 Yesterday I talked about this example, and the main theorem that classifies for all these cubortism categories, it classifies functions from them into group points, or more precisely it translates that question to a homotopytheoretic question, which a priori is often easier. I mentioned this example, finite subsets of R, and one dimensional cubortisms in R cross an interval. The offshot is the fundamental group point of loops of RP2. It's easy to see that's equivalent to a two-object group point, where both automorphism groups are z. So that tells you that functions out of this category are uniquely determined up to isomorphism by specifying two objects and one automorphism of each object, satisfying nothing. There was a question about what is this isomorphism, and there was some typos or whatever you want to call it in the problem set, but the two objects you can take to be the empty set and the one point set, and the two automorphisms can be a circle, and a circle with like a line under it. Okay, so today I'll try to get a little bit closer to what other people call field theories, namely symmetric monoidal functions into some symmetric monoidal category, COP-D or variance. Let me remind you what symmetric monoidal category means, or not in all detail, but at least as all definitions are some data and there's some conditions on that data. Let me tell you what all the data is, and then not all the conditions. Symmetric monoidal structure on COP-D. Symmetric monoidal unfortunately sounds like an adjective, as if it's a property or something, but the symmetric monoidal structure is extra structure. It consists of an object, which in this case is the empty D minus one manifold in D. Then it consists of a functor from the category across itself, back to the category. In this case, it's disjoint union of closed D minus one manifolds and disjoint union of D dimensional cobertisms between them. Then there's three natural isomorphisms. There's one, so natural transformations goes between functors. You can take disjoint union M0, disjoint union M1, disjoint union M2, or you could put the parenthesis the other way, the same for morphisms. That's two functions and they're kind of obviously naturally isomorphic. That's a preferred way of identifying these two ways of putting the parenthesis. They're likely not equal. Depends a little bit on how you implement things on the set level. Another natural isomorphism, D cross cop D, cop D. If you take disjoint union in this way, it's canonically isomorphic to taking disjoint union in the other way, both on objects and morphisms. The last natural isomorphism goes from M to Z, disjoint union M. This is one of the functions, cop D, cop D. You can take disjoint union with the empty set and then you could just take the identity. They're natural isomorphic and that's a preferred natural isomorphism. That's the structure you have to specify to say that cop D is a symmetric monoidal category. Then what? Four axioms. The most famous one maybe is called the pentagon axiom. Let me not write them. That's how it's a symmetric monoidal category. Topological field theory is at least one definition. It's a symmetric monoidal function to vector spaces or maybe modules. If you have a commutative ring, there's a symmetric monoidal structure on the category of modules. Specify the same data. The object you specify is just R itself. By the way, these all have names. This is called the monoidal unit. You'll put it here. The next piece of data is the tensor product. I don't know what that's called, maybe just the tensor or the product or something. It's often considered the most important part of the structure. Then all these natural isomorphisms are named. This one is called the associator. This one is called the symmetry. This is sometimes called the unitor. This is the standard way of identifying these two things. M0 tensor M1 tensor M2 goes to M0 tensor M1 tensor M2 and so on. Associator, the symmetry M0 tensor M1, M1 tensor M0. There are other ways to do it. You could try to put some signs in or something, but if you don't say anything else, I think the symmetric monoidal structure you mean here would just be the plain one. Then the unit or is the usual isomorphism from R tensor a module, so back to the module. Just put the name here. It's the name for the natural transformation. It makes this the monoidal unit. Let me just say one of the axioms in case you haven't seen this before. The pentagon axiom has to do with, if you put parenthesis this way, there are two ways of identifying it with, there are two ways of proving that these are natural isomorphic and they might not be the same way or I mean they are, but in the abstract that's an axiom of this associator. If you first use the associator to move the parenthesis there and then move it again, it's the same as if you do something else you could also do. So this is an equality of natural isomorphisms. Yeah, so for isomorphisms it's, there's no confusion, they're either equal or not equal. So pentagon axiom, it's an equality of two natural transformations between these two functions and similarly for the other axioms. Okay, so that's the structure these have. Then symmetric monoidal function is also an extra structure. I mean it's a function together with some extra structure. Again, let me just tell you what the data is and skip the axioms. So again the data is the function of underlying categories and then it's a natural isomorphism. So from the monoidal unit in the codomain, sometimes just denoted 1 to z of 1 where 1 is empty set and another natural isomorphism, z of m0, z of m0, etc. And they're required to satisfy various diagrams have to commute that have to do with the associator, the unisor, and the symmetry. I looked at a TS paper this morning, he just writes equal and doesn't keep track of the isomorphisms but as far as I could see. Okay, so that's the definition or half part of the definition at least. So you could take d to be r modules, maybe complex numbers and this tensor product. That would be an r linear. So this would maybe call that d-valued. Again, maybe you want to remember the dimensions, d minus 1 comma d, topological field theory, something like that. Such a functor. I don't have anything in this class, we won't talk more about them in this generality but again, I don't have anything in this class. In this class we won't talk more about them in this generality but again, if all morphisms go to isomorphisms, or we might as well just say if d is a groupoid, then we can again translate this to some homotopy theory. So the translation we already had had to do with spaces and categories, continuous maps and functions and I didn't emphasize it so much but here I could also put natural transformations. The corresponding thing to put over here would be homotopies but it's actually better to say that the corresponding thing is homotopies up to homotopy. So homotopy classes of homotopies between maps. So you have a homotopy between two maps, if you have two of them they can be homotopic rel and points of the homotopy. The translation here went, if you have a space, you take fundamental groupoid, category homotopy class of a homotopy is enough to get a natural transformation and in the other direction you take classifying space of the category and you get this data. A natural transformation actually gives a homotopy but composition of natural transformation only goes to composition to concatenation up to homotopy of homotopies. So I think this is the right thing to put here and if you go back and forth you always end up in groupoid functions and natural isomorphism. Now up here I could put symmetric-monoidal everything, symmetric-monoidal functions and symmetric-monoidal natural transformations whatever that is. Then there's a forgetful function here. So what I'll talk about today is what to put here, how do we translate categories with this extra structure to spaces with some other extra structure. So the answer is going to be of the form spaces with something, maps preserving that something and similarly back and forth here. Well there's a kind of answer that's easy to give but somehow doesn't seem to occur very often in nature. What extra structure do you need on a space to inherit a symmetric-monoidal structure on the fundamental groupoid? Say if you don't want to make any choices or anything this is answer one. You can just take the axioms of a symmetric-monoidal groupoid and just write the same thing over here under this dictionary. The axioms of a symmetric-monoidal category of groupoid is something that's expressed in terms of functions, natural transformations and equality between natural transformations. You just write the same thing over here. So there would be a space x, a point 1 in x, a map from x cross x to x, let me call it m, homotopy, x cross x cross x. Out of this map I could either take m of x0, m of x1, x2 or I could take m of x0, x1 then m of that, x2. Say name some new kind of structure in a space that is given by a point, a map, a specified homotopy class of a homotopy or erase the axioms of a similar homotopy between x cross x, m of x0, x1 that gives a continuous map like this, m of x1, x0 and homotopy of maps from x to x, one hand identity and the other one m of 1, x. If you have a space with all this data and you take fundamental groupoid, you get precisely the data you need for a symmetric-monoidal category like no more, no less and then you just take all the axioms and phrase them as axioms in terms of this structure. Like the pentagon axiom becomes some axiom out of this data you can extract two different homotobbies between maps from x to the 4 to x and then require them to be homotopic as homotobbies and similarly for the other axioms. Okay so that's in some sense a complete answer because that allows you to go back and forth here if this is your dictionary. I don't think this structure has a name because it rarely occurs in nature. What occurs is either less or more. That's the notion of an H space for example, that's a space together with a map, then there's a notion of a homotopy associated with H space, that's a space together with a map for which such a homotopy exists but it's not part of the data. Then there's something that's more rigid than this data I wrote here which also occurs in nature, that's the notion of an EN space. Let me talk about that and that fits well with the corporatism categories and so on. So to say what an EN space is again I'm going to skip a lot of the data but I first have to talk about little disks. So there's a space called dn of k. It's the space of maps from k carbons of the disk into the disk that are embeddings such that let's say they exist ti01, vi in the interior of the disk so that j of i comma x is vi plus tix. So embeddings that can be written as, embedding from k carbons of the disk back into the disk, that can be written as scale by something and shift by something else. I apologize to the experts in the audience that might have heard this before several times. So that's a space and you can see what I said using this. These are of course unique if they exist. So they determine j. No, this is just a space and they satisfy the condition that if you take this function then it should be injective. That's all. As I said it's a condition but absolutely. Yeah, so this is a proper subspace for that reason. This has the structure of an upright. Let me again say what the data is and not all the conditions. So this is a topological space to apologize it as a subspace of the Euclidean topology. When we say that these spaces form an upright. By the way, it's a non-negative integer. We mean that it encodes the action of a symmetric group. Then there are various composition maps. If I have an element of dn of k. I have an element of dn of k prime. I'm going to get a map of dn of k plus k prime minus one. Composition i. Where this means if I have two embeddings like this. Let me just say it in words. J and j prime. J circle i j prime. k plus k prime minus one. Carbous of the interior of the disk. You take the k inputs of j. And then you take the output of j prime and plug it into the i-th input of j. That gives you a map like this. You can write some formulas, but let me not do that. And it's continuous. And then one more piece of data, namely the identity in dn of one. This is one way of writing the data of an upright. And then there's some properties of which the most important is associativity. J circle i j prime circle i prime. J double prime is apparently equal to j circle i j prime. Circle i plus i prime minus one. J double prime. And some other properties. That's another associativity. Property then there's one saying that about composing with the identity. And then there's something about how the group action interacts with composition. I mean, if you know the image of this j, that's what you're saying. Yeah, no, that's the same. I mean, I think it's good to think of them as maps from multiple carbons from the same object into the same object, but I don't know. So you're pointing out that the image of this thing determines what the map is because there's only one way of scaling and translating as long as you know. If you know the image and like the labeling of which disk is the image of which copy. I mean, if you say it with maps, then somehow composition is composition. Otherwise, you can also say it with like plug one picture into the other picture. But yeah, it's easy to go back and forth between those two points of view. More properties. If x is a space, then there's another operator called the endomorphism operator whose case space is just the space of maps from x cross with itself k times into x in the compact open topology. And in that case, the i-th composition, circle i of x1 up to x k plus k prime minus 1. Is again, you take the output of f prime and plug it into the i-th entry of f. That also satisfies some associativity. And then a dn structure on x is a map from this piece of structure to the other piece of structure. Dn structure on x. This is continuous maps dn k to nx k for all k bigger than or equal to 0. Preserving the structure is k actions, circle i composition. That's one of these circle i. I can run from one up to the number of inputs of the left factor here. That takes a while to say all that, even skipping a bunch of stuff. The reason I'm saying this is two things. Maybe I should have called it dn structure. Okay, so an en algebra is a space together with an action. It also makes sense in other categories. You can talk about en algebras and so on and so on. But in spaces, it means a space together with this data. Okay, the simple versions. I talked about little disks. There's also a version with little cubes where you don't have round things but boxes. But let's stick to this one. So, one thing I wanted to explain is if n is at least three, there's a, let me say canonical, symmetric monoidal structure on the fundamental group point. And by canonical, I mean factorial. As you'll see, there's a choice you have to make. And yes, for, on this, yes, exactly. Thank you for so given structure on x. It's functorial in, okay, I didn't say that. But there's no maps of this structure compatible with all this. And this kind of induces a symmetric monoidal function. That's one thing. And the other is that, for the dimension of v being n, the equivalence that I mentioned last time, called the main theorem, b, c, d, v, equivalent to loops v of...