  günpunk  günpunk Gentlemen, I willCHEERING, and you'll find what I will say today.  soak in arat 푯 iich saits ta ports. So, I give scourses and Gengen and Kashyabara give three courses.controlled courses wll be more or less like that. I will speak of hand-shift on also a sub-category, the sub-analytic shift, then in order to define the hand-shift or sub-analytic shift of temperate holomorphic function on the operation and on it, and then I will explain the regular Riemann-Hilbert correspondence, so that's me, regular. But I will follow, because the way I will treat regular is very parallel, almost the same as irregular. So I will not recall the Keshore's proof of regular of this theorem, but I will explain it in the language of hand-shifts what we need. And after Keshore will do the same with E, which means enhanced. So enhanced and shift, OE, which is enhanced version of OT, and irregular or general Riemann-Hilbert, but with this enhanced setting. So that's why we organize the talk like that. And I think it's good to have the first three talks to understand the structure of the proofs. So I think I could do, but I think it's a bad thing to give history. And the subject, I will write a few lines of history, but it's not to the people who do something to write the history. It's dangerous. So maybe I skip this, and I prefer to start directly. So in my notes, you will find all the bibliographical references in particular about hand-shifts. Maybe hand-shifts, this is done in asterisk. The theory of hand-shifts is developed with all details in this booklet. So one, one. So I say I don't do history, but I do a little, a little history, not so much. So in 75, Kashiwara, constructed, shows that the function of solution are going from autonomic demodule to construct the function, shows that the function of solution is well defined with value in constructable shifts. So it's a very important result. And at the same time, people don't know that. I can say that because he's not here. He would be shy, but he gave the condition of perversity. Of course the world didn't exist at this time, but he gave the condition which are called perversity. And of course he has the question, what about the regular case? He defined that in 77 in some footnote of Rami's. It's written by Rami's. I don't want to make too much history. Anyway, what he did in 80 is very close to what I will explain now. It's a starting point of what I will do now. He defined what is called the function theorem. The theorem with value in OX, let's say on here you put F which is R-constructible. So it's a function from R-constructible shifts to demodules on the approved. So I say I will not do, but I give my version of Riemann Hilbert. So always, this is published by Ecole Polytechnique. It's a written seminar of Ecole Polytechnique on the approved Riemann Hilbert correspondence at this time. And the paper was published in 84. And at the 84 also there is another proof by Mebkut. I don't know. And that's for the regular case. And then came much later in, I don't remember, around 2000, I don't know exactly, around 12 or something like that. Came the work of Mochizuki. But before that there was some work by Sabah. And after Mochizuki there was some work by Kedlaya for the analytic case. So with this work, you understand more or less how it works for irregular. But then Kashiwara and Danielo Kashiwara, I think in 2013, gave a big theorem of reconstruction. They call it Riemann Hilbert, but I think it's not Riemann Hilbert. I will say it's a reconstruction theorem. Anyway, so that on the, the tool was this, Annanth and Schiff's, that Kashiwara will explain. Ok, so that... It's important, but it's T-Wachizuki, it's T-Tai-Ko-Po-Wachizuki. Yeah, I don't know which one. The young one. The young one. Ok, so I will speak of... Today I hope to explain to you the theory of Ann Schiff's, subanatic Schiff's, and to give a few examples. Ok, of course in this course I assume that people are familiar with the six operations on the old-fashioned theory, which is called D'Rave Categories. I don't use the luri and so things like that. So what is an in-object? So when you have a category, so of course we work in a given universe, you are the one necessary, but it's not our subject, we don't need to change universe. Except when you have C, you denote by C-Wedge the function from C up to set. So this is not in the universe anymore. On inside you have the category of Ann Schiff's and object, excuse me, Ann C. So what is an in-object? An in-object is in C if you can represent it by an inductive limit where XI is in C, but the inductive limit is taken in C-Wedge, of course. Because C has no limit, a priori. And you ask that I, the category on which you take the limit, is small. And I don't know, I use the name Filtrand, maybe we use that always. Anyway, that's the basic. So that's the theory of in-object, which is well known since Botendic. And we will be interested in the case where, so I don't give the theory of in-object. I will explain Ann Schiff's. So we will be interested when C is Abelian in this case and C is also Abelian. What can we say, so there are a lot of classical things. C in Ann C is exact, fully faithful. What else? There are a lot of results, but I choose whatever. So Ann C admits limits, young people say co-limit. I say inductive limit. And this is the same limit where it's commute. These limits commute with Ann C to C wedge. But by construction more or less, you have this formula. Ann C of, I don't know which notation, take F, Y. And limb of Xi is isomorphic to the limit, usually in set or in group, of C, Y, Xi. But here you assume, of course, Y is in C, not in Ann C. So you see this formula, classically, if you delete the quote, it's totally false. That's the main point of in-object. You also believe the axiom of five is exact? Yeah, of course, in this category, small filtrant inductive limits are exact. I don't give all properties because it will take the world out. But I will give examples which are more interesting than the theory. So the main point is the following, the danger, if assume that C admits inductive limit. Then the function C to Ann C does not commute with such limit. It does not commute because of this formula, which is false in the category C. If you delete, this is false, it's not the same limit. So I propose to give a short to take examples in dimension zero. So let's do and sheave over one point. So I take a, so it's an example, it's not a theory. So I take K a field for short. On what else? So mode K is the category of K modules or K vector spaces. And also I use mode FK, which means finite dimensional. And I will denote for short, it will be the minutation after, I of K, the category of an object of mode K. So of course you have a function for mode K to IK. If you have a vector space, it is an inductive limit of vector space. But you have two functions going by. So this one is yotta. And I have alpha on beta that I will describe now. Alpha of an inductive limit in quote of Xi is a usual inductive limit of Xi. And beta, let's say one way is like that. So V is a, excuse me, beta, it's the contrary. Alpha, beta, beta, excuse me, like that. Beta goes from sheaves to end sheaves. And beta V is inductive limit for W contained in V on W finite dimensional of W. So you can check that these functions are adjunct. So what is right and left. I have not written it. OK, we will come back for sheaves later on this. So another way to define beta is that beta of V applied to W. W is a wk tensor V. It's a contravariant function on W. So I want to show you some surprising things with these functions. For example, take V in finite, not finite. And look, so you have a beta of V goes to V. There is an, it's an epimorphism. So let's denote by N the kernel. So if I calculate M k beta of V, I find the inductive limit. Is it epimorphism in which category? Maybe I agree. The beta of V, it has to not be that I agree. Because it shows the bearing I between. I is in bidding, yeah. Yeah, I don't write I. No, but why is it in the epimorphism in the category N? Because beta V is like the limit of W. Then you have limit, trigger limit. So the kernel is limit of W, of limit of W, which is a non-privileged inductive system. I think it's injective and not subjective. No, no, it's not. No, no, it's like that, I'm sure. It's objective. So I want to show you that if I take the kernel, then I find something which I test the kernel to K. So I will find M k W. So I find V. And of course, M k V is V. So you have an example of an in-object such that M over K N is 0 and N is not 0. So if you want to have a shift which is 0 on every open set, there are not so many open sets. There is two open sets, empty set on the whole space, one point. And when you take the value... I think it can be really mistake. I think it's inclusion, but in the case, it's more of a finite rank, according to you. Maybe you are right, I don't know. Okay, anyway, if it's the contrary, it doesn't change. I think it's co-kernels. Okay, so it doesn't, anyway. It's still true that... It's still true that... It's still true that one from K to this is 0. It's because of using the formula that you have written. Yeah, you are right, the limit is exact. No, to be of great, it's our usual limit, it's finite rank, yeah? Yeah. So it's part of all names. Okay, you are right, it was a mistake. Okay, so anyway, nevertheless, you have an example of an in-object which is 0, like that, which is not 0, of course. And other things which is a little strange is the following. If you take the end-object of the finite-dimensional vector space, let's call it for short I, F, K, then it's more or less obvious that you have an equivalence of category by the function alpha, which alpha, I remember, sends inductive limit with quote to the usual inductive limit. And this is an equivalence of category. As well as, for example, the category set of sets is the category of end-object of finite sets. It's an equivalence. But you can send them both in I, K, by Yota and here by abuse from term. So what is dangerous? It does not commute. It does not commute because the inductive limits here and here are not the same. Okay, so it was a few examples not to discourage you, but just to tell you that the theory can be subtle. Another thing, which is a little depressing, another thing is that the category end, it was of mode K, what I call I, K, the category has not enough injectives. So this is in our book on category, on category and chips with cashier, but I must say that the proof is very, very difficult, too difficult for me. But it's a result, okay? But there is something which is... So whenever you put a restriction on the modules, like you take modules of some bound that you sleep on the dimensions then but I got the result, it ended up injecting. No, but here there is no bound, so I think it should work. If you have fixed the universe, there are not enough injectives. Of course, if you change the universe, then you can, okay? So, but there is a notion which is substituted to injective, which is a notion of quasi-injective and C, so C is abelian. So you say that A is quasi-injective and C dot A is exact on C, not on C, but on C. And in practice, quasi-injective is enough for many purposes, okay? So I don't say more, okay? Yeah, it's useful, but it's... Okay, but the theory is difficult. When you want to derive, it's maybe with the language of infinity categories. Easy now, but at this time it was not easy, okay? Anyway, so let's pass to... So now you know everything and sheaves in dimension zero. So now we'll define and sheaves in any topological space. There are other mild conditions that are always quasi-injective. No, yeah. If C is a grotendic category, then there are enough quasi-injective. That's the theory. If C is a grotendic category, okay? And C is not a grotendic category, but it has enough quasi-injective. Okay, don't worry so much with all these things because I will speak both of hand sheaves and also subanatic sheaves. And subanatic sheaves are much easier to manipulate, okay? So in all this talk, M will be what we call a good topological space, which means it's locally compact. So it's not for algebra. It's Hausdorff. It's countable at infinity and also finite, flabby or injective, I don't know, dimension. Anyway, we will work over a field so it doesn't change, okay? Okay. And I fix K, a field. After we will use the sheaf of rings, or sheaf of K algebra, okay? Most of what I will say can be extended by taking unital commutative rings with good properties, okay? But it's not a subject, okay? So I assume that, so yeah, my notation mod KM, so the sheaves of K modules on M on the derave, as far as possible, we use a bounded derave category, but sometimes it's no more possible. So that's our notation, okay? The derave category. And I don't, just one notation maybe, not all notation. A, or let's say Z, is locally closed in X, in M, excuse me. I always use the constant sheaf KZ, which is the constant sheaf we stole K and Z. Okay, so I don't say more on the classical theory of sheaves, on the six operations and so on. So what is A and sheaf? So for short, I denoted I KM, so it's the in-object of, so it's not what I'm writing now, it's not the in-object of sheaves, in the in-object of sheaves with compact support. So of course, over one point, it was not the same, but so why do we take the in-object of sheaves with compact support? Because there is a theorem that if you take U gives I K U, for U open in M, this pre-stack is a stack. So we denoted I, not the same, I, script I KM, okay? And if you don't take the in-object of the category with compact support, it will not be a stack, it's easy to have counter-example. So for those... How do you restrict an in-object of smaller open? If you have only sheaves with... Do you know that if the sheaves is compact support on U, then it doesn't restrict the sheaves is compact support on... You can code by an open set, smaller open set. It will have compact support. If... Something like that, I guess. To be honest, I forgot, but it seems reasonable, no? You just take extension by zero from I of code sets, something like this. Honestly, I forgot, but it's a general for stacks, for unstacks, you can do that. At this level, you don't need sheaves. Okay. I swear it's true. You have to check. Okay. So it's interesting to make the comparison for... I'm sure there are a lot of analysts in the assistance. So the comparison with analysis if you take distribution on M, it's not the dual of C infinity on M, what is the dual C infinity function with compact support. Okay. And in the object is a little like taking the dual. And that's the reason why distribution are sheaves, gives a sheave, and sheaves give a stack. Okay. So we have a stack and we have many operations on the stack. So, okay. So maybe I will take my time. So I recall IKM is the end object, category of end object of category of sheaves with compact support. Okay. So we have many operations, again, from mode KM to IKM as before, on... So I describe this operation. What is IKM of F? So you take, I don't know, inductive limit, I have not written it, for you relatively compact in M of FU. Okay. What is alpha of an inductive limit in quote of FI? It's the usual inductive limit of FI. What is beta? Beta is more difficult to explain. But as an object of sea wedge, as a functor, recall that this is contained in the category of functor of mode CKM up to mode of Z or set as you want. So if I take it, I calculate it on G, it will be HGH0 of R gamma M. So dual of G, that is GKM turns out F. Okay. I don't give... You have to prove that it's in the object. I give a short way to define it. Okay. Maybe it's not the best definition. Yeah, G has compact support. Now, in object is a functor, is in particular a functor and the category of sheaves with compact support. You have to think of and sheaves as follow. A sheave is a functor and the category of open sets. But now we take functors and category of sheaves or sheaves with compact support. Okay. So it may happen that there are and sheaves which are zero on each open set without being zero as we have seen in example. Because now the category of and sheaves is much bigger than the category of sheaves. That's why it's interesting. Okay. So what I wanted to say. Okay. So this notation looks too complicated because you are just doing H zero of this coefficient in the sheet. I am wrong. Why I am wrong? Because everything is positive. Wrong is a non-negative degree. Okay. So it should be just H zero. Yeah, you are right. Bad habits of derive category. Okay. I will describe this functor beta in a moment. So there are adjunctions. Where are they? My adjunction formula. I have to recall. So I have Yota. I put my adjunctions. Here. You have a pair of adjunct functor like that. There are adjuncts. Moreover, Yota M and beta M are exact. And what else? They are all exact, excuse me. But those one are fully faithful. And of course, alpha M is exact. And you can deduce the commutation with limits from the adjunction, of course. As usual. Okay. So now, I will define the operations. So first the internal operation. So maybe you will protest because it's not canonical, but nevertheless. If F is like that, G is like that, J, inductive limit of F i. Then you can define the transfer product as the inductive limit for i, J of F i transfer J G J. On the i M, which is not M, F G as. So we have inductive limit here. Inductive limit here. We first take off this one. So we find projective limit. So this is always true. So of M F i inductive limit of J J. And this is inductive limit with quote of M F i J J. So here it's, I define that. Okay. So this is an in-chief clearly on ear 2. Here there are usual sheeps. Okay. Home is the home sheep or home factor? It's the same. Excuse me, home sheep. It was badly written on ear 2. And when you take a limb of the eye, this would be in which category? In this... And sheeps. Do they have in those limits? Yes, they have. Yes they have. I don't give an exhaustive course and sheeps. I just give a few formulas and I will insist on the formula which are strange, which are not the classical one. Okay. And there is another home without eye because... And the Inverse Limits are computed in 0. In what? The Inverse Limits are in 0 for an... Ah. Inverse Limits does not change. It's the inductive limit which is dangerous. Projective limit is okay. And if you remember I told you that I K M was a stack. So a stack has an internal home. So what is the link with home of the stack? Let's say I K M. Then this is alpha composed with I M. Okay. So we have one new operation which is I M. So it's no more the sixth operation because we will have four others. Okay. So... Okay. So maybe... Okay. So now the external operations when we have a a continuous map F from M to N. So we keep the same notations when the functions that we define commute with Yota. Okay. But we change the notation if the function does not commute. So if it is an extension of the classical function we keep the same notation. So we can define F minus 1 of inductive limit with quote of J J. Okay. So J J the inductive limit of F minus 1 J J. Okay. So you will say this is not compactly support but it is a shift. We know that mode K M I don't write Yota. It's contained in I K M. Okay. Just a moment. I understand. So you write this formula if I am composed not on the previous blackboard if I am composed with this. Yeah. Okay. Maybe I write. This is an end. So let me write it. I don't see any particular area. Yeah. So you are from I K M up cross I K M to I K M this is I M and then mode K M mode K M So this is the usual shift. Usual arm. Yeah. That's the room. And this is the shift of morphisms in the stack. This is the idea. Yeah. What I said. The morphism of the stack. Okay. This is not well defined because I choose an addictive limit. So we should it's not very intrinsic as a definition but it works. No, we don't care that it's not compact supported because these are sheaves. Okay. And sheaves are contained in hand sheaves. Ah, by the previously explained by Yotta that I don't write. So I take the inductive limit it's perfectly defined. So now I forget compact support when I have a sheave I look at it as a hand sheave and I can say that a hand sheave is the inductive limit also of sheaves. It's not the definition. So the reason is here in F K M Well, the reason is T in the category because it's a hand sheave. What do you want? This is a usual sheave and I take the inductive limit it is an hand sheave. So the direct image so I give the formula which which is a little maybe I'm not obliged to give a formula because if I tell you it exists and it is adjunct of F minus 1 it's enough for the formula. So you cut F I by a compact set K takes the inductive limit and after you take the projective limit not before. So with this definition you can see that you have a pair of adjunct functions. So what is more interesting is a proper direct image because it will be different. What is different is does not commute with Yota we denoted by a double So what is proper direct image of A so maybe I make a parenthesis. What is a direct image usually what is a direct image with proper support of F it is inductive limit for you relatively compact in M of the direct image of F U that's one definition which is better than to say I take proper support so here we take the same definition so I assume that F I have a compact support so it's the same definition so what do I change the notation I change the notation because inductive limit are not the same so this function does not commute with Yota it's not the same with the function so maybe I will write a because you can have a compact there are many things about that does not come okay so let's maybe I summarize all my results because I'm afraid that some people may be lost so I summarize my result oh yeah maybe I summarize okay let's make a big a big step and we pass to the derived categories okay so you have to believe me because I don't prove anything on the derived function so I go to the derived bounded when it's possible derived category yeah so now I derive an exact function it's not difficult Yota M Alpha M so it was exact so there is no problem Beta M and after I will give a few result of commutation tons of product I am over a field Alpha goes to usual shift Alpha goes to start up no no no Alpha goes to usual shift okay oh you can replace B by other okay so this is I am so this one we don't know if it is bounded or not yeah maybe okay I don't write it's too much work so let's write the F-1 this one and also the proper direct image admit a right adjunct with the same notation because it commutes with Yota and I make a short tableau here I write my function F-1 F star or maybe derive I don't know okay so so it means it commutes okay so it's a little boring so let's do something more I think but in practice I tell you I will skip to subanitic shift but in practice we will use the end shift mainly in this situation we will look at our I am for example of F G where this one is a usual shift in practice it will be like that very rarely or maybe never we use our arm for both and shift and then you can have quasi injected yeah zero means commute yeah zero means commute dot does not commute okay but unfortunately we are obliged to to work with a shift of things because I want to apply all this theory to demodules okay commute cross does not but I will give you notes don't worry okay so in practice I will have something like that a is a shift of K M algebras for example M is in I KM so what does it means that M is a module so it means that you have a morphism of shift of things and maybe I will write M of in I KM of M M but this is alpha of I M M okay so it is the same as to have a map from beta by adjunction beta A to I M M so there are two point of view that we confuse maybe it makes things maybe difficult to follow we define okay I I A is a category Abelian category of A modules in I KM okay so in a module is the data of a morphism of algebra maybe it is better to write this like M you have a morphism of algebra from A to the endomorphism of M okay no I say N this is a usual shift okay which is alpha so M is alpha I M and beta alpha are adjunct okay so if I write alpha I M I put alpha here so I have a morphism from beta A here okay so but beta A is no more a shift it is an unshift okay so you have to look at beta A as a ring a ring object in the category of unshifts okay so beta A is a ring object in I KM so you have the choice but it's equivalent so you can take the category mode so you can define what is a module in the tensor category and you can define mode beta A and it's equivalent to what we call I A which are A modules in I KM what happens if you consider shifts of A modules instead of shifts of K modules no we can do also but that's another story so I don't want to it's a little confusing so I don't want to confuse more because we shall not use it okay of course we can define the unshifts of the category of A module with compact super and object of the category of A module with compact super no it's not the same no so I prefer to be keep this it's dangerous so in so and I want to give a formula which is totally not usual and it's different if I charge an field but just extension of fields if I do just extension of fields yeah but A is a where you see a field A is a shift of rings yeah but I can just take if I just take super things take constant shift of rings yeah if you take A equal KM then you'll find A is a ring usual ring and take the constant shift of rings A yeah so what happens then is the category of everything is the same as this I don't know if you take A equal KM for A KM it's okay that's why if A equal KM IA is IKM and if A is a constant ah anyway anyway it's sufficiently complicated it's not necessary to add other questions that we don't need in this story yeah but we shall use it only with D on the content on C in practice it will be CM on CX on DX okay so there are a lot of formulas and I think I have no time to to develop but yeah first okay just one because this notation will appear so I need to spend a little more time so we shall use Db IA so if M so if M is in here we'll use where is that for example we will have such notation where K also is in Db IA then this notation will be the same as R IM over beta A beta M K and similarly ah if N is in Db IA hop then N transfer A K will be the same as beta N transfer beta A okay depending if you look at things here or here so it's a little confusing and I prefer not to insist okay so what is a strange formula which will be very useful and which has no content part so M here is M up which is written it's an object of the dirac category of IA it's really an angle because you would like to apply beta of M yeah you can M is also an object of IKM so it's what I said you can look at it you have M which is Db IKM but you can also with an action of A in the endomorphism of M okay so you can look at or maybe okay if you don't like you put beta everywhere but as we did before in hysterics it's not so important I want to give a very strange formula you will you will not believe but it's true we have A as above the shift of K algebra and F is a usual shift why did you write the IA up in the work of the reason to because it's right right and left ah that's it don't call it activity not the opposite category no no no up is inside it's right and left we work with D module it's not coming you don't put beta beta of K no because no I think it should because beta of M no no no it's not not beta no understand K and M are elements of the same category yes this is my problem I think M should be in a smaller category I see it's a beta A module but it's not beta K so it's not beta A certainly not wait so I will finish this but with a formula that looks strange so I take K in DB IA up on L in DBA not IA then I have this formula R I F K transfer as you want with beta or not beta maybe I write beta beta L F K is right K transfer beta A beta L but with this notation you can also write the same beta as you want so if you want things it's equation of notation but it's a little confusing ok so that's very strange you can put inside when you have beta and maybe just to explain you what is beta to give you on after I stop let's come back to the beginning what is beta of K U where is that so it will be inductive limit so U is open for V relatively compact in U of K V and when S is closed beta K S is inductive limit for V contain S of K and to prove this formula in fact one reduce to the case where L is something like K U and then there is some work or A U ok so I stop 3, 5 minutes ok so after this next I will speak of the subanity topology which is much easier on the link with in the ships what is the can you before the home what is the like T underline I I what is the I kind of an I script I don't know ok 5 minutes so now I will speak subanalytic shields so M let's say real analytic manifold or more generally real how do we call it analytic space subanalytic space more generally subanalytic space and so all these things I should have said or I have said maybe are due to catch your eye on myself this part so so that's an idea that I explained to the analyst all the time but they don't understand that in analysis there are too many open sets very bad open sets that you cannot do anything with and it's better to restrict to good to a good family of open sets and then we take up so you have up M is a category of open subsets but it's better to look at the category of subanalytic so it will be subanalytic open subsets and it's convenient but to take compact so it's there are two approaches you can take compact or not but for what we I will say it's enough so it's a pre-site it's a category by inclusion of course and I will put a got an ectopology very simple M S A is a site such that the object are the subanalytic relatively compact on the covering are the finite covering to say that the covering is finite doesn't mean it is finite strictly speaking it means you can extract a finite covering you have a covering U I there exists J in I J finite such that U is a union a few J that's why I call a finite covering so it's a very easy things it's a got an ectopology you can check the axioms yeah the point we don't care who care of the point yeah when you teach shift theory in the first year at university is better to have point flat shift is flat if you don't have point then you will get but it's not very coherent or not the purpose is not very coherent still no idea because of the final covering there is enough point I know there are people working on that and you can add points to M more or less you take the boundary of all open subsets and then you have a yeah of course subanalytic I like subanalytic but you can say exactly the same thing with O minimal the only point is that I am not familiar with O minimal but you can extend the theory to O minimal without any problem okay so of course you have a morphism of sight from M the usual topological space to the subanalytic sight because there are less open set here and less covering okay so you have a pair of adjunct pointer as usual for shift theory and also there is also the another row minus one admits a left adjunct which is easy to describe and after I will make the link with and shifts of course but wait so maybe I don't write M always this is a shift on M subanalytic associated with U gives F of the closure of U so we have these functors which are take care the direct image maybe I skip M in the notation there is no row star is left exact of course but not right exact in general on beta of course row minus one is exact admits adjunct right and left on lower shriek row is exact on how to recognize that a pre-shift is a shift on the subanalytic site there is a very nice a criteria so it is a a pre-shift F on M subanalytic is a shift if and only if F of the empty set is zero and also for any U1 U2 subanalytic the sequence F U1 major-viatory sequence is exact so you have just to check that you can glue a section and two open set and you have a shift you can also prove but we shall not use it if moreover the sequence with zero here then F is gamma U acyclic for any U just a remark we shall not use it ok so we have a very easy way to construct shifts so my aim is to have a lot of hand shifts not so many in fact there is one which is interesting for me for us ok so here again of course this category mode KMSA is a grotendic category contrary to the category of ind object because this is a general result nothing to do with subanalytic you have a small site the category of shifts and this site is a grotendic category but the inductive limit we denote it like that ok it's a notation because it does not commute with rho it's an active limit well here this is actual limit it's the true limit yeah it's a true limit here because of the finiteness ok but since it does not commute ok you worry that yeah maybe it's not a good notation but let's say in this talk at least I prefer to make a distinction between limit here and limit in the usual category because right now I will have to confuse usual shifts on subanalytic shifts and to make a difference with inductive limit because now I take the category what are my notation I denote by RC KM the Abellian category of R constructible shifts so you know what means constructible it means there exist a subanalytic stratification such that the shift are locally constant on the strata on finite dimensional it's clear and here we have the subcategory of shifts with compact support so here is a CRM which says the following it says that the function direct image so I forgot it maybe so direct image this function is not exact but it is exact on the category of constructible shifts and of course it's fully phase so you can look at constructible shifts you can look as well as a subcategory of usual shifts or as a subcategory of subanalytic shifts so we don't make any for constructible shifts we don't care if we are on the subanalytic topology or the classical topology and now I define a new category RCKM as the end object of the category of constructible shifts with compact support so of course I have a function from this category of end object to the category of subanalytic shifts to limit with quote of FI I associate the same thing but with different notation if FI is a constructible shift with compact support here I have its inductive limit in the category of end object and here I have its inductive limit in the category of shifts on the subanalytic side so this defines this function on CRM this is an equivalent in other words subanalytic shifts can be viewed as the end object of the category of constructible shifts with compact support so you see we have this we have our constructible shifts I forget we can look at shifts this was YOTA M on here I call this YOTA tilde M on this commutes of course so on this is equivalent to mode KM subanalytic so this is clear but there is a a danger, a trap so you can go also you can take this diagram also the same diagram this one and it will be something to take the direct image here so take care because it does not commute it has no reason to commute and it does not commute and this function is not exact it is only left exact when you construct a end shift for example when you take maybe I give an example maybe here take a complex manifold X on the shift OX so I can look at at this at its direct image as the subanalytic shift and I can look at OX as a shift on here I go to the end shift then I will not find the same image in fact if I take the direct function here which is better because the function is not exact this function is not concentrated maybe I have to speak later of that I will say that next week but I say it already the direct image of OX is not concentrated in degree 0 and I explain you why because first what is the meaning of OX this it means you take the dolbo complex with infinity and I say that this dolbo complex if you work with a subanalytic topology will not be exact and it will not be exact because if you take for example an open set like that the dimension of X is bigger than 1 the difference of 2 2 bolts an open bolt minus a closed bolt something very pseudo concave here then it's pseudo concave and you don't have you are allowed to take only finite covering so if you take a finite covering dolbo complex will never be exact there is nothing to do because the open set with the usual topology you can take small bolts the dolbo complex is exact but here you have finite covering only then the dolbo complex is not exact so this is not concentrated in degree 0 that's one reason but if you take this function it's exact and then O as a hand shift is in degree 0 ok so I will come back on this kind of problem so what is the link so I say it again so subanalytic shifts are embedded in hand shifts because this is nothing because it's isomorphic isomorphic to equivalent to the hand object of constructable shifts with compact support so what is the link between all functions that we have constructed so you remember we have mod m yota beta m alpha m i km and also mod km direct image inverted image lower shrink mod km subanalytic so when you send this here you can see that alpha m corresponds to inverted image and beta m corresponds to row lower shrink it could be or if you want it means that the diagram I give one example mod m subanalytic i km mod km here for example here if I take beta here I take row lower shrink and it was called i tilde m then it commutes similarly with the other ok so you see ok so we can look at subanalytic shifts as particular case of hand shift so what happens with the derive category it is much easier to work with subanalytic shifts but for example it's why I give this example of oh if you look at oh as a subanalytic shift it's no more in degree 0 so sometimes sometimes you have no choice we will define the oh with temperate growth then we have no choice we cannot define it directly but if you can define it directly so since we work in the derive category let's call db i r c km as a full sub category it will be triangulated of db no i I forgot excuse me excuse me where is it written i ok i km so in the case this is the object f such that for any j 8j of f is in the category subanalytic the end object the object of the derive category of and shifts whose common logic is subanalytic and it is triangulated ok so in fact mainly in the sequence we will work in this category we will work in the sub category of and shifts whose common logic subanalytic and in this category there is a important trick that we shall use very often maybe it's a proposition it's more or less obvious amorphism u from f to g in db i r c i km is an isomorphism if and only if for any what are my notation u open subanalytic om k u shifted on for any shift n u shifted by n is isomorphic to om k u n g so to test that a ok so you remember I told you that for and shifts there are and shifts which are 0 and all open sets but here we don't work with all and shifts but with and shifts which are more or less subanalytic and then you can test them on subanalytic sets ok and in practice it will be very important so maybe I have time to to start some example because so the m was to construct and shift so now I will do some analysis sorry not so much so m is a always a real analytic manifold so there are classical shifts infinity or real analytic function infinity function or what else distribution on hyper function ok that's a basic shifts for the analysis it's unless infinity on distribution but real static hyper function are useful so I will define what is a temperate I don't know in English temperate or temperate infinity function what does it mean temperate infinity function ok so that's the of everything that which I'll do after so you take an open set so you take you subanalytic on relatively compact in m let's say for short on f infinity a few so I say that f has polynomial gross so you take a cart a distance so so if so you take a at some point p let's say so you choose coordinate a local coordinate system at p p point p is on the bottom of here or anywhere and you assume that there exist k compact it's more easy intuitively if you want to write express good definition but intuitively it's obvious a compact a bold of p and there exist on c any integer such that the sub for x after I will I will make a picture so you take the distance of x to k minus u over n f of x c so it means that f gross at the boundary less or equal to the distance one over the distance to some power f of x is the gross at the boundary is polynomial I think maybe otherwise the boundary is empty what is the distance to the empty set 0 I was sure that you first I wrote it and after I will erase it the distance is here the distance to the empty set is 0 maybe it's better to understand intuitively here is the open set u you have that the gross of the function is polynomial with respect to the distance that's clear no that's the beginning of the definition it's not finished so it's at p you say that f has polynomial gross if the same at each p and you say that f is temperate if all derivative have polynomial it's very intuitive it's easier than n-shift so maybe we can attribute it to but it was not formulated like that is that u the pre-shift u so maybe I give a name so I call c infinity t u the subspace of function with rate gross so if you take the pre-shift u gives infinity t u this is a shift on the sub-analytic site so I will explain the proof it's a corollary of the logicevich inequality you remember that for a pre-shift to be a shift you need to f of the empty set is 0 on the sequence f u1 union u2 f u1 plus f u2 f u1 intersect u2 is exact then f is a shift on the sub-analytic shift so why will it be true here because there is something called logicevich inequality so I give it in rn it says that let's say in rn if you take two open set relatively compact sub-analytic the distance of x to the complementary of the union v is less to some there exist n such that on c such that this is less or equal to c the distance of x to the complementary of u plus the distance of x to the complementary of v so something which is if you want an example where this is not satisfied you can take something like with an exponential here on another open set like that then the distance to the complementary of the union it exist to open set so the distance here here you will have exponential with respect to this distance but if if the if the contact is of exponential type it's not sub-anity so what can you tell what the open set is an example exactly oh you take the graph no no no just I didn't see the open that the thing above doesn't a drawing you take this set this set something like that maybe I have not prepared that ok so anyway it's easy to construct two such open set the distance to the complementary of the union is much bigger than the distance to each open set ok here you have the distance to each on the distance to maybe no you don't anyway in the sub-anity case the distance is a polynomial depends polynomial the distance to the complementary of the union is a polynomial with respect to each distance and it is not true in general of course so if you take close sub-anity sets the distance to the intersection will be a polynomial of the distance to each one ok ok so once you know that the distance to the union is a polynomial function of the distance to each one if if you know that if you have a function which is which belong to c infinity temperate a few one and in c infinity temperate a few two then it will belong to c infinity temperate a few one union u2 because of this inequality ok of course with this temperate can one also if consider derivatives the order of rows can one also order of rows but in principle it is possible to make it more but after we will apply dolbo and then the derivative will be controlled ok but at this level it is not necessary of course we could define we could ask that the derivative the distance of the derivative is one more than the distance of the function ok but it is not useful at this level it doesn't change ok for what we have in mind because of Cauchy's inequality for your holomorphic function we want to apply this to the holomorphic function we want to define the temperate holomorphic function that is the dolbo complex of c infinity temperate ok let me give another example very simple which follows also of Lojasevich's inequality the work of Lojasevich is very old but there is a book of Malgrange which is old also but not in 66 where all these things are very well explained at that institute conference but did people have subanalytic at that time or just semi-analytic? yeah at this time it was semi-analytic you need to know about subanalytic yeah you guess it works yeah yeah yeah of course but these things are true for subanalytics I'm not a specialist I'm a user of subanalytic but not a specialist but it works so another example is temperate distribution so what is temperate distribution and you so it's the image so the image so it means you can write an exact sequence as you want so this is so set of distribution on u such that maybe let's call it s such that s extends on to a distribution to a section of dbm okay so by duality so in fact we have this sequence which are exact in fact it's also there is a zero also here by a good partition of unity it's not so obvious also for db t yeah yeah there is a very very refined partition unity theorem in Hermander the best theorem you can imagine and moreover since db t if you is contained in v by definition almost dbt v to dbt u is subjective it's like flabby if you want so temperate distribution are flabby in some sense and it follows that db t is quasi injective not as a unshift but as a subanitic shift I have not defined what it means but you can guess or flabby if you want for subanitic shift but it's not soft it's not soft soft it's flabby or quasi injective let's say flabby so flabby means that kind of the chromology is zero yeah this is enough to have zero but dbt is better because you can define what flabby is yeah even quasi injective in the category it means that in the category of subanitic shift db m t is exact maybe I should write it will be better to write subanitic because it's subanitic is exact on the category of constrictible shifts it's not exact on all all shifts but when you restrict to constrictible shifts it's exact it's like quasi injective the second morphism u and v are again nice subanitic always okay so next time I will I will explain the operations inverse image direct image and also the dual shift just I will not insist the shift of Whitney functions which is dual to the shift of temperate function and mainly the direct image for temperate distribution and temperate allomorphic functions okay thank you