 Thanks very much. I'd like to thank the organizers for the opportunity to speak here. So what I'd like to talk about is a little bit more of a down to earth about some kind of geometry on a certain site. And I hope you'll see maybe later in the talk there's some kind of algebraic structures which arise which are maybe a little bit different. The idea is to view geometry as algebra in some way. So there's a lot of algebraic manipulations that we would like to do which are sort of inspired by the small object argument in some way. Although that doesn't necessarily play a big role in itself, in and itself. This is a part of an ongoing joint project with Ludmila Katsarkov, Alex Noel and Ponav Pandit. And the project concerns basically some stuff about Konsevich Soymelman, point of view on structure and infinity of cluster varieties, log-Kolabiyao things and stuff like that. And also Gaiotov-Mornaytsky, spectral networks. So last time I gave a talk actually about the same subject. So maybe you can see on the video probably something about the general framework of the project. So this time I'd like to talk a little bit more about the precise geometry. So the motivation, I'll describe this a little bit later. The motivation is that if we have a, yeah, andy-neight scan, we have a family of ordinary differential equations which we'll call nabla t, connections on a bundle. The bundle can maybe be fixed on a Riemann surface x. And we let rho t be the de-monitory representations. Then as t goes to infinity, say from pi1 of x into sl, uh, sl and c. Okay, as rho, as t goes to infinity, then the rho t, well it's going to be, n is going to be equal to 3 today, okay? Well maybe I'll start with n equals 2 to say something about n equals 2. As t goes to infinity, rho sort of approaches a limit which is an action of pi1 on a building. And here we're using parot and parot's theory. And the idea of our project is that we'd like to understand the building starting from limiting data for nabla t. And limiting data, this is a Higgs field. Roughly speaking, the Higgs field is sort of the leading term of nabla t. It's the term in which it has coefficient t. And our Higgs field gives a spectral curve. So we have the Higgs field in particular, the spectral curve of phi. So what's the spectral curve of phi? To just be extremely brief, it's just, um, it's a ramified cover of x contained in t star x. So for our purposes, we can just view this as a multivalued section of omega 1x, in fact an n value. So let's call it lambda 1, lambda n with the condition summation of lambda i equals 0. That's the trace, that's the condition that the trace of the Higgs field is 0 because we're looking at SLN representations. We assume that the determinant of the connection is trivial. So in terms of this picture of the spectral curve, the lambda i's are just the different values of the ramified curve inside the codangent bundle of x. So at a general point of x, we just have n different one form on x. How do you know they're different? And n one forms which, we make generosity assumptions so that, for example, we'll be assuming that the curve is irreducible. If they're not different, then it's more complicated. Okay, so, yeah, there's the eigenvalues of phi, yeah. So this is my question, maybe, does this Higgs field have a connection with a Simpson correspondence? Well, a little bit, but not completely. I mean, the last time I was here, I made a board with lots of different columns on it, explaining that a little bit. So let me just refer to the video of that conference. Which conference are you talking about? Sorry, which video of the conference are you talking about? The one for Maxime's birthday. Conceivage birthday conference. Let's see, so to get down to, now that's all the motivation stuff. So to get down to the basics, so basically, so these are holomorphic differentials on our complex Riemann surface, but we just take the real part, okay? So we get, these are real differentials. The fact that they correspond to real parts of holomorphic differential forms just means that they're harmonic, okay? And of course, I'm saying this away from the ramification points. At the ramification points, a lambda i can switch with a lambda j. We'll see the picture of that in a minute. And the sort of initial lemma, or proposition, let's say, which we did in our first paper with Kassar Kovnoil and Pandit, is that the differential, so let BB, the building given by Parrot, then we have a limiting map, the universal cover of X into B. Let me say this is R building, okay? We have a limiting map, harmonic map from the universal cover of X into B, which is equivariant for the action of pi1. And the main point here is that D of H is equal to this vector of form. Now what does that mean? The building is locally Rn minus 1, locally pieces in Rn minus 1. The Rn minus 1 is just viewed as the set of vector element points in Rn, whose sum equals 0, or equals some constant. So the condition summation of lambda i equals 0 means that the differential, a map into Rn minus 1, is given by n vectors whose sum equals 0. So this is a, you might say, a valid equation. And the reason for this is local WKB theory. Now this is actually, we discussed the proof in our paper, but I guess it was, you might say, probably would have been well known. In any case, for n equals 2, this whole theory is well known. So let me discuss that for now. So for n equals 2, then the Higgs field is just, so then the spectrum of the Higgs field, since it's lambda 1 and lambda 2 whose sum equals 1, whose sum equals 0, then lambda 2 is just minus lambda 1. So we just have a plus or minus a differential, and that corresponds to the, if we take the square, then we get a section of the tensor square of the canonical bundle. So that's a quadratic differential. So you've probably heard the whole theory about the relationship between quadratic differentials and maps to trees basically. That's what's happening here. B is, in this case, is the tree. Maybe I should say dimension of B equals n minus 1. So the map H, so in this case, the map, the building is a tree and the map from x tilde to t factors. This is in our paper we call this phi, maybe. Let's call this whole thing. T phi is the tree of leaves of the foleyation defined by the differential form real part of lambda 1 or real part of minus lambda 1. Remember that at a ramification point, those two guys can switch around. Let me just draw ramification point. So a simple ramification point of a quadratic differential is the point where there will be no choice of uniform choice of square root of the quadratic differential. So lambda 1 is going to be something like z to the 1 half dz, which is d of z to the 3 halves up to the constant. Now if you try to draw the level sets, so the leaves are sort of real part of z to the 3 halves equal constant. And if we try to draw this, we get a, we'll do this in the wrong direction I suppose, but this classical picture. So the foleyation, at a singular point like that, the foleyation looks something like this. And the space of leaves is a tree which you might draw in the opposite direction because points here correspond to leaves here. The origin corresponds to these three lines that come together. This guy is this guy and this guy is this guy. And the harmonic map is just, locally it's just a projection from x. This is in x tilde. This is in T phi. So what's being said here basically is that the size of, I'm not saying the details, but the size of the monodromy transport function for the connection, if the connection has a large term which approaches a quadratic differential, then the size of the transport from one point to another is sort of the, it's the integral of this quadratic differential from one point to the other. This, in Thurston's terminology, this is like a measured foleyation. There's a measure transverse to the foleyation and the distance in that measure is exactly the growth rate of the monodromy. Sorry, I should say the monodromy, the size of the monodromy is e to the t times that measure. Now, this, what we wanted to do in our project is to understand this for n bigger than 2 and see how it relates to Gaiotto-Mornitzky's spectral networks. And as, so when we were here last summer of 2014, we basically decided to concentrate our efforts on the case n equals 3 for maybe I'll explain the reason in a moment. Okay, so the goal of course is to understand the case of n arbitrary. So far we've only been looking at the case n equals 3. The point about this case is that in this case, dimension, the real dimension of x is 2 which is equal to the real dimension of the building. In the tree case, and that's why you have this quotient by the foleyation, in the tree case the dimension of the building is 1 and the dimension of x is 2. So the map has to contract one dimensional fibers. That's why we have these leaves. In the case n bigger than or equal to 4, then generically at least the image of the map is going to be some, you know, thin subspace of the building. So in that case it's much less clear how to get a hold of the points of the building. But in this case at least the dimensions are the same. So we can hope to get a hold of some points of the building at least starting from the points of x. Here we basically got a hold of all the points of the building. I mean it turns out that the map for x tilde to t phi is surjective. So you get all the points of the tree by just leaves of the foleyation. So here what's happening, and that's what I'm going to be talking about today, what's happening is that the image of the curve is some subset of the building but it's not sort of the whole building. Part of the question is how to recover the rest of the building. So now this is what leads to the notion of pre-building. So pre-building is the thing we get before the building. So this is basically the image of x tilde in B phi. So B phi is, we would like to have a similar factorization statement. B phi is supposed to be depending, we're supposed to have, we'd like to have a similar situation there. Let me just remark what's the, what's the characteristic of this situation with the tree that we'd like to replicate? So the point is that this map, x tilde to t, depends on some kind of complicated limiting process. I didn't say so but in Perot's work you need to choose an ultra filter. You take some kind of complicated limit on an ultra filter. So the building that you get there, in Perot you don't have any geometric control over what it looks like. Whereas in the case of the tree of leaves of affiliation we have a really easy, or you might say somewhat easy, geometric picture of what this tree, what points of this tree you represent. And that picture only depends on the quadratic differential. So we'd like to have a similar situation here. We'd like to be able to construct a building B phi with a map from x tilde to B phi with a property that really it only depends in some geometric way on the spectral curve, on these differential. The quadratic differential you mentioned is just a determinant of phi. It's a determinant of phi, yeah. Okay, so let's call this B phi pre. Now once you look at an example, so we did a basic example in our first paper. Once we look at an example we see that x tilde is not going to subject onto a building. So that's why you get this idea of a pre-building, which is going to be essentially the image of x tilde plus maybe a little bit of extra stuff. And so the audience is mainly, mainly we would like to construct this pre-building. Then maybe the building, to go from the pre-building to the building there's sort of a small object argument. But for one thing it doesn't seem completely unique and maybe it's not completely essential. So maybe the real statement is that we have a factorization like this. So here's our map to the building H. Maybe the real thing we would like to consider is a map from x tilde to the pre-building factoring into the building. But the pre-building should sort of contain the structure of the building near the points of x tilde at least. So now the question is, what is a pre-building? So there's lots of definitions of buildings. There's books and papers and everything discussing what are buildings and so on. But we can see from this example that we really want to understand subsets of buildings rather than just buildings. And the axiomatics for buildings are usually pretty strongly related to the axiom that any two points are contained in the common apartment. Which sort of implies that we're already looking at big R2s in our two-dimensional building, these R2s that are sort of going off to infinity. And some of those are exactly going to be the directions which are not hit by x tilde. So here's where we can get to the question of a site. So what is the basic question of what is a pre-building? And so we wanted to have an axiomatic notion of this type of object. And furthermore, so it turns out that there's a whole process involved in the construction of this guy, which I'll be discussing. And we'd like to view that process as something algebraic rather than something analytic. Because a lot of the points of view on buildings have a lot of metric stuff in them. Some of the metric stuff is obviously crucial. But maybe the idea is to try to stay in the world of algebras as much as possible. So let's consider the case. Let's consider the case n equals 3. Our buildings are modeled on the standard apartment, R2. R2, as I said, should be thought of as the points x1, x2, x3 summing to 0. What is x tilde? x tilde is the universal cover of x. So our buildings are modeled on the standard apartment, which is in R2. And the valgroup ax is just S3. And the affine valgroup is translation semi-direct product S3. So these give some kind of special affine transformations. Well, let's call this A. This is the apartment now. So now what is geometry inside the apartment look like? Geometry inside the apartment is controlled by three different directions. So there's three directions of reflection hyperplanes, separated by 60 degrees. So these are the hyperplanes of the reflections along which the transpositions in S3 are reflections. So these are the guys xi equals xj. Now let me comment that, in fact, we have a little bit more structure. So at any point, we have sort of three preferred, actually six rays. So there are six, but they're labeled into, they're divided into two groups. They have a parity label. So if you think about it, in S3, there's a rotation of order three, but there's not a rotation of order six. So there's nothing which is going to turn us by 60 degrees. The rotation in S3 turns by 120 degrees, which means that the different, it preserves the parity of the vertices. So the edges coming out from a point have a parity attached to them, which is preserved. Well, I mean there's two groups. There's the white dots and the black dots or whatever. An element of S3, S3 is the group which preserves, say, the origin. An element of S3 preserves the parity of the edges. There's no element that goes from here to here. There's only one that goes from here. You can rotate or reflect or whatever. You have to triangle, but not an hexagon. Yeah, yeah. Okay, so that's just a sort of side comment. Now we have half planes delimited by these reflection hyperplanes. The reflection hyperplanes in the affine valve group are sort of any translates of these guys. But you are not in the arithmetic situation. You take R2 as translation. We're taking R2, yeah, because this is due to the motivation here. The edge lengths in our building are going to be sort of exponential. I mean, they're going to be the exponents of the size of the monodromy. There's no reason a priori that those should be in some, I mean, they'll be in some subgroup of maybe probably rank N or something like that. Probably not. It's actually the periods of the spectral curve. I mean, there's kind of the real parts of the periods of the spectral curve, which are some subgroup of R, which are the translation lengths. But it was just a question of terminology. Usually people are called affined by who they are. Okay, that R affine valve group or something. Okay, so we have reflection half-planes. Okay, so standard half-planes. The half-planes are dominated by translates of these three lines. Now an enclosure is just any compact intersection of closed half-plane. So it's convex. So we can just draw these. I mean, a standard one might be a triangle or a parallelogram. There are intersections of standard ones. Standard ones, yeah. Any kind of hexagon. We also have a point. So let's call the point P. We have segments. That's about it. I mean, by these, I mean, fill the N, of course. Okay, so NK is going to be the category of enclosures. So NK is the category of enclosures with maps, which are given by the elements of the affine valve group. The inclusion, you don't consider inclusions of inclusion. Inclusion, yeah, an inclusion. Category of the maps. The maps just have to be linear maps, but they have to be linear maps that aren't, that don't. They're allowed to turn by an element of S3, but not. But then they map some figure into another figure. Yeah, yeah, yeah, a map into, a map E into F, E into E prime, you might say. Yeah, so for example, a triangle maps into a parallelogram. Inside, yeah, yeah. Now, one guy should map into the other, okay? And in particular, the point P maps into everybody in lots of different ways. And the segments, I mean, we have, I mean, an enclosure has edges which are maps from segments into the enclosure, okay? I mean, there's also a map, there's also a part of an edge. And also the directions for the... As long as they preserve, yeah. So they preserve the metric. They preserve the metric, yeah, yeah. I mean, there's a Euclidean metric on this guy. That's not actually the metric which is useful for our WKB problem, but still. Okay, now, so the point is that inclusion has a fairly simple structure of sight. This is, you might say, inspired by, once I heard people saying that rigid geometry, in rigid geometry, maybe Tate had a site where you had rigid open sets and the point was that you only liked finite covers by rigid open sets. I don't know if I can maybe confirm or deny that. Anyway, this is maybe inspired by that idea which is that coverings are finite coverings. Are surjections from finite disjoint unions. So you're not going to be fighting it back in the topology? No, because in the topology, you can take an enlarge of family with those to be covered in. The generating covering, okay. Generating coverings. Generating family. The topology is generated by coverings which are finite disjoint unions. The point is you're not allowed, so we're not allowed to, because we could try to do, we're not allowed to sort of subdivide the triangle into a whole sequence of triangles like that, okay. That would be a surjection. You would have a surjection from a union of infinitely many triangles onto the triangle. That's not a covering. But in your examples, it seems that the interior, otherwise disjoint, is this, can be overlapped? No, no, no, no, no. Maps from disjoint unions. So I mean, you mean like a scissors when you cut, oh. So for example, the map from, this would be a covering. They can overlap if they want. Of course, the point is not to have them overlap. The point is to have them intersect just along the edges. This guy is not actually going to be very helpful as a covering because. You can always refine into a covering where they don't overlap. You can always refine to a covering where they don't overlap. In fact, you can always refine into a covering which is sort of standard in the fact, in the sense that it's just cutting by, cutting by a bunch of lines like that and so on. Finite, everybody's finite, yeah. So now, what are the sheaves for this topology? The point is that this topology is not subcanonical. Because your maps are not piecewise. Yeah. So if we have an enclosure, then the pre-sheaf represented by that enclosure is not a sheaf. But if we let E tilde be the associated sheaf, then what is E tilde of F? So maps, F into E tilde, it's piecewise, piecewise affine maps. We call these maybe folding maps. So we can have a map from an enclosure or to a different enclosure which sort of folds, but we're only allowed to fold along these reflection lines. So it can fold things up and... So when you say piecewise, it is also finite. Finite, piecewise. That's why we wanted to say that we only have finite coverings generating our topology. So there is a one-dimensional version of what you're describing now, which is just somehow piecewise linear maps between the intervals. Yeah, yeah, yeah. That would be the case that you would use to do trees. That's what's catch up to on the RCS topic. At some point. Excuse me. Don't you mean maps from F to E, not E tilde here? Well, I mean, okay. From the space F to the space E. And in fact, maybe that's a good question. Maybe I should say that the sort of the underlying set of points. So F, so... My picture is wrong. I'm sorry. Suggested piecewise linear maps between the intervals, but I think the slope should always be one or minus one. Yeah, yeah. That's a bigger restriction. I mean, I didn't think about this too much, but I think that there should be like a piecewise linear version, which is probably actually well known or something like that, of this whole discussion. In this discussion, we're just limiting ourselves because we're interested in buildings for this value group. We're limiting ourselves to maps whose... which if it rotates, the rotation is only given by an element of S3. Or in the case of one dimension, S2. I mean, for a general league group, you'll have the value group of the league group acting on the... the carton algebra. And that's going to give you the transformations that you want to look at. The underlying set of points of E, of an enclosure E is E of P, where P is the point. And we can think of that... E of P becomes a metric space and so on. So, here I could have marked F. Now, the thing that we're interested in doing is we're interested in sort of gluing together the pieces to form something like our building or our pre-building. And for that, we introduced just a basic definition because let me just explain here's an example which we would like to sort of rule out. I might say some bad example. Ah, so, sorry. So, then our... the objects we would like to look at will be then view the sheaves on the side of enclosures. Now, a bad example. So, let me just... So, first of all, note, first of all, that if U inside E, E of P, if U inside E of P is an open set, so remember, E of P is just the usual space of points. Then it corresponds to a sheave. Let's call it U tilde. It's the set of maps. U tilde of F is the set of maps F into E tilde such that F of P maps into U. But this is not generally finitely generated. You might have a triangle. Just consider some open set. This open set is sort of tessellated by sub-triangles, but they just have to get smaller and smaller as you sort of go towards the boundary. Wait, when you say open set, do you mean a sub-sheave? No, no, no. This E of P is the usual topological space. Just the triangle. Okay, uh-uh. When we start talking about arbitrary sheaves, then it's not clear what we want to talk about for the topology, but for an enclosure itself, the topology is just the usual topology. Yeah, inside R2. E of P is containing R2. And you sort of look at this open set describing some sub-sheave of the... Yeah, it's just the sub-sheave of all the guys which go into that open set. It's a valid sheave in closures. It's just not going to be a very nice one. But, I mean, it's one which could actually happen inside our building because of the fact that we're talking about edge lengths, which could be arbitrary in particular arbitrarily small, uh, something like this could actually happen inside the building. But what we'd like to avoid is the following thing. So, um, we'd like to avoid having something like E1 Union E2 over an open set U when we glue, yeah. I mean, E1 might be equal to E2, for example. We might glue together two copies of the triangle over the same open set. Okay, so you don't want all the sheaves. That's what you want. Yeah. This is not going to be a very good object. Could you impose a fanatness condition? Yeah, so that's what I'm going to explain. At least hope to. Okay, so, uh, what we'd like to avoid this type of thing is, uh, we say that a sheaf F is finitely generated if there exists a surjection from a finite disjoint union of enclosures. I guess, like, you know, a surjection in the topological sin. I mean, a surjection has to be a surjective map of sheave. And you see a sheave at the corresponding interspace or not in a sheave can be represented by the interspace or both. Well, except that we have the S3 actions. I'm not completely sure about that. So it's a little bit like, uh, in Topol's theory, sometimes you have coheret sheaves or things of that sort. So could it be like the subtly Topol's of coheret sheaves that don't exist? Maybe. Is this something you'd prefer to present it in fact? Well, I'm going to get to that in a minute. As I said, this is not actually a condition which we usually want to impose because our building is not going to be finitely generated. I mean, it's going to have infinitely many. Notice, for example, the R2 itself is not an enclosure because I ask the enclosures to be compact. You could maybe discuss whether you want to do that or not. But in particular, to get R2, you'd have to, you know, cover it by infinitely many triangle. But we say F is finitely related to the fiber product. For any two enclosures mapping to F, the fiber product is finitely generated. I wasn't able to prove this actually, but we would like to show that you get a Hausdorff topological space if we have one of these guys. So this is like the notion of quasi-compact and quasi-separate. Quasi-separate, yeah. So it looks like are you working in something which is like here and topos in the sense of I thought that was a good question. That's a good question. Well, anyway, so the definition is that a construction is going to be a finitely related sheath. In the case of N equal to 1, the sheaths that you get are some kind of good sheaths or some kind of polygon. In the case of N equal to 1, I mean, I'm not playing, but I have figures like piecewise, some graph. I think so. But you can have infinitely many. I mean, it's like an R. The goal is to get something which essentially recovers the case of R trees and R buildings. So for the case of trees, the idea would be to recover the theory of R trees. You can have, you know, an arbitrary collection of medium. That's the hope in any case. So the idea is that a construction is, okay, a definition of construction is finitely related. And now we can do things with constructions. We can sort of construct things with constructions. So maybe an accessory definition is that a co-fibration, maybe an accessory definition is that and I think we need this, is that a co-fibration is a map of, say, constructions is an injection of sheaves such that for any enclosure mapping into F prime, the trace of the enclosure, the intersection with F is finitely generated. This is a sub-shef of E. So again, we would like to not have an inclusion of an open set or something like that. And then, for example, pushouts along co-fibrations preserve constructions. And this is the basic operation which we'd like to use. Well, let me just get back to the case of X tilde and the pre-building and so on. So getting back to our geometric situation, we can write X tilde as a union of small open sets and sort of define a we can kind of define an enclosure EI for each UI. So maybe modify the open sets so that they look in terms of the coordinate system given by the differentials so that they look like enclosures. The UI intersect UJ giving a gluing relation among the EIs. And from there, then we glued these together. Glue them together to get F0, which is an initial construction. So the first step to define our pre-building is going to be to glue together little pieces corresponding to small open sets of X tilde. Well, infinitely many to cover X but sort of locally finitely many. That's maybe I was thinking about the case of the case of a fallation in trees is actually more complicated than the case of n equals 3. Because in the case of n equals 3 since the dimension of X is equal to the dimension of the building we're not going to end up gluing together infinitely many pieces. If we try to do this for the fallation for a tree we might already get into trouble because this gluing relation could sort of glue together infinitely many things. If we take a irrational slope fallation on a torus and we try to do something like this we're going to end up gluing together infinitely many pieces. So in the case of a fallation you need some condition that says that the leaves don't sort of loop back arbitrarily close to themselves. In this two-dimensional case just looking at things geometrically you can just sort of glue together the pieces pretty calmly you might say. Now the only problem which is left and now I don't have time to discuss but now the basic problem is that f0 is not negatively curved. That's where the you might say some kind of higher algebra comes into play. So let me just explain why f0 would not be negatively curved. So there's a negative curvature condition this is seen if we look at points of our construction then there's a notion of local spherical construction which is a graph. And inside the graph if you have a hexagon in the graph that corresponds to something flat. It's kind of the local erase this. The graph will have vertices for each edge coming out of the point and edges for each two-cell. When you have a hexagon that's something flat you have say an octagon then that's going to be negatively curved. If we have something of a loop of length 4 you might have something like this. When we do this and generally this will happen all over the place when we start gluing things together according to gluing by little open sets of x of x tilde you might say pretty much anything can happen in particular a picture like this will happen so two parallelograms glued together along two edges. At this point at this point the graph looks like this the graph has only four edges and that's a positively curved point. You just geometrically can see it's a positively curved point. And so the basic algebraic problem is if you open it you don't follow your doctor and you don't cover the water. If you open it you only get two-thirds of the circle. So now the problem to define a pre-building. It's like to be negatively curved. That's kind of the essential thing which is halfway between the notion of a construction and the notion of a building is to ask for some kind of negative curvature and so the basic problem I'll just explain the problem here without explaining the solution the basic problem here is that when we have four-fold vertex like that well it's clear what we want to do we want to fold together two of the pieces so we're going to have to be gluing together some stuff and the difficulty is that we don't know which way to glue together. It's like this toy of children. You can glue together like that or you can glue together like this. So some additional algebraic data is needed to determine at each of these vertices which direction to choose. So there's a procedure of collapsing together positively curved vertices so you'd like to end up collapsing together these two parallelograms. There's a procedure of collapsing together positively curved vertices but you need additional information you need some information of marking of the whole picture telling you which direction you should glue at each point and that's the information which is going to come from the original Riemann surface and so on and so that's what we explain in our paper at least we don't actually have a it's not a theorem it's sort of a program of how you should be able to do this as you sort of follow through this procedure so we should think of this as sort of an algebraic procedure of forming a quotient of this construction according to some additional information just from the Higgs field or more than that? Well the map comes from the Higgs field and then we're using the geometry of the map basically but the data that you need to do is glue in choice comes from the Higgs field? Basically, yeah, yeah essentially I mean if we had a if the front parallelogram came from a piece of the Riemann surface and the back parallelogram came from a different piece of the Riemann surface for some WKB we know that we're not supposed to fold the Riemann surface itself so that tells us we're supposed to squish it like that we're not supposed to fold along this line but the problem is that once you start folding then you get you get edges which are priori don't necessarily come from come from pieces of the Riemann surface that's actually where the Gaillot-Mornitzky spectral networks come into play but I'll leave it at that You said at the beginning of your talk you said that there was something about the small object argument? Yeah, this is what I'm trying to say actually because you'd like to, I mean the original hope was just some kind of general procedure like the small object argument was going to allow us to map X tilde to a negatively curved space but it turns out that the small object argument needs you need to sort of control it as you move along in some way because you need to control which direction you collapse at each point so there's some sort of control of the argument that needs to be done as you proceed the very first idea I would like to say by a small object argument we sort of take the universal negatively curved building negatively curved construction which is a pre-building with a map from X tilde mapping to it with that given differential the point is that that universal guide doesn't actually exist a priori and in exactly the cases where there are spectral network where there's BPS states in the spectral network you can see that there there might actually be two possible choices now the point is that with a generic generosity condition we're hoping to prove that that doesn't happen basically so it's kind of small object argument but with piloting along the way so you need to sort of maintain the data that tells you at each step what you need to do sorry I wanted to ask the category of points of the topos you have introduced that's a good question what's the category of points of the topos there might only be one which would be the point P possibly only one point I'm not sure maybe like germs of enclosures at the point P probably maybe there's the point P there would be a point which is a germ of an edge and a point which is a germ of a triangle so maybe there's three points possibly but I don't know so you have the defining as far as I understand the defining of the topology it's defined by finite cover covering families and the site in question does it have fiber products or not it does the site have fiber products I think the site has fiber products you can intersect the enclosures one should be careful that products rather than fiber products don't have a good property product of construction so anyway there is this notion of a coherent topology the 5x5 is finite with the topology covered families so my question during the talk was whether your in particular whether your topology is a case of this kind of site or does it give this kind of topo so this it was not served to no I don't know maybe is it quasi-comfort or you don't I don't know you know it's a very simple it's a very simple minded you might say example so you take your defining object of those enclosures and the maps are given by things in their find variable sending one to another so okay so you maybe it's locally of this so is it really the fiber product of two the finitely related guidance are closed under operations like fiber I think so yeah I think so yeah in particular the fiber product the basic guys are finitely general okay the fiber products of basic of enclosures are finitely related that's true this starts to look right I mean I don't see any reason why that would not be the case anyway okay so it seems that there is no final generating family for the world in terms of the world you could have larger and larger if you have something smaller you can also cover any enclosure by copies of yeah but I think probably the problem is the smaller and smaller ones you need sort of arbitrarily small enclosures you don't need arbitrarily large enclosures but you need arbitrarily small enclosures in fact you need infinitely many because if you've fixed some countable collection then you're not going to get all possible edge lengths that way although I guess you can cover I just need that there is a generating family with fiber this is finite limit so you said you have fiber product the product in the topos of two so the product in the topos of two enclosures is not finitely no the product of two is not finitely generated no the product in the topos the product of two enclosures is not finitely generated so this means it is not Cartesian it's not a coherent topos because in a coherent topos whatever comes from the site is a coherent object and then the category of coherent object is closed under finite limits yeah so that's probably not true because the point is that we're looking only at two dimensional objects and if you tried to take the product of two different two dimensional objects it would sort of be a four dimensional object but it's not that's not going to exist I mean the product of the sheaves obviously exists as a sheaf it just doesn't have very good property now but the product looks more like an intersection well the fiber product a fiber product actually works pretty well but a product but not a fiber product yeah yeah among other things one should be careful that the point p is not at all the same as the empty product the empty product star the constant sheaf with values equal to one point is not a good object it's not finitely related for example so it doesn't have a terminal object that's the only difference well there is a terminal object in the turbos but it's not a construction otherwise it has a fiber product yeah okay so maybe so there's a also a locally queried and an algebraic yeah it's locally queried so much