 Thank you to the organizers for transitioning this from in-person to online. I have to say, I mean I'm a little disappointed not to be able to be spending part of my summer in Paris but I really appreciate all the work that goes into this and I'd also like to thank the staff of the EASHA as well. So my talk is going to be on Exadri for Elatic Sheaves. So I'm going to tell you what that word means. You might not know it yet, so I'm going to introduce it. And I want to emphasize that in so far as it's in line with the purposes of this talk, this conference rather, it's an indication of where things might move in the motivic world. And so maybe toward the end I'll give some indications of how that direction can go. But so right now I'd like for you to think of this as a particular kind of realization of some sort of more complicated theory of motives than any of the kind that I think exist today. Okay, so since this is a talk that is virtually at the EASHA, I thought I'd go with a theme. And the theme that I thought I'd start with is a sort of growth and dekean theme. And the theme that I thought I'd select is that sheaves or the category of sheaves determine a homotopy type. And then conversely, that homotopy type has the ability to reconstruct the category of sheaves. This is a very growth and dekean idea and there are lots of instantiations of it about which I'm going to talk today. And I think the places where I mean this is sort of imbued in different forms of algebraic geometry these days, but the two places where I feel like it really came out most strongly are in growth and deke's pursuing stacks is letter to Quillen that then got amplified into a long sort of text. And Toen's habitatacion thesis. So I actually grew up reading Toen's thesis. Pretty much all of my graduate education and my postdoctoral time was spent sort of carrying this habitatacion around with me and trying to understand the things that he was telling me about the future. And so for those of you who were raised only on the texts of Lurie, I suggest going back and looking at this habitatacion thesis and understanding really how good Toen's vision for the future really was. It's really quite remarkable. Okay, so how does this theme sort of actually get instantiated? So here's the first example of this. So let X be a connected manifold. It'll be connected for right now. So I'm going to write LC vect C for the category of locally constant sheaves of vector spaces. So these are locally constant sheaves of vector spaces on X. And so if you have this category of locally constant sheaves, well, what can you do with it? Well, I've given a little example here on this side of the page. I have this point X here. And well, I've got my that's supposed to be a drawing of the Mobius band. I did my best. I'm going to drive around my little loop there. And while I'm driving around, I'm going to hold in the air a vector and that vector is going to be a vector is going to be a section at that point of that line bundle. And I'm going to carry that line bundle around. And then I'm going to discover that as I've carried it around, I've reversed orientation, right? And so what you see there is you're seeing a representation of pi one of the circle, which is of course just the integers, you're seeing a representation of that thing. And the point is, is that if you choose a point, if you choose a point X and X, then that will determine an equivalence of categories between the locally constant sheaves, in this case of vector spaces, and representations of your fundamental group. So this isn't a homotopy type yet. This fundamental group here isn't a homotopy type yet. But what it is, is it's an approximation to a homotopy type, right? It's giving you some information about the homotopy theory of your manifold. And it's giving it to you as it relates to the category of locally constant sheaves. So this is an example that I want to be inspired by, but first I want to generalize it as many times as I can. So let's try and generalize it as many times as I can. So first off, well, you might not have been very happy with the fact that I had to choose a point. And you might feel as though well the manifolds that I'm interested in, they're not connected. So, you know, can you do something better for me? And indeed, we can do something better for you if X is any manifold whatsoever. And C is really any category. It doesn't really matter what the target category is at all. Then you again have, now it's a canonical equivalence. I don't have to select a point. So you don't have problems with functoriality. I don't have to select a point. And I've got the category of locally constant sheaves valued now in just the complex numbers. And on the other side, I have, well, now it's not representations of a group, but it's representations of the fundamental group void. So this is the fundamental group void attached to your favorite manifold X. And now with this information, you can actually recover the category of locally constant sheaves. And what's the story here? Well, the story here is that, well, if I've got my manifold, which maybe I'll just, this is just a chunk of my manifold here, and I've got some points on it. Then what will I do? I'll take the locally constant sheaf, and I'll take the stock at this point. And then I'll watch for any path inside my manifold. I'll watch as that stock varies from point to point. And that provides me with the functor. And because it's locally constant, I actually have an honest functor from paths into this category. And in the reverse direction, I can just go back the way I came. And that's why this thing is really a group void and not just some arbitrary category. So by the way, I should have emphasized this earlier. I hope that people feel free to ask questions. This is meant to be a summer school. So I have some slides here, but if I don't get to everything, that's fine. I feel like these things go well if I'm interrupted frequently. So please ask away. Yeah, absolutely. I relay the questions. Thank you very much. Okay. So this is the story. So we have this nice equivalence between locally constant sheaves. So sheaves that are constant on some open cover of my X and representations of the fundamental group void. And that's completely canonical. There's no funny business there. There's no choices. It's just a completely canonical equivalence. Of course, you might be a little dissatisfied with this result because, well, if I've got a locally constant sheaf, I'm not just interested in it as a sheaf. I'm also interested in things like its co-homology. So how can I extract things like its co-homology? Or more generally, how can I extract things like the category of locally constant sheaves, where the target now isn't just, ah, so what is a locally constant sheaf valued in a category? So let me answer that question. So by locally constant sheaf, let's see, I'll just add a page underneath this, by locally constant sheaf. So we're going to have a sheaf F on this manifold X, which is valued in a category C. And to be locally constant, what do I mean? I mean that there exists an open cover, which is going to be an open cover of X, such that when you take our F here and you restrict it to each U alpha, you get the constant sheaf is a constant sheaf. So if you have a sheaf, and this doesn't matter, it doesn't know what C is, right? It's just an arbitrary sheaf valued in C. But the point is, is that according to some open cover, all those pieces of that open cover, it's actually a constant sheaf. Okay, so you might be unsatisfied with this because you might want to say, okay, well, I don't want to just contemplate sheaves of vector spaces. I want to do something more sophisticated and capture comological behavior, which means that I need to pass to something like the category of sheaves valued in complexes, right? And so if I think about sheaves valued in complexes, then how do I do that? And this is this beautiful theorem of, it really goes back, I think, to Toen and Toen Vazalsi. I mean, this was sort of suggested in both of these pursuing stacks, but I think the first sort of complete proof is in Toen and Toen Vazalsi's papers from around 2002, which is that if X is any manifold, you can be more general than this, but manifold is enough for our purposes. If X is any manifold, then the category of locally constant sheaves valued not now just in the category C, but in fact in any infinity category C is the same thing as functors out of pi infinity of X. So pi infinity of X this is the fundamental groupoid, but fundamental infinity groupoid attached to X. So we have our manifold X, but we can form the infinity groupoid. Someone, let's see, I have a couple questions here. So constant sheaf does not mean constant functor. It means the sheafification of the constant functor. That's the first question. So the constant functor isn't necessarily a sheaf, it's the sheafification of the constant functor. I'll get to the talk about the tall fundamental group in a minute. You'll be very happy with my answer, I hope, fingers crossed. And let's see, so what kind of restrictions do we have on the category C? Very few, basically none. You know, in order to state the sheaf condition, I need to have enough limits lying around. So as long as I have enough limits to state my sheaf condition, then I'll call it good. And so, for example, if C has all limits, then I'm in great shape. Similarly, in the infinity category world, if C has all homotopy limits, then I consider this a victory. And the theorem applies. Okay, so the fundamental groupoid, infinity groupoid attached to X, this is the this infinity groupoid here. Well, this is, this is, this can be thought of literally as taking the singular complex, the singular simplicial set attached to your manifold, right? So the in-simplicies are literally just what you get by mapping in, the in-simplicies of this thing are literally what you get by mapping in the geometric realization of the standard simplices, so the ones that sit inside coordinate space. And that defines for you not just an infinity category, but even an infinity groupoid. And there's this remarkable fact, which I guess is called growth in the homotopy hypothesis, which is that spaces in their homotopy types are completely captured by the theory of infinity group oids. Is the manifold hypothesis on X important? A hypothesis is very important, but manifold is too strong. So the correct condition is probably something, well, there are topological conditions that you need to have that basically say that there are enough sheaves for the space to be reasonable. But I've just chosen manifold to be kind of easy with it. So you need, on the one hand, you need a homotopical condition to, to, to reflect the fact that the, the homotopy type is correctly classifying the weak homotopy type. But then on the other hand, you need a condition that says that there are enough sheaves lying around. So as long as you have those two things, you're good. So I just chose manifold because it's easy to get to. So this is all motivation at the moment. So you have this fundamental groupoid and there's this remarkable fact that the homotopy theory of spaces is actually completely captured by infinity group oids. This is called the homotopy hypothesis. These days we have a particular instantiation of the homotopy hypothesis, which is an equivalence of homotopy theories between spaces and infinity group oids. And by infinity group oids, I mean simplicity satisfying the con condition. And so we have this remarkable fact that if you want to understand locally constant sheaves on your favorite manifold, then what you have to do is understand the entire homotopy type of that manifold. And if you organize that data in a particular way, then locally constant sheaves are nothing more than funkters out of that homotopy type. And so that's the sort of perspective that I want to chase up because I think that it's a particular case of a much more general picture. Okay, so the theme is that sheaves, in this case locally constant sheaves, determine a homotopy type, which we expressed as the infinity groupoid. And that infinity groupoid then in return meant allowed us to recover the category of sheaves. So just a moment ago, someone asked about what happens if I try to work with the scheme. And so let me answer that. It's conveniently already here before me. Both of these showed that if X is, well, let's say it's a connected scheme, we'll again start off with a connected case. And let's give it a geometric point. So geometric point for me means that the residue field of little x here is a separable closure of its image point. That's what I'll mean by geometric point. For the purposes of what I'm saying now, I don't really need that much condition, but more generally, oh, Mark Levine asks whether or not the identification of LC, XC and maps from M infinity to see if that's a topology. It's not a topology, at least not in the topological world. You have to prove something. And the thing that you have to prove is you have to prove that why, for example, are locally constant sheaves a homotopy invariant. So one thing that you'd have to prove, for example, is that there aren't any interesting locally constant sheaves on the interval. And actually, that's sort of the key point, right, is that if you think about the locally constant sheaves along the interval, there aren't any interesting ones. They're all trivial. And that's basically the little theorem that you have to prove. So it's not a topology. I'm not saying it's the deepest factor in the universe, but it's not a topology. Okay, so if we have a connected scheme with the geometric point that we'll call X, then we have, again, the same kind of relationship here. We have this, the category of locally constant sheaves. Because we're doing algebraic geometry, I'm stuck in this world where I have to say finite locally constant sheaves. So there's always going to be a finite miss hypothesis once we move to the world of algebraic geometry. And locally constant sheaves with finite stocks are the same things as pi one sets where those things are finite pi one sets. And here, what I mean by this, let me emphasize that what I mean by this is I mean that these things are continuous pi one sets. So one of the important facts here is that this guy, this is a tall fundamental group, is not just a group, but it's a profinite group. It's a profinite group. And because it's a profinite group, we think of that thing as a topological group. It's a group equipped with a topological structure. That topological structure is a very particular kind of topological structure. It's a stone topological space, which is to say that it's compact house dwarf and totally disconnected. So it's a very particular kind of group. It's very nicely behaved. And I look at the continuous finite pi one sets. This is a tall fundamental group. And how does this thing define? Well, roughly speaking, what growth and discovery was that you could just sort of backfill this equivalence to actually use this as a definition for pi one. That is to say that you basically set it up so that covering space theory works correctly in the world of schemes. And if you do that effectively, then you end up with this beautiful profinite group and you discover this exact theorem. Now let me mention that, well, we can extend this to a sort of linear setting before we are talking about the representations of the fundamental group valued in the complex numbers. Here, if I take a finite ring, and again, the finiteness is important for the sentence I'm saying here, if we take a finite ring, then locally constant sheaves valued in, say, finitely generated lambda modules, lambda will be the name of my ring. Those are the same things as lambda linear representations of the etall fundamental group. So this is often called, by the way, the monogamy representation. So the monogamy equivalence, I guess, is stated here. The idea is that to every locally constant sheaf, you have a particular representation. That representation could be called the monogamy representation. Why monogamy? Well, dromos is, I guess, the Greek word for running or an avenue, I guess. And mono is referring to the fact that you're running around a loop once and getting back to where you started and seeing what happened to your vector. And so that phenomenon is exactly the phenomenon that we're trying to capture here. Now here in this context, we don't have literal etall paths. But nevertheless, we're going to work with this. So this goes back to SGA1, for those of you who are keen to look this up, I can recommend reading SGA1 about this. And he sets up the entire theory of the etall fundamental group. And it's a remarkable story. Yeah. Okay. So, but if we're working with sort of more sophisticated kinds of co-emologies, this restriction that we had here, let me go back to the page, this restriction that we had here on lambda that would be a finite ring, that's a heavy restriction for us. So if you remember the story from grade school about the vague conjectures, the way the story of the vague conjectures emerged was that we wanted a co-emology theory, we, they both and DeLene wanted a co-emology theory with something like rational coefficients. And Sarah told them that you couldn't have something with rational coefficients, that wasn't going to be allowed. But nevertheless, you wanted something that had the ability to extract bedding numbers. So you really wanted something that looked a lot like rational coefficients. And so the idea was that, well, and this is inspired by earlier work of Tate, the idea was that, well, q rational coefficients isn't good enough. But if you sort of extend to a suitable completion of q, you can't go to r, you can't go to qp, but you could go to some ql. So if you go to a suitable completion of q, that there you might actually be able to see a working co-emology theory. And so this is what inspired eladic co-emology. And so the story with eladic co-emology is that you're not just taking the etal co-emology of zl or ql. You're taking the, instead, you're going to do something quite a complicated. You're going to look at the category of locally constant sheaves with values in z mod l to the n, and you're going to take this limit, that limit right there. And you're going to let your n's grow large. And in the limit, you're going to get a category. And inside this category is where you can spy your co-emology with zl coefficients. Okay, so that might seem a little unsatisfying because this isn't just the category of locally constant sheaves on the etal topology. So I have to do something more intelligent. And so the more intelligent thing actually came very, very recently. So this is in, I think, 2013, if I'm not mistaken, somewhere around there. Actually came very recently, which is that you can instead pass to what's called the pro etal topology. So the pro etal topology is a version of the etal topology where you're allowing sort of infinitary covers. And those infinitary covers give you extra added expressive power. And what Barton Schultz approved in 2013 is that, well, if I've got myself a geometric point, again, I'm going to be working with these geometric points. If I've got a geometric point, little x of x, then I can extract a new monogamy equivalence between the category of locally constant sheaves of zl coefficients and the representations of now a new kind of thing, the pro etal fundamental group. This is the pro etal fundamental group. And the remarkable fact is that while I'm looking at representations of this pro etal fundamental group, and they're going to be valued in zl, zl has a topology now. So I want to think of that topology on zl. And pi one here, pi one pro et, is not just a profinite group. So it's not just profinite. It's a much more complicated group. It's something called a new heat topological group. So this is a much more exotic kind of topology. It's not profinite anymore. It doesn't have these sort of nice structure. It's, it's so new he described these things as pro discreet, but they needn't be pro discreet either. They need not be the pro objects in groups anymore either. They're a little more complicated than that. So we're going to see a different instantiation of this story in a minute. So if this isn't familiar to you, just understand that there is a kind of topological group that you can attach to the sort of infinite version of the etal site. And when you do that, what you get is something that's not just profinite, but something that has this sort of funny new heat topology. And representations of that fundamental group, continuous representations of that topological fundamental group with zl coefficients actually extract your category of locally constant sheaves, which is defined now in this kind of formal way. But then using the pro etal topology, you can actually define it as locally constant sheaves to the pro etal topology valued in zl. Can I give a couple properties contained in new heat? I can actually give you the definition. So, so if you look at the two sided uniformity, I guess this is the official definition, but this is equivalent to the official definition. If you, if you look at the two sided uniformity on a topological group, and you require that the topological group be complete with respect to that uniformity, that's equivalent to being a new heat group. So that's the, that's the condition writ large. L does not need to be invertible right now. For the purposes of what I'm saying right now, L doesn't have to be invertible on X to produce a vague co homology theory when I tensor up. I'm going to want this to, I'm going to want L to be invertible. But right now, for this sentence here, absolutely fine. Q, L can be equal to p and X can be a variety over fp. It will work perfectly fine. Okay, so the same story happens with Q, L coefficients. And once again, we can't use the etal topology and just define what it means to be a Q, L chief just using the etal topology. Instead, I'm going to look at this category of locally constant sheaves. And the category of locally constant sheaves with Q, L coefficients is going to be this category that I have from this previous page defined by this limit. This gadget here defines a Z, L category. And I'm going to tensor that Z, L category up to a Q, L category. David Corwin asks what a two-sided uniformity is. If it's okay with you, David, I want to come back to that at a different time. I just want to try and, I'm going in a different direction. So don't worry. I'll give a different description of this group in different formats. So but let me just emphasize that this Q, even tensor it up to Q, L, we can still have the same theorem. That is to say, locally constant sheaves with Q, L coefficients are the same things as continuous still representations of this topological group, this topological fundamental group with Q, L coefficients. Okay. So this really does capture the full Q, L theory. And it is very good in that regard. So this is now this wonderful topological fundamental group. Let me take a moment to emphasize the fact that you might be familiar with the sort of SGA3 version of the atoll fundamental group. And that atoll fundamental group isn't pro-finite either. This is still different even from that. So for example, if you take a curve of genus 2, say, and you identify two points in that curve of genus 2, even over the complex numbers, then this group will coincide with the group even from SGA3. So if I didn't write, let's see, I'm looking at this now. If I didn't want to take the limit, could I just take the pro-atoll sheaves with Q, L coefficients? Yes, that's exactly what I could do. So the pro-atoll topology allows me to do exactly that. That's right. And I can do the same thing. Yeah, I can do the same thing for both Z, L and Q, L. I can just take sheaves for the pro-atoll topology with these coefficients. You have to be a little cautious about how you define that ring of coefficients, but as long as you do it correctly, everything's fine. Because again, you've got to embed the topology as part of the structure of the sheaf of pro-atoll rings that you're writing down. Okay. So okay, so this is the story so far. So now this is, I'm only talking about sort of locally constant sheaves of ordinary categories right now, right? So these are really locally constant sheaves of Q, L vector spaces. No complexes. I'm not picking up anything more complicated than that. So for example, I don't have any access right now to say the category of complexes whose co-amology is locally constant with Q, L coefficients. I don't have, I'm not having a theorem that says what kind of object those things are representations of. So it's analogous to the situation that I had before where with manifolds where I understood how to understand locally constant sheaves of sats or of ordinary categories, but I don't necessarily have access to locally constant sheaves of something valued in infinity category. And to get that, I'm going to need much more structure than just a single group. I'm going to need the whole structure of a homotopy type. So let's get at that. Well, first we have the sort of original art and mazer homotopy type. And the version of the story that I'm telling now is actually sort of the more sophisticated version that Toa and Vizalsi showed us. And so what's the story here? Well, the story is that if I take a Scheme x and I want to understand, I'm going back to the finite world here, if I want to understand locally constant sheaves valued in s pi, what is s pi, s pi is the infinity category of pi finite spaces. So pi finite means that it only has finitely much homotopy. The spaces only have finitely much homotopy. And all homotopy groups is finite. And also pi zero is a finite set as well. So that's the infinity category. So this is sort of like a canonical example of a finite category. I could make that a little more precise if you want. But I'm going to look at locally constant sheaves valued in pi finite spaces for the atoll topology just I'm not doing anything fancy yet. And what we discover is that the that's equivalent to the category of funkers from an infinity category but or infinity groupoid. But what kind of infinity groupoid? Well, it has to be profinyte. So this is a profinyte infinity groupoid. So what does it mean you might ask to be a profinyte infinity groupoid? Well, I literally mean that it's going to be an object of the pro category of pi finite spaces. So I have this category of pi finite spaces, and I'm going to look at this pro category. And that's got to be that. So what does that mean? That means that I'm going to be looking at spaces now that are profinyte. So they might have infinitely much homotopy. And each of those homotopy groups are themselves going to be profinyte groups. And so that's the story. So if x is a scheme, then I can have a monodromy equivalence where if I'm looking at locally constant sheaves valued in something homotopical like the category of pi finite spaces, then I can recover that whole category from a single homotopy type. And that single homotopy type is the homotopy type of the at all profinyte homotopy type of x. How do I define this category? So somebody asked how I define the profinyte at all homotopy type. I'm going to define it for the purposes of our conversation right now, just using this formula right here. I'm going to say that I'm interested in trying to fill in that blank right there. And there's only one group point that can do it. And this is that group point. So that's the definition if you like of this object. Now that's not a very satisfying answer, but I will give you a more satisfying answer soon in the form of a theorem. So right now for the purposes here, that's just the definition. Again, our theme here is that well if I have a suitable category of sheaves that I'm interested in, I'm going to use that to determine the homotopy type in this way by using these kinds of formulas. And as soon as I have a homotopy type that is sufficient to reconstruct my category of sheaves, then I declare that to be the homotopy type. And so I don't give you a more explicit description of this thing. Now I can, I'm just not doing it right now. That was a complicated way to not answer your question. I'm sorry, I will answer it. I swear. Okay, so there's a linear version of this. Once again, if I've got a nice finite ring lambda, then you again have this equivalence of categories where on the one side I can look at locally constant sheaves valued in now, well oops, now valued in perfect complexes over lambda. I'll look at locally constant sheaves for the etal topology. And those things are now not representations of a group or a group void, but of an infinity group void. And in fact, now it has to be a profinite infinity group void. And so on this side of the formula, let me emphasize this. I really have to say continuous here. I can't get away with less. I really have to talk about continuous functors from this profinite infinity group void into the infinity category of perfect complexes on lambda. Does any of this work for more general objects like derived stacks? So it certainly does. So derived schemes, I should emphasize the fact that derived schemes from the point of view of the etal topology are not particularly exciting, because the derived direction doesn't contribute anything. That's essentially a nilp of thickening in the derived direction of your scheme. And so therefore it has no effect on the etal topology. These in the stack direction, it's a little more interesting. And you can indeed tell exactly the same story for the derived stacks. And if you like, I think that person was anonymous, but whoever it was, you can get in touch with me and I can show you how that works. It's very easy. It's not easy to compute. I want to emphasize that. It's easy to define, given current technology. It's not easy to compute. Okay. So we have this profinite infinity group void, and we're going to use that profinite infinity group void to recover the category of locally constant sheaves. Is LC a perf lambda linear category? Yes, it is a perf lambda linear category, absolutely. So the pullback functor from perf lambda, the constant sheaf functor from perf lambda to this is a symmetric minoidal lambda linear functor. The CTS have an explicit meaning. Yes, it does. So as a profinite object, you can write this thing as a limit of pi finite spaces for some inverse system of pi finite spaces. And what this literally means is that I'm going to take the cold limit over the same alpha of now just ordinary functors from that k alpha to perf. So it has a very precise meaning that's not pick not at all. Does this work for stacks if it's not tame? The pi one for stacky curves with wild ramification is not homotopy invariant, right? I'm not sure if I understand that question. I mean, pi one is homotopy invariant because it's an invariant of the homotopy type. I'm not sure how to answer that in a better way. Maybe we should talk after. What does it mean for a functor from the infinity a tall groupoid to perf lambda to be continuous? It means literally exactly what I said here. It means that it's a cold limit. This functor category is the cold limit of these things. And right now, that's the only definition we have at our disposal. I'll give you a better definition actually in the next lecture. But for now, this is what we've got. Okay. So good. So now here's our central question. So we have some theorems. Those are great. And now we have some central questions. The first one is, can we replace lambda with ZL or QL or QL bar with the relevant topology on these rings? So that is to say, if I look at this formula here, well, I've got the category of perfect complexes over lambda. And now if I take lambda and I endow that with the topology, I'd like to endow that with a similar topology on perf lambda so that this formula can still be true. Where I hear I'm really going to take again some sort of continuous functors from this thing. That's the first question. The second question is, well, you know, locally constant sheaves are great and everything, but, you know, they're not, they're not the end of the story. If you take a push forward of a locally constant sheaf, it's unlikely to be locally constant again. So the question is, what do I do to deal with the story where I'm going to pick up the kinds of things that can be constructed out of locally constant sheaves. I'm going to take my locally constant sheaves, we're going to push them forward and pull them back and tension them together and stuff. And I'll get different kinds of sheaves. Those sheaves are called constructable sheaves. That's a bigger category of sheaves than just the locally constant guys. And the question is, can we recognize those locally constant sheaves or the constructable sheaves as functors out of some more exotic now homotopy type. So there are two ways in which I want to try and extend this theorem. The first way is I'd like to replace lambda with something that can have some topological structure. And the second way is I'd like to pick up not just locally constant sheaves but constructable sheaves. And so the questions are, can I actually do that? And the answers are yes, we absolutely can. So this is the topic of my paper with my former PhD student, Saul Glassman and my current PhD student, Peter Hain. The archive reference that I've given to you here is going to be updated soon with all of the stuff that we're going to talk about today. So the versions on our websites are current, but this needs to be updated. But we'll get there maybe even this week. So here's the theorem. Let me tell you what the theorem is. The theorem is that if you want to look at the category of constructable sheaves, now let's think about it for the pro etal topology so we can get those nice coefficients like ZL and QL and QL bar. Constructable sheaves are equivalent in a canonical fashion to continuous funkers from a certain category that I'm going to define for you in a minute called Gal X into the category of perfect complexes. And that category of perfect complexes is going to have to come with some additional structure that will allow me to make sense of this word continuous. So I've written continuous here, but I have to have some justification for what that means. So lambda here can be a finite or a pro finite ring, or it can be an algebraic extension of QL. Perfect complexes, well perfect complexes over lambda if I'm thinking about it with lambda being say QL, QL has a topology and I need to find some formalism where I can talk about topologies on categories or on infinity categories. And we have that formalism now, and that's the formalism of picnotic or if you like condensed infinity categories. So the category of perfect complexes, the infinity category of perfect complexes we're going to view as a picnotic or condensed infinity category. And part of the what I'm going to tell you about next time is the story of of picnotic categories or condensed categories and I'm going to go into some detail about that. And I'm going to show you how to see perf lambda as one of these objects. Okay, and now what about gal X? What kind of thing is that? Well, it's not just a homotopy type in the usual sense. It's not a pro finite homotopy type. It's not even a picnotic homotopy type. It's got to be something a little bit more. It's going to be a pro finite stratified homotopy type. And if there's anything that I want you to get out of this talk, it's the following sentence, which is that stratifying your homotopy types gives you more power over those homotopy types. It gives you a lot more information over those homotopy types. And this is a very particular example of this. In fact, you know, I said a minute ago that I sort of ducked the question of how to define in an explicit fashion the atoll homotopy type of a scheme. I avoided that question. But here I'm not going to avoid that question at all. I'm actually going to tell you very explicitly what precisely gal X is as an object. Can I work with formal power series over Fp as my ring of interest? Do you mean for the coefficients lambda? If you do, then the answer is yes. Well, I mean, I guess if you mean as the base field, the answer is also yes. But in particular, lambda really could be something like Fp double brackets t with its topology. That's absolutely fine. Double round brackets t, yeah. Okay, so here's the theorem again. I'm just restating it for you, but I want to tell you precisely what this object is. So I said that gal X was going to be a pro finite stratified space. And so I will tell you how to regard a profite stratified space as a category. So a profite stratified space in this case is going to be incarnated as a profite one category. So it's not even an infinity category or something fancy. It's literally just a one category. And I can tell you precisely what the objects and the morphisms are. The objects are going to be geometric points. And the morphisms, the morphisms are going to be specializations of those geometric points. Okay, so how do specializations work? Well, I'm going to take two geometric points X and Y of my X. So there's one geometric point and there's the other geometric point. And well, what's a specialization? What's a map from X to Y? Well, what can I do if I've got a geometric point of my X, then I can look at the a tall local ring of X at little X. That is to say I can look at the local ring of X at X. I can look at a strict hand salientization because I've got a geometric point. And I can look at speck of that. And that's what I'm going to call that's what I'm going to call X of little X. This is literally speck of X little X strictly hand salientize. And now what's a specialization? A specialization is nothing more than a map that goes like this. So it's a point of that a tall local ring or speck of that a tall local ring given by Y that covers the original geometric point of Y. Someone asked what about constructible complexes of finite torque dimension? I'm talking about constructible complexes valued in perfect. These are constructible complexes. This is the derived category of complexes with constructible co-amology. This is equivalent to that category. Good. Okay. So this is the category. It's a perfectly good category. It has this. I will emphasize that this category is not just a category. It's also coming with a profinyte topology or a stone topology. So plus a topology, which is going to be a stone topology. And that topology, the way you can think of that topology is that it's a kind of globalization of the topology that we usually have on absolute Galois groups. So this is a sort of globalization of that same idea. And I'll be more precise about that soon. Don't worry. Okay. So here it is again. So the objects are geometric points. A little X of my scheme X. And the morphisms are specializations. So morphism from X to Y is a map fee that goes like that. Someone asks whether the theorem behaves well under incompletion of the target category. The answer to that question I will answer in great detail, I think, but the short answer is it behaves reasonably well under incompletion. So to have it work really well within completion, you really need to have X be an aetherian in some strong sense. And there's some cautions about how big the target category can be. So I can show you how that works. But for now, let me just kind of dig into this Galois category a little more. So this Galois category, you know, it's a relatively easy category to understand when your scheme is spec of a field. So when the scheme is spec of a field, then I have this equivalence here between the Galois category and the classifying space of the absolute Galois group. So I'm supposed to think of a single object here, which corresponds to the almost unique algebraic closure or separable closure of K. And then here, I've got a whole bunch of automorphisms of that, and the automorphisms are profinite, and they're the group GK. That's the group of automorphisms of that. So one thing that you immediately learn from that is that, well, if I choose a geometric point of my scheme X, then, well, I can compute the endomorphisms of that point inside this Galois category. If I compute the endomorphisms, well, they're the same as the automorphisms. The endomorphisms and the automorphisms are the same. And that's going to be a trend that we're going to see a lot of in this series of lectures. The endomorphisms are the automorphisms, and the automorphisms are exactly the absolute Galois group of the residue field at X naught. Matt Booth asks if there's some subcategory of the Galois category, parameterizing complexes with locally constant poemology. The answer is it's not a subcategory, but rather a quotient category. We're going to take the category Gal and we're going to invert everything. So I'll show you that in a minute. Are specializations invertible? Very much not. Specializations are not invertible at all. So a standard example of this is that if you think of a DVR for a moment, so if I think of a DVR, then I have a little point and a big point, a closed point and an open point. And if I think take two algebraic closures of each side, or separable closures of each side, then I have a whole lot of specializations from the little point to the big point, but I have no specializations going back. So they're not invertible at all. It's very much not invertible. Does this correspondence specialize to some categorification of the subcategory of perverse sheaves? Yes, we should talk about that. I can show you how that works. How can I see a telepi two if it's a one category? So I have a one category, but if you have a one category and you take the classifying space of a one category, then I can get any kind of space you like, right, including things that have pi twos. So if I take a, I can, I can represent any homotopy type that you like using, this is without me, topology or picnotic stuff. If I just take any homotopy type you like, I can always represent it as the nerve of a category or one category with no fancy stuff. That's always possible. And that's exactly what this Galois category is doing for you. So even though the atoll homotopy type, and this is a very important point that David Corwin, this is David Corwin's question, is a very important point that he's bringing up, which is that, you know, the atoll homotopy type can have tons and tons and tons of homotopy. For example, if I take P one over the complex numbers, that has the atoll homotopy type of the profanate completion of S two, which has arbitrary much homotopy type. So that might seem like, wow, how could I ever get that out of something that's sort of reasonably small, right? It's going to have to be a big infinite thing. But the amazing fact is, is that that is actually just the nerve of the category. And I can tell you which one it's exactly the Galois category. And so even though it's just a one category can give you arbitrarily much homotopy. And in fact, I'll give you a couple of examples of that. Well, maybe next time, but we'll see. Okay, so, so I'm just trying to give you some examples of how to think of Gal. Let me give you one more example, which I'm really quite fond of. So let's look at the Galois category of spec Z. So if I look at the Galois category of spec Z, well, then what do I have? I claim this is the story of knots and primes that was developed by Mumford and Mazer in the 60s. So what does it look like? Well, the points are geometric points. So if I take, oh wait, so I see that there's, oh, Mark Levine is complaining that the specialization that he knows goes in the other direction. And that's right. So that this is a specialization that goes like that. That's right. So for those of you who are confused, that's the direction of the specialization. X is a specialization of Y for there to be a map from X to Y. That's the way round it is. Okay, so the objects of this thing, well, what do they have to be? They have to be geometric points of spec Z. So that means that they're all of the form spec of Fp bar or spec of Q bar. They have to be algebraic or separable closures of some of the residue fields. And they're going to be either spec Fp bar or spec Q bar. So those are the kinds of points that we have. We have all those. And now let's look at what we've got. Well, we've got automorphisms. And we just learned how automorphisms work. So the automorphisms of spec Fp, those are just going to recover a nice Z hat for me generated by our friend, the Frobanius. The automorphisms of spec Q bar, that's going to give you the absolute Galois group of Q, which is a pretty complicated object. And now this is going back to the point that I was making earlier. There are no maps whatsoever from spec Q bar to spec Fp. There are no specializations that go that way. There are, however, specializations that go the other way, specializations from spec of Fp bar up to spec of Fq. That is to say spec of Fp bar is a specialization of spec of Fq to go back to what Mark Levine was pointing out. And the space of those, well, how can you compute that thing? Well, you take the absolute Galois groups of the two end points, and then you actually quotient out by the decomposition group. You product those together and quotient out by the decomposition group. And that's the set. This is just a set now with actions. This is just a set of maps from spec Fp bar to spec Qp bar. Okay. So how do I want us to visualize that? So I'm going to go into detail, I guess, next time about how stratified topology works. But I want to draw this picture here because this really is the nuts and primes picture that we know and love. So here I've got a knot sort of sitting inside the sort of three-dimensional cube. And in this knot, well, I've got the sort of central knot itself. And I want you to think of that as spec Fp sitting inside this three-dimensional space. So spec Fp there is sitting inside the three-dimensional space. And that, how does it look? Well, it's the classifying space of the profinite integers, which means it's a profinitely completed circle sitting inside this three-dimensional space. That the compliment, or sort of, I have this sort of bulk piece. So this is a knot sitting inside the thing. And I can look at all of the knots sitting inside the thing and look at the compliment of all of them. And then what I have left over is just a copy of the classifying space of the absolute gawa group of Q. Right? So that's corresponding to the generic point, which I didn't know how to draw, so I just left it blank. And then, well, what else do I have? Well, I've got the knot, and I could look at the deleted tubular neighborhood of that knot. That's what I'm supposed to be pointing out there. It's in purple. And what does that look like? Well, that's really a speck of QP, regardless of the deleted tubular neighborhood of that knot. And so all of these things are fitting together into a coherent picture. But I claim that that coherent picture is really organized by this category, so that you can actually see these knots sitting inside as closed subs. And that's precisely what this picture is doing. Okay, so let's look at the theorem again. The theorem is that the constructable sheaves for the pro etol topology, landing in perf lambda, and remember, lambda can have topological structure or not as you see fit. So it works for finite rings. It works for pro finite rings. It works for extensions of QL that works for FP double round brackets, T, those kinds of things. Constructable sheaves are the same things, just as functors from this Galois category, right? So the Galois category has as objects, geometric points, and it's morphisms specializations of those geometric points. So functors from that into perf lambda, but I have to say it with this caveat that those functors have to be continuous. They have to have the structure of continuous functors. So I have to make that precise. I haven't made that precise yet. That's what I'm going to be doing in this next series of in the next couple of talks. So I have to do that. Let me emphasize, this is a question that came up from here. What can I do? Well, I can look at those constructable sheaves such that the specialization maps that I get are all invertible. Those are necessarily going to be locally constant sheaves. And so then I arrive at the following theorem. I arrive at the theorem that locally constant sheaves, landing in perf lambda again, are the same things as functors from now. This new object right here, the pro et al homotopy type of X. Now I have to regard this thing with this sort of topological structure that I'm going to be talking about. But I can tell you precisely what it is. It's just the nerve of a category in this sort of continuous world. I'm going to take this category gal, and I'm going to invert all the morphisms in it in the correct homotopical way, which is to say I'm going to take its nerve, but I'm going to take it in the world of pro, or of a picnotic spaces, which I'll define. And once I do that, then I have a perfectly good pro et al homotopy type. So for example, pi one of this pro et al homotopy type will recover the bot Schultz of pi one that we saw last time. Okay, so I'm rapidly approaching the time. So I think now is a good time for me to just start fielding questions. So you've been great about asking questions, and I'm really grateful for that. And I hope you still have more for me. Okay, you did a great job for answering your question on the fly. Okay, so now you're finished. You'll go on afterwards, and we go to other questions. That's it. Yes, thank you. Okay, so thanks a lot for our talk first. And now you, okay, so you can see your first question. Yeah, where's the knot coming from? So that's an important point. So let me go back to that. So the knot is coming from the following place. So I'm looking at this speck of FP sitting inside here. So speck of FP sits as a closed inside speck of Zed. And the question is through the eyes of the etal typology, how does that look? And the claim is that it looks like a knot. It looks like an embedded circle inside speck Zed. So I'm also claiming something extra, which is that speck Zed looks like a three-manifold, but that's a longer story. But let me just convince you that speck FP really does look like a knot embedded inside there. So what has to happen? Well, first of all, it's connected. So then I need to find out what its homotopy groups are. So what are its homotopy groups? Well, they're exactly given to you by the automorphisms of a point, right? And what are the automorphisms of a point? Well, they're just Zed hat. There's nothing else going on there. So this thing really is a BZ hat, which I feel entitled to refer to as a circle. And in fact, it's an embedded circle, and therefore it's a knot. And so if you're familiar with the story about knots and primes from, say, the book of Morishita, everything is actually contained inside this Galois category. You can interpret all that stuff using features from this Galois category. Is there a way, given an infinity category, to recognize it as the constructable sheaves on some scheme? Gee, that's a really good question. Let's see. I don't think I have anything interesting to say, except that's a really good question. That's a really good question. That's a great question. I don't know the answer to that. I have a description here of constructable sheaves as functors. But I don't see how that could give you a recognition principle for when a category is a category of constructable sheaves. Okay, so maybe I can say something. That's sort of a Tanakean question. And I think that's a great question. I'm going to stick with that answer. Yeah, I think that's exactly the right question. And whoever had that question, I want you to talk to me because I have some ideas, but I don't have answers for you. Is there a precise condition on the pycnotic categories of good coefficients? So it depends on what you want to say by good. So I'm comparing the category of constructable sheaves, as it's classically defined, for QL, QL bar coefficients, those sort of classical definitions of constructable sheaves to this functor category. This functor category makes perfect sense with values in perf lambda for any pycnotic ring lambda. And moreover, it has the full complement of the six functor formalism and all the good things that you can think of. So it works perfectly well as a theory of constructable sheaves. It doesn't compare to anything as far as I know that's known for just some random pycnotic ring. As we'll see, the world of pycnotic objects is very, as many exotic beasts in it. So it works insofar as you have a working theory. Whether or not it agrees with something that someone might have defined elsewhere, I can't say. I've only concentrated on the cases of ZL, QL, and extensions of QL, those kinds of contexts, where I really understand the existing definitions correctly. So maybe I have a question about this finite tall dimension. So it seems that what you did is adapted for the constructability condition, which can be tested on fibers on specialization maps. But what if you consider over finiteness conditions? So for example, tall finite dimension, I think it appears when you want the tensor product on the category, on bounded category. So in this statement here, we do have a tensor product on this category here. Yeah, okay. In fact, on this side of the equation is really just the object-wise tensor product. So there's nothing. Then maybe it's closer to the finite or the CTF, like Deline said, the constructable finite tall dimension condition. So I have to remind myself what CTF means normally. So let me just see if I can understand. So it means the category of sheaves with constructable co-amology. And then is the statement that the complexes themselves are globally a finite tall dimension? Or is it another statement? Yeah, no, it's that, I think. So what you can have tensor product on? Yeah, there's a perfectly good tensor product. So I guess what I want to say here is that for this theorem to be, so first of all, I did hide some conditions here. So my X's do have good quasi-compactness properties for this story to work as desired. So the difference between global and local tall dimension is not so serious. How do I say this correctly? I think the locally finite tall dimension condition is, I mean, so my literal definition may I'll just say exactly what the definition of this category is. And maybe that'll help us understand. So the definition that I have for this category, say QL coefficients for the moment. Well, I'm going to cheat first. I'm going to look at constructable sheaves. This is an infinity category with ZL coefficients. And I'm going to invert L. And well, so how is this defined? This thing is defined first as just the limit as n goes to infinity of constructable sheaves X at perf Z mod LZ to the n and then inverting L. So there's no condition explicitly made here for finite tall dimension. And I guess it seems like I could have things that don't have finite tall dimension here, unless I misunderstand. But if you realize, does it mean that all complexes are bounded co-mology? That's what you said? Yeah, they have bounded co-mology. Yes, sorry. Yes, that was important. Yes, bounded co-mology. Okay, let's dig that later maybe. Yeah, I do want to understand if there's something to be refined at that point. That'd be a good point. Can one define X pro at like a Simpson shape? Yeah, so there are, and we're really just starting to define these things correctly, there are stratified schematic homotopy types. And so you really can define things like, you know, X Betty XDR and similar gadgets like that. But we're only just starting to think about those things carefully. Right now, this thing is really just an abstract thing. It doesn't have a kind of schematic description at the level of specificity that I've given so far. So, can I clarify the global structure, the global geometric structure of Gal Zed? When I say objects are all the form of speck fp bar and speck q bar, do I mean there's just one of each type? I mean, there's only one of each type up to isomorphism. Right, so I mean, I only have speck f bar, and I have all the things that are isomorphic to speck f bar. And I have the space of those things that are isomorphic to speck f bar and those things form this knot. Right, so all of the, you should think of each point of this knot as a certain, so if I take a point here, of that knot, that is a choice of an algebraic closure of speck fp. Right, so that's the way you should think of it. And when you think of the deleted tubular neighborhood of this knot, and you think of a point inside that deleted tubular neighborhood, that is again a chosen algebraic closure of qp. But you have a whole bunch of them, and they're all isomorphic, and that's represented by sort of producing paths between them inside this connected space. So this knot here is just a single speck fp. There are other speck fp, so they link in interesting ways. Right, so for example, their mod two linking number is exactly, there's a little genre symbol, at least if your primes are congruent to one mod four. Right, so this is part of the motivation for trying to describe this knots and primes picture. Okay, no more question it seems, or maybe it will be for later for your overtalks. So I think the day is over now. Thanks a lot, Clark, again for your talk.