 Okay, thank you. The main object yesterday was the cobaltism category. And the main thing I discussed was if D is a groupoid, we started functions from the cobaltism category into D, the category of functions, and argued that precomposition is a canonical function from any small category to the fundamental groupoid of its classifying space. Precomposition with that gives an equivalent of categories from functions out of pi1 b cop d into D. I didn't emphasize that so much, but this is also equivalent. If I have a function, I can take b of it and get a map from b cop d into b d. This gives an equivalent of category from the function category to the fundamental groupoid of the mapping space. Natural transformation goes to a homotopy. And also, if I have a two-connected map, b cop d, then we could also precompose with that. Pi1 x into D. First, I could go this way. And similarly, I could replace this mapping space with maps out of x. That's a quick summary of what. So pi1 means the fundamental groupoid. That's my notation if I don't put a base point. So it's a groupoid. And the set of isomorphism classes is then pi0. And the automorphism group of an object is the usual fundamental group. Today, I'll talk about how to lift the cobaltism category. It's an object in the category of small categories. How to lift that to a topologically enriched category. I don't know what the standard notation is. I'm going to call it g-cat, category of topologically enriched. I'll say what that is in a moment. And there's going to be an object here. There's going to be a construction that associates an ordinary category to a topologically enriched one. And there's going to be something here that goes to the cobaltism category, as I defined it yesterday. So a topologically enriched category is just a category where all the morphism sets are equipped with a topology in a way that's compatible with composition. So it still has only a set of objects. But for every pair of objects, there's a space of morphisms. And the obvious condition, composition C, X, Y, C, Y, Z. I compose, I get into C, X, Z. This should be continuous for all the X, Y, and Z in the set of objects where the source is given the product topology. And a functor of topologically enriched categories, let's call it topologically enriched functor. Also sometimes called a continuous functor. It's just the usual thing. And the only extra thing to say is that the induced map of morphism spaces should be continuous. So that's what the category of topologically enriched categories is. And the proposal for how to lift this to a topologically enriched category is to, there are many ways to do that. But the one I'll talk about here is embedded carbordisms. And they're going to be embedded in vector space, Euclidean space, but I'm just going to call it V. So a finite dimensional real vector space. So this category of topologically enriched category, the object is the set of subsets that are d minus one dimensional closed manifold. So the good thing about subsets as opposed to abstract sets is that being a manifold is now a property. If you have an abstract set, you say it's the underlying set of a manifold, means you're specifying all kinds of extra structure. If you have a subset of Euclidean space, being a sub-manifold is just either true or not true. So this, as a set, is a subset of the power set of V. We don't have to mumble anything about making it small. The morphisms, I can say there are ways to say it, but the definition I'll take are pairs of a t, w, where t is a positive real number. And w is a subset of the interval 0, t cross V, which is an embedded carbordism, or let me just say a carbordism. But again, it's going to be a property, not extra data. Being a carbordism for a subset means, let's see, it should be a compact. That's a property just of the topological property of the space. And then it should be a smooth sub-manifold with boundary. And sorry, this was the morphism space from M0 to M1. I'm writing. There should be a space for each pair of objects. And the condition is that there's some epsilon, so that w intersect 0 epsilon cross V is equal to 0 epsilon cross M0. And w intersect t minus epsilon comma t cross V is t minus epsilon comma t cross M1. Equals, again, is much less confusing if you have a sub thing of equal. Just means it's equal as a subset. Doesn't mean that there's some specified anything. So for example, if V is equal to R and D is 1, an object is a zero-dimensional subset of R. That's a sub-manifold. Enclosed means compact, no boundary. So that's just a finite subset. If I have another one, this direction is 0 t. This is R. I should have an embedded carbordism. That's a subset of 0 t cross R that has this property that they access some epsilon. So as a subset, it's equal to M0 cross 0 epsilon. And there's another epsilon here. So it should be like product near the boundary. So again, a property that replaces the extra structure of having this color near the boundary I had in the abstract setting. That's the set. Oh, sorry, I forgot one thing. And then the boundary of W should be equal to 0 cross M0 union t cross M1. I mean, this has to be bounded, but there can't be any more boundary than that. Again, there's a version with orientation or any other kind of structure. But if you come to the problem session, there's a question about working out how that definition is. But I'll just talk about on-oriented, but everything makes sense with any kind of tangential structure. So this is still just a category. I didn't talk about a topology. So I should explain how to topologize this set of carbordisms. Let me not do that in all detail, but sort of say where it comes from. So the Ts in R, I just use the usual topology. One from the usual metric space, Euclidean topology on R. These manifolds, you can topologize the following way. If I have an abstract carbordism, you can take the embedding space of that abstract carbordism into 0, 1 cross V. An abstract carbordism with these C in and C out from last time. Then you can take embeddings. So if this goes from M0 to M1, you can take embeddings which near the boundary sort of is compatible with this product thing here. I'm going to call that embeddings boundary. And then I can write the map that sends an embedding. To its image. Let me not write out, you have to say, bunch of epsilon and so on, to say this boundary condition, but you set it up so that this precisely gives a point there. And then you also take a T in R bigger than 0. And you take T comma this, except you stretch it in the R direction. The notes I call it lambda T. The obvious defiant morphism from 0, 1 cross V to 0, T cross V. That sends S comma V to T S comma V. And then take lambda T of that. So for each abstract carbordism, this is not subjective because the image is always defiant morphic to the W, but it sort of precisely surjects onto the embedded carbordisms that are defiant morphic to the given abstract one. And then we give CDV, M0, M1, the cross-interpolarity from, OK, shouldn't just fix W because this is not subjective, and the cross-interpolarity is going to be a little bit weird. But just take the all possible Ws. So if you take the disjunction union of all possible Ws, you get something that surjects to this set of embedded things. And then, oh, I didn't say that. The embedding space has a good topology. That's the point of this discussion, namely the so-called c-infinity topology. If you, for example, have a sequence of embeddings, you declare that it converges to some other embedding, if and only if it should converge uniformly, but also all derivatives should converge uniformly on compact subsets of any charge, something like that. So OK, the intuitive idea is that if I wiggle this a little bit, it's going to be kind of close to the other one. But if I wiggle it a lot, it's going to be less close. I mean, I'm not saying it's a metric. So I only want a topology. And secretly, I only really want, I mean, this is sort of a model for a homotopy type. So who cares? In other words, I also forgot how analysis works, but I don't think it's an important question right here. Good point. Thanks for reminding me. There are no identities the way I said it here. So this defines a non-unicycle category. So OK, you can imagine what the definition is. Just don't have that data and don't have the condition that. You could put it in as the limit of sort of cylinders that get smaller and smaller, but let's just say that it's a non-unicycle category. So in my diagram, as you'd have put non-unicycle topological inverse categories. Let me say a few words about the homotopy type of the morphism spaces. Again, slightly sketchy. But this has sort of an interesting homotopy type that has some geometric meaning. Namely, it's a kind of classifying space. There's a good interpretation of what it means to map into one of these, from a smooth manifold, at least. Let me first say if, OK, there's some technicalities about this t-stuff. Let me first ignore that and discuss the related question of if you don't have comportism, but if you have a closed manifold. And then let me say something about closed manifolds. Then you can again give the embedding space of w into b, the c-infinity topology. And then you can take the quotient map to embeddings of w into b, quotient by the action of the diffeomorphism group of w, which acts by pre-composing. So this one has the quotient topology. As a set, embeddings modular diffeomorphism, you can again just identify, I mean, two things are related if and only if they have the same image to embeddings because, so you can say this is the set of subsets of v that are diffeomorphic to w. I mean, that has smooth closed, smooth manifolds having that property and being diffeomorphic to w, because that means being the image of an embedding from w into v, which is unique up to diffeomorphism of w. So that's kind of similar to these morphism spaces, except I don't have the added complication of these t's. So let me just tell you how to interpret this space here up to homotopy. Namely, it's a classifying space for embedded bundles. So if x is a smooth manifold, let's say without boundary, but not necessarily compact. A good source of maps from x into this space is to start with a sub-manifold of x cross v with a certain property, namely that if I take the inclusion and then the projection, then first of all, e should be a sub-manifold. Then it makes sense for, OK, then pi is going to be a smooth map, just because this is smooth and that's smooth. Then it should be a submersion. I mean, that's the requirement. The differential should be search-active at each point. And then it should be proper as a map. Inverse image of compact is compact. That implies that the inverse image of each point is going to be smooth manifold by the inverse function theorem or something. It's going to be a smooth sub-manifold here, but it's really going to be x cross a smooth sub-manifold of v. And then let's also put the requirement that for all x, this inverse image is diffeomorphic to w, meaning access to diffeomorphism. If I have such an e, let's call that an embedded w bundle over x. It's again just a subset with a property. I can write down a map of sets from x into embeddings of w into v, quotient by diffeomorph w, using this identification, which was really a bijection, I suppose, for each x, just pick an embedding whose image is this thing. And I'm not making any choices because I question out by this thing. So this is a function that only depends on e, and that's going to be continuous in this topology. And moreover, any continuous map is going to be homotopic to one that arises this way. And we get a, it's like a grass manion, or that models vector bundles, homotopic classes of maps into embeddings w, comma v, what diffeomorph w. By using this construction, it's a natural bijection when x is a smooth manifold to the set of embedded w bundles up to an equivalence relation called concordance. I wrote that in the notes, let me just say it. Concordance is a smooth w bundle over embedded w bundle over x cross r, whose restriction to x cross 0 is one thing, and to x cross 1 is another thing, and then you declare those two things concordant. So that obviously gives some homotopic maps. But so it's a classifying space for certain bundles, and it's supposed to be quite similar to the way that grass manions classify vector bundles. This is like a non-linear grass manion. So using the same ideas, there's an interpretation of all these home spaces, at least the homotopy type, that smooth maps into it up to homotopy are the same thing as embedded bundles of cobaltisms with experiments of families of cobaltisms between M0 and M1. OK, so that's a space you might be interested in, whether or not, I mean, that's an independently interesting space. If you don't like this finite dimensional v, if you're the kind of person who wants to make the most universal things possible, you could take the co-limit. You could pick some infinite dimensional space, take the co-limit over a finite dimensional subspaces, call that cd, and the object should take co-limit of sets, and morphism should take co-limit of spaces. So that's sort of, OK, I'm not going to say much about that, but if you do that, the morphism spaces have a different description up to homotopy. You can write them as the so-called classifying space of the diffeomorphism group, overall abstract cobaltisms from M0 to M1, and take the diffeomorphism group, take the diffeomorphisms that are the identity near the boundary, take the classifying space, and take the disjunction on a wall w. So that's because embeddings into 0, 1 cross r infinity is contractable, and the projection map to embeddings mod diff is a principal bundle. So that's one definition of what the classifying space of a topological group is. So, OK, obviously not being very precise here, but that's sort of the infinite dimensional version of this discussion. Again, a space you might be interested in, whether or not you're interested in field theories. OK, so that was this particular topologically enriched category. Now I had this little diagram. Let me get back to that. Watch the connection to define something here. But I did not talk about how it's related to that thing. So that's a very simple construction. If you have a topologically enriched category, there are several ways of getting just an ordinary category. For example, you could forget the topology, but that's kind of a weird thing to do. It's better to take pi 0 of all the morphism spaces. So let me talk about that sort of abstractly for the moment. So if I have a topologically enriched category, I can associate an ordinary category. The definition is just it has the same object set, but the morphism set from x to y is pi 0 of the morphism space in C. So identify two morphisms if they're connected by a path. And then composition is just induced from the one in C, of course. This is a way of getting is almost left adjoint to something going the other way. Namely, if you have an ordinary category, you can give the notes I call that iota. If you only have a set of morphisms, you can just give that the discrete topology. But if you have a C there and a D there, and you ask for topologically enriched functions from C into iota D, it's almost true that this is the same thing as ordinary functions from H, C, into D. The function means on the objects you're specifying the same data. Each object there should go to an object there.