 Yeah, so thank you again to the organizers and to the Iyasha S. It's been really wonderful. I mean, you know, it's very difficult for me to extract myself from bed at 4am, but it's really wonderful that the Iyasha S has made these sort of videos available for me online and so that I can watch them later on and reflect on them. So it's been, that's a really wonderful contribution and I'm very grateful. All right, so I've repeated the theme here. I'm kind of aware of the fact that it's less of a theme and more like a five note melody and a Philip Glass piece of music. I've just been repeating it again and again and again. But I'm going to say it again because it seems to bear repeating that if you take a suitable category of sheaves, that's going to determine for us a homotopy type and then we're going to use that homotopy type to recover the category of sheaves. And we saw this in the first lecture. We saw this with the story of monogamy where we're sort of extracting a certain kind of locally constant or lease sheaves. And then in the second lecture, we started to see this with what we were calling exoddermy for topological spaces. And so the story is now that what we're trying to do is we're trying to construct a stratified homotopy type attached now to a scheme, X, that will do for X what the exit path category did for a stratified topological space. The exit path infinity category will do for a top a general stratified topological space. So what's happened so far? Well, I've shown you how things work in the sort of topological setting or at least I've outlined how they work in topological setting. I've constructed for you the exit path infinity category of a stratified topological space as long as that stratified topological space is sort of well behaved and well behaved. Includes things like if you take manifolds with singularities and you take a stratification where you stratify out each singular locus of each piece from the previous stage and you sort of produce a stratification that way. Then this is well behaved enough for this exit path category to work. And what does work mean? Work means that the exoddermy theorem is true, which is to say that constructable sheaves are the same things as just functors from that exit path category into whatever your target category is. So now we want to do the same thing with schemes. And the question is how do we do that? What do we how do we orient ourselves? And the fact that I want to kind of use to orient ourselves is one that I stated last time, but I'm going to repeat it here. So we're going to allow X to be a scheme. And again, for the technicians in the room, I mean quasi-compact and quasi-separated. All of my schemes are going to be quasi-compact and quasi-separated for the entire of the lecture. There are things you can do to get away from the quasi-compactness, but I don't want to do anything to get away from the quasi-separatedness. So this is just going to be a fact of life for us for the next little while. And so what's the fact that I want to share? The fact that I want to share is that the Zariski topology is actually the limit of its finite quasi-compact stratifications. That is to say that in the category, this is happening in the category of topological spaces, in the category of topological spaces, the Zariski topology is the limit of the finite quasi-compact maps from XR to a finite poset P. And so what does that mean? That means that if you want to contemplate constructible sheaves relative to a single poset, you're allowed to do that inside the world of algebraic geometry. But you could also just sort of pass to the limit. And that limit will actually give you all of the information that the Zariski topology has to offer. In other words, the Zariski topology is nothing more than a pro-finite poset. And by the way, I should mention something. This is due to Hoxster and his thesis. If you've never looked at Hoxster's thesis, it's kind of an amazing read. It's a very, it's an amazing read. OK, so I'm going to now use this definition. And in an algebraic geometry, what happens is that you don't work with a fixed finite stratification. Instead, you try and work with all of them simultaneously. So you're going to take this limit over all possible stratifications. And you're going to combine them all into a single object that we call the category of constructible sheaves. So let me define all the relevant terms for you now. So again, X here is going to be our friendly neighborhood scheme. And we're going to be looking at sheaves. And we're going to be looking at sheaves for the etal topology. And a tall sheaf F. And these are going to be valued in spaces. So when I say sheaf, I mean sheaf in the sort of homotopy theoretic sense. And these are going to be valued in the infinity category of spaces. And it's going to be declared to be lease if it's locally constant with pi finite stocks. So let me break that down. So I mean that there is an etal covering of my scheme and a bunch of pi finite spaces, one for each member of that etal covering. Such that, well, when I take the pullback of my F to each Ui, I get the constant sheaf at the corresponding Ki. That's the thing that happens. So I have an equivalence like this. And I don't specify the equivalence as part of the data. It's just stuff that I'm capable of doing. So this is locally constant, but it's locally constant with pi finite stocks. So I require that my stocks have only finitely much homotopy. So after some stage, I only have zero homotopy groups. And furthermore, each of those homotopy groups is itself finite. And pi not is a finite set as well. OK, so that's what I'm going to mean by a lease sheaf. So what's a constructible sheaf? Well, OK, so first let me tell you what the classification theorem is for lease sheaves. The classification theorem, I mentioned this in the first lecture, is that if you have a scheme, then the lease sheaves for the etal topology, that's what this is. This is the category of lease sheaves for the etal topology, are the same things as functors from a certain infinity groupoid. And in fact, it's not just an infinity groupoid. It's even a profinite infinity groupoid. And I'm going to look at functors from that profinite infinity groupoid to spaces. But of course, because the source is profinite, I have to mean continuous functors. And at this stage of the conversation, continuous just means that, well, if I write my profinite infinity groupoid as an inverse limit of finite infinity groupoids, pi finite infinity groupoids, and then the functors just means the co-limit of the corresponding functor categories. And that's all that's happening. OK, so that's the story when I'm just trying to pick up lease sheaves. Why include art in Mazer? I don't know if that's a pun. I'm sorry, I don't know what that. Art in Mazer are the authors of the text that I'm describing. If you can, I'm not really sure what to answer with that. Sorry, I'll press on. So OK, so what's the definition for constructible sheaves? Well, the idea here is that, well, lease sheaves aren't very well-behaved under functors that we might have lying around. So for example, if I want to talk about push forwards along open immersions or something like that, then we can talk about push forwards of locally constant sheaves, but they won't be locally constant anymore. OK, so let me write down the definition. Let XR to P be a finite quasi-compact stratification. So remember, so I'm working with only those kinds of things. Then I'm going to look at a tall sheaf, F, and I'm going to say that it's P constructible. If for any element P, the sheaf, F restricted to the stratum, XP, remember the inverse image of little P under this map, is itself lease. So along the strata, I have locally constant sheaves. And so P constructibility means that I just am locally constant with pi finite stocks along each of the strata. And we say that a sheaf is just constructible with no reference to a particular poset, is just constructible. If it's P constructible for some finite poset P. Harry asks, if we take the category of sheaves on the site of lease sheaves, what is the infinity category of points, functors from spaces to the effective for sheaves for the effect of epimorphism topology? I think he's referencing the fact that, well, there's a way to describe this profina infinity groupoid that's pretty efficient. And this is sort of a glimmer of the future here. So if we take this profina infinity groupoid, how could we describe it? Well, one concrete way of describing it is as follows. We could take this category of lease sheaves. That isn't itself a topos or anything. There's just not nearly enough limits and co-limits for this to be a topos. But we can kind of generate a topos out of it. So if you take the topos generated by it, and I guess Harry wants me to write sheaves for the effect of epimorphism topology, well, then what can we do? We can look at the category of points for that, the infinity category of points for that. That infinity category happens to be a groupoid. And that groupoid happens to be, well, I should be a little more careful. So this is the infinity category of points. And the thing that I actually get out, well, is this. But without its topological information. So let me emphasize that I'm getting this without its topological information. So I take the materialization of that profinite stratified space, or sorry, profinite space. We'll get to stratifications in a minute. OK, right. So a sheaf is constructible for just constructible full stop if and only if it's P constructible for some finite poset P and some quasi-compact finite stratification. And so what does that mean? Well, that means that when we talk about the category of constructible etol sheaves, we're actually just taking the co-limit of the category of the P constructible sheaves over all finite posets P. So we're just going to take that co-limit and see what we get. And now, so what's the theorem? The basic theorem here, and this is the theorem of me and my former student, Saul Glassman, and my current student, Peter Hain, that if we fix a finite quasi-compact stratification, if we fix a finite quasi-compact stratification, then there's a profinite stratified space. And I emphasize the word stratified. So remember, how are we thinking of stratified spaces now? Well, in the previous lecture, Peter Hain's theorem gave us permission to think of stratified spaces up to homotopy as infinity categories along with a conservative functor to a poset. So I'm going to do that. Right now, when I say a profinite stratified space, here I'm taking a infinity category with a conservative functor to a poset. In particular, that source there, that pi infinity x that over p, that's an infinity category. And it has the property that, well, it has a bunch of objects. Those objects go to things in p. But if I have a morphism that goes to the identity of one of the objects in p, that morphism will have to be an equivalence. So the fibers are all infinity group voids. And what's the rule going to be? The rule is going to be that there's this profinite stratified space with the property that the continuous functors from this profinite stratified space are the same things as p constructible sheaves to the etal topology. So that's the way this profinite stratified space is going to be set up. And so how do we prove this? So if we have one of these finite constructible stratifications, if we have such a finite constructible stratification, how do we produce such an object? So let me tell you. Basically, what we're going to be doing, oh, so Sean asks, can I explain how to get from x zar to p to x at over p? To x at over p. Did I define x at over p? I don't think I did. So x at over p right now, this is one piece of notation here. I didn't define something called x at over p. I'm defining the entire notation pi infinity of x at relative to p. So this is a relative notion of the etal homotopy type. So before we had the etal homotopy type, which is given to you by some infinity group void, a profinite infinity group void. And now I'm doing something kind of strange. I'm taking that, but I'm taking it relative to a map to a finite poset p. So this is a relative form of the thing that already existed. I haven't defined anything that's by itself called x at over p. And I'm not going to. There is something that you can define, but I'm not going to. Instead, what I'm going to do is I'm going to say, well, I'm trying to capture the p-constructible sheaves. And so I'm going to define this object. And if you like, I could even define it using this representability condition. That would be a really annoying way to define it. I'd like to be more precise. But that is a valid definition. And so I'm trying to define this entire object. I'm trying to define the single object. And now the observation about czar being the limit of its finite stratifications, what that's supposed to suggest to you is that if I do this well enough for each p and I take a limiter co-limit along the p's, then I'll get what I want. So I'm going to give you this profinite stratified space for each p. And what I'm going to try to do is I'm going to try to do a good enough job of it so that if I take the limit over all the p's, then I'll end up with a profinite stratified space that classifies all constructible sheaves. That's the program. So first, I have to work with a fixed poset. And then I'm going to let my poset vary. So first things first, I need to define this gadget so that this equivalent holds. So it's excited to be equivalent holds. OK, so what's the idea here? Well, this is something I mentioned last time, which is where if you're trying to think about a stratified space over a fixed poset, it's often convenient not to think about the entire structure of that infinity category, but to break it up into pieces. And those pieces are the strata and the links. So I need to be able to tell you here what the strata and the links are in this p-stratified infinity category, or p-stratified profinite homotopy type. And so I need to tell you what the strata are and I need to tell you what the links are. And what we know basically what the strata have to be, the strata have to be the things that control the least sheaves on the corresponding stratum of my XP. And so that's exactly what I do. I'm just going to have this be the homotopy type of the corresponding stratum of XP. So that's fine, but then the question is, what do you do with the link? And that's actually the hard problem that we solve in the paper. This is in some sense, the very heart of the issue. And the heart of the issue is that well, we need to understand how to pass from one stratum to the next in a way that respects all of the information that the etal topology is providing us with. And the answer to this question goes back to a really remarkable construction. And I guess it was originally written down by Girod, but it was really implemented for the first time by Deline. And it's the oriented fiber product of Topoi. The oriented fiber product of Topoi is Deline's purpose and it was to sort of understand how vanishing cycles work in higher dimensions than just one. So in one dimensional situations, you can write down a good definition of vanishing cycles in the etal topology with sort of schemes that you already have or formal schemes in some case that you already have lying around. But if you want to do this in higher dimension, you no longer have access to that. And so you have to use something more exotic and that's where this oriented fiber product comes in. And so what's remarkable is that if you have the following situation, let me just add a page here to sort of indicate the situation. The situation is the following. The situation is that we're gonna have an X and we're gonna break it up into two pieces. It's gonna have a closed piece and a complimentary open piece. And this is the situation that's sometimes called, in the world of the etal sheaves, this is sometimes called a recourement. But the question that we need to answer here is how do we define, how do we take these pieces Z and U and reconstruct X from some additional piece of information? In other words, how do we construct a link that's gonna be doing the right thing? Someone asks, but that profinite space is not constructed with the Tanakhian kind of argument, right? Like showing that XP constre is of some kind, hence there is a profinite space such to the equivalence. I'm so sorry, I don't know if I understand. All of these things, I don't know if this is gonna help or not. Let me try my best. So all of these things are constructed with a Tanakhian kind of argument. What I'm saying is that you can break up this sort of hard question of how to construct a piece stratified space into the questions about how to construct the strata and the links. And for each of those pieces, you could just use ordinary homotopy theory. So here, this is really honest to God, just the ordinary etal homotopy type. And this is, well, it's the homotopy type of a topos. That topos isn't just the etal topos of the scheme in general, it's something more exotic, but we can say precisely what that exotic thing is. And that's what I'm trying to do right now. So I'm trying to show you what that exotic thing should be. And then my claim is that because of the sort of magic of stratified spaces, you can take a stratified space and decompose it into its constituent pieces at strata and links. And analyze those and then reassemble the result. And that's what we're doing. So we're taking it apart, we're analyzing it and we're reassembling it. And those are the things that we're doing. And when we take it apart, we can actually analyze it using traditional tools. That's the whole point is that here, we're really just taking the sort of homotopy type of a topos or infinity topos. So the question is, how do we do this? How do we actually extract these kinds of pieces? And I just wanted to give a special case of this in the situation where you have a closed subsets, a subscheme of X and a complimentary open subscheme. And the question is, if I just had Z and U, what other information would I need to give you in order to reconstruct X? That is to say, well, I'd like to produce X down here or at least the atol topos of X. I don't really care if I construct X or just it's atol topos. It's not, I'm not gonna notice the difference for the purposes of this homotopy type. I wanna reconstruct this piece, but from something back here. And while that's something, isn't just some kind of arbitrary question mark, it sits in a diagram like this that doesn't commute, but it commutes up to a two natural transformation that goes like this. And the remarkable fact is that in the world of infinity topoi, if you write down the universal gadget that makes this diagram exist, then you've written down the correct thing. So in other words, I wanna write down here the deleted tubular neighborhood of Z in X. And what that turns out to be is it turns out to be the universal thing that occupies that blank. It's the oriented fiber product of Z and U over X. And so really a huge amount of our work in exotomy is in analyzing these oriented fiber products. In particular, one of the sort of remarkable theorems of Hain, maybe I'll just mention this now. One of the remarkable theorems of Hain is that oriented fiber products satisfy a good base change condition. So there's a base change theorem for oriented fiber products. And this it turns out is absolutely critical for the entire development that we produce. Question, with taking a formal thickening of Z and then taking a more classical fiber product, not work. That's right, it would not work. I could show you precisely why some of their time, but yes, that's right, it doesn't work. There isn't a, at least to my knowledge, there isn't a classical way to thicken up Z and then remove Z from that thickening that works in all cases. So one example of this kind of phenomenon, well, so that there is something you can do in the case of curves, let me emphasize that. So in the case of curves, you really can do the kind of thing that you're suggesting. So for example, if I take spec Z and I take P, then I can thicken spec Fp to spec of Z localized at P or Zp, and then I could take the complement of the Fp, spec Fp and that, and that will give me the right answer for this WD-2 different approach. But such moves are not always possible. Do you and Z have to be complements of each other? Or is there some more abstract formulation? Like do you require the homotopy pullback square also be some sort of push out? So right, so they, for a gluing square of the type that I'm saying, I am talking about the situation which are complements. So the situation I'm trying to get you to think about is a situation in which you're stratified just over the poset zero to one, which is a rather mundane poset. And the fiber over one is U and the fiber over zero is Z. And then I'm trying to figure out how to reconstruct the whole stratified thing from just the piece. And so in that situation, yes, this is both an oriented pullback and an oriented push out square. But you're right that in the other sense as well that there really is a definition of an oriented pullback for any two maps of infinity topoi. That's right. So the stratification is precisely sort of controlling that open closed complement data. And the whole sort of yoga of the sort of decollage type perspective is this functor from the subdivision op to whatever your target is spaces, I guess. That whole yoga is that you could take something that's stratified over a big general poset and reduce it to its constituent pieces which are the strata, which are just singletons and the links which are things that are indexed over post sets that look like that. And so that's our whole goal here is that if we can understand well enough and in functorial enough and away the strata and the links then we'll be in good shape and we can continue. What other questions are there? Oh, there's a few, good. Is there a version of the oriented fiber product construction for motivic spaces which are not topoi? So in order to do that, I mean, you really need to know what a two morphism is to make a meaningful oriented fiber product. So if you have access to a non-orientable, a non-invertable, not orientable, a non-invertable two morphism in motivic spaces that yes, I'm not aware of such a thing, but if you know something, yeah, you have it. Can I state precisely the universal property of the oriented fiber product? Can it be understood as some lax limit? Yes, absolutely. So what I'm saying is that if you've got a morphism of topoi, let's see, maybe I better just keep the same notations. If I've got morphisms of topoi that go like this, then the oriented fiber product is the universal thing with morphisms W to U and W to Z and a two morphism going in that direction. So it's a, oh, Harry Gindy says it's not a lax limit, it's a weighted limit. I'm not, okay. Is this just the comma category? I never know what people mean by comma category, so I assume so. It's the thing that I just said, it's that universal property. So I think there's a whole lot of language that category theorists go back and forth about what? Yeah, oriented fiber product seems clear enough to me. I mean, the thing with that universal property. Okay, so the point that I'm trying to get at is that understanding this in the category of topoi in a very explicit way is a really powerful objective. And actually, this is why it's so important in our story. It is exactly providing the links. And this in particular is exactly providing the deleted tubular neighborhood of a closed inside some general scheme. And having access to that is a really important, having access to a tall sheaves on that is a really important tool. And that's actually what we're gonna be doing. Okay, so what's the story here? So the story here is that we're gonna take our scheme, which is stratified over a finite poset, and we're gonna break it up into pieces. And we're gonna break it up into pieces that consist of the strata and the links that connect two different strata. And the strata are gonna be described in terms of the usual at all homotopy type of the corresponding strata of the scheme. And the links are gonna be understood in terms of this oriented fiber product. It's just gonna be the homotopy type of the oriented fiber product. Does this fiber product actually depend on a choice of compactification X of U? Well, it depends on this whole piece of the diagram. So yeah, you need X as well. Yeah, absolutely, absolutely. I mean, you know, yeah, totally. Yeah, and so now that that's a good story for a finite poset P. So then what happens if I want to sort of take the limit over all of these posets and try and understand the general category of constructible sheaves. So let's see how this works. So I've got this sort of basic result that I can understand P constructible sheaves for a fixed finite constructible, fixed finite stratification P. And more general constructible sheaves are formed by taking the co-limit of these categories of P constructible sheaves. It's the union of P constructible sheaves. So I'm gonna do that, but I have a formulation for what this thing means for each individual poset. And so I'm gonna take the co-limit over all these things. And then, well, when I say continuous, what I exactly mean is that I can push that, pull that limit out as a co-limit or push that co-limit in as a limit. And so then that's gonna define for me this thing here, which is now a pro-finite stratified space. And I'll just call this pi infinity X at XR. So at the moment I've solved the problem. I've found a pro-finite stratified space that definitely classified as constructible sheaves. But if you remember from the first lecture, I give you a very explicit description of a category that I called gal. And I said that that was gonna be our answer. So what does this have to do with the category gal? And that's where analyzing these oriented fiber products becomes so important. So this is a pro-finite stratified space. So I regarded over the pro-finite poset XR, which is something that I'm allowed to do. And so let's look at the links and the strata. Or sorry, strata and links. I don't know why I wanted to do it in the wrong order. Oh, there's a question. Pi infinity X at P was already pro-finite. Yes, it was already pro-finite, but I'm taking the limit in the pro category. So I can have, if you have a pro-object or pro-objects and I take the limit of the whole thing, I could just have a single pro-object. There's nothing, no danger in that. Okay, so let's look at the strata. So I'm gonna look at the strata now, but I'm gonna look at the strata of this guy. So this is mapping down to XR. So I'm gonna take a point of XR. So it's a point of the underlying couple logical space of my scheme. I'll call it X zero. And well, what do I get when I form this thing? Well, I should see the homotopy type of spec of the residue field of that point. And that homotopy type is exactly the classifying space of the absolute Galois group of the residue field. So in particular, it's just a group void. It's just a one category. Okay, so what happens for the links? The links are a little more complicated to understand, but what happens is that you start looking at oriented fiber products that look like this. These oriented fiber products, they're trying to, so the situation is that sort of X naught is an element to the closure of Y naught. This is inside the Zariski topological space of X. And I'm trying to form the oriented fiber product of these two things over X. And this turns out to just be the space of specializations geometric points X and Y that cover your fixed X naught and Y naught. And so the link here is really the sort of category of, or not category, but group void, of specializations of geometric points that cover your favorite Zariski points X naught and Y naught. What happens? So what's the upshot of this? The upshot of this is that this sort of very complicated looking profite topological space or profite stratified space becomes reduced to something that's actually just a single one category with some additional topology with a topology. And by topology at this point, I don't mean anything terribly deep. You could encode this if you wanted to as a topology on the objects and morphisms of your category on the sets of objects and sets of morphisms of your category. Or you could choose to encode this just as a sequence of finite one categories that converge to the one category Gal X. Okay, so how does this work? Let's see some examples. Well, so the first example, I guess I could have put it in here. I could have emphasized that Gal of spec K is just the classifying space of BGK. That's a fun story. But the second, I think, example is a little more indicative of what's going on here. So let's look at the situation which A is a DVR. And I might be interested in a picture of Gal of spec A. So A being a DVR, that means that when I look at spec A czar, this is an exceptionally boring pro-finite stratified space. It's really just a finite stratified space. It consists of two different points, the residue field and the fraction field. And the residue field is an element of the closure of the fraction field. So that's it. It's just a Sierpinski space, as they say. Okay, so I've just got up to isomorphism these two points, the little spec little K and spec big K here in purple. And well, so what are the strata? The strata look like the following. They're on this side here. I've just got the classifying space of the fundamental group of the absolute Galois group of the residue field. And on this side here, I have the classifying space of the absolute Galois group of the fraction field. So I have each of these. And then I have to try and figure out what the space of specializations is going in this direction here. So I have to figure out something that sits between BG little K and BG big K. So a particular example of this, the yet sub example is if I look at Z localized at P. You know, on this side, I'll have the absolute Galois group of Q. On this side, I'll have the absolute Galois group of FP. And in between, I have something. So what should that something be? That something should be the absolute Galois group of QP. And more generally here, what we get in this blob here is we get absolutely the classifying space of the decomposition group sitting inside GK. Now notice that the decomposition group itself, as you might recall, is only defined up to choices, right? It's only defined up to conjugacy inside the absolute Galois group GK. But the map from BDA to BGK, that's a completely canonical map. There's no base point issue there because we're just ignoring base points when we take the classifying space. So this map is induced by the inclusion DA into GK. And this map here is the map that identifies the decomposition group, not the inertia group with the absolute Galois group of the residue field. So Stephen McCain asks, for knots and primes, the tubular neighborhood of the knot is the link from the knot stratum to the knot complement stratum. And I recover the tubular neighborhood algebraically in terms of the oriented fiber product. Am I understanding this correctly? Boy, are you ever understanding this correctly? That's exactly right. Yeah, that's exactly the picture that's being described here. So that's exactly right. So we have the absolute Galois group of Q. So this is the sort of, this is the situation of the single knot, right? I've got the absolute Galois group of Q, which is sort of a bulk. And inside that bulk, I have a single knot which is given to me by BGFP. That's the circle, the profoundly completed circle. And the deleted tubular neighborhood is exactly this guy up here. It's BGQP. And that's that two dimensional deleted tubular neighborhood. So it's a knotted torus, or at least it would be if we didn't have all the orientation problems that we have. Yeah, that's exactly right. Good, so this is the situation. This is the kind of category that we're extracting from this thing. And so what's the theorem now? The theorem is that the entire category of constructable sheaves for the etal topology can be described as these continuous funkers from Galax which now has a pretty explicit description to the category of pi finite spaces. So that's just with pi finite coefficients, but you can easily extend this to the situation of, if you take a finite ring lambda here, you can look at constructable sheaves valued in perfect complexes on lambda. And those are gonna be the same things as continuous funkers from Galax to perf lambda. So remember the story that we were telling ourselves is that we wanted to have this kind of exoddermy, but we wanted to have it for all different kinds of coefficients, a whole wealth of different sorts of coefficients. And so this is the story for finite things. This is the story for finite rings. But the question is, what happens when we want to extend this to sort of more exotic coefficients? That is, we don't just wanna think about things like torsion coefficients when we think about etal topology. We wanna think about coefficients and things like ZL and QL. QL in particular is a really important example because that's what relates to the Betty numbers of your favorite algebraic variety, say, over a finite field. So the question is, how do we pass to coefficients like ZL, QL or QL bar? And along with that is the fact that these rings here they have topologies. And in this story, if you look, I'm constantly using the fact that I have access to a category. And I'm talking about the continuity from Galax into that category. So somehow the topology of the ring needs to give me something that resembles a topology on perf lambda. And that topology on perf lambda needs to interact with the topology or the profinite structure if you like on Galax. And I need to be able to speak about continuous functions from Galax into perf lambda. And so the question is, how are you gonna do that? So how do you take the infinity category of perfect complexes and endow it with something like a topology? And for that reason, Peter Hain and I produced the theory of picnotic structures. And at around the same time, Dustin Clausen and Peter Schultz produced the theory of condensed structures. And the definitions are roughly the same. For our purposes here, they're exactly the same, but there are some slight set theoretic differences. So let me tell you what a picnotic thing is. It's quite easy to define, although it's kind of difficult to work with if you're not prepared. So let me just say what it is though. So a picnotic object of an infinity category C is it's a sheaf on the site of compact house door spaces, compact house door topological spaces. I'll call that category comp and I'll often refer to these things as compacta. And sheaf, well, what do I mean by sheaf? Well, I guess I mean sheaf for the epimorphism topology. So cover is just gonna be something that is an epimorphism inside comp. However, I need to caution that I really don't just mean sheaf, I really mean hyper sheaf because here I'm talking about things that are landing in infinity categories. So I really need to be able to talk about not just descent, but in fact, a hyper descent. And that turns out to be a key issue in this case. There's a big difference between the category of sheaves and the category of hyper sheaves. And so we have to be cautious about that. Okay, so this is fine as far as it goes but what are some examples? How can you relate to this thing? So let me just emphasize for a second that well, this is just a different way of building up things with topological structures for the ones that you might be used to. You might be used to sort of building up topological spaces as kind of co-limits of compact pieces that are growing bigger and bigger and bigger. The kinds of topological spaces you get from that are called compactly generated topological spaces and they include most of the topological spaces we think about day to day. In this situation, we're also gonna do the same thing. We're also gonna build up things out of co-limits of compacta, but the co-limits are gonna be described in a different way and in particular they're gonna be described in such a way that this works for any target infinity category C. Okay, so what am I telling you here? I'm telling you that if you just give me a topological space, topological space Y, then if I take maps from blank into Y, that gives me a functor in the correct direction from comp up into sets and this turns out to be a pick not accept. So that's all well and good, it's perfectly, and this works for any topological space all no matter how absurd. But if you wanna restrict yourself to sort of the more respectable topological spaces that compactly generated topological spaces, then we actually got a fully faithful functor. And why is that? Well, that's roughly because if you want to know what it means to map out of a compactly generated topological space, it suffices to know what to do for all the compacta that map to your compactly generated topological space. And so that's why this becomes a fully faithful functor in the pick not accept. So this category of sheaves or of hyper sheaves on comp actually already includes all of the topological spaces that we think about on an ordinary basis. It includes all the compactly generated topological spaces. So that's good. So it's containing most of our friends. But now I'm permitted to do all kinds of constructions at a very formal level here because this is just a category of sheaves. This is really just nothing more than a category of sheaves. So I can really work with this thing in a very explicit fashion. Okay, so let me give you an example of another way to construct a pick not a copjects. So let's suppose that I have some random infinity category, C. Well, the first thing I can do is that if I have a pick not a copject, I can take its underlying object of C. And that just means that I'm gonna take this thing. This is a sheaf on the site of compacta and I'm gonna evaluate it on a point. So that's gonna give me an object of my infinity category C. And that's what I call the underlying object of T. Sean asks, is there a theorem that says that pick set is a co-completion of these old fashioned convenient categories of spaces? I mean, it is true that every pick not accept can be written as a co-limit of compacta, right? If I have a, I mean, and that's just because every pre-sheaf on the category of compacta can be written as a co-limit of compacta. But I think the key point here is to understand how those constructions are done, how that co-limit works. And that co-limit works slightly differently from what you might be used to in topological spaces. It's a more kind of formal construction. Yeah, I'm not sure if I've answered your question precisely, but it is a co-completion in that sense. Do we think of comp as a one category when we think of a pick not a object in infinity category C? Yeah, yeah, absolutely. I mean, it's comp is still just a one category. I'm not doing anything fancy there. I'm just treating it as a one category in the usual sense. We're something like Dougher's universal homotopy theories result. Pick set is gotten by joining certain constructions freely. Yeah, I mean, maybe I can say something about that. So let me see, I think I'll just add a page. Yeah, maybe this is what you're getting at, Sean. And I'm not sure you can tell me if I'm not from staring you wrong here. So, but, you know, so what I said was that these things are sheaves valued in whatever on the site of compacta. So I can be a little more efficient in my description. In that case, this is roughly how we actually really work with it. So inside the world of compacta are very, very small objects which are the projective compacta. These are the compact house door spaces they're all totally disconnected and they have the property that the closure of any open subset is all is still open. So these are very, very unusual topological spaces. So the stone check compactification of any set is an example of one of these projective compacta and retracts of such things are projective compacta. And that's it, that's all you get. That's all this category is. So it's a very strange category but they're projective objects in a very serious sense. They're really projective objects in this category of compacta. And so what's the story here? The story here is that we're gonna look at the projective compacta and we're gonna look at functors from there into sea. And it turns out that the sheaf condition here actually becomes a really, really simple condition here. So what am I telling you? I'm telling you that pichnotic objects of our sea are the same things as functors from proj op to sea that carry finite co-products of these projective compacta to finite products. Okay, so what does that mean for the purposes of Sean's question? What that means for the purposes of Sean's question is that if I look at pichnotic spaces now you can think of this as the non-Abelian derived category derived infinity category of proj. So if you like, it's really that I'm taking proj and I'm adding on all sifted co-limits in a completely free way. And so that's really the kind of construction that we're doing here. It happens to have this nice way of describing it in terms of comp, but in actual practice this is actually the description that we use a lot. And so it's that kind of formulation that we're using. Yeah, these are good questions. Some of them are too hard for me. Okay, so if we have one of these pichnotic objects of the infinity category, we can extract its underlying object. And its underlying object is just what you get when you evaluate that sheaf on the point, right? So it's the global sections, right? And this is the underlying object. And then left adjoint to that, we have the discrete objects attached to any object of A. So for any object A, I can talk about the discrete thing attached to it. And so this is gonna be set up exactly so that the adjunction rule holds. And so this allows me to talk about discrete things. And then once I've got a whole bunch of discrete things, I can start doing manipulations with them and then I take me out of the discrete category. So let me show you an example of that. But first let's see, is there a way to define PICC directly as the category of beta modules or algebras and C where beta is the stone check compactification? So yeah, so roughly speaking, this is the category of algebras for an infinitary Lavera theory. I'm not sure I'm using that phrase right correctly, but yes, yeah. It's not a Lavera theory in the usual sense. It's like a Lavera theory where the underlying category can be way too big for its bridges. I'm not really sure, but the right term for that is, but yes. There's actually some interesting questions surrounding that that we can talk about if you're curious. But let me show you how I want to use this for this particular purpose, for the particular purposes of exoddermy here. So I can talk about discrete things, that's what this example allows me to do, and so I'm gonna do it. So this is how I build ZL. Another way to build ZL is I could have just remembered that ZL is a topological space and then made it into a pignotic set. And well, it's a topological group, it's even a topological ring, so I can turn all those things into pignotic groups and pignotic rings. That's all formal. As such, it's actually the limit of the discrete things. That's good because that's the way it is in topology, the same thing happens in this context. Similarly, QL, when I wanna regard QL as a pignotic field, I can take ZL and I can invert L. In other words, I can form this co-limit. And that, again, is a perfectly good way to construct QL. So now what we're able to do because we're in this pignotic world is that we're able to just freely categorify those sentences without fear of reprisal. We can just categorify those sentences and I don't have to worry about how the categorification interacts with topology or anything else because I'm working in this setting where everything is just a sheaf. Right, so since I'm only just working with sheaves, I can talk about this thing, the category of perfect complexes in ZL, as a pignotic category or a pignotic infinity category as a limit of a bunch of pignotic infinity categories and which ones are they? Well, I'm gonna take the discrete ones on perf of Z mod L. So this is now an object in pignotic infinity categories. So I'm gonna talk about this category of perfect complexes on ZL and now it has a topology, quote unquote. It has some sort of topological structure. It somehow remembers the information that built up the topological structure on ZL and it's carrying that with it. So the same thing is gonna happen now with perf QL. I'm gonna take a co-limit of these categories and these are all pignotic categories, perf ZL, perf ZL, perf ZL with multiplication by L everywhere and that's gonna provide for me another pignotic infinity category. And again, this is just happening because I'm speaking about sheaves of categories on compactive. And so I can just say precisely what I mean by that. Okay, so now I've actually given this thing definition, right? I now know what I mean by perf QL and know what I mean by perf ZL. And so I need to now tell you how I'm gonna regard Gal X as a pignotic category. And well, remember what kind of object is Gal X? Well, I constructed Gal X as a profinite stratified space, right? So it was a sequence of high finite stratified spaces and I took the limit of it. And so now I'm just gonna do the same thing but I'm gonna do it in the pignotic world. So if I have an inverse system of high finite stratified spaces, then the profinite space that I get by taking the limit of this, I can do that in the pignotic world. And one of the things that we prove is that doing that in the pignotic world gives you the same structure. In other words, profinite stratified spaces in the sort of formal sense embed fully faithfully into pignotic stratified spaces. So that gives me permission to think of Gal now as an object of pignotic stratified spaces. So remember, what does that means? That means that I'm talking about a pignotic category with a pignotic functor down to a pignotic poset, right? And the pignotic poset in question is this XR and the pignotic stratified spaces is Gal X. Now that's quite abstract, but in fact, I can be extremely concrete with this. It turns out that Gal X, the K points, if K is sort of a random compactum, the K points of Gal X are actually just the geometric morphisms from the category of sheaves on K into the etal topos of X. So this is some kind of elaborated form of the category of points, where I allow myself now K points for any compactum K. Someone asks, is perf ZL, as you defined it here, equivalent to dualizable objects of mod ZL, where ZL is the limit of Z mod L to the n and pignotic rings? Yes, yes, absolutely. You're also asking about solid things. I'm not using any solidification here right now. There's no solidification in this picture yet. What you say is correct. But the important point that I wanted to make is that perf ZL, it's really, I mean, this is a pignotic category whose underlying infinity category really is just perfect complexes in ZL, as you know and love them. There's no funny business there. There really are just perfect complexes in ZL. And so as an underlying category, it is the thing that you think it is. But then I'm giving that thing a sort of topological structure or precisely a pignotic structure. Okay, so I'm about out of time. So maybe I better state the theorem one more time and now it really all makes sense. Everything here makes sense. We're now talking about constructible sheaves with perf lambda coefficients. And what do I mean here by continuous? Well, I don't really just mean continuous. I really mean pignotic functors from Gal X, which I just gave a pignotic structure to, to perf lambda. And now this is at least a sensible theorem. And the way in which you prove it is you're gonna take the theorem that you had for finite rings and you're gonna extend it to this case first by taking limits and then by inverting L and those are the two things. So there's two more questions here. What is the version of the earlier observation that Gal X is a profite one category in the pignotic setting? It's just that it can be expressed as an inverse limit of finite one categories with the discrete pignotic structure. Oh, and Mark Levine asks, in my description of Gal XK, what happened to the stratification? Yeah, there is no stratification anymore. I've already taken in Gal X, I've already taken the limit over all the possible stratifications. So at that point, once I've gotten to Gal X, I've actually gone through all the finite constructible stratifications and I've got myself just as pure profite category. Yeah, so it's quite surprising though. I mean, this isn't, well, I can say that it took me a long time to notice this sentence. I don't think this is obvious or if it is, it certainly wasn't to me for quite some time. So if this seems surprising to you, surprise the hell out of me. So, no, it's actually quite delightful that it's so simple to state. If C is a closed symmetric manoidal category, will PICC have a closed symmetric manoidal structure too? Yes, absolutely, absolutely, definitely. And it's just the usual one where you take the tensor product and you sheafify. Tensor product object-wise and then you sheafify. What is the K twiddle? K twiddle here is the category of sheaves on K. So I'm gonna take, think of K, K is a compact house door space and I'm gonna take sheaves of spaces on it in the good old fashioned sense. Federico asks, back to the theme sheaves versus homotopy types, can one pin down a more specific class of allotic sheaves? Suppose that X is defined over a finite field and I'd like to study least QL sheaves with fixed rank and bounded ramification, maybe up to a twist, what would be the homotopy type? Good question. That's a really good question. So you can cut out some kinds of interesting homotopy types here. Oh, I see, you're thinking about these skeleton sheaves. I don't, I think that's a great question. I don't have anything cool to tell you about that. That sounds like a fantastic question. I don't know the answer to that question. There are some things that you can do. There's a, I mean, so I think maybe something that is pursuant to the question that you're asking is, how do you control certain aspects of the sheaves using just the Galois category? And I have some information about that but I don't have complete information about that. So for example, I pretty well understand how the six functors work with respect to these, to operations on Gal. But I may not have access to sort of some of the, in particular bounded ramification, I'm not sure if I know how to access that information using Gal yet. I think I don't yet. I would like to talk to you about that. Another question. So what happens when you replace picnotic perfect complexes with solid ones? Do they happen to be the same for QL? Yeah, so those are the same in that case, right? I mean, if I take, if you just think about vector spaces for a minute, if I think of vector spaces over QL, but I think of the finite dimensional ones, there's really only one way to give that a reasonable topological structure. Now, that's not true as soon as I become infinite dimensional, that's not true anymore, but for finite dimensional things, that's true. And so that's why in this case, just using the sort of raw picnotic structure is fine and you don't need the solidification story here. But in general, you, I mean, to do something fancy or still, you absolutely would. But I wasn't trying to be all that fancy, I was just trying to keep it simple. David Corwin asks, is the fact that it's a one category related to the fact that the Zariski topology trivializes all higher etal homotopy groups beyond Pi one? No, I don't think so. Well, it depends on what you mean by that. Oh, which comes from art in good neighborhoods. Yes, yes, absolutely. So if you take, right, so that's right. So I constructed the sort of approximation to Gal. I finally understood what you were asking. I took this approximation to Gal, which I called something Pi infinity X over P somewhere. Here it is, this guy. So this guy here. And a statement that's true is that if you're working with a reasonable variety and you take a stratification that's sufficiently fine, still finite, then you'll actually get something that's just a one category already. And that comes down to the existence of these strongly hyperbolic art in neighborhoods. That's right. Can we interpret class field theory in the picnotic point of view? I believe so. That's kind of a long story that I can share with you if you like. So interpret yes, prove, at least for me personally, no. Although the person that I would ask about that is Dustin Kloss and who probably understands a strict superset of what I know about class field theory in the picnotic setting. Joshua asks, but he says first some very nice things. Thank you, Joshua. Are there any limits to what pro-finite stratified homotopy types can be represented homotopically as Gal X for a scheme or stack? Do I understand this question? Any limits to what pro-finite stratified homotopy types can be represented homotopically? Oh, do I understand? I mean, so right, so there are, I'm not sure if I'm answering the right question, Joshua, but there are, I suspect, well, I guess that now I even know, there are stratified homotopy types that are more of a linear variety in the sense that there are, this is some kind of stratification of what you might call Gal Wajin duality. And one of the things that we've been working pretty good ideas in that direction, and I suspect there are a lot of examples of that kind of thing. I don't know if I'm answering your question, but I think such things are possible. Is there a model structure on picnotic sets that restricts to the Quillen model structure on topological spaces? I don't know of one. I don't know of one. It would surprise me a little bit, but I don't rule it out. Let's see, could I explain again the definition of commutativity in the diagram for the oriented fiber product? Does the oriented fiber product satisfy associativity? I'm not sure. Oh, yeah, I do know what you mean. Yeah, it doesn't satisfy associativity in the sense that you might mean it at first. Oh wait, so sorry, the oriented fiber products back this way. Where'd you go, oriented fiber product? There you are, oriented fiber product. Here it is. Yeah, so the oriented fiber product is kind of strange. So here, so this is, I'm gonna say it again, so this is the universal gadget that makes this oriented square not commute, but it's the universal thing with a map to U, a map to V, and a two arrow filling in that thing. So that's what that thing is. And no, it doesn't satisfy any associativity in the obvious sense. So let me kind of give you an example of that. This might kind of creep you out a little bit, but I'm still gonna show you. So this may be very uncomfortable the first time I saw it. So if I have, let's say I'm trying to do an oriented fiber product where I'm trying to go back to some Zed Prime further. So I do an oriented fiber product here, say. And then if I want the oriented fiber product here, I actually just take the ordinary fiber product here. There's no two arrow in the square. This is an invertible square. And that really upset me the first time I saw it, but I've gotten used to it since then. So, yeah, that's the nature of the sort of, so that's showing you that, I mean, another way to describe this is that you're taking the oriented fiber product of Zed with U over X is the same thing as the product of, okay, actually I'm not gonna write this down. It's this sort of orientation is that, I don't wanna get things confused, but it's just this sort of orientation. If we consider the oriented fiber product of all strata, out of the least sheaves, of all strata. Yeah, so if I take the oriented fiber product, maybe this is the question that you were asking. So if I take the oriented fiber product, let's take three strata. I misunderstood your question maybe. So let's take three strata. So I suppose I've got Zed and W and U. And I try and take the oriented fiber product of all three of these. So Zed oriented fiber product over W with, or sorry, over X with W and oriented fiber product over X with U. And this does break down into Zed oriented fiber product over X with W crossed over X. This is just the ordinary fiber product now of W oriented fiber product with U. You do have formulas like that. And so you can, this is the Siegel condition, right? This is the Siegel condition kind of in some sort of funny way. Let's see. Sean asks, like an inverse Gaoua problem. I'm not sure what that is a reference to Sean. I'm sorry. Michael says, back to nuts and primes. Is there any hope that this point of view might actually recover any classical invariance that we weren't able to see very, very before like a crossing number of polynomial invariance? Yes, that's my hope too. Yeah, I absolutely hope so. I don't have anything, I don't have any concrete progress on that to report to you yet. I'm just hoping. Does the pognotic formalism give a simpler construction of the QL homotopy type of a scheme? Yes, yes, absolutely it does. So this is, so that's part of the upshot of this is that this is giving to you a QL homotopy type of a scheme by taking, you can roughly speaking, take the stratified homotopy type, tensor it with QL and this is gonna give you a stratified version of the usual QL homotopy type. Absolutely. Yeah, given the analogy between motives over a base X and constructable sheaves, do you think one should be able to get a motivic Gaoua category so that functors out of it give back motives over X? Wow, you guys ask such good questions. In the spirit of motivic Gaoua groups. Yeah. Yeah, I hope so. I don't have any concrete, again, I don't have any concrete progress to report on this. I really just have this sort of eladic story here. But I agree with you. I mean, I think the first place where I would like to see something like a kind of example along the lines you're talking about is taking this category of constructable isocrystals of lestume and constructing a kind of Gaoua category and it won't just be a category anymore, but some sort of object attached to that that classifies those in the same way. And I think that would show us where we needed to go in the motivic story. If you like the eladic story by itself isn't gonna be enough to show us where we need the point. So let's see there are questions on this other side too. I should have been reading this. Sorry. Already what fundamental groups can, this is a question from Remy. What fundamental groups can occur for varieties? It's a widely studied question with positive and negative results. Oh, sorry, sorry, sorry. I think this is a reference to a previous question I'm just understood. Okay, all right. Sorry. I'm doing my best to keep track of all the back and forth. Are there other questions? I feel like I might have missed one. Did I miss one? I didn't really end. I just sort of tapered off with questions, but I prefer it that way. Here we go. Can I take an oriented fiber product for the Rekolma of G spectra along a family or a Mackey Functors out of a sieve? What does that look like? This is a question from Andrew Smith. So I know what to do for G spaces. G spaces fall naturally into the stratified perspective. So the post-set that stratifies the infinity category of G spaces is the post-set of conjugacy classes of subgroups, where sub-conjugation is the relationship. For G spectra, so I haven't thought of, I don't think I have anything concrete to say about what happens stably. It's, I don't have anything to say. Oh, David Corwin says, sorry, at the beginning I was just wondering, did Art and Mazer talk about homotopy sheaves valued in spaces? I was wondering why they were included in that. Well, they're included in that because they didn't talk about homotopy sheaves of spaces, but they're included in that because they constructed the atoll homotopy types for the first time. And they did construct what was amounting, you know, what they constructed classified simple sheaves, and so therefore the, which is related to what was done later. So I'm not sure if I'm answering your question, but they did construct the atoll homotopy type. So it seems like a good idea to include them. Okay, that's wonderful, Clark. Thanks for- Stastic series of lectures. I think they're all very happy with you.