 Thank you for this wonderful introduction. So recap. OK, so this was a category of stable maps. So it comes with a grading. It's the number of floors minus 1, so European notation for the top floor. Then there are stable maps that have outgoing, ingoing ends. So we have two kind of evaluation maps, which are functors. Then having this structure here, we can build a weak fireboard product. Then there are these sort of chopping functors if you take a building with top floor i plus j plus 1, then you can chop it at a particular place indicated by these two numbers, and you get two buildings connected by a morphism in p. Then trivial cylinder buildings, let me point out, are never objects by themselves because they don't satisfy the stability condition. But if you have a stable object, you can add one to them or more. And then one can take this joint unit of two objects, which produces a finite dimensional family of objects because you can sort of shift them against each other. So then is what I call the more sophisticated Facebook functor. So in order to distinguish subcategories, so the subcategory associated with this would be just all objects where this is positive. But each object would have a weight. In the particular case where it's 0 and 1, then 0, you wouldn't take this object and 1, you would. So then we had that func- in this category, we can talk about pseudo-holomorphic object. And that was sort of the functor which takes value 0 if it's not pseudo-holomorphic and 1 if it is. And it had this property, and we want to perturb this here into some other functor sort of close by. But of course, we don't know yet what that means, which is then in the associated subcategory will be sort of the thing we are going to work with. But it should also have additional structures which we will introduce during this lectures. So that was one of the properties which we wanted perturbations to have. And this is satisfied in particular for this one. OK, so this was so far algebraic. There was nothing, not topology, nothing. So now, so whatever object I introduce will survive the whole lecture, but it will during the lecture having more and more properties because of additional structures one can put on the objects. And the structures actually turn out to be more or less natural things. So what you see here are more or less natural constructions after you fix some discrete set of data. Then it turns out then, of course, the invariance might depend on it. But if you take a different set of data, there's actually co-partisan between one choice and the other. And it's independent of these choices as well. So I don't know what a good, how I should call this thing. So GCT is easy to say. GC group portal category in T stands for topology. So it consists out of a group portal category. So the stable maps was an example of this. So by definition, this means that between any two objects, you have finally many morphisms. At most, finally many morphisms. Each morphism is an isomorphism. And in addition to this, I require that this, so we have only isomorphisms. And I require that the isomorphism classes, so if I identify isomorphic objects, so that could a priori be a class, is actually a set. And on this set, I want to have a metrizable topology. So if you look at the stable map, so look, for example, in Gromov-Witten, at the modelized space. You could make a completely similar setup like this. Actually, if I replace r cross v by a symplectic manifold, you just allow nodes. I get a category of stable maps for Gromov-Witten. Then you have pseudo-holographic objects. And then you take the equivalence class of pseudo-holographic objects. Then that is a Gromov space. Now it has a topology. And in fact, it comes from a topology defined also on other objects which are not pseudo-holographic. So you can imagine how that looks like with stable maps. So I didn't say which quality I take, but I take a quality of function for which I can do analysis. So that would be h3, away from the punch, h3 locks, and the sublap space, h3 away from the punctures. And at the punctures, if you introduce the nodes, introduce polar coordinates you want. That in this polar coordinates, it's h3 with weights. So that's the quality that you can do analysis there. The weights have to be adjusted to spectral gaps associated to the periodic orbits. So then you can imagine what the topology is away from the nodes. So you have a sequence of such things. You go to isomorphism classes. Then you can replace this isomorphism classes. So something should be the limit. Then you can use a limit thing by plumbing at the nodes and so on. And away from this, it would converge in h3. And you have to say something, as already pointed out last week, complicated. So now it's, yeah. Why h3? Well, so suppose I would have take 1 billion 1. Then he would ask me, why do you take 1 billion 1? OK, so I can take everything greater equal 3. So first of all, I want to work in the Hilbert space and have partitions of unity and so on. And then I need an embedding to C1 and two domains to C1. Because when you construct sort of the smooth structure, you have to actually to divide out by group and have to take sometimes also constraints. So h3 is sort of the smallest Hilbert space I can take. But it doesn't really matter. It works in anything. Sorry for the joke. So now, first theorem. So after we made our choices, one can show that there's a natural metrizable topology. So there's no, I mean, you don't have any choices. That is, give you a recipe how to construct the basis of the topology. Of course, different person might construct a different basis. But the associated topology is the same. So it's natural. So always when you do mathematics and you don't base it, I mean, and everything is natural, it usually looks to me like you're going in a good direction. So that's natural. I didn't understand the orbit space of what? Stable maps. The orbit space of stable maps. So when I define the stable maps, I have to say what quality of functions I take. What is the word orbit space? Isomorphism classes, which is a set. So and in fact, if you would know what that means, what the topology is, then if you look at the subset generated by pseudo-holomorphic objects, the induced topology is precisely, which we define for SFT with several authors. OK. And if you replace this category with stable maps in the Gromov Witten context, then it would be sort of the conceivance version, which is how stuff, so that would be precisely topology induced by the natural topology, which you have there. How important for your theory is that you have finite stabilizers? Not really, but it makes. So actually, when we started this project, we had a lot of energy. So we wrote a lot of pages about we could have Lie groups, compact Lie groups as stabilizers. So that definitely, you can generalize the theory to this. And then during the course of the things, I thought I might even have infinite dimension groups. So SC smooth groups, but I don't know. For this, I don't know a good application yet. But the language is extremely flexible. But for compact Lie groups, you could see if you have other group actions on the space and so on, this kind of stuff might arise. So now, so this will be already introduced last week. So assume this is a 10M polyfoil. So this is sort of the next thing to a manifold with boundary and corners, or Banach manifold with boundary corners. Suppose you have a finite group, a group homomorphism, so that we have an action by different morphisms. And then the associated translation group is the category where the objects are actually this points here. And there's a morphisms are all those pairs where g is a group element and q a point. And they are being viewed as morphisms from q to g star q. So this is also a category. So if you have group action, you get a category where the morphisms are basically the group elements together with the information of the starting point. And the end point is g star. It is the point obtained under the action. So how much are you saying you think you could replace g by a new group without requiring that? I mean, so what would be the theory here if you have a Lee group? So in this category, so you usually would look at, so what are interesting functors? You go into the scoop, and then from this thing, you could go by a morphism somewhere. But let's not go into this. Let's discuss it. So maybe I'm already running out of time here. Yeah, OK. OK. Is it just a re-packaging of the data of the interaction? Yeah, yeah. But you just view it as a category. I want to bring it on a footing that I can compare it now with my category. So the uniform miles are now, so this is now an abstract category with similar properties like stable maps. So groupoidal is a metrizable topology in the orbit space on the set of isomorphism classes. And so a uniformizer at the object alpha with isotropy g is a functor from this group point, from this category, from such a category of that, which looks like this, which has a following. So it's injective on objects and fully faithful here. So that means injective on this clear. Fully faithful means that if you have two objects and the morphisms there go to the morphism set of their image points bijectively. Then if you assume that there's a point which is actually mapped to this object, you assume that the image of objects and you go to isomorphism classes is open in C. And you require that if you pass to isomorphism classes, you get a homeomorphism onto that open set. So you have seen such things if you are given an orbifold, then the neighborhood of a point in orbifold, you parametrize like this here as well. But here we just have this requirements that have been. So here's of the picture. So we have this enormous category here, which is not a set. But assume that most isomorphisms go like this. It's something like if you look at the orbit by isomorphisms, view this as a group action. Now you want to take a sort of a transversal slice to this, to the isomorphisms. But you can't, obviously, in general because of isotropy. So you try to be as good as possible. Transversal, taking into account there is isotropy. Then here is the orbit space, which is a topological space. You have this thin thing here, which is the isomorphism classes associated to this thing here. And you map this here into this category. And so if the isomorphisms go sort of in this direction. So that this thing here would sort of, let's say, collapse everything if I go to isomorphism classes to that thing. So it's something. And so we have to talk about compatibility of this object. But it's something like this. So if you have a lot of isomorphisms in this direction, so the differential geometry of this category is something like this. So I put a uniformizer in and look at the trace of a functor. Now I have a smooth structure on this. If I see many faults, if I have a subcategory and it hits my hand in many faults, that's already some structure. If I put this hand in, I see also many faults. And if there is a morphism from here to here, I have a local change of coordinates which explains what I see in this picture on that picture. Is that? I have no idea what you just said. OK, good. So thank you. Thank you. OK, so I just want to say what will be the idea of a smooth structure on this category. So I want to ultimately study a functor on this, like our Facebook functor. What should be a smooth Facebook functor? So suppose it's the original one, 0 or 1. So it likes some objects, it doesn't like some other objects. So let's assume here in this room, the isomorphisms go mostly in this direction, and I put a uniformizer in this like this. So now I look at the trace of the liked objects on my hand. And suppose they line up as a manifold. So I see pieces of manifold. And if I put my hand in here, I see pieces of manifold. So let's assume there went through one object here, alpha and alpha prime, which are sort of related by isomorphism. Then what you want to have is of a change of coordinates from here to here, which explains what I see in this picture to that picture. And you can formalize a lot of things like you can give conditions on this functor that what you see here is actually is a topological subspace, a compact. And from the smooth structure here it will inherit a smooth structure and will be a manifold. So you can actually formalize everything on the language of functors, which is rather easy. And then there are some abstracts here which tells you that there are a lot of interesting objects you can create from this. So that actually in the approach here you work on a level of a sort of the language is easy, and there's a certain amount of additional infrastructure somewhere which tells you you can basically construct whatever you want out of this by just basic, I mean. It's abstract business. So rather than proving things about modellized spaces where at each point you have to work very hard because you're on the wrong level. So OK, so this is what I'm explaining now. This was just the preview. In terms of what you said on the last slide right now, we only have topological jokes and we have not. Yes, I have not explained yet. Of course I could say if I have a subcategory here and I look at the trace of this, I could say it looks smooth now with respect to that chart because I see the objects in O and so on. But of course I have to compare them on different things. So that's the next. So suppose I have two such uniformizers, then I build the weak fiber product which is the following. You go from O, you go into this category, we get this, and you go with the other into this category. And if you look at an image point here, an image point here, you assume they are connected. You look at only those pairs of points which are connected by an isomorphism. So Z phi Z prime so that the image under psi is connected to the image under psi prime of the point. Why do you call this B product? Because of this here. It's a static fiber product. Yes, the usual is a usual fiber product. No, but the isomorphism. You're a topologist. If you're used to... Topologist, I'm just a modelist. It's a usual part of products that if you're a topologist. I mean, sometimes real fiber products also occur. So weak fiber product, we even categorize, okay, so. I'm coming from a different planet, so I can't say. Okay, so this here we call the transition set. And now of course, what we would have to know of this is a smooth structure on this. So here, let me point out to you what the structural things are. So if you have this, you can go to... If you look at this data, you can take the projection to Z. You can take the projection to Z prime. So projection onto the first thing we call source map. So these are all rather standard things if you do Lee group holds or so. Target map, then there is a map which associates to an object here. Just this tuplet here, where this is the one map, the one isomorphism at the image path Q. You have an inversion map. Just interchange the role here and just replace this by the inverse. Then you build here. This is the real fiber product with automorphisms in the image. The source and the target. So you require that the target of that one is the source of this one so you can actually compose this. Which means this actually can be composed. But application map. And this, so this is bailed. I mean, it's nothing deep. It's just the only interesting things you can do. And the maps above are called structure maps. Okay, so now, here's what a polyfoil structure for this is. So it's a construct. So in real terms, the following. So with all these kind of categories which appear in real life, the objects in this category are usually are geometric objects and they have enough geometry to build something. Like, if you do the Lin-Mampho theory, you could look at the category of stable remanin surfaces and you just take one object and then by cutting and pasting and gluing, you can actually construct a neighborhood in the isomorphism classes. So the point is that these categories which occur in symplective geometry, for example, but others, the objects are not just only points but they are geometric objects and then you can actually give in general a recipe. You just say, take an object, do this, do that, blah, blah, blah. There might be several choices involved but at the end of the day, for each object, there's a set of constructions you can do. So here, a polyfoil structure. So it would be a functor from C into set, category of sets, which associates to an object, a set of such uniformizers. Yeah, so for each object, there's a construction. I get a set of such uniformizers. Then if you have isomorphism between two such objects, you get a bijection between the construction here and the construction there. So in general, you can say a lot about it. So it's all the equivalence of constructions. If you know one object and do the other, then usually the data which you choose here by this morphism corresponds to a precise set of data on this slide. So this, in general, in application like stable maps and everything, this bijection can be very precisely explained. And then there is a construction which associates to two such uniformizers, a tame and polyfoil structure on this transition set so that these things are true. Like the source and target map are local difumofisms. The unit map is a smooth. The inversion map is an AC difumofism. And multiplication is smooth. So think of this as a maximal recipe at last for this category. So for Gromov-Witton, for SFT, and for others, there's a set of rules. You write it down, maybe as a page, what you have to do. Take an object, do this and this and this. And then you get F. And then usually this comes out implicit from the construction. So in general, the hard part is actually in these concrete cases to define the F and come up with the right set of conditions. So now, if you have this, so let's me. I'm sorry, I'm thinking this first thing. Why is Iota a scale difumofism, not a local scale difumofism? Because it's an inverse, I. Sorry, I was just saying, except for the number of letters. So this is just the inversion. So when I have one slice here and one slice here, and I have a morphism, and this point here is psi of z. And this is psi prime of z prime. And this is phi. And that's one of the important conditions here. So then S would map this to z. So this here is a point here. And this would be mapped to z prime. And then what does that mean? S and T are locally acidifumofisms. That just means if you start moving this point here, the source would move, and this endpoint would move smoothly. So if you have a slice here, and my functor would cut out a manifold here, and I have here the isomorphic, then actually I could, near this object, take this isomorphism, and it would actually isomorphically, difumofically identify it with that manifold here, for example. So it's actually pretty amazing just having this definition what you can do. You can write a whole book about it. It's like when you introduce the notion of a manifold, we know you can write books about it. So if you start here, you can do as well. And why can you write books about it? Because there are a lot of constructions you can do on that level without going to local coordinates. Of course, when you do the proofs, you have to. You have to go back to basics for certain things. But you can do a lot of constructions. And of course, I think you're always happy when you study a concrete problem where usually this is the moment where you have to get your hands dirty. And then you creep up, you go up, and you come to some level where you can apply some theory, which means there's a lot of implicit construction, given constructions you can apply. And then usually things progress quite fast. So it's principally here as well. So OK. So this is the basic notion. So let's see what we can do with this. So let me first say, for example, which maybe comes up later, but not now, if you have such a smooth structure defined, so that's the notion of a smooth structure here. Actually, if you have such a structure, then there's a tangent construction, for example. You can associate to such a category a tangent. You can associate, then you can take witness sums of this tangent. You can look at particular functors which resemble to be differential forms. So you take a direct sum of these things. Then it turns out to such a psi, then there's a T psi. It goes from the tangent space of O into the tangent category. And then, of course, you can pull back such a functor. And when it looks in local coordinates smooth, it's a smooth thing. But you don't have to check it for each object. You just have to check it for an isomorphism class of an object, for the functorality, if it seems smooth. So you see already, so you can do a certain number of things. So it turns out that in the case of stable maps, you have the evaluation functors at marked points. If you pull back a differential form, it will actually be a differential form on the tangent, on C. So it's all compatible with the usual stuff. So here are certain things which are immediate also and important. There is a degeneracy functor. So if you have this smooth structure on this category, this polyphonic structure, there's a degeneracy functor. Namely, it just defines the degeneracy of the object alpha to be that of an object representing it. That's independent because the degeneracy is being preserved by the transition maps. So if I take two representatives, then I have this thing here, and it preserves the D. So it's well-defined. And then the objects have level structures. You can say, for example, alpha has regularity k. If you can write it like this, and that lies actually in OK on the level k. So it inherits everything you saw for an M-polyphon. OK. Now let's talk about the smooth versions of subcategories. So here is a nice-looking polyphold here as square. So let's first say what we can say about submanifolds in this case. But the picture is the same in general. So general position would be something like this. If it's, say, one dimension, it shouldn't hit in the corner. So general position and transversely at the boundary. Good position. Well, that still is OK here to come out like this. But of course, it's not general position because this one dimension should hit not a corner. But you might have stupid things like this, but there might still be retracts. So these are possibilities for submanifold. So now the general definition is the following. So if always an M-polyphon, an M-subset, then we have heard what a subpolyphold is of this. This is a subset which is locally a retract. But now I require more. I require a retract such that if I lift the index by one here, that this is SC smooth. What does it mean? This means the differentiability property is stronger. If you go from you to you, we have SC smoothness. If I go one up, if I shift the index by one and require differentiability, I'm actually dividing by a smaller norm. So differentiability is a stronger condition. And it turns out that if you have a set which locally is a retract of something which has this property, then, first of all, it's called a submanifold. And for good reason, namely, the induced structure from the M-polyphold on it is a subpolyphold structure which, however, has an equivalent set of charts which are actually usual manifold charts. So you mean finite dimension? Yeah. So why is it finite dimensional? So first of all, if r goes from you into u1, u1 goes to u2, goes into u3, and so on. And r composes r as r. So the image lies in u infinity. So such retracts can only retract onto smooth points. So then, when you are at a smooth point, so you need just level one, then you can take the tangent map of r. And at each smooth point, you have a tangent space. So the tangent of r on that tangent space is, of course, the identity. So tr on a point in this thing is the identity. But because of this one in compact inclusion, the identity is compact. The identity is compact has to be finite dimensional. So if you have this here, you see that on the tangent space, the identity is compact. So then it's actually a good exercise like this. So then you take the tangent space and, actually, nearby, you can represent by an implicit function theorem the solution space as a graph of the tangent space in local corners. OK. So this we already know. A weighted subcategory was this functor into 0, 1, which associates to an object irrational number in this range. Then the support, so the support are the objects we want. But they have a function, a restricted function, which is a weight. So now what is a smooth version of this? A smooth weighted locally finite dimensional subcategory is a weighted subcategory, so that the following holds. So if I take an isomorphism class in C, I pick an alpha in that isomorphism class and take such a uniformizer, which mapped that point q0 to alpha. Then you can find an open neighborhood around this and properly embedded finite dimensional submanifolds here so that this functor is written in this way. So it's an indication. So if this is positive, it means it lies on a smooth submanifold and the number of occurrences where that is, each of those manifolds calls 1 over the number of elements in your family. And this just counts on how many manifolds you lie, divided by the number of manifolds you have. So for example, if theta takes the value 0 or 1, you would just see one manifold here and it would just be one if you lie on the manifold and otherwise you wouldn't. So trivial. Are you guys with me? Is it like being a constructable function? Don't ask me this. I don't know. Why, sure. In general, no. So then, of course, you can have different flavors like locally equidimensional. They are of all the same dimension. In general, position to the boundary, in good position to the boundary, this thing could be tame, orientable, oriented, and so on. So you have all these different flavors for these things. So here I give you some exercises, how you can consolidate this thing with what you know. So then, if this is oriented, and we have a differential form, which I have introduced, you can actually integrate this over that. And how do you do this? Well, there is an interpretation of the stuff above by data on the orbit space. And the differential form defines your assigned measure on the orbit space. And you have the weight on the orbit space. And you just integrate the weight with respect to the assigned measure. And it turns out you have stock, cerium, and this kind of stuff. Can you explain how it could be that you have this description of data locally, but you can't globalize it to a whole category? Well, you have these bits and pieces lying all over. I mean, let me give you an example. Then you'll see how that would work. So let me first give you a picture. So in local coordinates, you'll see all these MIs flying around here. So on O, you have the action of G. Then you have all these manifolds lying here. And that is sort of the local picture you see. If you have a smooth functor like this, that is a picture you see if you stick your hand in. And look on my hand. I see the green stuff. The green stuff with the MI? This is all the MI. That is what I see. The red stuff is the neighborhood U of Q. And O is the domain of the uniform. So I stick my hand in. That is, I see the green stuff. If I stick my hand in, you'll see some other green stuff. And if there's something isomorphic, then this S is affected in the structure with S and T are local diffeomorphism. It maps a point here diffeomorphically to the picture which I would see. In this picture, however, you might, for example, it could be so if I have a certain number of manifolds doing this, I could, for example, take another copy of each manifold and just degrease the weight by half. That would also be a representation here, so you can change this. So here are some exercise and remarks. So assume you have a smooth weighted subcategory like this. And assume theta only assumes values 0 and 1 between two objects. And between two objects, the social subcategory, in the social subcategory, those where theta is positive, there's at most one more fism between any two objects. And first, assume that all objects have degeneracy 0. So in some sense, they lie on the interior so that I don't have to talk about boundary. Show that the support of this is, in a natural way, a smooth compact manifold without boundary. So what you have is that the orbit space is compact. You have these local pictures. You have the diffeomorphism between it. You just glue this stuff together, the local pictures. Yeah? So that's a, so now let's do it. You're rid of the boundary in that picture. Because there's degeneracy 0. Oh, degeneracy 0. Because your boundary is always compact. Yeah, so if I wouldn't say this, then what would happen is I get a sub-manifold in the sense of the definition which I have, which might, however, have horrible boundary things. But you would have tangent spaces at this manifold at the boundary. And if they would lie in good position to the boundary, then actually you can show that this would be manifold with boundary in corners. Each of the MIs, this one. What is the definition of this good condition I think? By any good condition? Or a good condition just basically means that something like that. So the definition in general is a bit more complicated. But that the tangent space doesn't like like this. Like it just goes off nicely in. And if you wiggle a little bit, it would still go nicely. And that's the minimum condition you want. This is one of the possible things you can require. I mean, smooth thing just means you have this smooth representation like that. But then you could start requiring all kinds of things of your manifolds. Like orientable, all having locally the same dimension, on the boundary lying in good condition to the boundary, lying in general position to the boundary, and so on. So these guys occur in all kinds of different flares. And here are two of them. So if, say, everything is as degeneracy zero in between any two objects as at most one morphism, and the orbit space is compact, then this thing has a natural smooth manifold structure induced from the polyfoil structure on the category. If I then require that, again, I require that you all have only values zero and one, but assume that I don't say anything about how many morphisms I have between two objects, then actual support is a smooth compact orbiform. So with this language, then two possible things happen. So it describes manifolds and orbiforms. And then the most general thing is that you can have rational weights, and it looks a little wider. So that's the generalization. So there's one overall more general notion. And if you specialize and require additional properties, they reproduce things which you know. But the whole formalism is actually much closer to the constructions you do if you want to construct a modernized space. So the level on which I talk is sort of streamlined for saying the least amount necessary to construct something. And then everything, as you say, sort of abstract stuff you can do. And not so long, there will be actually a book about all this abstract things, construction there. For each individual case, presumably, you only need a fraction to do. So you can integrate and all kinds of things. So basically, the construction of a modernized space comes up, you have to get a good recipe. If I give you an object, there has to be a recipe what to do with this object. And there's a local construction, you're an object. And as we see, then, we also have to introduce bundles over this. And then we can talk about the Kusch-Riemann functor. And then when I extend the smooth notion to this, we will actually have a perturbation theory for functors, for certain classes of functors. OK, so this is already said. So there's a tangent construction. And then if this is oriented, you have actually a formula like this. So this is what you see if you construct this transverse and modernized spaces. And you've worked out the orientations. You integrate forms over this. But it also tells you that if you view things from this general perspective, so if you want to do SFT, the only differential forms which ever matter would be pullbacks of forms by the evaluation map at a mark point and forgetting the map and collapsing unstable structure and go into the little hemomphode space. So this evaluation may actually turn out to be smooth functors. And then if you pull this back, this would be a particular type of this differential forms. But there should be many, many others which perhaps see somewhat the topology of the function spaces itself. Then the data you get doesn't have so much compatibility. So it cannot be represented as SFT, but there might be a representable then in some other way. Do you have some interpretation of what would go on complex and use? Yeah, I don't have an interpretation of what it is. So it just exists. And I think my feeling is there are many, many forms than we just ever used. And it must have to do something with, so if you fix the women's surface and look at all maps into the manifold, and then you divide out by being isomorphic things. So it sees on some level, I think, the functions of the function spaces from women's surfaces into the manifold. And there's more topology. So but if you use such forms, I think, then it restricts you in what kind of invariance you can get. So if you pull back the other forms, you have a lot of symmetries and so on, which you wouldn't. But who cares? But I think there should be other theories as well. Excuse me. Can you give an example of a scale-smooth, weighted, locally-fined, or whatever word for it, where the local scripting can't be globalized? What does it even really mean? It does not exist a finite list of sub-categories. Well. No, so how about anything that's an immersion, but not embedded by the notion of space? That at least answered your question previously, right? If I have just a figure eight, I can't write this as a union of embeddings. Yeah, OK. So something like this, yeah. That would be an example of something which requires just local description. It's locally-fined union of embeddings but it can't be globalized and finite union of embeddings. But that figure eight is a thing which can occur in this thing. Sure. I would get a much worse thing if I could. Oh yeah, figure eight, yeah. So in the transversality results, during homotopies, bad things can happen. So the c-tas you construct are usually not everywhere smooth, but they can have singularities and singularities sets which you have to throw away and which you can distinguish and so on. So theorem has a natural metrias of a topology. We already knew this, turning this. And then has a natural polyfoil structure after fixing a mon... It's a minimal amount of discrete data, actually. But the construction is so general that it actually... You can use the same construction and then fix the minimal amount of data. So whatever comes out of it doesn't depend on it. What is S and what is SP? SP? And this is the category of single-laps and SP is the weak fiber product over this. So then S of p is a weak fiber product which we introduced last time to say how c-tas should behave and that also has an induced structure then. And then here, the nice thing... So there's sort of a natural construction for a smooth structure. So then the nice thing is some kind of reality check. The degeneracy index from the polyfoil structure agrees with the grading factor. So when we did the algebra, the grading factor was the height of the building minus one. So now if I have a smooth structure, I have a grading factor coming from that and they are the same. So the structure takes building height into account and preserves this quantity on a geometric level. So now you see now having this topology, a lot of structure happens. So we have the orbit space. We can talk about the connected components in the orbit space, pi zero. We can talk about connected components there. Well, this here is just the orbit space of that. Then if I have a connected component, I have an associated category for this, namely all objects which are isomorphic to an element in A. So the isomorphism class lies in A. Then there are phases. Actually, all the combinatorics now comes from this in SFT. So a phase, you take a connected component in that topological space in the subspace where D equals 1. So you just take honest boundary points. You take buildings of height 2. So top floor is 1. And it takes a closure of this. So here's an example in dimension in differential geometry. So if I have this here, this square, then I would have to take this point first away. Then this here would be a connected component of D equals 1 on its boundary points. Then it takes a closure. Then I get that, including this points. Here's another example. Here's another connected component. It takes a closure. I get that point again. So I get four phases in this case. And they overlap at those points. And the number of overlaps at a given point is actually the degeneracy index. So then you can think about this here, if you have that. And then you get three phases coming together. Interior points here. Closest up, you get sort of this sector here and in front. Now, so it's a fact that the degeneracy index is precisely the number of phases you live in. If an interior point up is now phase 0, honest boundary point, one phase. Simple corner point, you line two phases. You can have four phases, though, in this picture, potentially if you have four lines coming together. Yeah, but that wouldn't be allowed in that here. But I don't allow that in my structure here, because I have boundaries corners always. But we had a discussion of this last week. You could rather than taking quadrants, partial quadrants, I think you could replace it by convex sets with non-empty interior. So then you could generalize this somewhat. So in particular, you could have polygonal boundaries and so on. What do you mean with right arrow S sub A? Right arrow, red stuff. There's not S sub A. Given a connected component, there's an associated subcategory. Namely, all objects whose isomorphism class belongs to A. Also given a phase, there's an associated. Yes, then given a phase, there's an associated category. So if the phase lies in A, it's a subcategory of S A. So if this here is A, then this could be an S theta, which is a subcategory of that thing, which would be the closed thing here. And this awful Z sub P, can you maybe say how is the realization of S sub P connected to the realization of S? Well, so locally, because it sort of goes in this discrete category, it is a product of things in S. Once you fix your arrow. So Z sub P is something like a fiber product of Z? Yes, so Z sub P is a union of fiber products starting with at least two factors. And this also has, you just forgot the market. I mean, it has a topology as well. So let's say I have a building of height 5 and Z. It sort of happens, I don't know, 25 times in Z sub P. Yes, because you could shop actually five times, or you could shop at different places. Just shopping once, or can I shop several times? Well, you can shop several times, and it lies in a longer product. So I could take a building of length 25, and I'd shop it into five buildings of length 5, of height 5. Of 4, actually, yeah. So that is where all the compatibility comes from. That when you see a stable thing, you could shop it in different ways, which lie in some sense below and were perturbed. But when you take the different combinations back, it should be the same perturbation on the next level. So I don't understand the second to last line, because if I take a teardrop in R2 with the right angle at the origin, then the origin is the genesis 2, but it's only in one phase. Are you demanding phase structure? What is the phase structure? Are you demanding phase structure? Oh, here, OK, so not at this point, so here what could happen is, I think that is what I want to say, if I have something like this, then that would be the phase here. But OK, so at this point. But what will turn out so locally, so ah, OK, what I said before, the degeneracy index is the number of phases you lie in, is only true with local phases. So here is actually one phase, but locally there are actually two. So thanks. So your S sub theta then does not really have the structure of it? No, no, no. So then it turns out that that picture actually never arises for stable maps. And also, so your S sub A there, does that contain broken buildings? Oh, yeah. So it contains, so if you take any unbroken building, you take sort of all the limits you can take. All right, so it's not just the interior, you mean? No, no. So that is actually where the whole structure comes from, that an S A has boundaries connected, which consists out of broken things. And an S B has also this, but they have sort of common things also. That is what intertwines all things. You might have a disk with two more points in the boundary. And this one can fly into this one from this side, or it can go around and fly into it from that side. That sounds like it's that picture. Yeah. But you said that picture doesn't. No, so it does not happen for stable maps. I mean, in this case, for the closest, in the relative case, I don't know, but here in this case. You've got no boundary function. Well, there's no boundary. So now, in our particular case, actually, the faces containing a point actually are ordered. And you can see it like this. If this is a building, and these are the periodic orbits, the interface period, so this stands for the top period, for the collection of top periodic orbits, bottom periodic orbits, first interface, second interface, third interface. So then, this here, things which contain this, this is a face, and the closure of this. So if you have this stuff degenerate, it might degenerate into something like this. Then if you break a little bit higher, it's the same, and you break here the same. Because in our theory, there's more energy associated to this, or there's different complexity if I go up, if I have to. So I measure if I have branched cylinders and so on, there's some complexity, also some energy associated with this. So here, the closures of these things are the faces. They all contain this thing here. And I order them in such a way that if I break at the lowest thing, it's the first face. If I break higher, it's the second, and so on. So if I take the closure of this, it's smaller than that face, it's smaller than that face. So they're all of the same energy. No, they're all in the same space where that lies. I mean, there are different spaces, but they intersect all in that space. But a face, in our case, is you just give outgoing, ingoing, and where it should break. This set of time you're describing, that's a face. Right, so you said something about energy being different. Yes, this breaks higher, so this thing has more energy here. Of the lowest piece, the lowest piece has more energy. The lowest piece has more energy. I see, but those are all co-dimension one faces. Yes, they all have co-dimension one. Yeah, so you have an ordering on this stuff. OK, so start now with a face, and then look at the map F for the interior. So on the interior of this face, of course, it's not interior in the whole space, these are the unbroken things. So you can actually use the functor F00, which we introduced in the first lecture, to go from here to here, to cut this thing, which has one interface, precisely, at the interface. So you look at building of high 2, if you're in S theta 0. This is degeneracy 1. This means each thing has height 2, so top floor is 1, number is 1. And you can only chop it at a one point in the middle. So and the chopping was the F00, so get this. And then the theorem is, so you study the topology, and so on, this map is continuous on orbit space. So if you pass the orbit space, this is continuous. And it has a continuous extension to the closure. Chief, can you maybe, in the big picture, I want to define, say, flow theory and maybe prove D squared equal to 0. Tell me what we're about to try and do here. Well, whatever theory comes from the interplay of the phi, I mean, which I explained, and I think I precisely explained in the first lecture, the covering functors associated between the boundary of S and the fiber product. So I'm using now them and show that they actually line up continuously by creating this kind of thing. So it's a continuation of this discussion by putting now the topology in the game. So why would I care if all I want to do is define flow theory? Well, if there wouldn't be no continuity, you wouldn't define anything. So what does the continuity, where does that come in? Wait for a little while. Yeah, so how, if I would be on a manifold and they would just consist out of points, I would not be talking about curvature. So I have to have a notion of topology and smoothness to be able to talk about it. So I'm explaining to you now that these covering functors actually are continuous. F0 is continuous on orbit space. It has a continuous extension, I think. That says that you can extend, but the inch. Something to do with D squared equal to 0. Yeah, I thought that is why we are here. I thought I was implicitly understood. But how exactly? I'm in lectures number two. But what are you trying to add? OK, but I explain to you precisely that among these conditions, algebraically, d theta equals f star theta p. That is where the stuff comes from. So that's what you want to achieve with this? No, this we already know algebraically. But I have to perturb this function, so I need more structure to be able to do so. So now I'm studying this relationship and show that certain things are continuous, even smooth. So that you'll be able to perturb it. Yes. OK. So this has a continuous extension. And the image of any theta is a connected component in that space, in this fiber product, in the orbit space of this fiber product. So also every connected component comes from a phase. So you see already topological data, connected components in this fiber product correspond to phases and so on. And here is an explanation for how this extension here, the continuous extension of this, comes by putting several of the previously algebraic defined things together. So if I'm on the interior of a phase, the only function of the structure which I have before, which I can apply as f00, then I go to the closure of this one broken things. And there might be many things which are very often broken. Is this the way of obvious? No, it's not that difficult, but you have to write down what the topology is and look at the pictures and you see this. So somehow this is just, if I think about FLIR theory, this is about putting, viewing boundary of trajectory spaces as sequences of broken things. Well, if you just would look at zero one-dimensional components or zero, then it's all trivial. So I'm discussing how it's a higher-dimensional model of spaces. This is the game where I have a three times broken thing I can view. So if the three times broken thing lies in the closure of a specific phase, so these are of one times broken thing which broke over certain things, then I have a continuous extension of that function on orbit space over this. And the extension depends from which phase I come. That is where the algebra comes from. So if I extend that phase, there's a particular extension and if I do this. So here, so if you look at S theta, so on S theta zero, it's just a functor F00. But on the boundary, it actually looks like this. These are the algebraic guys which we saw. So this continuous extension can be represented as a functor and pieced together from the Fij's which I had. So that means, so the Fij's were only defined on certain strata, namely on things which were i plus j plus one times at this height. Can you remind us what Fij was? Yes, so Fij was just forgetting at the right spot the asymptotic markers. Where is something non-trivial happening in this theorem? You're taking things that you knew were continuous. You're putting them together by algebraic operations and then you're forgetting a finite amount of data. So the first time, I just was talking about the algebra of things. Now here, and this was in some sense like Fij was defined on a completely different strata of the category than something else. Yeah, so algebraic strata. So now what this says is actually you can piece them continuously together. That is what this says. And the interesting thing is that this functor now, which goes from the category of phases to this fiber product with two factors over the component associated. So the image of theta is a connected component. So this functor here is an SC smooth covering functor. So since I have a smooth structure now, I have a smooth structure for the fiber product. And I can look at, then it turns out that this induces a smooth structure on this as well. And then this thing is actually a smooth functor. And you can imagine how a smooth functor is almost like discontinuity. So it means you can, if I have an object here and an object here which projects on it, I can take two things so that that functor actually maps this slice into this slice and the local representation is smooth. I'd say that we have a wonderful candidate for your explanation this evening. You can give us an example of this where you have a very simple example. The local direction would say, sweet breaking or something. Show us what the app, the content is. It's actually very obscure to me. I have to say. I mean, I believe the content, I trust you, to that extent. But where it's hidden is... Well, it just says that this family of functors define on different strata. Just a line up to something continuous on this thing and something smooth. OK, good. OK, here is a picture to explain that to you. It's really the end of this. Yes, OK. So consider here this object. So this stands for periodic orbits here and here. And it breaks twice. OK, so if I look at this kind of thing here which would not break here, that would be in a phase. And if I would look at this thing here and it would break here, it would be another phase. And this red stuff lies in the intersection of these two phases. So if it takes a closure of things like this, this clearly lies in there. And if it takes a closure of this, that lies in there as well. So it lies here. So now you see already that three different connected components in orbit space come into play. This one, that one, and that one, so A, B, and C. Now, you also have this one comes into play, which is Q, a connected component Q. And this one comes into play, which is a connected component P. And all this stuff might be in a connected component E. Is Q equal to sigma 1? No, but sigma 1 of the sigmas is a phase in this. So these are all connected components. E, Q, P, C, B, and A. So now if you look now at the category theta 1 intersect with theta 2, that's the category associated to the intersection of the phases. And you take this function of theta 1, then it would. So what would it do? It would just go, it would preserve this here, and it would chop up this one. So you go into SA, then you get this thing here. But this thing here is actually a phase in S. This is a phase in SQ. So if this is Q here, then this one would lie in Q, but it would be broken one. So it would be a phase in SQ. So this object lies here in the intersection. What is sigma 2 equal to Q, right? No, it's a phase. Phase. It's a phase. So sigma 2, so this thing here lies in a phase of sigma Q called S sigma 2. What is the difference between sigma 2 and Q? It's a phase. Sigma 2 is a phase in Q. And Q is a point in that phase. But there, A is a point. No, no, no, no, no. So what I would describe here, so the connected component like this is C. The connected component containing this is called B. And the connected A. The connected component calling this is Q. The connected component containing this one is E and so on. So now, if I have an object in here, these are this type of broken things. So now, f theta 1, that is the lowest one, would forget data here. Would forget. I have equally confused this guy. What does sigma 2 mean? So sigma 2 is a phase. S2. So it consists of things which lie in C and B, but have gluing data at B. That's what it consists of. And the things which are in C to 1 and C to 2 are twice broken, so they have things in A, B, C, and you remain, keep all the gluing data. And what he's talking about is f C to 1 is he forgets the gluing data. So here, you would take such an object, forget the gluing data here. Then you get this thing here which lies in the connected component A, and you get this broken thing here lying in the connected component Q. But since it's broken, it has to be lying in a phase. So now, if I do the same here, forget this here. That would be going down here. I would get this thing here lying in a phase sigma 1 of P. And I would get something in the connected component C. Then, of course, I can chop again. The things, and then I'm ending up in the triple-fibre product. And that's commutative. And these are all smooth-covering functions. They're all smooth, and it's commutative when they're smooth-covering. And they're smooth-covering functions. That's the important thing. So these are actually, for the smooth structure, they are smooth objects that are covering functions. And then when you pull back data, it will be smooth and so on. That is how the whole stuff starts. So ultimately, the geometric input there is smooth to the C gluing map? What is the? So, of course, algebraically, it's sort of obvious already. It's sort of something covering. But that is actually compatible with the gluing and so on. And it's nicely compatible with our notion of smoothness. And then, actually, I'm done. I'm done. You see? OK, so I was actually pretty good. So in my estimate. Look, why don't we start? I think we have enough of that. Why don't you start with this slide? You see, there's something like sublime would be happening if I just say this. It's just a summary. So what we want to see is our deformations, we want to have smooth functors in the sense how I explained it. They should satisfy this identity. And this identity here is the algebraic identity reorganized. But now all the ingredients you see here are smooth functors. And again, that is what I've already found in the first lecture, is we want to find such things which are smooth, satisfying this identity and having all these properties. And these things will be constructed as a solution of a first-time problem. Thank you.