 I should also that this kind of category, where you put this condition of having inclusion of the outgoing boundary being zero connected, was, I think it's occurred several places in mathematics, but it was introduced for this purpose of studying kind of infinite genus surfaces in Nulrigus, Tillman's paper, in 95. Oh, five time, I'll say a little bit more about that later. So we have this functor that sends M zero to a co-limit of C two zero M zero S one, and then co-limit over gluing a cylinder, connect some RP two. If I do it on the space level, it's better to say homotopic co-limit, so also known as maving telescope. That goes into spaces, and it's contra variant. This sends all morphisms to homologous morphisms. That's what we stated here. Then there's a completely formal argument that this co-limit homology of C two zero M zero S one. So the kind of common value of all these functions is homology of the loop space of the classifying space of the category. Via this map I have erased now that you have two zero M zero S one. Call it alpha two of the space of paths from M zero to S one in the classifying space of the category. So if I take induced map on homology, then take direct limit. Or in this case, I'm just concatenating with a path. That's a homotopic equivalence. So on this side, I don't need to take this co-limit. And then, just have this thing here. Okay, so I still wanna read this as, it's the left-hand side I'm interested in. This is some homology of some morphism space, not on the nose, but in the direct limit over multiplying, composing with the same element infinitely many times. It's a little bit like localizing a ring or something. The oriented version of this is due to Herra much earlier in like 85 or so. He proved homological stability for orientable surfaces. And there's a version of this argument as well there. Right. I didn't understand that, but it did not sound relevant. It sounded complicated in a different way from what I was saying here. In some ways, a lot of things become kind of a trivial when we take this direct limit. Or, yeah, so that's one thing about this thing. Then, there's another, okay. So now suppose I understood this right-hand side, I would have calculated the homology of classifying space of different morphism groups of non-orientable surfaces, at least after connect summing with RP2 infinitely often. And that's in fact the case. So there's another theorem, which in dimension two is also in this paper with Matz and Thurman and Weiss. That the inclusion, I mean, there was a subcategory. I just imposed some condition. So there's an inclusion function. This is a weak equivalent. So even though it's kind of a somewhat drastic operation throwing out a bunch of path components, you don't see that after taking classifying space of the category. So then we can combine that with what I said before. So using that here, I get instead the loop space of the full two-dimensional un-arranged cabotism category, of which we at least have a name for the homo-tipetype. And here the rational cohomology is this polynomial ring. So this is the, so a corollary of that. The cohomology here turns out to be a free commutative algebra, a polynomial ring on infinitely-managed generators, one in each degree divisible by four. That that's what the co-limit or the limit I suppose if you take of b diff of g-manicarbis of rp2, say with a disk removed or connect some with d2. Take the rational cohomology in the limit where g goes to infinity. So if you take co-homology, it becomes a co-limit. Has the same homology as a path component of this space, which is this thing here. So that's the kind of statement you can get out of this machinery. It's not particular surfaces, but it's typically in some sort of limit where you keep gluing something onto the surface. Right, so this squiggly arrow here. So precisely this statement that if you have a, I think I wrote something slightly more precise than the, well, I wrote something in the notes. You have a pro-object, I guess you can call it. You have this, I'm using this fact that first of all, I'm taking homology of morphism spaces. Taking harm into something fixed. And then I composed with some fixed morphism infinitely many times. Then I'm using that after doing that, gluing something else on the other end induces a homologized morphism. So you can actually monetize it if you have a topologically enriched category, there should be some filter direct system of objects so that if you take harm into that, take homo-type of co-limit, it should, as a function of the remaining variable, it should send everything to homologized morphisms. So I'm using something slightly weaker than what Natalie proved. Anyway, yes, they're kind of on-oriented versions of that. Oh, I mean, they come from integral homology. I just, this was a nice morphism, that's what I meant to say. But these classes are integral if you define them the right way, yeah. Okay, the only reason I said the non-oriented version is because I've been not emphasizing tangential structure so much. In this surface case, most things were done in the oriented case before the on-oriented. So as I said, the analog of Marx's theorem is due to Harrow. The analog of this statement is due to Matzner and Weiss. Used to be the Mumford conjecture. The answer looks a little bit different if you have a class in each event degree, but otherwise it's always similar flavor. If you have connected surfaces. And before all that, or before the Matzner and Weiss theorem, Ulriga Tillman had already proved that Ho-Colem be def of a genus G surface. Take the direct limit where G goes to infinity, has the same homology as an infinite loop space. And the methods she proved, this sort of was precisely using this condition on cabotism, so that including our going boundary should be zero-connected. And I'm not going to go into that too much. And, okay, that's of course a lot more to say, but that was sort of the main object, precisely this category. So all this also works in higher dimensions, at least if the higher dimension is even. So there, most of it is joint work with Oscar Randall Williams. So this subcategory, the inclusion is a weak equivalent, provided K is, oh, what is it, D over two? That's what it is if D is even. Forgotten if it's this, or just look up the statement. If I wrote it down, I wrote it somewhere. Okay, okay, I think this is true. So as long as you only impose roughly up to half the dimension connected, you don't change the homotopy type of the classifying space of the category. I guess if you want to, you could think of that in terms of field theories as like functions out of this category into a gruboid in a homotopical sense, can only extend to this bigger thing. Anyway, I'll just use this statement. If D is two N, which, so even, then the furthest this applies is N minus one, two N. We'll prove that. But also the analog of homological stability also applies as long as you use this category. So let me state it slightly imprecisely. There exists some direct system so that if I send, if I take co-limit of homology of the morphism space in this category from M zero into a thing in the notes, I call it X of J. But also there's some, I think I put it in the wrong entry, so I'll have to update the notes. It should be in this entry. So that says some direct system. So there's some indexing category that's filtered. So I suppose that's called an end object. Okay, it's filtered, but you can actually always use the natural numbers with the ordering. Maybe I should just have said that. You take harm into something and you keep composing with something that behaves a little bit like this RP2 in the non-oriented surfaces case. After doing that, you still have functoriality in the first entry and this functor sends any morphism to an isomorphism. So this is again a homological stability theorem. If you like, you can put the co-limit inside as a homotopic co-limit. You have a functor into spaces that sends any morphism to a map of spaces that is a so-called homology isomorphism, induces isomorphism on an integral homology. This is again a kind of homological stability. And this same formal argument tells you then that this must be isomorphic to the homology of a homology of the loop space of this category, which by this theorem is the same as this one. And this is a homology of what's then called MTO2N. The co-limit over V of omega VT, V and the rational, oops, D2N, the rational co-homology of each path component is known. And it's what I said before. It's a polynomial ring on monomials and pontragon classes. So, okay, so we didn't quite calculate co-homology of harm spaces in the category, but we did that after gluing on something kind of infinitely many times. If not just existent statement, it's kind of something explicit and that behaves. So for eight manifolds, for example, you can use infinitely many copies of HP2. And let's see, what's the more I wanted to say? Ah, yes, one more thing. It's also known how this, or two more things. There are versions of both with tangential structures. For example, orientations, you can get formulas for homology of orientation preserving diffimorphisms. And they're also on a slightly stronger assumptions, but not too strong. They're also some stronger homological stability theorems that, in fact, if you fix the homological degree and stick to one path component, then this direct system stabilizes. So after a certain point, gluing on, for example, more copies of HP2 or the analog of that in higher dimensions does not change the homology. Okay, let me stop there. In the oriented surface case, you glue on that. Depends a little on what structure you have. So in general, this system will depend on what the structure is. The basic, I mean, yes, I guess you can always use, let me answer that later, let me think for a moment. Other questions? Yes, the only thing, I mean, everything is kind of integrally, actually, except for the calculation of what you get at the end. You get some infinite loop space, you have to work out what is the homology. And at that point, that's sort of a simple answer rationally and with some other coefficients, you probably have to do a lot of work. That's a lot more, yeah. Yeah, so the, if you have an infinite loop space, likely the Mod-P homology is quite complicated. So that's so-called dial-as-off operations that you can apply to stuff and, yeah, quite complicated. Yeah, yep. Like for example, if you have a six manifold and you do S2 cross S2 cross S2, I don't know the answer to that. I've gotten this question quite often, but yeah, there seems to be, I mean, there are many things we don't understand yet, but for example, we don't exactly know how the story goes in odd dimension. Certainly middle dimension plays a very special role as it's often done in high-dimensional manifold theory. And it's kind of seems important that what you're stabilizing with is sort of adding middle dimensional handles. Yeah, good question, I don't.