 Thank you. Thanks for the kind introduction. So today's lecture was gonna be about cobaltisms and cobaltism categories. I hope it will be quite elementary. So let's start from the definitions. I think cobaltisms have been mentioned earlier in this summer school, but let me just tell you what I mean, so it's the usual picture. There's an incoming and an outgoing boundary, and then the cobaltism itself. So just to emphasize what's part of the data, cobaltism from M0 to M1. So these are, should be closed, D minus one manifolds. Closed in these lectures means compact and no boundary, not necessarily connected. Cobaltism is a smooth manifold, smooth compact D manifold, and then the boundary should be identified with M0 disjoint union M1, and the identification is part of the data. For technical reasons, it's better to have a little color around the boundaries. So let's say, let's write C in for the incoming color. So zero infinity cross M0 into W, and C out for the outgoing boundary. So two smooth embeddings with a property that the boundary of W is what it should be, C out, and these should have disjoint image. So the official definition is these three pieces of data, this one, this one, and that one. By the way, I was promised that the notes for this talk should be up on the app, I haven't checked it, but like as of 20 minutes ago or so, they ought to be there. So all of what I say should be in there. So it's this triple, W, C in, C out. A more classical notion is being co-bordant, M0 and M1 are co-bordant if there exists a co-bordant. And just to say where this whole theory came from, in case you hadn't heard it, this is a famous paper of René Tom, defined notation of his paper, Curly N subscript D, the set of co-bordant classes of closed D manifolds. Maybe my indexing I should like D to be the dimension of the co-bordism, so we should say it like this. Then if you take the direct sum of all of those, you get a graded ring. Addition comes from disjoint union, product comes from a Cartesian product of smooth manifolds, and also two is equal to zero. The unit in this ring is the class of a point. If you take the disjoint union of that with itself, it's co-bordant to the empty set. The empty set is an allowed manifold, and an interval is a co-bordism from two to zero. And Tom's famous theorem is that this ring is isomorphic to a polynomial ring over the field with two elements. In one generator in each degree, except those that are of the form two to the k minus one. So that completely determines this question of classifying manifolds up to co-bordism. Okay, so that's where the whole thing came from. But what we'll talk about in these lectures is not so much whether things are co-bordant, but if you have some co-bordant things in how many ways are they co-bordant? So call that cop d m zero m one. Well, okay, you can't say the set of co-bordisms because that's not a set. You can say the set of diffeomorphism classes of co-bordisms from m zero to m one. This is gonna be empty if that is not co-bordant, but otherwise they're presumably more than one way to be co-bordant. For example, in this dimension two picture, this would be a different reason. Then take this up to diffeomorphism of co-bordisms, which means diffeomorphism preserving, okay, so let me say what that means. So if I have w c in, c out, and w prime, c in, c out, then I wanna declare them diffeomorphic as co-bordant, as co-bordisms if they exist at diffeomorphism, w to w prime, that preserves these colors at least kind of if you restrict up to like epsilon. So let's add phi of the color, say the incoming at t comma x is c in prime, t comma x for all t comma x in say zero epsilon cross m zero, some epsilon and the same condition at the outgoing. Of course at some point you might then go on and keep track of the reasons that diffeomorphic, co-bordisms are diffeomorphic, but right now we'll just take the set of diffeomorphism classes of co-bordisms. It's important that we don't say diffeomorphism classes of objects, but on the co-bordisms themselves, we say diffeomorphism classes. This set has some structure, namely you can compose co-bordisms, which is incidentally the proof also that being co-bordant is an equivalence relation. Again, as usual, if I have a co-bordism from here to there and another one that starts where this one ends, then I glue them together to a co-bordism from there to there. So in the notes, I write this as, okay, so I mean it's often written something like this. So if w is a co-bordism from m zero to m one, w prime from m one to m two. Okay, so I'll probably start omitting these incoming and outgoing parametrizations from the notation, but it's part of the data. So I might write just w for this dribble w c and c out, but I mean slob in notation. So this means take the disjoint union and then identify the outgoing boundary of this with the incoming boundary of that. So that's what this gluing means. So that's what you do is, so I do this on the set level. So this is what I do for the underlying tribological space. And then you have to explain what the smooth structure is and you use the colors for that to say what you do at the gluing points. So, or I mean there are many ways to say something different morphic to what I say, but what I said in the notes is you do this on the underlying tribological space. And if you want a chart around a point that's glued, you take a chart in m one and then you glue the color in this direction to the color in this direction. So if this is parameterized by u, that sits inside our d minus one, then you get something parameterized by r cross u. And that's what you do on the glute points and outside the glute points you just take a chart on either w or w prime. That kind of sits canonically inside this. And that gives you smooth manifold with boundary. Then you forget what the chart you had in the middle and then use c in and c out. And then you take that's a cabotism from m zero to m two. Call this w double prime, m s c in, m s c out, prime. Cabotism from m zero to m two. Okay, so that's how to prove that being cabotined is an equivalence relation. But it's also, I mean, it's kind of obvious that if I replace w by something that's diffimorphic as a cabotism, then the glute manifolds are again diffimorphic as cabotisms. So that's a well-defined composition map. Cobb d m zero m one, cross-cobb d m one, m two. Two-cobb d m zero, m two. Okay, it's of course important that I say diffimorphic as cabotism's here. Otherwise, this would not be well-defined. This is a well-defined map of sets. It's also associative. So this is about composing three cabotisms. If you have w, w prime, and w double prime, then the class of w glued along m one to w prime glued along m two, w double prime. This cabotism is diffimorphic as cabotisms so if you do it in the other order, you can read this, especially if you're in the back. Okay, associativity, you know what that means. Okay, let me write it. So to check this, you have to check these gluings are diffimorphic rail endpoints. They most likely are not equal cabotisms because the underlying sets are not equal, but they are diffimorphic in a canonical way. That's associativity. The last thing is you have identities for this composition, namely the diffimorphism class of say an interval is a cabotism that acts as an identity. If you have any cabotism and you glue an interval cross either incoming or outgoing boundary, you get something that's diffimorphic to what you started with. So that's the axioms for being a category. That's the cabotism category. The d-dimensional on-oriented cabotism category. Have to note a cop d. You should also say smooth. Can imagine other variants with topological manifolds and whatnot. Okay, a tiny comment, something I won't dwell on. If you know what a small category is, it means there should be a set of objects and a set of morphisms. If I really take all, okay, so the objects here will be all smooth closed d-minus one manifolds. It's probably better to say that the underlying set should be a subset of some big set so that you have a set of objects. But up to equivalence of categories, that does not matter. Okay, that's the cabotism category. There are lots of fancy versions of this where you have to talk about higher category theory and whatnot. I might briefly touch on that at some point, but not really. I think most of my lectures will be just ordinary categories with, I think, fancy. And this is the kind of most down-to-earth version of the cabotism category. The topic of these lectures is what can you say about a category, a small category, like the cabotism category, from the classifying space of the category. So a classifying space of a small category, C. So small as a set means a set of objects, you can. Okay, probably heard that before. Like the category of sets is not small because there's no set of all sets, but. So you have a small category. There's a topological space that I write BC. That's usually defined in two steps. First you take the nerve of C and then you take the geometric realization of that. Where the nerve of C is a simplicial set, the P-simplicers, we didn't heard it, NP of C. It's the set of P-tables of composable morphisms. So the target of F1 should be the source of F2 and so on. Let me abbreviate that to F1 composed with up to Fp. Exists. And zero just means the objects. And one means the set of all morphisms in C. So on. And geometric realization means disjoint union NP of C cross the standard P-simplex, modular sum equivalence relation. But okay, it's, you start with a point for each object. Then you glue on an interval like a one-simplex for each morphism, attach it to the start and end points. Whenever you have a triangle, F, G and H, you fill it out if H is equal to the composition of F and G. If H is equal to G plus F, et cetera. So G comma F, in that case, is gonna be an element in N2 of C, and that gives a two-simplex. And then you keep going, filling out higher and higher simplices. So that gives a topological space associated to the category. Eventually I of course wanna take B of the cabotism category. But let me first discuss in the abstract how much of a category can you recover if you know the classifying space. I want something like how much of a category up to equivalence of categories can you recover from knowing the classifying space up to homotopic equivalence. Something of that flavor. It's a topological space. Yeah, likewise C is not an equivalence class of categories. So okay, so this is how to turn a category into a space. In the other direction, I can turn a space into a category, namely the fundamental group point that I'm gonna denote pi one of X. So it's a category, it's actually a group point. The objects are just the points of X and the morphisms from one point to another are the homotopy classes of paths starting here and ending there. X, X, Y is paths up to homotopy rail end point. Composition is concatenation of paths which is associated up to homotopy rail end point and all paths are invertible up to homotopy rail end point. The automorphism group of a point is just the usual fundamental group based at that point but this kind of encodes. It encodes both all the fundamental groups and pi zero X into one object. Consider this as some sort of partial inverse to the other process or that kind of point of view that will be helpful in these lectures. More precisely, if I start with a small category turn it into a space and then take fundamental group point of that space and there's a canonical functor from where I started to where I ended. So by definition of BC it contains as a subspace the set of objects. So objects of C by definition is the zero simplices of BC that sits inside as a discrete subspace. So that's what this functor does on objects. Send an object there to the corresponding point in BC regarded as an object in the fundamental group point. On morphisms, so if I have F and C, X comma Y. By the way, I'm using this notation to mean the harm set in C from X to Y. The way BC is constructed, the one simplices are precisely the morphisms. So for each morphism there's a canonical path from X to Y by definition. So I send this to the corresponding one simplex. So I guess it would be, so this sets inside N one of C. So strictly speaking, the notation would be something like F cross delta one. And then you parameterize this as interval. This canonically maps to BC. The path starting at X and ending at Y. So that's what this canonical functor does on morphisms. Then to check that it's a functor, I have to check that it preserves composition. But that's what the two simplices do. If I compose these, I get that one. And this concatenation is homotopic to that one. The two simplices is what makes gamma functor preserving composition. Okay, now it's likely not an equivalence of categories. For example, this thing is always a groupoid. So if this thing is not a groupoid, then it can't be an equivalence. For example, if you have like the most trivial example of a non-groupoid category, two objects and one morphism between them, one non-identity morphism, then BC is homomorphic to just an interval. So the fundamental groupoid is gonna be trivial, whereas this is not a trivial category. But that's in some sense the only problem. There's a, okay, not prove it, but it's kind of a standard proof. If C is a groupoid, then gamma, an equivalence of categories. So no information is lost in that case. By going from C to BC, you can get C back up to equivalence by taking fundamental groupoid. Sure many of you have seen that. Especially in the case if the category has only one object, then being a groupoid is the same as being a group. Then BC is the usual construction of Islandburg McLean Spaces, K-Pi-1. And this just becomes a statement that you can get the group back by taking Pi-1 of.