 Thank you all for returning. So I want to continue from where we got to last time. And I want to just remind you of the theme that we introduced. The theme that we introduced was that if we have a category of sheaves that we're interested in, that category of sheaves is capable of determining a homotopy type in many good situations. And that homotopy type is then capable of recovering the category of sheaves. And so what we are going to do is we are going to understand that theme and we were going to apply it in a particular example. And the example that I wanted to think about was the example of constructable sheaves with any kind of reasonable coefficients. So this is the theorem. And I'm in the process of explaining this theorem and sort of showing you how it works. And well, let's look at some of the pieces. Well, I told you what this thing is as a category last time. And I introduced that object. I told you what kind of lambdas I was going to allow. I didn't yet explain. And I won't today. I'll do that tomorrow on Friday what CTS means with this continuity condition amounts to. And I actually exhibited a little bit of confusion. I wanted to correct a thing that I misstated last time. Somebody asked about whether or not this was the same thing as the derived category CTF with this torfinitinous condition of Deline. And it is. It is the same thing as this torfinitinous. I'm really requiring that the stocks be perfect complexes. So the torsion finiteness is there. So this is the same thing as that guy. OK, so I'm going to tell you more about how to prove this theorem. But I want to spend some time trying to motivate the definition of this Galois category. So if you remember, this Galois category was defined, it's just a one category. And the objects are geometric points. And the morphisms are specializations. And one of the interesting properties that we saw about this, if you recall, was the fact that every endomorphism was an automorphism in that category. But there weren't any endomorphisms that weren't invertible. And that's something that we're going to see more of in today's lecture. So the motivation for this, why would you ever contemplate this thing? And why would you expect a theorem like this to be true? And the motivation comes from an interesting piece of topology that's been kind of percolating over the past 30 years or so. So today, I'm going to tell you about that piece of topology. And I'm going to tell you what it gives you permission to do and why we can then move to the algebra geometric setting. And folks were really great about typing out questions last time. And so I hope you will continue to do that this time. So please, I've got my, I'm ready to go. I'm ready to answer your questions. So if you have any, even if they're left over from last time, I'm happy to answer. OK. So the question that I want to try to answer today is what kind of homotopy type is this category, is this Galois category? I want to think of this thing as a homotopy type. But what gives me permission to do that? And that's the question I'm going to answer today. Well, why don't I go ahead and answer it then? So it's a stratified homotopy type. It's a stratified homotopy type. So I'm going to tell you all about stratifications and stratified homotopy types. So here's the basic definition. So if you have a topological space X, then a stratification of X is a continuous map from X to P. And P here is going to be a post set with the Alexandrov topology. So with the Alexandrov topology, in case you've not taken general topology in a little while, probably I need to remind you. So let me just remind you. And a subset U of P is open if and only if the following condition is satisfied for any X less than or equal to Y. If X is in P, then Y is in P. Oh, sorry, U. If X is in U, then Y is in U. In other words, the opens are the upwardly closed subsets of your post set. You've all, I assume, seen an example of this kind of thing. So for example, if I just take that post set, which I guess on some screens, that's not as visible as I want it to be. So maybe I'll write this elsewhere. Let me just add a page here. So if I look, for example, at this post set here, zero less than one, that post set has the property that that point is open and that point is closed. So open, closed. And you can see how to generalize this kind of story. Okay, so that's the definition of a stratification. And so if you have such a stratification, then you can talk about the strata attached to a stratification. And those are just the inverse images of the points in your post sets. So P is a point of P, and this is called the P stratum. So for us to reflect on this kind of definition, let's have a look at a few examples. So I've drawn out a few examples here in the hopes of giving you some intuition about how these things work. So here in the first example, I've just got our friend, the interval. The interval, I'm gonna stratify it so that the closed stratum is zero and the open stratum is the interval from the open, the half open interval from zero to one. So it's everything but zero. And that's a stratification over the post set that I identified last time. I can take both the, if I have a stratification X to P, I'm allowed to take say X squared and P squared. I can product those things with themselves and that'll again be a stratification over a new post set. So here's an example of exactly that. So here's the square, the closed square. And I'm gonna stratify it over these four points in the stratification. So one, one, it's a big open stratum. One, zero, that's just this leg here and one, zero and zero, one. Sorry, zero, one is one leg and one, zero is the other leg and then zero, zero is the origin. Let's see here. So some of the other more interesting ones. So there's S one here and I've stratified that in two different ways just to kind of give you an idea. So for example, I've stratified S one where I just identify a point and I've also stratified it in another way where I identify two points. So here this is S one stratified in two different ways. I can do this with spheres. You see these different kinds of stratifications on these spheres. Here's an interesting one. Well, I don't know if it's interesting but it exists. So here, what have I got? Well, I've got two points for my stratification and then I've got two halves of the equator as my next piece is actually the two halves of the equator. That's the next piece. And then the last piece, the big open piece is just the north and south hemispheres. Okay, so that's the idea here. That's the way these different kinds of stratifications work. And I'm gonna look at, we're gonna revisit this table in a little while once I introduce it in variant and we're gonna stare at this again. So this is all gonna come back to us. Okay, so if you've got one of these, if you've got a stratified topological space, if you've got a stratified topological space then how should we speak of its homotopy type? I feel like, is someone's mic on? Okay, so Sean Tilson asks whether this kind of formalism is useful for thinking about manifolds with corners. Yes, absolutely. So this is the sort of formalism that one uses to deal with not just manifolds with corners but also more generally manifolds with very kinds of singularities, various kinds of singularities and stuff. And so what you do is you typically wanna take your manifold with singularities and stratify out those singular loci and sort of arrange things exactly for that purpose. And yeah, and that's exactly one of the first sort of objectives of stratified topology was to deal with exactly that kind of circumstance. So yeah, that's a great question. That's exactly right. Okay, so but I'm just gonna be interested in the homotopy type. I'm not gonna do all that fancy geometry yet. I'm just gonna try and extract a homotopy type of these stratified topological spaces. But the question is, is that if I have a stratified topological space how should I talk about its homotopy type? What should I mean by that? And again, I'm gonna go back to our central principle which is that I'm gonna let the sheaves do the determining of the homotopy type and then the homotopy type will recover the category of sheaves. And as long as I have that arrangement then that's gonna be, I'm gonna consider that a victory. Okay, and so what kind of sheaves should I choose in the case of stratifications? And that's the thing that I wanna get to now. Well, if I've got a stratified topological space then the kinds of sheaves we're gonna be talking about are constructible sheaves. So a constructible sheaf on X is a sheaf F such that, well, for any point of the poset, when I take my sheaf and I restrict it to the stratification corresponding to that point of the poset, that thing is locally constant. So I no longer have local constancy over the entire space X. Instead what I have is I have a way of kind of carving up my X and then on each of those pieces of that carving up I have local constancy. Okay, so the kind of picture that you should have in mind here is that, well, if I take say zero one stratified in this way then for example, I could have a sheaf that is actually just constant here and just doing nothing on the rest of the interval. It seems that the stratified homotopy is equipped with an orientation as that manifests some topology of the sheaf. It's, there's an orientation in the sense that you know which is closed and which is open. That's true. I mean, the poset flows in a direction. That's certainly true. But that's really, it's, it does manifest some topology of the sheaf in the sense that it's the relationship between open and closed, right? So for example, when I take the, when I take this poset, when I take something stratified, for example, over this poset here with the Alexandrov topology, well, I have the first stratum which is open and I have the zero stratum which is closed and that's it, right? I just chose a closed subset and it's open complement and that's all I've done. I haven't done anything more intelligent than that. So there's an orientation in the sense that the closed open duality is the orientation that I think you're referring to. Okay, so we're gonna try to understand these constructible sheaves. That's gonna be our objective. We're gonna try and use these sheaves and we're gonna try and use these to determine for us a homotopy type of stratified topological spaces. And we're gonna see that stratified homotopy type is sufficient to reconstruct this category of sheaves, these nice constructible sheaves. Okay, so let's start. And so we're gonna start the same way we did last time. We're gonna start working our way up to the Posnikov Tower but we're gonna get bored in the middle and just pass to infinity so that we finish the job early. So, but we'll start off with the first stage and this goes back to a definition. I think this is unpublished of McPherson. So if I have a stratified topological space then I'm gonna define what he called the exit path category. So the exit path category, I don't know what happened to my notes there but anyway, I'll write down the objects and the morphisms for you. So the objects are just gonna be points of my X. They're just gonna be points of X but then the morphisms, the morphisms are gonna be more interesting, the morphisms are gonna be exit paths taken up to homotopy. So what's an exit path? And this is a point that I wanna try and be a little careful about because it's not as obvious a notion as it might seem. So an exit path, how does that look? Well, I'm gonna, it's gonna be a commutative diagram that looks like this. So here I'm using the stratification that I described before. This is just the stratification that goes like that where I have zero as the zero stratum and everything else is the first stratum. So I have that stratification and I'm gonna write down a map where on the bottom I just have really a map of posets from zero one into P and on the top I have a path inside X but it has to preserve this structure that's given to you by these posets. So what does that mean? Well, that means that if I'm inside X here and I have a stratum, then if I want an exit path, I have to get out and stay out. As soon as I've moved from one stratum into another stratum, I have to do that immediately. First of all, I have to do that instantly. And then once I'm there, I can't go back in. I'm not allowed to go back into any other strata. I have to stay in the strata that I'm in. And so that's the definition of an exit path. But I'm gonna take these things up to homotopy and that's what's gonna allow me to compose these things. So let's look at a couple of examples of this of exit paths and not exit paths. Can we view exit paths as morphisms of stratified spaces from zero one to X? Absolutely, yes. Great question, absolutely. So this is a stratified space and this is a stratified space and I've written down a map going across there from the first stratified space to the second. Absolutely right, yeah, exactly. Okay, so I've drawn some examples of exit paths. Oh, what structure do the homotopies preserve? I'll go into great detail about that in a second. I'll be happy to tell you that. Is the exit path category a one categorical notion or is there also an infinity categorical notion? Again, fantastic question. I'm just about to answer it. I'll show you exactly how all that works. Yeah, these are the right questions. This is good. It turns out I've already got it all prepared, but yeah, those are the right questions. So I wanted to draw some pictures of exit paths and non-exit paths just to get these things in front of you. So in the exit path, you see the idea here is that you really are getting out right away and going from a stratum to another stratum, you go immediately and then you're free to do whatever you want to do in that stratum. But the thing that you can't do is you can't go down a stratum. You can't go from the open stratum to a closed stratum or anything like that. And the other thing that you can't do, I mean, notice the orientation on this thing here. I am in some sense exiting, but for the purposes of our work here, I don't wanna consider this an exit path because I spend some time in another stratum before I go into the final stratum that I pass between strata somewhere in the middle of my inner one. I don't wanna do that. Now, so this thing is however, homotopic to an exit path, right? Which goes like that, that is an exit path. But this thing right here before I do all that is not an exit path, okay? And of course I'm allowed to take loops. I can take loops that stay inside a stratum, but I can't take some loop that sort of dips into a previous stratum and then comes back. That's not the sort of thing that's permitted. So if I'm in a stratum and I feel like looping, I'm happy to do that, but I just need to stay where I am. I can't go out to some other stratum. So when you're exiting, the door's locked behind you and the doors are also locked in front of you. You can only go through one door at a time as it were when you're taking an exit path. So I thought I'd draw some pictures of the exit path categories attached to these examples because it seemed like fun. So in some of these, I think you'll see that these things are kind of interesting. Some of these have properties that are a little surprising. So let's look at them for a second. So here I've got the interval and well, I hope you might have predicted that indeed I'm just getting this thing as a poset regardless of the category now. Nothing really exciting very happened there. That's quite dull. Okay, but I could take zero one squared and well the exit path category is the product of the two exit path categories. So that worked out very nicely. Here's an interesting one. Here's zero one squared stratified over just the poset zero one two, so I just have three strata, the closed stratum, the open stratum and then in between them is the locally closed stratum and the exit path category just recovers that same poset. So the first three examples that I've drawn here up to this line right here, the first three examples that I've drawn are all half the property that the exit path category happens to coincide with the poset over which you're stratified. That needn't be the case. However, as we seen this example right here, if I take the exit path category of S one stratified in this way, well, then what can I do? I can start at that point and I can exit one way or I can exit the other way. But as soon as I do, I only have a contractable space in which to stratify. Right, if I look at these two strata, I've got a contractable stratum, which is the point and I've got another contractable stratum, which is everything that isn't that point. So I have these two contractable strata, but the exit paths I have two distinct ones. And one thing that you can notice if you're looking at this and you're thinking from the point of view of nerves of categories and things, you can notice that the nerve of this category exactly recovers S one. That isn't an accident. That's gonna be a general phenomenon that we're gonna witness a little bit later, but it's something to think about right now. Another situation in which you can get the nerve of the category exactly recovering S one is this example here. So in this example here, I have two different points. And well, what can I do? I can flow into the top stratum or the bottom stratum or the top half of the stratum or the bottom half of the stratum in two different ways. And so this becomes that poset there. Sean Tilson asks, when I get to a suitable stratified space on this page, can I give another example of a constructable sheaf? Sure, yeah, absolutely. Well, actually I can give you a whole pile of examples. So the idea behind the definition of a constructable sheaf is the following. If you look at locally constant sheaves, one of the things that's a little unsatisfying about them is that they're not closed under push forward, right? So if I take the push forward of a locally constant sheaf, it won't be locally constant anymore. An example of this, for example, is if you have an open immersion, if you have an open subset of a topological space, and I try and push forward a locally constant sheaf, it won't be locally constant anymore. It'll be locally constant on the locus of that open subset, but it won't be locally constant anywhere else. And so the idea behind constructable sheaves is that you're just gonna put together all of the different ways that you could try and take push forwards of things from constant or locally constant sheaves. And that's the idea. So for any time you see a set with an open subset, you could think about just push forward the constant sheaf along an open immersion. And that gives you tons and tons of examples of constructable sheaves. And there's a sense in which you can kind of build all of them from that kind of operation. And so that's a really good sort of way to generate these examples. There's tons of constructable sheaves out there. Okay, let me give you a different kind of example here on this side here. Let's look at S2. So if I look at S2 and I'm gonna stratify it where the only closed stratum is gonna be at the point and then everything else is gonna be the open stratum. And I look at the exit path category and well, what can happen? I can exit from this point out to somewhere on the sphere that isn't that point. And they're all homotopic, aren't they? They're all homotopic in a way that sort of respects that stratification. So this is it. That's the exit path category, zero is less than one is that post-app. You might be a little unsatisfied with that answer because this thing doesn't look so good from the point of view that we were expressing down here. Over here, we were saying, look, that category is nice because the nerve of that category recovers my space. And you might be a little angry that that hasn't happened here. We're gonna address that concern in a moment. Now, if I add a little more, if I put a little more effort into stratifying my S2 and add just one more point to my stratification, then I can get a category that will work great. So this was my best attempt to draw a picture of, this is a category where I have three objects up to isomorphism. I have three objects, zero, one and zero prime. I have unique maps from zero and zero prime into one, but then one has a big pile of automorphisms indexed by the integers. So they're supposed to represent all the automorphisms that you have of one. And now, so what happens if I take the nerve of that category? If I take the nerve of that category, well, then I'm suspending BZ. In other words, I'm suspending the circle, so I'm getting S2 as you'd like. So that one seems good, whereas the previous one doesn't seem so good. And that's an interesting question. In the next example here, I've got S2 and I'm stratifying it how? I'm stratifying it by taking a point on the equator, the rest of the equator, and then the rest of the sphere. And then if you sort of check it out, then you get this exit path category here, where here the, I'll just emphasize if I just, this highlighted piece here is equal, the map from one to two equalizes the two maps from zero to one. The two composites are supposed to be equal, but the two maps from zero to one are distinct. Here's yet another stratification of S2. So now I'm taking two points on the equator, the rest of the equator and the rest of the sphere, and then you get this pleasant little shape. And that's a nice post-set, that post-set, the nerve of that post-set is again, recovers the sphere, so it's quite satisfying. And then there's the torus, and the torus has just two copies of the circle. So if you take the two copies of this circle here, stratified in this way, then I'll get the torus. David Corwin says, is the exit path category always represented by a quiver? I beg your pardon, I don't remember what the correct definition of a quiver is. David, if you tell me what a quiver is, then I'll be able to answer your question. The example with the S2 and the equalizer, it's a quiver with relations, it's not a quiver. Because a quiver doesn't have relations. Okay, all right, thank you. Yeah, I'm not up on my quiver knowledge. The thing that is true about these is that they're always categories with the property that every endomorphism is an isomorphism. But that's the only restriction that you have. So I don't know if people normally count that as a quiver, but I guess Mark is telling me that the answer is no. Okay, so these are just some examples to kind of get our intuition flowing here. And so, but one of these examples is a little strange, right? We wanna think about this again, because there's something kind of funny there and we'll come back to that funniness. Okay, so but here's the theorem. Why did I introduce this silly thing in the first place? The reason I introduced this silly thing is that if X is a reasonably stratified topological space, so a reasonable stratification, the correct condition is a conical stratification, but unless someone insists, I'm not gonna go and get into details about what reasonable means in this case. You can think of this as a kind of fibrancy condition, it's just making sure that your stratification is not too wild. If you have a reasonable stratified topological space, then you have an equivalence of categories between the constructible sheaves of sets, here they're sets on X, and functors from the exit path category into the category of sets. And I prefer to call this the exodromy equivalence. Why exodromy? Well, what's happening here is that, when we're talking about monogamy, remember we were talking about taking a loop and we're talking about traveling around that loop once, right, so that was monogamy. Here, we're taking exodromy, which means that we're watching what happens to our section as we exit, as we follow an exit path, hence XO. Okay, so this is a good story, but again, we had this little slightly unsatisfying problem here, which is that, well, it was a little strange to think about S2 stratified by this stratification here. This stratification seemed a little confusing, so I just do an example of an exit path just for fun. I sort of spun around the thing a little bit. Oh, Mark Levine asks, what's the composition in Pi 1? So I can compose exit paths. And if my stratification is good, then when I compose to exit paths, I'll get another exit path, or up to homotopy, but it's only defined up to homotopy. So that's why I'm working with exit paths only up to homotopy. So one thing that's a little unsatisfying here is that, well, I've got this exit path, but it seems like it's a little strange to say that there's only one exit path out into the sphere. That seems odd, because after all, if I'm thinking about my presence right here on the earth, and I think about exiting this room into the world at large, I sort of have the idea that I could go that way or I could go that way or I could go that way or I could go that way. And those all seem to be different in some sense. They're homotopic, but I wanna try and keep track, maybe, of the homotopies between them. And so a better solution, this was suggested by David Truman, was that I should define instead a two category, so that when I look at the, and this is the sort of exit path two category, so that the hom, now groupoid, from zero to one should be an S1, that my path should go that way or that way or that way. Joe Michel asks, does the conical condition ensure that the composition is well-defined? It does indeed, that's exactly the point. If I don't have the conicality, then I don't really have a well-defined composition. That's exactly right, yeah. That's right, so okay, and the same thing is gonna be here in this two category as well. I wanna be able to try and think, not only about the fact that I can go out in all those different ways I could go out are actually secretly homotopical or homotopic to one another, but I also wanna try and keep track of that homotopic. So for example, active going around this way, and then sort of spinning around, that has some kind of content. And I wanna try and keep track of that content. So I really want this to be, oops, I really want this to be the set of maps, now set groupoid of maps from zero to one. And what Truman showed in the early 2000s, I think, is that if X is again this kind of reasonable stratified topological space, then we have a sort of higher categorical version of the exodgermy equivalence. I have the exodgermy equivalence now with constructible sheaves of group oids, and I could compare that to functors, just plain old functors from pi to XP to group oids. And again, how does the functor work? Let's think about it. Well, if I have a constructible sheaf F on my ax, valued in group oids, then what can I do with it? Well, I can try and extract a functor defined in the following way. So here's what the functor is gonna do. Well, the objects of this two category, the objects are just points, little X of X. And what am I gonna tell you? I'm gonna tell you to take your sheaf and take its stock at that point. That's what this functor does on objects. And then on morphisms, well, you have to do a little work to say that you've actually got a well-defined functor, and that's why this is a theorem and not an observation. But you do indeed have a well-defined functor when you're sort of passing out from one stratum to another. And so this is quite inspiring, but again, you start to have the same feeling again when you think about more general stratifications. We said we passed to this two categorical thing and that looked better, but you start to have this same kind of sinking feeling that something's gonna go wrong if I take an N sphere. So if I take an N sphere stratified again with just two strata, the closed stratum being say the North Pole and the rest of the sphere is the open stratum. Then, well, I have this problem that I'm gonna start to get these contractable mapping spaces. As soon as N becomes at least three, I start to have these contractable mapping spaces. And clearly this kind of game is pretty unsatisfying. So we need to go ahead and cut to the chase and start defining ourselves an infinity category. So we're gonna define exit path infinity category and that exit path infinity category is gonna be defined. So the mapping space in this thing from zero to one is gonna be the N minus one sphere. In just the same way that when I was trying to exit this room, I had a circle of ways to do it. Now when I'm gonna exit the North Pole to the rest of the sphere, I'm gonna have an S N minus one ways of doing that. So let me give you the definition of this thing precisely. So I'm gonna give this definition to you in the following way. I'm gonna try and tell you what the in-simplices of the exit path category are. So these are the chains of maps zero, one, two, three, four, five, six, seven, up to N sitting inside your category. And I'm gonna tell you what the set or space of those things should be. So how am I gonna do that? Well, I'm gonna start off by taking the in-simplex and I'm gonna stratify it. And the way I'm gonna stratify it is kind of the way that I drew in one of the pictures. I'm gonna stratify it in such a way that, well, if I look at the first K strata, then that is the case, the delta zero through K sitting inside delta N. I'm gonna stratify it over the post set N. So I'll just emphasize this. It's over this post set here. I'm just gonna stratify it in that way. And it's going to be stratified in that way for all N. So it's gonna look sort of like that. That's my two-dimensional picture, but the same kind of process works for all dimensions. And now what am I gonna do? If I take a reasonable stratified topological space, then I'm gonna take the exit path category or exit path infinity category to be the infinity category who's in-simplices. Well, they're morphisms in just the same way that the questioner suggested. They're morphisms of stratified spaces from this stratified space labeled lambda here to your favorite stratified space X over P. And that's what the in-simplices look like. So what's happening here? Well, here I'm allowing myself to sort of, I'll take an exit path, so that goes out. And if I have another exit path, then I'll allow myself to consider a homotopy that connects the two things. And that's a two-simplex in my exit path and infinity category. So I'm hoping that this answers, I think it was Remy's question. I hope this answers his question. And so then I also have three simplices that will allow me to sort of talk about a homotopy of homotopies and homotopies of homotopies of homotopies. And I don't get to stop at any finite stage, so I have to just keep doing that. And so this gives me access to now this beautiful exit path category, exit path infinity category. And this exit path infinity category, I really want us to think about this exit path infinity category as sort of giving us a homotopy type of a stratified topological space. So if that's gonna be the case, then our principle is that homotopy and sheaves determine each other. So then we have this theorem of blurry, which is that X over P is again gonna be reasonable in the same way as before. It's gonna be a reasonable stratified topological space. And then we have an exoddomy equivalence between constructable sheaves valued in any infinity category C and functors from the exit path category into that C. And again, on objects, what happens, it's the same thing. You take your constructable sheave and you take its stalks at all the different points of your space. Okay, so the question that we have to ask now is what kind of justification do we have for the idea that this is a good homotopy type, is pi infinity of X over P, this object is a good homotopy type. And furthermore, what kinds of structure does this thing have? How can we think about this category in a way that sort of seems familiar to us? So here's an observation. And this is an important observation for our purposes especially. This infinity category, pi infinity X over P comes equipped with a functor, right? If I take pi infinity X over P, well, that's the objects are points and the morphisms are exit paths and the two morphisms are homotopies of exit paths, et cetera, et cetera, et cetera. So I've got this functor here. And well, what can I do? How does this functor work? I can take point X and I can just tell you which stratum it's in, right? I can just tell you which stratum it's in, like that. If F is the name of my stratification. So I have a perfectly good functor. And well, if I think about what this functor is, I can look at the inverse image of any point in my poset and what am I gonna get? Well, when I look up over a point in the poset, I'm gonna have just the stratum sitting over me and I'm gonna have the whole homotopy type of that stratum, right? I'm allowed to put any simplex. There are no conditions on how this simplex back here behaves if my n here goes all the way to a single point, is a constant at a single point of the poset. There are no conditions on this whatsoever. So that means that I'm allowed to look at any in simplex I like of my X, which means that when I look at the inverse image of a point under this map, then I'm exactly getting here the infinity groupoid attached to my space. So remember what the situation is is I'm looking at the, and this is regarded as the singular simplicial set. This is the singular simplicial set of that stratum, right? So, oh, Sean Tillson says, can I go back two slides? You're confused about the stratification on delta two. Sure, of course. So here's the stratification on delta two. There it is. So it goes zero, one, two, it's a stratification. What's the edge between zero and two? The edge between zero and two is part of the second stratum, right? Everything except for the edge between zero and one is in the second stratum. It's in stratum two. So stratum zero, which just consists of zero, zero. There's stratum one, which consists of everything except for the vertex, and there's stratum two, which consists of everything except for that edge. That's the stratification that we're taking here. Is reasonable the same condition as in the classical case? Yes, yes, yeah. I'm just talking about what I would call conical stratifications. In the classical case, you can actually get away with something slightly more general, but it's not important. You really want to have this much. You really want the reasonable condition, yeah. So that's just being conically stratified, which basically I can just tell you. Basically what it means is that in the neighborhood of every point, that stratified space looks like a cone with its stratification. And so if you want me to say more precisely how that works, I can, but let's leave it for now. Okay, so what does that mean? So that means that this functor here has an interesting property. This functor F here has an interesting property. I have this category. It maps down to a post stat and the inverse image of a point is a groupoid, right? All the morphisms are invertible. Can you realize this sort of as a join of stratified spaces? So I get the stratification of delta N by taking joins of delta zeros. Yes, yes, yeah, that's exactly right. Yeah, you can just keep joining on a new delta zero each time. That works out fine, yeah. Absolutely. So but I just wanted to say this one word here which is conservative. This functor here is conservative. What does that mean? That means that if I have a morphism in pi infinity of X over P and it becomes an isomorphism in X and P, well, what does that mean? That means it has to be the identity because P is just a post stat. Then it already was an equivalence to an isomorphism upstairs. So in other words, I've got this functor. It's a functor from an infinity category to a post stat but it's also conservative. The inverse image of any point is actually just an infinity groupoid and it's the infinity groupoid attached to the stratum XP. And so this is a very special class of categories that have conservative funters to post stats. Sometimes in the literature, you'll in at least in ordinary category literature you'll see these referred to as EI categories. I've never liked this terminology but I guess that the statement's supposed to be that every endomorphism is an isomorphism so it becomes EI categories. Okay, so here's the theorem. The theorem here is that the assignment that carries stratified topological spaces to these kinds of infinity categories, infinity categories with a conservative functor down to the same post stat. That's the assignment that carries your stratified topological space to this exit path infinity category. This is actually an equivalence of homotopic theories. So this is a theorem of my PhD student, Peter Hain but there are versions of it that sort of have been circulating prior to this. So one is this Ayala Francis Rosenblum paper and another one is, and there's some joint work of Nandlal and Wolf that also sort of points in this direction. But let me take a minute to sort of emphasize what this is telling you. So, oops, so there we go. So if you recall, there's this sort of homotopy hypothesis of growth and deek and the homotopy hypothesis, what does it say? It says that, well, if you wanna talk about topological spaces up to homotopy, then topological spaces up to homotopy can be dealt with algebraically in the form of infinity group voids. Nowadays, we tend to think of infinity group voids as just a simplicial set satisfying the con condition. And that's just now these days maybe even our definition of infinity group void, at least for some practitioners. And so this homotopy hypothesis is really just this equivalence of model categories between topological spaces and simplicial sets which goes back to con. So this homotopy hypothesis of growth and deek, what does it really permit you to do? It really permits you to say, oh, let's see, can I build up any linearly stratified space using iterated parametric joins where I can specify the homotopy type of the link as well? Yes, that's a good question. Yes, so the question was, so suppose that I have, so let me actually take a moment to answer that question because it's a really neat question and I wanna do a fair job of answering it. So suppose that I'm trying to build up a stratified topological space X and let me just do this with two strata just for a second just to give you a clear idea of how this thing operates. So suppose that I have my two strata X0 and X1 and I want to have X0 be the closed stratum and I wanna have X1 be the open stratum. And so the question is what other information do I have to give you in order to construct this stratified topological space here? And the answer is, well, I have to tell you about the deleted tubular neighborhood of X0 inside the thing that you're gonna construct. I have to provide you with the information that the link is, it's called. And so how does that work? Well, I have to give you some sort of information about something that I'm gonna call X01 and it needs to come with maps from X0 to X0 and X1. And so this thing is gonna be, it's gonna be the deleted tubular neighborhood of X0 in the thing that I'm gonna construct in X, in the union of these two things. And then well, what am I gonna do? I'm gonna put these two pieces together to give me this X, right? So what am I telling you here? I'm telling you that actually stratified homotopy types, another way to reconstruct them, I'm still just working with the stratification over P equals 01 right now. But stratifications over just P equals 01 are given to you precisely by diagrams like this. And now, so the question is, well, what happens if I take a more exotic post set? And the answer is, well, I have to choose a more exotic diagram and the exotic diagram I choose is by taking the subdivision of P op. And so if you take a diagram satisfying certain properties, basically a sequel condition, indexed on the subdivision of P op, then I can actually reconstruct the entire stratified space. And that's actually a key component in how Peter proves this theorem here. One of the things that he does is he doesn't just work with these categories with a conservative functor down to a post set. He also replaces these, he also considers at the same time what we started calling decolages over P. And these are functors from the opposite of the subdivision of P into spaces satisfying a sequel condition. And these things turn out to be the same. So that's how you can build up a stratified space by just taking the strata and all of the sort of iterated links. Do I have time to remind us what the sequel condition says? Sure, I got time for anything. Let's tell you. Yeah, so what does the sequel condition say? Well, so if I'm talking about a functor, maybe I shouldn't call it X. Maybe I called too many things X. If I've got a functor from the subdivision op to spaces, what's the sequel condition gonna look like? Well, let me tell you. So what are these things? The objects here are, they're linearly ordered subsets of your post set P. They're linearly ordered subsets of your post set P. And so to any sort of linearly ordered post the subset of P, I can associate the space, which is this guy here. And well, I've got a functor from SD op into spaces. So what does that entitle me to do? Well, it entitles me to write down a map from D applied to P zero through P in to D applied to any of the individual pieces. So this is P zero less than P one, right? So that's the link between the strata P zero and P one. And then I'm gonna talk about the link between the strata P one and P two. And I'll just keep doing that. Clark, I don't think we can see what you're writing at the bottom of the page. Maybe you can scroll. Oh, I see. Yeah, I see it. It becomes too far down. I'll try that again. Thank you. Thanks. Here, I'll just erase this and write above it instead. That's right. I always forget that there's that weird band that Zoom puts on everything. Thanks, Paul. So what can I do here? I can write a map down to D of P zero P one cross over D of P one with D of P one P two, et cetera up to D of P N minus one P N. And so I'm perfectly allowed to consider this map and the Segal condition says that this is an equivalence. And that's the Segal condition. Does the target of Hain's equivalence behave well as we change the post at P? Yes, it absolutely does. This equivalence is completely functorial in P. So I can write a map down to D of P N minus one P N. I've written this down with a fixed P, but there's a statement that works for all P simultaneously. And yeah, absolutely, yes. If you like, this is a fiber of a parameterized equivalence of homotopy theories. If you want to say it that way. Yeah, so this is what the decalage perspective allows. It says that if you want to understand these things by kind of decalage the idea of that word, there was that I was going to take the strata and the links and just sort of disconnect them all and then use that to be able to reconstruct the thing that I actually want, which is this infinity group order, this homotopy type, this stratified homotopy type. And this is telling you exactly what the rules are for regluing your thing from the individual pieces, the strata. And so what have I got here? Well, I've got, we had this homotopy hypothesis of growth than the, which says that spaces and infinity group loads should be the same basic data. Spaces considered up to homotopy and group loads should be the same basic data. And what Peter is telling you, Peter Hain is telling you is that stratified topological spaces should be a certain kind of infinity category. And in fact, this thing really is an equivalence of homotopy theories. By the way, for people who like this kind of thing, you might be amused to know that there isn't or doesn't appear to be an actual model structure on the left-hand side. It appears that you have to work with a semi-model structure in order to get this off the ground. But nevertheless, this does happen and it is a correct theorem. Okay, so what's our moral here? The moral is that just as homotopy types are infinity group loads, that's what the growth index homotopy hypothesis, stratified homotopy types are certain kinds of infinity categories with a conservative funk to a poset. So what does that give us permission to do? Well, growth index homotopy hypothesis gave us permission to say, well, if I wanna understand spaces, for example, if I wanna understand things like homotopy types of schemes, I'm allowed to construct those things in an essentially algebraic way by building simplical sets with certain properties. And so that's exactly how you construct the atoll homotopy type. That thing is constructed as a simplical set with this and that property. And so what we're doing now is we're asking ourselves, well, what happens if I introduce a stratification? So now the idea is that if I take a scheme and I wanna associate to it a stratified homotopy type, something that's capable of detecting its constructable sheaves, then I should be trying to construct that stratified homotopy type as an infinity category with the conservative funk to a poset. Now, interestingly, if you work with just a single poset, you end up with something that's got quite a lot of homotopy. But as David Corwin pointed out on Monday, it turns out that if you take all of the posets simultaneously and you sort of take the inverse limit over all those things, then you're gonna get yourself a profinite stratified homotopy type. And that profinite stratified homotopy type is gonna be exactly the Galois category that I defined last time. So now what's our plan? Our plan is to, and I'll get to the fun fact in a minute, the plan now is to pass to schemes. That's our next objective. So we're gonna pass to schemes. And the way in which we're gonna do that is we're gonna use this fun fact here, which is that if you've got a scheme, and for technicians out there, I mean coherent scheme, if you've got a scheme, then when you think about it, this is a risky topological space. That is a limit in the category of topological spaces. This is really happening in the category of topological spaces of posets with their Alexandrov topology. In other words, the Zyrzky topological space is really just telling you that you have a profinite poset on your hands. And so now what are we gonna do in the next lecture? What I'm gonna do is I'm gonna start defining the notion of a stratified, I'm gonna stratify the etal topology over the Zyrzky topology. And I'm gonna look at the corresponding exit path category and that will turn out to be exactly Gal. And that will be just what we need to prove all of the theorems, including the main theorem of the mini course. Okay, I think this is a good place for me to stop. So please ask your questions. I know you have them and I'm happy to answer. Okay, well, first we all wanna thank you, Clark, for a wonderful talk, so thank you. Mark Levine asks, oh, thank you. Mark Levine asks, do I have perverse sheaves in that category, in the category of Constructible Sheaves? Let me switch back to that page. But the answer is yes, let's see. My funny diagrams, it's gotta be here somewhere. I didn't state a theorem in this talk, didn't I? Okay, there it is, yes. Yeah, so there are perverse sheaves here. It's actually, this is one of the things that isn't written down yet, actually, interestingly, is how you sort of cut out the perverse sheaves. It's a fun exercise. The first step, as you might imagine, is to correctly identify the six functors from the sort of this functorial point of view, the point of view of this category on this side. And actually writing those down is actually with all their sort of available properties as a fun exercise, but in a sense, it's an exercise in sort of weird, pure category theory. Oh, Mark asked, in the general setting of a stratified topological space, yes, absolutely, yeah. You absolutely have a category of perverse sheaves there too. In fact, I mean, the answer is the same from this sort of exoddermy standpoint, I'm still just talking about functors of different kinds. Here I have to do it with some continuity conditions, but in the topological setting, I can work with just honest categories. And again, it's, you know, the six functors are defined in sort of categorical terms in that way. So Lemmy asks, what does the modifier cons in Hain's theorem mean? It just means conservative functors down to the poset. So let me go back to Hain's theorem and emphasize that point. Thanks, let me see, it's somewhere around here. There it is, yeah, so here I wrote cons and Lemmy asks, what's cons mean? It means conservative functor down to the poset. So I'm looking at an infinity category over a poset and I insist that that functor, that structure functor be a conservative functor so that the inverse images of points are all infinity group points. Is the category of all good stratifications always filtered for any manifold? Yeah, so you're saying the category under refinement as I take further and further refinements, that's right. So if I have any two stratifications, they have a common refinement, that's right. I'm absolutely allowed to do that. I said the excited me equivalence is a fiber of a parameterized equivalence. Yeah, I did say that. So I think the way to think about this is that, I have a category strat top, which is just all stratified topological spaces and all posets. And I have a forgetful functor down to the associated poset for the stratification. And on the other side here, I have, well, let's see here, I'll call these, maybe I'll call these EI infinity category since I used that term before. I could talk about these EI infinity categories. These are infinity categories that have the property that if I, that have the property that every endomorphism is an equivalence. And that has a forgetful functor down to the category of posets as well. These two are just equal. And the claim is that these things are equivalent in a way that preserves the poset structure like this. So that's the, and then if I take fibers over P, then I get the equivalence that I wrote down in Haynes theorem here. Do exit paths correspond to field extensions when X to spec K? So yes, that's right. So for X being spec K, well, it depends on what you mean precisely, but an exit path, if I'm just talking about spec K, then I only have one stratum to work with. So all of my exit paths are invertible. And what is that invertible thing? Well, if I've got two separable closures of my field, then I can talk about an isomorphism between them. So the only exit paths are isomorphisms of those field extensions. That's all I have. More generally, if I have two different points of my scheme, that an exit path is gonna be thought of as a specialization. And I'm gonna have that specialization relationship. So I'll talk about that in more detail next time for sure. Remy asks, how is the reasonable stratification hypothesis captured in Haynes theorem? That's a very fine question. Oh, wait, wait, sorry. There's a question before that. What is the problem with putting a model structure on the category of stratified spaces over a particular P? This, the problem is the following. This is hard. So the problem is the following. The problem is that it doesn't appear that there is a, how do I say this so that it comes out correctly? It doesn't appear to be a, you know, not every, you know, in the category of topological spaces, every object is fibrant. And we use that a lot when we prove the equivalence between stratified topological space or ordinary topological spaces and simplicial sets. In this context, not every stratified topological space is fibrant. And so you have to, roughly speaking, what happens is that if you want to do a fibrant replacement, you first need to do a kind of co-fibrant replacement. I can get into more detail about that at a different time. That's a very technical issue, but I can tell you what the issue is. It could be that there's a different model structure that we didn't think of, but it doesn't look good. No, but so Remy asked, how is the reasonable stratification hypothesis captured in Haynes theorem? And that's exactly right. So really what happens on this side here is that you have actually a model or semi-model structure on these things. And the fibrant guys include the reasonable spaces. And it's the fibrant guys that you show correspond to infinity categories with a conservative functor to P. So this is an equivalence of homotopy theories, not of abstract spaces or of abstract categories. So for a manifold M, if I take the limit of all of its stratified homotopy types for all good stratifications, what do I get? You get something kind of, it depends on what you do, what you mean by a good stratification, but you end up with something a little unsatisfying. So let me give you an example. So I suppose that I take a complex manifold. So I suppose I take a complex variety and I look at all of the finite quasi-compact stratifications of that complex variety. And then I think about the complex manifold attached to that complex variety. And I think about the limit of all those things. If I take the limit of all those things, then I have the map from the complex points of my variety with its topological structure down to the Zarisky topological space attached to the variety. So it's a little bit unsatisfying in that regard. Taking that limit is really, taking that limit without sort of taking the homotopy type first is actually a little unsatisfying as a process. And that's actually, there's a big frustration there that I can tell you about. Let's see, what else have I got here? Is there any new insight into Riemann-Hilbert-type correspondences in this language? Yes, I think so. So that's actually another thing that I'm working on with my current PhD student, Harry Gindy, my former PhD student, Jay Shaw, and well, anyone else who wants to be interested? Yes, I think so. It's, you know, one of the interesting issues there is that, you know, the Riemann-Hilbert correspondence includes this sort of very, there's an analytic part to that story in the usual Riemann-Hilbert correspondence. And what we're trying to do is we're trying to sidestep that really analytic part of that story. So I can tell you more about that if you write me. I'll be happy to talk about it. Is there a version of Whitehead's theorem for stratified homotopy groups? Well, so what there is, is there's a Pocnikov Tower for these things. So I can tell you what it means to be connected or truncated as a stratified homotopy type. So for example, being truncating, being in truncated just means you're in N category. And so there is a Pocnikov Tower and it does converge. How excited that makes you, I'm not really sure, but there is a convergent Pocnikov Tower at least. Or the theorem that a map is an equivalence on fundamental group points and cohomologies. Yes, right, so right, so there's this fact that if you have a space, any sort of space and you have a map of spaces X to Y and it gives you an equivalence on an isomorphism on the fundamental groups and on the cohomologies of both spaces with all local systems coefficients, then you have an equivalence. And the same thing is true here. The equivalences in these stratified topological spaces are equivalences of categories of constructible sheaves. I guess I should say infinity categories of constructible sheaves. So in other words, yes, that's the short answer to that question. Let's see, someone asked, what if I take the limit of the homotopy types? I'm not sure if I follow. Oh, this is in reference to the Zariski topological space thing, right? So you can do that, you can take the limit of the homotopy types, but really what you should do is you should first profinitely complete and then take the limit of the homotopy types. And if you do that, you'll actually recover gal of the original complex algebraic variety. So that's a sort of stratified Riemann existence theorem. And that's in our paper on exotropy. Does this perspective allow one to write down a Dirom complex for a manifold with corners? That sounds like a good question, Sean. I don't know how to answer that question. No, that sounds like a great question. Unfortunately, I don't really know the answer. It sounds really interesting. Oh yeah, Peter Hain is reminding me that another form of Whitehead's theorem is that a map of stratified topological spaces is an equivalence, if and only if it's an equivalence on all strata and all links. And so that's another way to sort of characterize the equivalences on the left-hand side here. Yeah. Yeah, so Peter Hain's in the chat. He should answer your question better than, he can answer your question way better than me, Michael. It's a, yeah. Other questions? I know you have them. So you know you have Pocnikov towers. We could talk about Eilebert-McLean spaces, right? So, well, I have Pocnikov towers, but I don't have fibers so very much. I mean, I have fibers, of course, but they're strange, right? I mean, I don't know how to take, what do I mean by taking the, you know, I have the sort of, I have layer N and layer N minus one, but what does it mean to take the fiber of that? I could take it at a chosen base point, but that's not gonna be good enough for the purposes of writing down something, like a homotopy type invariant, like a pi N or something. So I don't know how to address that. Might it be interesting to see how the etal topology is stratified over the Nizhnevich topology? Oh, yes, yes, it is very interesting. So, yeah, maybe I'll, just as a little preview, just because I love this example. Let me give you just a quick example just to show you something kind of fun about this. So let's take, this is my favorite example. So I'm gonna take the nodal cubic, and I'm gonna take the local ring, the usual local ring at the node, and I'm gonna think about the Galois category with that thing. And the Galois category, it depends on what field I'm working over, but it's pretty easy to write down, you can understand it. But let me go further and actually think about something that you might call the Nizhnevich Galois category. So the Galois category has to do with the etal topology. I'm really taking the points for the etal site, but I could try and take the points for the Nizhnevich site instead and work with that. And it's very nice to see that what you end up with is, well, you end up with the node point, and you end up with the other point, and you end up with two maps that go just like that, because you have two specializations. If I take the Hensel local ring at that point, I have two specializations from the node to the generic point of this. And so what does that mean? That means that I've got two of these maps here, two specializations at the Nizhnevich level. So this is Gal Niz, as I like to call it. And, well, if you take that space and you geometrically realize it, what do you get? You get exactly the circle. You don't get the profinitely completed circle, you actually get the circle. And so this is a situation that I find very appealing, because, well, this is just a finite category. It's finite in all the conceivable senses. It's a very finite category, but it produces an honest S1. It doesn't produce the profinitely completed S1. And so that's why you're getting the actual fundamental group, not this profinitely completed fundamental group out of this kind of example. And yeah, there's a whole lot to say. I mean, the difference between Gal Nizhnevich and just the Zyrissky topological space of X is entirely about the question of whether X is geometrically unipringe. So there's all kinds of fun questions in here that you could ask yourself. Yeah, I think there's a whole interesting world here that I haven't thought about very much yet, but it seems promising. Is there any kind of story for the FPPF topology or Lise et al? So for Lise et al, yes. So if you're thinking about stacks, then there's a story for stacks. There's a Galois category attached to any stack. And that's really just there's nothing kind of deep about that. I mean, you're just going to take the stack and present it as a simplicial scheme and then take the Galois categories of those and take the cold limit. So it's a little dull in that regard. For FPPF, I don't know. I mean, I think if you take the, I don't know whether to expect something new for the FPPF topology. I don't really understand. I mean, so part of the story is really about the et al topos. And one of the advantages that we have is that we know very well what the points of the et al topos are. And the points of the FPPF topos are a lot harder to understand. So there's a paper by, I'll probably mispronounce this name, but it's, I guess it's Shoeir, Stefan Shoeir, about the points in the FPPF topology. And they're funny. So I don't really have anything interesting to say about that, but that's a good question. I do think that thinking about other topologies might be promising. The et al topology is quite special because it has the et al topos is what we call a spectral topos. And that's the analog of this condition right here, which is one of the reasons that I wanted to, oh, my pen has stopped working, just in time. This condition right here in the topos theoretic world is what we call a spectral topos. And that's quite special. But I don't know, I don't think the FPPF topos will be. Is there a theorem that shows that the et al topos is the finest one that is spectral? Now you're asking hard questions. I don't see why. No, I wouldn't. I can't, I guess you mean like the finest subcanonical topology that's spectral or something. I don't see why that should be. I think it might, there could be other things that are special, that are spectral. I don't know. For Gal niz, is that et al fibered over niz or niz fibered over czar? It's both. So if I understand your question correctly. So what you have is you have this sequence of maps. So you have czar, which just for the minute, let's just pretend like it's a poset. So you have czar, which kind of looks like that in this case, because I took a local ring of a curve. And then you have gal niz. And then you have gal x, which sort of has all the same stuff that gal niz has, but it also has automorphisms corresponding to the absolute Galois group of the fraction field. Do I get the SGA fundamental group from gal niz? I get the SGA fundamental group from gal x, but not necessarily from gal niz. Gal niz is too coarse to know anything about, say, complicated field extensions or arithmetic. I'd have to pass all the way up to gal. But then absolutely, yes, I get the SGA fundamental, SGA 3 fundamental group. That's right, the non-profinite one, yes. So that's actually a theorem of Peter again, that if you take the, and I'll talk about this next time as well, that if you take the Galois category, which is a profinite category, and you take its nerve in the protrunkated space sense, then you actually recover the protrunkated Homo tapi type. And that's one of the many excellent theorems that Peter proved in Exandria. Doesn't seem to be more questions. We should just all thank Clark again for a wonderful talk. And the next talk is at 6 PM.