So, recap. Okay, so this was a category of stable maps, so it comes with a grading. It's the number of floors minus one, so European notation for the top floor.

Then, there are stable maps that have outgoing, in-going ends, so we have two kind of evaluation maps, which are functors. Then, having this structure here, we can build a weak fiber product.

Then, there are this sort of chopping functors, if you take a building with top floor i plus j plus one, then you can chop it at a particular place, indicated by these two numbers, and you get two buildings connected by morphism in P.

Then, trivial cylinder buildings, that we 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 zero and one, then zero, you wouldn't take this object, and one, you would. So, then, in this category, we can talk about pseudo-holomorphic objects, and that was sort of the functor, which takes value zero, if it's not pseudo-holomorphic, and one, 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 associate subcategory, where we sort of, this thing we are going

to work with, but it should also have additional structures, which we will introduce during these lectures. So, that was one of the properties, which we wanted perturbations to have, and this is satisfied in particular for this one. So, this was so far algebraic, there was nothing, no 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 is actually a cobordism between one choice and the other. And, it's independent of these choices as well.

So, I don't know how I should call this thing, so GCT is easy to say. GCT stands for topology. So, it consists out of a topological 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, so we have only isomorphisms, and I require that the isomorphism classes, so if I identify isomorphic objects, that could 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 at, 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, just allow nodes, I get a category of stable maps for Gromov-Witten. Then you have pseudo-holomorphic objects, and then you take the clearance class of pseudo-holomorphic 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-holomorphic. 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 puncture, H3 locks in the sublapse space, H3 away from the punctures. At the punctures, if you introduce the nodes, introduce polar coordinates, you want that in these polar coordinates, it's H3 with weights, so that's the quality, then you can do analysis.

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 the 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, suppose I would have taken 1 billion 1, then you would ask me why do you take 1 billion 1.

So, I can take everything greater than 3, so first of all I want to work in a 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 the smooth structure, you have to actually divide out by group and have to take some constraints. So H3 is 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 is a natural metrizable topology.

So, we don't have any choices, that is, give a recipe how to construct the basis of the topology, of course, a 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 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 variables or functions I take. What is the word orbit space?

Isomorphism classes, which is a set. 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 what we define for SFT, with several authors.

Okay. And if you replace this category by stable maps in the Gomov-Witten context, then it would be sort of the conservative version, which is Hausdorff, so that would be precisely the topology, induced by the natural topology which you have there.

How important is your theory, is it that you have finite stabilizers? Not really, but it makes sense. So, actually, when we started this project, we had a lot of energy. So, we wrote a lot of pages about that we could have compact Lie groups as stabilizers.

So, that definitely, you can generalize the theory to this. And then, during the course of the thing, I thought I might even have infinite dimension groups. So, SC smooth groups, but for this I don't know a good application yet, but the language is extremely flexible.

But with compact Lie groups you could see if you have other group actions on the space and so on, this kind of stuff might arise. Okay. So, now. So, this we already introduced last week.

So, assume this is a 10M polyfold, so this is sort of the next thing to a manifold with boundary and corners, or Banach manifold with boundary and corners. Suppose we have a finite group, a group homomorphism, so that we have an action by diffeomorphisms and then the associated translation group,

and the associated point is a category where the objects are actually these points here, and where the 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 a 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 the point obtained under the action. So, are you saying you think you could replace G by a Lie group without requiring that sort of…

I mean, so what would be the theory here if you have a Lie group? So, in this category, you usually would look at what are interesting functors.

They're functors, you go into the group and then from this thing you could go by morphisms somewhere. Let's not go into this. I'm already running out of time here. Is it just a re-packaging of the data of the group action model? Yeah, yeah. But just viewed as a category. I want to bring it on a footing that I can compare it now with my category.

So, a uniformizer now. So, this is now an abstract category with similar properties like stable maps. So, a groupoidal with a metrizable topology on the orbit space on the set of isomorphism classes.

So, a uniformizer at the object alpha with isotropy G is a functor from this groupoid, from this category, from such a category which looks like this, which has the following. So, it's injective on objects and fully faithful. So, that means injective on objects is clear. Fully faithful means that if you have two objects

then the morphisms there go to the morphisms set of their image points bijectively. Then if you assume that there is 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 homomorphism 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 these requirements. So, here's the picture. So, we have this enormous category here, which is not a set, but assumes 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 as isotropy.

Then here is the orbit space, which is a topological space. You have this sense thing here, which is the isomorphism classes associated to this thing here. And you map this here into this category, and so the isomorphisms go 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... So, you 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 manifolds, if I have a subcategory and it hits my hand in manifolds, that's already some structure. If I put this hand in, I see also manifolds, and if there's 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're saying. Okay, good. So, thank you. 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 and it's 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 an 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 a topological subspace, they're compact, and from the smooth structure here 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, actually, in the approach here, you work on a level where 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 modelized spaces,

where at each point you have to work very hard because you're on the wrong level. 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 tracks and we have not yet. Yes, I have not explained yet.

Of course, I could say, if I have a sub-category 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 a weak firewall product, which is the following. You go from O, you go into this category, you 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 points. Why do you call this weak fiber product? Because of this here. It's a static fiber product. As usual, it's a usual fiber product. I think you're a topologist. If you're used to...

It's a usual fiber product, but if you're a topologist... I mean, sometimes the real fiber product is also a curve. So, weak fiber product, we're in categories. I'm coming from a different planet, so I can't say. 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 Lie group holds or so. Target map. Then there is a map which associates to an object here. Just this tuple here, where this is the one map,

the one isomorphism at the image path Q. You have an inversion map. Just interchange the row here and just replace this by the inverse. Then you build here. This is the real fiber product without morphisms in it. 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. Multiplication map. So, this is nothing deep. It's just the only interesting things you can do. Maps above are called structure maps.

So, now, here is what a polyfold structure for this is. So, in real terms, the following. So, with all these kind of categories which appear in real life, the objects in this category usually are geometric objects and they have enough geometry to build something.

Like, if you do the Lin-Mamford theory, you could look at the category of stable Riemannian surfaces and you just take one object and then by cutting and pasting, gluing, you can actually construct a neighborhood in the isomorphism classes. So, the point is that these categories, which occur in symplectic geometry, for example, but are 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 polyfold structure. So, there would be a functor from C into category of sets, which associates to an object a set of such uniformizers. So, for each object, there's a construction, I get a set of such uniformizers.

Then, if you have an 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 sort of an 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, in general, an application like stable maps and everything, this bijection can be very precisely explained. And then there's a construction which associates to two such uniformizers,

a tame and polyfold structure on this transition set, so that these things are true. Like, the source and target map are local diffeomorphisms, the unit map is smooth, the inversion map is an SC diffeomorphism, and multiplication is smooth. So, think of this as a maximal recipe atlas for this category.

So, for Gromov-Witten, 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... Why is iota a skeleton morphism, not a local skeleton morphism?

Because it's an inverse i. So, this is just 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?

That just means if you start moving this point here, the source would move and this end point 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 identify 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 a 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, to the 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, 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.

Actually, if your category... So, 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 have to check it for an isomorphism class of an object, for the functoriality,

if it seems smooth. So, you see already, 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, or also important. There's a degeneracy functor. So, if you have this smooth structure on this category, this polyphore 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-polyphore. Okay. Now, let's talk about the smooth versions of subcategories. So, here is a nice looking polyphore,

it's called your 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 in transversally at the boundary. Good position. Well, that still is okay here to come out like this,

but of course, it's not general position because this one dimension should hit another corner. But you might have stupid things like this, but they might still be retracts, so these are possibilities for submanifolds. So, now the general definition is the following.

So, if O is an m-polyphore, an m-subset, then we have heard what a subpolyphole 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 SCsmooth. What does that mean? This means the differentiability property is stronger. If you go from U to U, you have SCsmoothness. 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 like this property, then, first of all, it's called a submanifold, and for good reason. Namely, the induced structure from the m-polyphore on it

is a subpolyphore structure which, however, has an equivalent set of charts which are actually usual manifold charts. So, you mean finite dimensional? Yes, so why is it finite dimensional? So, first of all, if R goes from U into U1, U1 goes to U2, goes into U3, and so on,

and R composed with R is 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, R on a point in this thing is the identity. But, because of this one in compact inclusions, the identity is compact. The identity is compact, it 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.

So, this we already know. A weighted subcategory was this functor into zero one which associates to an object a rational 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 that's at the following hole. So, if I take an isomorphism class in C, I pick an alpha in that isomorphism class and take such a uniformizer, which map 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 one 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 zero or one, 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 this like being a constructible function? Don't ask me this, I don't know. Generally, not now.

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've 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 Stokes' Theorem and this kind of stuff. Can you explain how it could be that you have this description of theta locally but you can't globalize it to a whole category?

Well, you have these bits and pieces lying all over. Let me give you an example then you see how that would work. So, let me first give the picture. So, in local coordinates, you see all these Mi's 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 or the Mi? 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, I see the green stuff. If I stick my hand in, I see some other green stuff. And if there is something isomorphic, then this S, the fact that 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, here are some exercises and remarks. Assume you have a smooth-weighted subcategory like this and theta only assumes values 0 and 1 between two objects. And between two objects, the associated subcategory, in the associated subcategory, where theta is positive,

there is at most one morphism between any two objects. And further 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 them, you just glue this stuff together, the local pictures. So, now let's... Get rid of the boundary in that picture. Because there's degeneracy 0. Oh, degeneracy 0. Because your boundary is always...

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 these manifolds at the boundary and if they would lie in good position to the boundary, then actually you can show that this would be a manifold with boundary and corners. It's the MIs.

No, this one. What is the definition of this good condition thing? Lying in a good condition? A good condition just basically means that something like this... So the definition in general is a bit more complicated but that the tangent space doesn't lie like this. It just goes off nicely and if you wiggle a little bit it would still go nicely.

And that's the main condition you want? This is one of the possible things you can require. I mean, smoothing just means you have this smooth representation like that. But then you could start requiring all kinds of things off 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 degeneracy zero

and between any two objects there's at most one morphism and the orbit space is compact, then this thing has a natural smooth manifold structure induced from the polyfold 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 orbit fold. So with this language then two possible things happen so it describes manifolds and orbit folds and then the most general thing is that you can have rational weights and it looks a little bit wider.

So there's one overall more general notion and if you specialise 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 modalised space. So the level on which I talk is sort of streamlined

for saying the least amount necessary to construct something and then everything else you say is sort of abstract stuff you can do. So in not so long there will be actually a book about all these abstract things, construction, 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 modalised space comes up, you have to get a good recipe, if I give you an object there has to be a good recipe what to do with this object and there's a local construction on the object. And as you see then we also have to introduce bundles over this

and then we can talk about the Cauchy-Riemann functor and then when I extend the smooth notions to this we will actually have a perturbation theory for functors, for certain classes of functors. Okay, so this is already set so there's a tangent construction and then if this is oriented you have actually a formula like this.

So this is then what you see if you construct these transversal modalised spaces and you work out the orientations, you integrate forms over this. So this 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 marked point and forgetting the map and collapsing unstable structures that go into the Lilhamford 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 these 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 they might be representable then in some other way. Do you have some interpretation of what the Perron complex can 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 that must then be just ever used. And it must have to do something with so if you fix the Riemann 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 Riemann surfaces into the manifold and there's more topology. 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 but 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 fine or whatever the words are where the local description can't be globalized?

What does that even really mean? Does it not exist a finite list of subcategories? How about anything that's an immersion but not embedding?

Did that at least answer your question previously? If I have just a figure eight I can't write this as a union of embeddings? Yeah, okay, so it sounds like this. That would be an example. Which requires just local descriptions locally finite union of embeddings but isn't globally a finite union of embeddings.

So that figure eight is a thing which can occur in this thing. Sure. Oh yeah, figure eight. In the transversality results during homotopies bad things can happen. So the thetas you construct are usually not everywhere smooth but they can have singularities and singularity sets

which you have to throw away and which you can distinguish and so on. So theory has a natural metrizable topology we already knew this, turning this and then has a natural polyfold structure after fixing a, 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. So what is S and what is SP? SP? S is the category of single maps and S and P is the weak fiber product over the whole of this. So then S and P is the weak fiber product which we introduced last time

to say how the thetas 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 then the nice thing is of course some kind of reality check. The degeneracy index from the polyfold structure agrees with the grading functor. So when we did the algebra the grading functor was the height

of the building minus one. So now if I have a smooth structure I have a grading functor 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 by zero. We can talk about connected components there where 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. The isomorphism class lies in A. Then there are faces actually all the combinatorics now comes from this in SFT. So the face you take a connected component in that topological space

in the subspace where d equals one. So you just take honest boundary points. You take buildings of height two so top floor is one and you take the closure of this. So here is an example in dimension

in differential geometry. So if I have this here the square then I would have to take this point first away then this here would be a connected component of d equals one honest boundary points. Then I take the closure then I get that including these points.

Here is another example. Here is another connected component takes a closure I get that point so I get four faces 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 and then you get three faces coming together the interior points here closes up you get sort of this sector here and in front. So it's a fact that the degeneracy

the degeneracy index is precisely the number of faces you live in. The interior point of it is now face zero. Honest boundary point one face. Simple corner point you line two faces and so on. You can have four faces though

in this picture potentially if you have four lines coming together. Yeah but that wouldn't be allowed so in that theory but it would I don't allow that in my structure here because I have bound obvious corners always. But we had a discussion of this last week. You could rather than taking quadrants partial quadrants you could I think you could replace it by convex sets

with non-empty interior. So then you could generalize it 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. I'm also given a face there's an associated Yes then given a face there's an associated category.

So if the face 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 of the subcategory of that thing which would be the closed thing here. And this awful Z sub P

Can you maybe say what how is the realization of S sub P connected to the realization of S? Well it's so each so locally because it sort of goes in this discrete category it is a product of

of things in S. Once you fix your arrow So Z sub P is something like a fiber product of Z? Yes so C sub P is a union of fiber products starting with at least two factors. And this also has

you know 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 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 So that is where all the compatibility comes from and if 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 think like the teardrop in R2 with the right angle at the origin

then that right angle then the origin is the genocid of 2 but it's only in one phase. Very demanding phase structure. What is the phase structure? Very demanding phase structure. Oh, here 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. So at this point but what will turn out so locally

so 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 your S sub theta

then does not really have the structure of 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 if you take any unbroken building you take sort of all the limits you can take. 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 which consist out of broken things

and S B has also this but they have sort of common things also that is what intertwines the whole thing. So you have a disc 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. You said that picture is not something. So it does not happen for stable maps

I mean in this case for the close in the relative case I don't know but here in this case you have no boundary functions I think. 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 this are the periodic orbits, the interface periodic orbits so this stands for the top periodic for the collection of top periodic orbits bottom periodic orbits first interface, second interface, third interface. So then this here

sinks 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 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 so I measure if I have branched cylinders and so on there's some complexity also 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 they're different spaces

in our case. But a face in our case is you just give outgoing, in-going and where it should break. This set of periodic orbits prescribed, that's a face. Right, so you said something about energy being different. Yes, this breaks higher

so this thing has more energy here. The lowest piece has more energy. I see. Those are all co-dimensional one faces. Yes, they all have co-dimensional one. So you have an ordering on this stuff.

Okay, 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 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 height two if you are in s theta zero this is degeneracy one

this means each thing has height two so top floor is one number is one and you can only chop it at one point in the middle and the chopping was the f00 so you 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 fluid theory and maybe prove d squared equal to zero tell me what we're about to try and do here Well, whatever theory comes from the interplay of

I mean which I explained in I think I precisely explained in the first lecture the covering functors associated between the boundary of s and the fibre product so I'm using now them and show that they actually line up continuously

by doing this 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 fluid theory well, if there would be no continuity you wouldn't define anything so where does it continue where does it come in wait for a little while

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 I'm telling it to you now that these covering functors actually are continuous

f0 is continuous on orbit space it has a continuous extension that says that you can extend but the something to do with d squared equal to zero yeah, I thought that is why we are here I thought that was implicitly understood but how exactly

I'm in lectures number two but what are you trying to okay, 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 functor 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, yes okay so this has a continuous extension and so

and the image of a connected 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 corresponds to phases and so on but here is an explanation for how this extension this extension here the continuous extension of this comes by by putting several of the previously algebraic

defined things together so if I'm on the if I'm on the interior of a phase the only functor of the structure which I have before which I can apply is f00 then I go to the closure which is one broken thing and there might be many things which are very often broken

is this the end of obvious or is this no, it's not that difficult but you have to write down with the topologies and look at the pictures and you see this so somehow this is just if I think about theory this is about putting viewing boundary of trajectory spaces as sequences of broken things

and if I look at one dimensional components then it's all trivial so I'm discussing how it's a higher dimensional modelized space so if the three times broken thing

lies in the closure of a specific phase so these are of one times broken thing which break over certain things then I have a continuous extension of that functor 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 comes from the fij's which I had 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 this was just for getting just for getting at the right spot the asymptotic markers where is something non-trivial

happening in this thing? so this continuous you're putting them together by algebraic operations and then you're forgetting to find out about the data so the first time I just was talking about the algebra of things and this was in some sense like fij was defined

on a completely different strata of the category than something else so algebraic strata so now what this says is that this functor now which goes from category phases two

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 think 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, a flow trajectory, let's say, sweet breaking or something. Show us what the content is.

It's actually very, it's 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 defined on different stratas

just align up to something continuous on the thing and something smooth. Okay, good. Okay, here is a picture to explain that to you. Okay, yes. So, consider here this object. So it has periodic, this stands for periodic orbits here and here,

and it breaks twice. Okay, so this, so if I look at this kind of thing here, which would not break here, that would be in a face. And if I would look at this thing here and it would break here, it would be another face. And this red stuff lies in the closure, it lies in the intersection of these two faces.

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 you have A, B, and C.

Now, you also have, this one comes into play, which is Q, the connected component Q, and this one comes into play, which is the connected component P. And all this stuff might be the connected component E. Is Q equal to sigma one? No, but sigma, one of the sigmas is a face.

So these are all connected components. E, Q, P, C, B, and A. So now, if you look now at the categories theta one intersect with theta two, that's the category associated to the intersection of the faces. And you take this function f theta one, 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 face in S, this is a face 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 face in SQ. Yeah, so if I look, so this object lies here in the intersection. What is sigma two equal to Q, right?

No, it's a face. Face. It's a face. So sigma two, sigma, so this thing here, this thing here lies in a face of sigma Q called s sigma two. What is the difference between sigma two and Q? It's a face. Sigma two is a face in Q.

And Q is a point in that face. But there, A is a point. No, no, no, no, no, no, no. No, this, so what I've got to 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, if I have an object in here, this type of broken things. So now, F theta one, that is the lowest one, would forget data here. Would forget. I'm equally confused as captain. What does sigma two mean? So sigma. Sigma two is a face in Q.

Of SQ. So it consists of things which go, which, things which lie in C and B but have gluing data in B. That's what it consists of. And the things which are in C to one intersect theta two 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 theta one is he forgets the gluing data at. Yeah, 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 face.

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 face, sigma one 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 fiber product and that's commutative. And these are all smooth covering functors. They're all smooth, it's a commutative and they're smooth covering functors. That's the important thing. So these are actually, for the smooth structure, they are smooth objects, they're covering functors 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 for the gluing map? What is the? Yeah, so of course, I mean, 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 nicely compatible with our notion of smoothness.

And then, actually I'm done, I'm done, you'll see. Okay, so I was actually pretty good in my estimate. Why don't we start, I think we have enough for now. Why don't you start by proceeding 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 we 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 photon problem. Thank you.