth. So lg is not smooth, this is the problem. Okay, and then sometimes I'm gonna say this has a stratification, and then there is by gluing, we are done, and might be one last thing. What I want to say, let me, I will just one last sentence, I will say what is the affine spin per shift, and say that what is, yes, let us say affine spin per shift. What is affine spin, or grotenic spin per shift, of course. What you do is the following, you can see that it's i over i, where i is the corresponding eva-hori, or might be l plus of i. There is a map from lg to lg, there is a projection. Now, the claim that this map is in improper sense. Yes, actually, it's better n, because it's compact element, so you get up here, and actually you need regular semi-simple. You remember somehow n was regular semi-simple condition. So you want, so then first of all, I say that this has a left adjoint projection, and you define s is the projection of the corresponding ql bar, shift here. And the assertion is that first of all, it's possible to define a t-structure. Namely, it has a stratification by the guys which we defined before, almost after some nil-potence that I didn't have time to discuss. And then it's also in, so the assertion is this perverse and equipped with an action of the fine-bile group. Thanks. I've been already many questions, so more questions. I mean, how critical is it that you use shriek instead of stop for that to define everything, because in this so smooth setting, the two don't really differ, right? It's interrupted a little bit. Critical. Well, okay, the question, first of all, the question is, what does it mean critical? It's more convenient, okay? You can ask whether they can define everything. Okay, most of the, okay, many of the things here, it's, let's say, the situation is false. You should decide whether you want to work on arbitrary pretexts or on pretexts with some condition, because our pretexts are essentially somehow finite dimensional up to some pro-unipotent. Pro-unipotent actually does not matter. For those, probably, it's not very much different. In general, it's more convenient to work on this one. But on the other hand, many of the results can be translated. And of course, I did not talk about the proof, but actually in some point, for the perverse, you need the shriek stock and the start stock. So in some sense, we need some kind, you need more operations. I say that the start shriek pullback, sorry, exists always, or the others exist only under some assumptions. So the answer is, let's say, I don't, okay. You can ask whether I can do it in the, for start. I did not check it. Possible, the answer is yes. But then there is the indirection when you start, this is one problem. No, no, no, no. But then of course, you have, instead of taking the shriek shift, you have to start shift. Of course, okay. There is a good chance that everything would work. In general, shriek is more convenient and, okay. By the same reason that Dennis explained it, there is a difference between right adjoint and left adjoint. But I'm not sure, let's see. So I invite you to continue the discussion with me.