 of the k pi one space. So we'll be most interested in equivalence of categories as opposed to isomorphism of categories. But nevertheless, let me make one comment about that. If we also remember the subspace of zero-simplices, so think of, temporarily, think of BC as something that spits out a space with a subset, then we can reconstruct the category up to isomorphism, just by taking, by construction, this canonical functor does not hit all objects in pi one of BC. It only hits those that live in the subspace. In the notes, I wrote this as pi one BC comma N zero C. And this just means the full subcategory of the fundamental group point, full subcategory on objects. So objects are points and I only take the points in N zero C. So it lands there by definition. And the same proposition shows that this will be an isomorphism of categories, if C is a group point. So, okay, and this thing is then an equivalence, that inclusion. If, okay, more interestingly, if it's not a group point, then you still have this natural transformation from C to a group point. And the statement in that case is that, if C is not a group point, by which I mean not necessarily a group point, I still have this functor from the original category to the fundamental group point of the classifying space. Let's say it hits only the subcategory on the objects here, but that's equivalent to the full fundamental group point. I still have this functor. And it, in that case, is the universal functor from C to a group point. That's functor from C to a group point. Let me say what that means. It can mean two things. So, so given any small group point D, and a functor D to D. If I have a functor from C to a group point, then there's a unique factorization like this through a functor. Okay, if you have to say it that way, it's important I remember this. Otherwise, I would have to say something unique up to isomorphism of functors. But here, I mean this is a correct statement. This is a unique functor. So, wait, I mean there's a statement. You have to prove. No, that was the comment I just made. I mean, this is one correct statement you can make. But there's another one. I mean the problem with this is that it's, it's a universal property of this thing up to isomorphism of categories. Like it's not a, okay. This is a universal property for this one, but not for this equivalent category. There's another statement you can make where you say something up to equivalence, but let me say a few words about it. See, probably have time for that. Okay, so these have the same objects. Let me see. I don't wanna say. Okay, so first of all let me say this is a true statement. Okay, so because it's a universal property, it characterizes this pair of this group point. The fundamental group point of the classifying space. The full two subcategory on points and n zero. Together with this functor, characterizes this up to unique isomorphism of categories as any universal property. If you don't like characterizing categories up to isomorphism, then you can find another universal property that characterizes it up to equivalence of categories, but I think this is maybe a simpler statement. The proof is kind of easy. First of all, these two have the same object set. This is a bijection on objects, object sets. So, certainly this thing is like uniquely given on objects. Just do whatever you do there. On morphisms, uniqueness on morphisms. So what do morphisms in the fundamental group point of the classifying space look like? What are the paths for interval into BC? Say zero goes to x, that's some object. One goes to y, that's some other object. Then, up to homotopy real end points. I can make this path, I can homotopy it into the one-scalism. So, up to homotopy is a concatenation of paths in the one-scalism, but the one-scalism are precisely the morphisms in C. So, up to homotopy and the one-scalism are another way of saying that is some morphism in C is regarded as a path. It's like gamma of that morphism, except I might run some of those in the opposite direction. So, morphisms have a direction, but paths can go in the kind of wrong direction. So, gamma of f1 to the plus minus one, composed with gamma of f2 to the plus minus one, and so on. So, homotobbing a path into the one-scalism writes an arbitrary path as a concatenation of things that come from here and the inverses. But then, if this is supposed to be a functor and this is a gruboid, it must send and the diagram is supposed to commute. Gamma of f must goes to capital, so if I have an f in C say x comma y, this goes to f of f. So, gamma of f must go to f of f, but gamma of f inverse must also go to f of f inverse, which makes sense because this is a gruboid. So, since any morphism is here, is a composition of things of this form. There's at most one option here, and then you have to check that that works. Okay, so that characterizes this one, uniquely obtuse morphism, and therefore this one obtuse equivalence of categories by a universal property. In the notes and in the problems this afternoon, there's some examples of what happens in this two specific examples. This construction has a name, I mean this universal property. It's called the localization of this category. Localization of categories. There's a general construction that starts with a small category and a subset of the morphism set, and produces a category denoted C brackets W inverse together with a functor from C, which is a bijection on object sets. Since morphism in W to isomorphisms is unique, sorry, is a universal with that property. If you have any functor from C into an arbitrary small category that sends morphisms in W to isomorphisms, then it factors uniquely through this one. Okay, such a thing exists, and therefore characterizes this again up to unique isomorphism of categories. You have to prove existence, of course, but that's not too difficult. In that notation, what we've done is the special case with W is all morphisms. This could be W is equal to all morphisms of C, and then we get the universal functor from C into a group point in this notation. Can be written this way. So in sum, I mean with that notation, the calculation is that the fundamental group point of BC, full subcategory in these things, is a particular model for this localization of categories. So IE, just to continue the statement, let's see. So that's a kind of concise statement about how much can be reconstructed of a category from the fundamental group point. It's precisely the localization at the set of all morphisms. Okay, I have 10 minutes left. Let me say a little bit more. Ah, okay, let me say a few kind of corollaries. So if you don't know C, but only know BC, you still know this localization up to equivalence of categories, and therefore you know what functions C has into group points. D is a group point, fun C comma D, this is the same thing as fun C, C inverse D. It's then the same thing as, that's what the universal property says, fun pi one BC and zero C D. This is up to isomorphism of function categories. And finally, the last step, if I only know the fundamental group point. So this is the function category. So true functions, I mean, function categories are object functions and morphisms, natural transformations. The last one is an equivalence, and the other ones are isomorphisms. And maybe in practice you kind of want to, maybe in practice the last, I mean maybe in practice you want to work up to equivalence, but yeah. So if you know BC up to equivalence of categories, you know what functions C have into group points. Just rephrasing what I just said. So maybe I can now make a preliminary definition of field theory. I don't really know what an actual field theory is. I mean, I think the one I consider the actual definition is probably also preliminary from a more fancy point of view. So in the notes I call this a poor man's field theory. I don't think this notion is actually useful for very much, but it's kind of a toy example of the definition I'll talk about later. So if D is a small category, then poor man's version of a field theory is just a functor from cop D into D. The only field theory aspect of this is that I call it Z. So this would be a D-valued, maybe D minus one comma D theory, because this category involves manifolds of dimensions D minus one and D only. What I consider the actual definition, but which I'm sure is also extremely preliminary, you would have to say something about a symmetric monoidal function. So, okay, I don't have very much intelligence to say about field theories in that generality, but so field theory is invertible if it takes any morphism to an isomorphism, that's the property it could have. For example, if D is a group word, so if Z of any morphism, we use this notation D with a tilde. So this is the same object, but only the isomorphisms. So subcategory of D, this is all co-bordisms to isomorphisms in D. So you might as well just forget about D and only work with the tilde and then rename it D. So say an invertible field theory is one whose target category is a group word. Again, very preliminary definition. To classify invertible field theories by this general discussion about small categories, it suffices to understand the homotopy type of the classifying space of cop D. The vacation of invertible field theories suffices to understand homotopy type of the classifying space of cop D. This version of the definition, I mean D is just an ordinary category, maybe group word. So we don't even need quite the whole homotopy type, since what we're gonna do is take the fundamental group word. Okay, so this suffices, as I said. It actually suffices to find the space X and a two-connected map from X into B cop D was suffices. What I'm implicitly saying is that you wanna find a space X that you understand better than B cop D. And then maybe you have some argument that this is two-connected. Then you just take the fundamental group word of this one instead. Two-connected means, okay, you know that, means bijection on set of path components and isomorphism on fundamental groups with any base point, surjection on pi two with any base point, and no further conditions. Even forgetting the surjection on pi two, that's still sufficient to say, equivalence of fundamental group words. So this implies pi one X pi one B cop D is an equivalence of categories. Okay, so that's what I'm gonna do in the lecture tomorrow. I mean, explain what this X is gonna be. And state the sense in which we understand there's gonna be an X where we in some sense understand the whole homotopy type of X in a way that we in practice can work out what the fundamental group word is. And then with classified invertible field theories in this poor man's version at least. Let's see. Lecture three, I'll talk a bit about, I mean, removing some of this. Lecture three, I'm gonna talk a little bit about symmetric monoidal structures and then change this definition to say something like symmetric monoidal functions. The answer doesn't change very much. I mean, you still have to understand the homotopy type of this thing. Just have to understand it as a space with more structure. That kind of mirrors the extra structure that we put on cop D. So even though this is kind of a toy version of the question, it's sort of quite close to the question I'll talk about on Wednesday. Let me see who's the more I wanted to say. Let me stop here. Yes, let me repeat the question. This is an unoriented, or it was not a question, it was a statement, but just repeating it and agreeing. This is an unoriented field theory because the manifolds are just kind of naked manifolds with no further structure. I will briefly mention that. I mean, I put it in the notes at least. You can put other structure like an orientation, spin structure, and so on. The general, or the level of generality I took was that you can put any tangential structure on the manifolds by which I mean an aquevarian map from the frame bundle into some specified space that I call capital theta. That's a GL, DR, aquevarian map. So whenever you have a space with an action of GL, DR, you can go back and change the definition and put this sort of structure on all the manifolds inside. And then you get a new cabotism category. For example, if this space is like plus minus one and the action is by the sign of the determinant, then an aquevarian map is the same thing as an orientation, but everything I say will have versions in this generality. I just take longer to state, and it's not a whole, on the level of ideas, there's no kind of new ideas, but it includes Riemannian metrics. This could be, right, all the non-trivial things I have to say are homotovian variant and theta, and up to aquevarian, I mean, Riemannian metrics is sort of a contractable choice, so the statements I'm gonna make, Riemannian metrics, you might as well not have any structure because it's a contractable structure, but yeah.