 something. Let's say it can be upgraded or, okay, let me say it in a sloppy way first, is an equivalent, is an equivalence of EN spaces. Okay. This is the kind of thing people will write, but it's a little, maybe unclear the first time what do you mean by that. What I mean is that I've specified some models of both sides that come with an action of this op-rat, like with an EN structure. And I've given a zigzag of maps that are both weak equivalences and strictly compatible with the structure. A zigzag means, yeah, okay, let me, that's a good point. That's kind of actually, okay, it's kind of jargon in homo-tobby theory. So often when you have a weak equivalence, you don't actually have a map. You just have something else that maps to both of them, and they might not have homo-tobby inverses, and that's what I mean. And you can have longer things. So first of all, I have to explain in what sense does, do both spaces have an EN structure? That's part of making sense of such a statement. And then saying there is an equivalence of EN spaces means that there's a zigzag like this, where all maps are weak equivalences, and there are given EN structures in all the spaces, and all the maps are compatible with the EN structure. Now, n is equal to the dimension of the ambient space. You can probably replace it with just one thing, but this is still good enough. If I take pi1, all of these by this, if this dimension is at least three, fundamental group point becomes symmetric monoidal, and then you have a zigzag of equivalences of those, and on that level you can invert them. So fundamental group point there will be equivalent to the fundamental group point there as a symmetric monoidal group point. Okay, we'll see how much I have time to say, but that's how this structure is relevant. I'm using this as a clock. Okay, so let me start with this thing. Why does an EN structure on a space induce a symmetric monoidal structure on its fundamental group point? So an EN structure is all these maps, a little bit sketchy, but just give you the flavor of the argument. Okay, this has to do with the homotopy type of these spaces, dn of k. Let's take just off and draw it like this. 1, 2, 3, 4. Here I'm just drawing the image of the embeddings. Up to homotopy, I just need to specify where the center point goes, because the radius is some continuous parameter. So this is in fact homotopy equivalent to just injective maps 1 up to k into dn, also known as the configuration space, the ordered configuration space of dn. The only way n bigger than or equal to 3 is used is that this is simply connected. Like dn of 2 is the space of up to homotopy is what, n minus 1 sphere. So if n is at least 3, then that's simply connected. Yes, thank you. Important point. Conf k of dn, yes, interior. So this is simply connected. Now I just take fundamental group out of this thing. This goes then to fun, fundamental group out of x, cross fundamental group out of x into fundamental group out of x. So on the object level, for each point in dn of k, I get a functor fundamental group out. And then the morphism level for each path, homotopy class of paths, I get a natural transformation. That's what the morphisms are here. So we get a functor like this. Well, now I just start, what am I looking for? Symmetric monoidal structure here is a, well, in particular, this should be some monoidal product. So that's something you have to pick once and for all. Pick an m in, I mean, in this translation, dn of 2, which is also the objects of pi 1 of dn of 2. Under this functor, we call it star. There's one of them for each k. This goes to, I don't know, we call it m again. M from pi 1 of x plus pi 1 of x to pi 1 of x. And then use that as the monoidal structure. Let me say maybe one more thing. Where does the dissociator come from? Let me say that right. m circle 1 m and m circle 2 m. m takes two inputs. I could take the output of m and plug it into either the first or the second entry of itself. That gives two points in dn of 3 that I can regard as objects in the fundamental group point of dn of 3. Well, the space is simply connected. So any two objects are canonically isomorphic. There's a unique isomorphism between them. Then take that unique isomorphism. That's a morphism in dn of 3. It goes to a natural transformation of functions. And use that as the associator. And so on. I so m circle 1 m, m circle 2 m in fundamental group point of dn of k. And this operation goes to a natural transformation of functions between pi 1 of x 3 times to pi 1 of x. And then use that as associator. Whenever you have to check any actions like the pentagon action, you're checking that two natural transformations are equal. That translates to checking that two morphisms here are equal. But since this is a simply connected space, then all morphisms between the same two objects are automatically equal. So one, the only thing that takes a while is to write out all the things you have to check. Everything is kind of obviously works once you've written them. Okay, but you have to make this choice. So that maybe sounds a little non-canonical, but that's how monoidal structures are. Like if you check the distant union of two sets, you also have to choose a way of doing that. Nobody ever emphasizes that, but maybe you implement it by saying 0 cross 1 set, then 1 cross the other set, someone else does something else. The reason we don't talk about that is because two choices are canonical isomorphic and the same is true here. If I pick a different name, I get canonical isomorphic symmetric monoidal structures. Okay, enough about that. That was this thing. How do I get EN structures on these two spaces? Let's start with the right-hand side. This is kind of also historically where operands came from. I have an embedding of open manifolds. I can get a map in the other direction of one-point compactifications. The one-point compactification here is SN. I want to think of that as DN interior, union infinity, one-point compactification. If you have something in the image, you just take the inverse, otherwise you just send it to infinity. That's continuous. The one-point compactification here is the Wetz product of k-carb is always in. I don't know what the standard notation is. Let me temporarily call it j-arvastar or something just because it goes the other way. If I now have a pointed space, x or something, I can take mapping space out of this. This sends, if I have an x here, it goes to either j inverse of x in 1 up to k cross DN interior. I regard this as the one-point compactification of that. I either take j inverse if that makes sense, or otherwise I take infinity if x is an image of j. Then you have to take it continuous, but if you sort of stop being in the image, it must mean you get further and further closer and closer to infinity. Now, if y is any pointed space, and x is the info loop space of y, which means based maps from SN to y. By the way, it's a based map. Then pre-compensation with this j-star gives a map from based maps out of this to based maps of SN to y. Based maps is by definition the loop space, info loop space, so this was x. Wetz is like the co-product in base spaces, so this is just x to the n, no x to the k. For each point in DN, this was a point in DN of k. I've gotten a map from x to the k back to x, and that's what I need for the data for an E-instructor on x and of k to the kx. This was the k space in the endomorphism of rad. Compensation goes to compensation and so on, so this is in fact an E-n structure that's canonically associated to writing x as the info loop space. We have that here because the n in this statement was precisely the dimension of v, and I remind you by loops v of anything, this was pointed maps out of SV, where SV means the one-point compactification of v. So if v is R-n, one point compactify all of R-n or just the unit disk, of course it's homeomorphic. So that's the E-n structure here. What about here? So E-n structure on B, C, D, V. If n is the dimension of V. Okay, maybe I shouldn't have written it this way. Maybe I should have said given a specified isomorphism to R-n. Maybe that's a better statement because it's going to depend on that anyway. The idea is very simple, and it almost works, and then there's one thing that doesn't work, and then there's an obvious fix to that. So I'm supposed to give a point in dn of k, which is an embedding like this. I'm supposed to give a map from this space, k times, back to this space. Well, okay, let's pick once and for all. So this thing, but pick a defiomorphism from V to the interior of the n disk. So that's possible on that assumption. Then I can think of this as 1, up to k cross V, and this as V. This is now an embedding of smooth manifolds. Okay, maybe I kind of want to think of it as the same as this one, but strictly speaking it's... Okay, just change the names. Then there's an obvious function from CDV, cross CDV, CDV function. CDV was the cobaltism category whose objects are d-1 dimensional closed sub-manifolds of V. And morphisms are cobaltisms inside V cross an interval of some length that you specify. Maybe I can draw this case d being 1 and V being r. The objects then are zero manifolds inside r, also known as finite subsets. And morphisms are cobaltisms, and then there was some fine print about them being kind of product. So this would be a morphism from 3 points to 3 points. So this was a picture of 0, t cross V, and we remember this positive number in order to have composition be strictly associative. We compose by just taking these next to each other. It's a non-unitural category. Anyway, if I have an embedding of k-carb is a V back into V, I just apply that to if I have k either objects or morphisms. So the objects are just d-1 dimensional sub-manifolds of V. So m1 up to mk where mi sits inside V is a d-1 dimensional sub-manifold. Just send this to the ith copy I apply, kind of the ith copy of V. So this would go to the union of j of i, mi. And these are destroyed, so this union is still a manifold. That's where you do an object, and on morphisms you do the corresponding thing. Just cross with this in total. And that's a function, and it's continuous on morphism spaces, so I can take b of it. b of, I think in the notes I again call this j star. b of j star goes from bcdv to the k back to bcdv. I have one of those for each point, j, in the space. So that's the data I need to give an en structure. The one little problem is that this is not a continuous function of j. Because this was a typologically enriched category that has a set of objects. And if I wiggle this j a little bit, the image looks like it wiggles a little bit, but the objects have the discrete topology, like n0cdv has the discrete topology. So this is not quite continuous. Then you fix that by topologizing the objects. You no longer have a topologically enriched category. You have what's called a category internal to topological spaces, where you have a space of objects, and source and target maps are continuous. So in the notes I used something that looks, I mean intentionally it's going to look almost the same, because they're meant to be kind of two instances of the same, I don't know, platonic idea of a corporatism category. Here was a topologically enriched category with a discrete set of objects. Here is what's sometimes called a topological category, or a category internal to spaces that has a space of objects. It's apologized in a similar way, where objects can kind of move continuously. And if you use that one instead, then this idea works. And you can also prove that if you take b of this, you get a weak equivalent. So this is a weak equivalent, and this has the end structure. Using precisely this idea, this was kind of the only reason for this discontinuity. Yes, kind of. If you have a continuous, you have a path in the object space over there, like why does that live to a path? You just take a morphism instead, use that as a simplex, yeah, exactly. I mean the actual proof is a, there's some general fact about simplexial spaces. If you take the zero simplices and make them discrete, it's called the Bausfield Friedlander theorem, that's probably the cleanest way to write an actual proof, but that's the idea. Okay, so now, oh, I raised that. I've explained why both sides in this equivalence, at least if your handwriting is ugly enough that you can't tell the difference, how to give both of them, I mean, the heavy end structures, but I mean, given the ones I just explained. Then it's a theorem. This weak equivalence, that was, as I said, the direct limit where V kind of goes to infinity, it was first proved by myself, and we'll retell them in Ibn-Ahten and Michael Weiss. Then there was a finite dimensional version. The proof there does not talk about the end structures or anything. Then, like I can guide you, proof that you can realize this as, in the direct limit, it's compatible with the upright structure, and then Christiana Mauprise two years ago, proved that you can realize this weak equivalence by, can be realized by zigzag, the references in the notes, by zigzag of EN maps. Therefore, that implies, if you take fundamental groupoid, it implies an equivalence of symmetric-monoidal groupoids, where on this side, you have something you might be interested in. The universal groupoid comes with a map from the compotism category, and on this side, you have something homo-theraetic, where the symmetric-monoidal structure just comes from writing this as an internal loop space of something else. That's where I wanted to end. If D equals 2 and V is our infinity, or 3, what it says, other than just the statement, I don't know that it says something, or can I calculate it? In that case, I should also set the diversions with orientation and so on, you would first calculate the homotopy type, I suppose. You would have loops 3 of some space, which has to do with, if V is our 3, then this is our 4, then you have the grass-manion of two planes in our 4, and then you take the tom space of the perpendicular bundle of that, then you take the 3-fold loop space of that, and if you're interested in the fundamental groupoid here, you are interested in pi 3 and pi 4 of this space, so you could, I mean, I'm sure you could do that, sufficient motivation. Surfaces in our 3. Surfaces in our 3. Okay, objects are, yes, you're right. Yep, in our 4. Right, pi 0 of the kind of groupoid completion. Yeah, knots up to whatever that's called. It's not concordance because that means it's like a surface of any genus and could be non-arrangeable, but yeah. Yeah, so it's not concordance that it answers. It's like a bombardism of something that's not necessarily a cylinder.