 Thank you very much. It is a great honor to speak at the conference dedicated to Masaki Kashivara. I have, since I was a student, I was exposed to his work and his influence indirectly while in Moscow. And I will talk about two subjects which were he contributed on a fundamental way, which is, first, perverse sheaves, as Pierre said. And second subject is categorification. So the talk is based on joint work with Tobi Dikkerhoff, Vladimir Schechtman, and Jan Sodimov. So both of these concepts and subjects are, by now, classical. Let me just fix notation here. So if you have the category Perf XS corresponding to a complex stratified space. So this is the category of certain constructible complex as F. And the conditions, let me write the conditions in the most suggestive way that the i-th caramology of the sheave is supported on co-dimension i. So the caramology kind of dies down as we go further. And an example is, as Pierre said, is the complex of solutions of a d-model, our home over d from n to o. So if we take this as an example of perverse sheave, that this is exactly the normalization. So in the meaning that it is perverse means that, at least generically, the correspondence between models and sheaves is exact. So whatever obstructions there are, they kind of sit on higher co-dimension. And this is a such a fundamental concept that appears in many areas of mathematics. And it seems that it will appear also in many other areas. So and here, the second subject, categorification, it is very prominent now in representation theory. It is a passage, so saying briefly, it is a passage from vector spaces. And the relations in those vector spaces of the form a equals b plus c to triangulated categories. So suppose vector spaces v and triangulated categories v. v script, and it can be appeared as grotting the group of the category, say, tensor our field, so vector spaces over the field. And the basic identity is lifted into something like this into an exact triangle, b, a, c. What's remarkable is that there are so many situations when algebraic structures for vector spaces are naturally lifted into data of triangulated categories in which the identities are lifted into triangles. So those structures, they basically ask, please categorify us. If you look at some point, your eyes are opening and you see that all the identities has to be lifted. And the theory of perverse sheaves seems to fit into this subject. So let me discuss, first of all, the absolutely classical example of this, and then our approach to extending this to a geometric theory of shift-like objects, but of this type. So of course, the classical example is perverse sheaves on the disk. So we have a disk. We categorize perverse sheaves on the disks with one singularity at zero. And as well known, it is given by two, such a perverse sheave can be described by two vector spaces and two maps, u and v, such that the transformation is 1 minus uv and 1 minus vu. Suppose this is transformation for one psi, and this is transformation of phi, are invertible. In fact, the vector space is a finite dimensional that only one, if only one is invertible, then the other is invertible. This is kind of a conceptual refinement of Picard-Lefschitz theory. And this categorifies to the concept of a spherical functor. And this was the starting point of our approach. And this is due to unknown logunyanka. So instead of two vector spaces, we have two triangulated categories, g0 and g1. Now when I say triangulated categories, I need to be more precise, because in order to perform the constructions, we need to do enhancements. We need to, let me just write, say enhanced. And there are two ways of enhancing by DG structures or by stable infinity categories as proposed by Luria. And in fact, more systematically, it's actually better to use this formalism. But I try to not go into those details so much. Suppose we have a functor, f, so that the triangulated categories, f and the exact functor. And suppose it has an adjoint, f, the right adjoint, then we have the adjunction maps. We have something like the identity of d0. Well, the initial transformation is called the unit of adjunction, I think, to f star circle f. And the co-unit is f composed f star to the identity of d1. And if we have sufficiently flexible formalism, then we can take the cone of those natural transformations of functors. Cone of this one and cone of this one. And call this t1, and call this t0. So those are, again, self-functors of those categories. And the functor is called spherical. I simplify just a little bit, because in the correct definitions I also another adjoint appears. I simplify just a little bit to the comparison. So f is called spherical if t0, t1 are equivalences. So this is completely parallel to this algebraic identity. And the adjointness condition means that we have the maps which sort of materialize those identities. So the goal of my talk would be to kind of develop this idea to a more geometric theory. So here I must say that, again, everybody knows this very well, that this phi and psi are not canonically determined by the perverse sheaf. We need to choose certain data. So this depends on choices. So for instance, we should choose a direction, or we should choose a coordinate, or something like this. And in reality, those are local systems, not just vector spaces, but local systems on the parameter spaces of those choices. So my goal would be to say things cleanly enough so that one can pass to the categorical analog. So the goal is develop a geometric formalism, because this sort of generalizes a particular way of describing perverse sheaf, the quiver description. And so define an analog of the global sections of a categorical perverse sheaf, which then would provide as a sort of candidate for fukai categories with coefficients. And this idea of introducing coefficients in fukai categories was suggested by Maxime as a categorification. So I don't think. What am I writing here? As a candidate, fukai category was suggested back in search. So why is it natural here? Because fukai categories at a very rough level is categorification of homology. And this is sort of allow ourselves to switch between homology and co-homology and be with coefficients so we can ask if there is something similar there. OK, so now let me just stick on this board. Oh, there's a third one. OK, so what I will do in the next maybe five or 10 minutes, I will just pound on this example. This is a very, very classical example. However, since I said the concept of perverse sheaf is so important and there are many consequences. And even this classical example, if you look at it carefully and systematically, it reveals more than we expect. So that's what I want to do. Let me call it. This part perverse sheaves on a disk invariantly. Perverse. You may ask what is there left to say about this, but it seems that the subject is completely inexhaustible. So we can look at this for many, probably many years. So let's consider a part of the structure. Suppose we have just a map like this from Pc to Phi. So this data, well, it can become, it is just a linear operator. Or we can look at the two-term complex. Now it is a classical construction of the grotating school of algebraic geometry that with two-term complex, we can associate the Picard category or Picard groupoid. I write it like this, Psi to Phi Phi. It's very common in the theory of stacks. So Picard, let's say Picard category. It's a category of which objects are formed by elements of Phi, so objects equals Phi. And morphism, home between two such objects. Yeah, all those Psi which map into the difference. Set of all those Psi, for which V of Psi equals Phi, say Phi prime minus Phi. So it's a Picard groupoid in the category of vector spaces. Now if we write all morphisms, so like this, that's all more of this category which is this joint union or Phi Phi prime, home, Phi Phi prime. So in doing this, we'll count some of them multiply. So every Psi will enter several times and it's clear how many times it will enter. This is going to be Psi plus Phi. Now it's a category, so let's form the nerve of this category. It's a simplicial object. In this case, it will be a simplicial vector space. Simplicial. I'll just call it N. So the Nth set of Nth simplices of this, it would be, as again, everybody knows, set of composable arrows. And if you write it like this, it would be Psi to the N plus Phi. So this is a simplicial object. So there are face maps, the N and minus one, the del i, the degenerations, S i, satisfying well-known conditions. So it's all very easy to write in terms of this map V. So now, it's a simplicial object. It's a factor from simplex category delta, with the contra variant factor, to the category of vector spaces. Now the category delta is embedded into various categories appearing in cyclic homology. And this turns out to be relevant to the problem of perverse sheaves. So let me say include this category delta into what's known as a paracyclic category. It's a version of the cyclic category of Alamcon. Paracyclic category. So here there are faces in the generation and there is a new, so let me just say that there are objects named by, labeled by natural numbers here. So new automorphism. Automorphism will be Tn from N to itself. Which kind of rotates faces and degenerations around the circle? The conjugation with this rotates, let me just write rotating del i, S i. So it's written in any book on cyclic homology. And the actual cyclic category lambda is obtained by imposing the condition that Tn to the n plus one is the density. And here we don't impose this. So there is a bigger category which sort of involves cyclic symmetry of things allowing us to moving them around. And I want to formulate the following proposition. Let's suppose we have this which is a part of a data of a perverse sheaf. Just one map V. For a given V, we have a bijection. So the following two things, sets I in bijection between, first, all the possible ways that the maps in the other direction which define a perverse sheaf. They do define a perverse sheaf. Maps, maps U from V to psi. Such that this diagram, U V, so corresponding to a perverse sheaf on the disk. And second, ways of extending this simplicial structure to a bigger category, to the data of a simplicial vector space to a paracylic vector space. So ways of extending, extending. I'll just write N dot to a paracylic. So in this situation, so in cons category, we have TN to the N plus one equals identity. Here TN to the N plus one for all N, they form a central system. It's an automorphism of the identity function. It's commute, they commute with everything. And this thing would correspond to the monodrome. Somehow correspond, it combines all the monodromes. So the corollary of this is that the category of perverse sheaf on the disk is equivalent to the category of paracylic vector spaces which obtained as simplicial objects, they are nerves of something. So paracylic spaces which are as simplicial ones are nerves of something. And this being nerve of something is a well-known condition on a simplicial set or a simplicial vector space known as the Siegel condition. And it's convenient to sort of qualify it and say that this is one Siegel condition. So basically the data that the Simplex is determined by its sort of outer skeleton. Okay, here's only two of them. Okay, I see. Okay, now this can be understood a little more geometrically. And this is where the point of view of around spaces and around categories comes into play. So consider the circle. So the whole somehow drift of the cyclic homology is to work on the circle. So consider the circle as one. And let's consider the, let me call it the run poset, partially ordered set of S1. It's a version of the category of exit paths of the run space. So it's a set of posets of disjoint union of arcs. So disjoint, A, which is disjoint union of closed arcs with a natural inclusion. Ah, not empty, yeah, not empty. Not empty. So on this poset, it is a category. In particular, the morphisms are embeddings. So there are morphisms which are homotopy equivalences, which sort of biject, so it's homotopy equivalence is bijection of pi zero. So, and let's call it run of S1. So this is this poset, and we invert weak homotopy equivalences. So in this, it is not difficult to prove. It's the same as the paraciclic category. Find it, find it, find it. Yes, not empty, and find it, yes, yes, yes, yes. It's a paraciclic category, and this way is sort of geometrically clear. So now we can, if we have a perverse sheaf, we can geometrically produce a factor on this set. So we have, suppose we have several arcs. So let's form a sort of cone over this set of arcs, some kind of propeller. So it's a K of A propeller. So if now we have F, a perverse sheaf, to function, which takes A into the first homology, relative k-perc homology, that K support K of A of disk. With coefficients in F. So in this case, only one, so this is the only one non-trivial, so all the other hyper-chromology will be zero. So this also shows that with respect to F, it's an exact function. So in this one, this how one can see that thing directly. So in this description, we already don't use any choices at all, or we sort of take all of them simultaneously. So you have, I want to know, so of course you cannot take everything. Yes, yes, that's also not allowed. Yes, it should be contractible. You cannot take everything, you cannot take the empty set. Yes, but points are allowed, close points are allowed. Points are allowed, it's okay, it doesn't really matter. Yes. Yes, points are okay. Okay, so now let me discuss the next part. We'll be still about perverse sheaves, but let's describe them invariantly on a surface, on a topological surface, again, invariantly. So we have S, an oriented surface, possibly with boundary, with boundary, okay? So something like this, and we have, so a stratification would consist of a finite set of points, interior points, and inside the stratification, and we have the category of perverse sheaves on the surface with possible singularities at this end. So we want to get a description of this category in a shift theoretic way. We want to describe, sort of say, perverse sheaves as sheaves. Of course, one can say, let's use the demodule language, but this is harder to, it's not clear how to categorify it. So let's sort of play a game similar to this one, but allow ourselves to move the central point, the center of the propeller a little bit on the surface. So let me first, or rather let me make a definition. Let's call basic, instead of basic pairs of open sets. Open, opens. So it's a open set U inside U prime, like I said, an open set U prime, with the following property. Let me first make a picture. It would be a sort of curved version of this. So such that U is topologically a disk, or maybe we can say that the closure of U is topologically closed disk. Second, U prime is topologically disjoint union of disks to find it, non-empty. And the claw, actually U minus U prime is contractible. Let's call it K. So it will be this part, it's the curved analog of the propeller. And it meets the singularities in at most one point, may either meet or not meet. So this is a version of this picture. And in this case, then this implies that hypercromology with coefficients, degree is not equal to one of U modulo U prime with coefficients in F is zero. And F lies in pair. So it's the same situation, except now it is a poset with respect to embeddings of pairs. Again, it's a poset. So the disks are open or open? Open, open, yeah. So this is an example, this is what we are allowed. So this is the disk, and those are topologically open disks, so it's an example. Or maybe I'll do it here. We can prove, if you want, it's a theorem, but it's really a proposition. On the same time, it's a clear description of perverse sheaves without any choices. And as such, it is a peeling pair of Xn, as equivalent, to the category of contravariant functions. That's called E. From this poset, so let's call this poset, let me call it run of Xn. It's a version, kind of global version of the run category, Xn, two vector spaces. Satisfying, the following two conditions. First, homotopy invariance. So basically if we have an embedding, so when we have v prime embedded into u, u prime, such that, well, embedded in the obvious sense, such that v to u, sorry, u is homotopy equivalent, v prime, v minus v prime to u minus u prime homotopy equivalent. And the same, let me just add v minus v prime intersected with n equals u minus u prime intersected with n. Homotopy equivalence, first condition. It's kind of clear that it would, for this fun car out there, it would hold. And second, the exactness, let me do it properly. So I simply write the exact sequence of triple, but in the case when all three terms are admissible. So let me write it here and then explain. E of u, v intersected with u prime, E of u, u prime, E of v, v intersected with u prime, zero, this is exact whenever all three, whenever all three triples basic. So a good example of such v would be, suppose this is u and this is u prime, this is u, this is u prime, and suppose we have v like this, then if all three are admissible, then it would be like this. So I should say that this is kind of mindful in the particular case of sheaves on the surface. Right here to the approach of McPherson to perverse sheaves via the concept of what we call fairy functors. Except his approach is more general and it's not precisely at the level of vector spaces. But it's a similar sort of general approach. Anyway, it's just a completely shifteretic description of this, of course, a very elementary category. No, no, no, if all three are basic, so if all three pairs, sorry, suppose you have all three pairs are basic. So in the definition of rank category, you don't require that they're basic. Rank category just keep- No, no, no, no, no, no, so only basics, sorry, only basics. The morphisms in the rank category are in which direction the inclusion goes exactly for morphisms. So if we want this to be contravariant functors, so that's actually embedded like this. So we want this to be the functor. We want this to be the functor. F would associate this functor H1. Yeah, so then- What's embedded on the U and the U prime? Yes, yes, yes, both of them. Well, let's write U prime less or equal than V. V prime means that U is contained in V and I think U prime is contained in V prime. Now in the homotopy invariance, you stated the condition of the complement. And then for the complement, first of all, you don't have a MA. Yes, yes, we need to say, sorry, about V prime, yes. But for the, for the, but this condition, we need anyway. This condition is important. So the complement, you don't have a map, directly a map between V and V prime and U and U prime. Yes, but if it's already sort of homotopy invariant in the naive sense, then we have this map. So if V prime and U prime are homotopy invariant in the equivalent, if this is homotopy invariant, then you can by looking at it, get a map up to homotopy invariant. Yes, okay. So why don't you just say, whenever all three pairs are basic, if you require the basic homo-basic pairs, then you get it. No, no, no, I'm saying if I have U, if you have U inside U prime, suppose this, and we have an open set V, such that, if you just drop superimpose on this, an open set V, that it not necessarily will be like this. But if they are like this, yeah. Okay, so now having such a clean description, we can with more sort of, more ease, define categorical analogs of such data. The next part will be called Schrober's on surfaces. So first of all, let me discuss the analog of the nerve of the Picard group point. So and this is a categorical construction known as the Waldhausen construction, Waldhausen S construction. It actually appeared in K theory a long time ago. It's relative Waldhausen S construction. So we start with an exact function B to C of pre triangulated DG categories, or just DG function, sorry, DG function of pre triangulated categories. Or we can do it in the other formalism. So we have a new DG category called SN of F. So let me write how it appears. It appears as a fiber product SN of B, SN of C would be SN plus one of C. It will be a fiber product, which is the same in this case as homotopy fiber product. So SN of B is by the, so the simplest way to say is a category of representations of A and quiver in B of representations. Something like this B1, Bm. Or it can be also viewed as a version of exact triangles. For N equals two, we'll have B1, B2, but we also can think of this as an exact triangle. So anyway, this is this category. This is the induced map. And this is the same category with label N plus one. And this map del takes, let me say C0 to CN, like this, diagram into the diagram of quotients, which really should be understood as cones. I'll write C1 over C0, CN over C0. And those are really cones, not quotients, just for simplicity of notation. So what's important about this category? So first of all, they form a simplicial object. So the simplicial object. And there is complete analogy with the formula Psi to the N plus Phi in the sense that this category SN of F has a semi-artogonal decomposition. It has N copies of B, that's semi, which means that have several copies of categories, isomorphic to B, it's kind of clear out of this presentation for the ion quiver. And one copy of category equivalent to C, they form in one direction, but not in the other direction. And everything is generated out of those copies by forming the cones. So on the level of growth in the groups, we get this. Similarly to the theorem about parasyclic vector spaces, we have a result that if the function is spherical, so the theorem is if F is spherical, then S dot of F, a priority simplicial object becomes parasyclic, if formulated properly. Yes, and I'm about to say some particular case here. So this sort of, as I said, indicates rotational symmetry with respect to two dimensional geometry. So let me just, let me consider the simple example. Suppose N equals to one. Then S one of F, it has semi-arthogonal decomposition made out of B and C. It's simply, so this is actually a canonical, particular case of the canonical way of gluing semi-arthogonal decompositions. If we have two categories, B and C, and the function from one to another, we can produce a new one in which there is a B and there is a C, and somehow the natural pairing between them is given in one direction, is given by this factor. So if you have the semi-arthogonal decomposition, presuming that the factor has adjoins, then we can form, so C is B-arthogonal. We can form a B double-arthogonal, which is C-arthogonal. So in such case, there will be an economical equivalence from B to B double-arthogonal, known as the mutation. We can continue and form a C double-arthogonal. It was a B triple-arthogonal. So it begins to look a little crazy if it starts doing those orthogonals. But there is, where was to say it? Yes, there is a theorem of Harper and Leitster and Schiffman, Schiffman, sorry. That if the factor is spherical, or if the factor is if and only if it is spherical, then the fourth orthogonal will be equal to B. So f-spherical, the orthogonals are fourth periodic. If you look into this, it is not hard to prove this, but it's a remarkable statement. So if there's a fourth periodic, there is a mutation from here to here, and there will be a mutation from here to here, and the composition will be self-equivalence, which will be exactly what happens in the context of a spherical factor. So already in this case, we have this situation. So now we can get a definition of a perverse Schauber. How much time do I have till 50? Yeah? Still five minutes. Ah, okay. So I still have 12 minutes. Yes, still. Oh, I still have 12 minutes. So a perverse Schauber. Something is wrong. A perverse Schauber on Xn, which is a categorical analog of a perverse sheaf, is a contravariant factor called S from the run poset of Xn to dg-categories. Or one can do the stable infinity-categories. Satisfying. So first, homotopy invariance. Second, the analog of exactness is, for a categorical level, is the concept of Rekolmann or a semi-artisanal decomposition, if you like. Exactness. So exactness one was exactly right there. So for, let me say admissible as before. Admissible u inside u prime and v. We have three categories. Sigma of u, v united u prime, S of u u prime. So it's, sorry, intersection, yes, here intersection. And actually I think it's united. It should be united in both here and here. And I think it's like this, yes. So basically it's the condition in which you can write the exact sequence of triple. So there is no way to, v is a subset of u, yeah. So an example would be this. So suppose this is v, this is u, this is u prime, this is u prime, this is u prime, this is u prime. Yes, I think it's okay, yeah, I think it's okay like this. Yeah, anyway, so it will be, it will correspond to sigma of v. Very difficult, it's okay, yeah, if you take union of v. Union of v, no, no, yes, yes, yes, yes, yes, yes. Yes, you probably should touch bound, yeah. Anyway, so there is only one way of doing this, just to write the, yes, we should touch bound or something like this. To write the long exact sequence of triple. v intersected u prime. So and this would be, would be a recalman or semi orthogonal decomposition. So that would be this category, we embed it here, it will be projected here and there's some other map, naturally we'll embed it back. This is property number two, and it's almost everything, but there is one more property which we need to impose, which does not follow in categorical case, which is the Myer-Vittorius property number gluing. Gluing, so when we have, when the u is equal to u one, united with u two, so and if, and u i, u i intersected with v two, and u one two, which is the intersection, u one two intersected v are basic, therefore we have, then we have the pairing, then we have the right sigma of u prime, we have sigma u one, u one intersected u prime, sigma of u two, u two intersected u prime, this will be a homotopy pullback. So in this, actually is a version of, can be seen as the next higher version of the single condition. It's like, we can call it the two single condition. The typical situation would be when we have something like this, would be something like this, and then we have the two single condition. So the typical situation would be when we have something like this, and the sort of, maybe represent this as a union of two things, so this one and this one, and this sort of corresponds to a polygon subdivided in the dual language picture, corresponds to a polygon subdivided into two sub polygons. So then it's a, this is the definition, it's a local data that can be glued, and for a disk it coincides with the concept of a spherical functor, so it's a flexible local definition. So a functor, when you say contrival, functor to the g-categories, you mean in the, not in the same sense, in the sense where you, the composition doesn't go exactly with. Well, there are several ways of handling this. We can sort of extend this, or in the, something over the groten-dick constructions. So if you write it properly, then there are several levels of doing this, or we work in the model category, when we do it on the nose, but sort of assume that such things can be done on the nose as needed, and then sort of proceed to homotopy data. So probably the most, again, the way the most free from various reproaches like this is to do, is to work with stable infinity categories, but it will be just more difficult to give a talk about this, it will be. Okay, so now close to finishing, let me just discuss the last part, that's called topological Foucaille category, category with coefficients in the shoulder. So then again the picture, which is the approach, which is more natural, is simply to extend. So we have some data on pairs of open sets, which have the meaning of sections, or cohomology of the pair. So once we know it for the basic pairs, we can simply extend this by universal categorical construction to more general pairs by con extensions, by the procedure of con extension, REN of XN, those embedded in simply a set of pairs in X. So pairs of all open set V in the prime. We can, so let's say, let's call it con extension, which means that we say S of more general pair V in the prime. This would be homotopy limit over the category REN of XN, over category S of UU prime. So it's like we define cohomology with coefficients in a sheet, which is a priori defined only on small open sets. So what's important here, that the system of this basic pairs, they don't form a groten-dictopology. However, it may be useful to compare somehow. So if you think of what is the analogy between meaning for this REN category, so we can say that REN of XN so it's kind of similar to what appeared in the study of vanishing cycles with higher dimensional base, some kind of vanishing site. It's somewhat similar to the vanishing site of Deline, Lamont, and Orgogosa. For vanishing site for, so this theory is about vanishing cycles with higher dimensional base. They have some Y to X and this is not necessarily one dimensional. It consists of certain pairs. So here is somewhat similar for the identity X from X to X. So it's kind of similar and the construction is more or less like version of vanishing cycles. It's relative cohomology in such a setting when there is only one relative cohomology. So one can think of this as being analogous. Although it's not, of course, completely the same. So now I'm almost finished. Okay, let me raise this. So let now assume that if X has, let's call it the corner structure, which means that on the boundary, we also introduce some set of corners. Will be something like this. Then we define the Foucaille category of X. X in the sense with the structure with coefficients and S, S of X U prime, when U prime is a small neighborhood of actual boundary. So small neighborhood dx minus corners. So in this localizes on, so this is kind of a priori definition, but this localizes on a spanning graph on any spanning graph. So if you have a graph, which sort of legs terminate in the corners, will be something like this. Then this kind of automatically defines a covering of this nature, of the nature that I raised here, to which intersections are of this kind. And we get a presentation of, in this case, it would be, suppose graph, let me call the graph K, will be sort of argama or homotopy limit of K, with certain shape of, did you categories are K of S. And this is the shape in which Maxime proposes to localize the Foucaille category, as we would now say with constant coefficients for a surface. And finally, maybe I would say as an example of this construction, let me say it here, example the Foucaille-Zeidel category, category of a holomorphic Morse function, say W from some complex menu from M to the disk. And we think of disks with one corner, so one corner. So there will be critical values inside the disk. And such a data defines a shubber, let me call it the left shubber called LW on the disk. So in the Foucaille category, Zeidel category is the same as the Foucaille category of this disk, with this structure with coefficient in LW. So it shows that the concept of perverse sheaves is such a universal all permeating concept and probably there should be still further developments in understanding the very basics of this concept. We have time for a few quick questions. And please wait one second, the time I reach you in order that your question is, oh, okay. Yeah, the question, do you have any idea what to do in high dimension like you have C2 and a point? Well, we don't have a clear idea, but this sort of thing that we're working with pairs is the right way to approach it. Yeah, so appropriate category of pairs from some more steritic reasons or from micro-local reasons. So what's sort of special about this thing? If you write something like this, it's almost if we write something like this and something like this has the vanishing cycle with respect to function sum is equal to L plus one of F. So vanishing cycle with respect to any function, not necessarily function with non-vanishing differential, still preserve perversity. So if you sort of idea that we should look, how should we call it? Possibly, let's call them miller pairs, something which mimics the miller fibers for functions with, say, either at a singularity. Yeah, I mean, it's kind of general idea, but final answer, what should be category of, is the service should be critical? No, we don't have an answer. No, that we don't, yeah. There is a simpler equation about the curve, not just the curve. Okay. Can you say something about the construction of this left-shut shrubber? Well, yes, so it has, so locally, we have it, it's kind of implicit in, or what has been discussed in the category, in the literature. So near a singular point, we have a spherical factor. Outside of those points, we have a local system. So question is gluing them, so it's, once we have a local. I mean, but it's not perverse sheep. What? It's not perverse, right? What do you mean? Or, okay, okay, okay, it is perverse, yeah. So, let me say it like this. If we have a holomorphic, so proper holomorphic, so maybe say proper, proper, proper holomorphic Morse function, then we can take the direct image of the constant sheep. It's a complex. We can take, there is one perverse homology of this complex, which is a perverse sheep. So our left-shut shrubber is a categorification of this. So you think it's any perverse homology or something? No, no, this one, which is the most interesting. The middle one. Yeah, of course, the one of which, the sort of the, so outside of the singularities, there'll be a local system, what local system? Well, the middle cardiology of the fiber. Yeah, so near a singularity, that will be a flip side data. What is the fee there? It's the vanishing cycle, the most classical vanishing cycle of left-shuts. Do you have something to say about mixed perverse sheaves from this RAN point of view? Mixed you mean with a hodge structures imposed? That has been sort of, we've been discussed like various people, but we don't have any clear ideas. One idea that was sort of aired was that one can look for analog of the hodge filtration to stability conditions. But again, it has never been formalized. But it's something kind of, if you have a sort of system of categories, and if you have an aboriginal stability condition, look a little bit like a hodge filtration. But that's where this idea is, we don't have any more. Any other question or remark? Okay, if not, let's thank Misha again.