3/3 Exodromy for ℓ-adic Sheaves
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SpacetimeFunktorAlgebraic structureMany-sorted logicTheoremState of matterRing (mathematics)Category of beingTopostheorieFreezingLimit (category theory)Topologischer RaumComplex (psychology)Point (geometry)Object (grammar)Perfect groupMultiplication signInfinityRight angleCompact spaceCoefficientModulformSubsetSet theoryOperator (mathematics)Goodness of fitNeighbourhood (graph theory)HomotopieRankingKlassenkörpertheorieSocial classLatent heatObservational studyAnalytic setAbsolute valueSolid geometryProendliche GruppeRamificationTensorproduktGalois-FeldDescriptive statisticsFinitismusHyperbolischer RaumExistenceVariety (linguistics)ApproximationMonoidSymmetric matrixSheaf (mathematics)Closed setHomotopiegruppeZariski topologyComplete informationVector spaceNetwork topologyComputer animation
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Natural numberSquare numberProduct (business)Fiber (mathematics)Orientation (vector space)Arrow of timeHomotopieNumerical analysisAssociative propertyInvariant (mathematics)Point (geometry)FundamentalgruppeConjugacy classPrime idealInfinityCategory of beingInverse elementSpacetimePerspective (visual)Algebraic structureModel theoryMany-sorted logicDuality (mathematics)Variety (linguistics)Right angleProendliche GruppeLine (geometry)Network topologyArithmetic progressionp-adische ZahlFunktorGoodness of fitTensorAbsolute valueAnalogyMereologyPolynomialFormal power seriesCondition numberFamilyTrailMultiplication signDiagramCommutatorResultantNegative numberPosition operatorRule of inferenceObject (grammar)Computer animation
01:16:38
HomotopieMeeting/InterviewDiagram
Transcript: English(auto-generated)
00:17
Yeah, so thank you again to the organizers and to the IASH-AS.
00:23
It's been really wonderful. I mean, it's very difficult for me to extract myself from bed at 4 AM. But it's really wonderful that the IASH-AS has made these videos available for me online and so that I can watch them later on and reflect on them. So that's a really wonderful contribution,
00:42
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
01:00
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 monodromy, where we're sort of extracting a certain kind of locally constant or least sheaves.
01:21
And then in the second lecture, we started to see this with what we were calling exodromy 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
01:45
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 general stratified topological space.
02:12
So what's happened so far? Well, I've shown you how things work in the topological setting, or at least I've outlined how they work in the topological setting.
02:22
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,
02:43
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 exodomy theorem is true, which is to say that constructible sheaves are the same things as just functors from that exit path
03:02
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? 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.
03:23
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.
03:43
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.
04:01
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 X-ZAR to a finite POSET P. And so what does that mean?
04:21
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.
04:41
In other words, the Zariski topology is nothing more than a profinite POSET. And by the way, I should mention something. This is, I guess, due to Hockster and his thesis. If you've never looked at Hockster's thesis, it's kind of an amazing read.
05:04
It's an amazing read. Okay, so I'm gonna now use this definition and in 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 gonna take this limit
05:22
over all possible stratifications and you're gonna 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 gonna be our friendly neighborhood scheme and we're gonna be looking at sheaves and we're gonna be looking at sheaves
05:40
for the et al topology. And et al sheaf F, and these are gonna be valued in spaces. So when I say sheaf, I mean sheaf in the sort of homotopy theoretic sense and these are gonna be valued in the infinity category of spaces. And it's going to be declared to be lease
06:01
if it's locally constant with pi finite stocks. So let me break that down. So I mean that there's an et al covering of my scheme X and a bunch of pi finite spaces, one for each member of that et al covering
06:21
such that, well, when I take the pullback, when I take the pullback of my F to each U I, I get the constant sheaf at the corresponding K I. 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.
06:42
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
07:00
is itself finite and pi not is a finite set as well. Okay, so that's what I'm gonna mean by a lease sheaf. So what's a constructible sheaf? Well, okay, 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,
07:21
then the lease sheaves for the et al topology, that's what this is, this is the category of lease sheaves for the et al topology are the same things as functors from a certain infinity group void. And in fact, it's not just an infinity group void, it's even a profinite infinity group void.
07:42
And I'm gonna look at functors from that profinite infinity group void to spaces. So, 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 group void as an inverse limit of finite infinity group voids,
08:01
pi finite infinity group voids, and then the functors just means the co-limit of the corresponding functor categories. And that's all that's happening. Okay, so that's the story when I'm just trying to pick up lease sheaves. Why include art in Maser?
08:25
I don't know if that's a pun. I'm sorry, I don't know what that. Art and Maser are the authors of the text that I'm describing. If you can, I'm not really sure what to answer with that.
08:45
Sorry, I'll press on. So, okay, 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 wanna talk about push forwards
09:08
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. Okay, so let me write down the definition.
09:24
Let X Zar to P be a finite quasi-compact stratification. So remember, so I'm working with only those kinds of things. Then I'm gonna look at an Atal sheave, F, and I'm gonna say that it's P-constructible.
09:44
If for any element P, the sheaf F restricted to the stratum XP, so that's the room of the inverse image of little p under this map, is itself lease, right? So along the strata, I have locally constant sheaves.
10:02
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.
10:24
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?
10:41
I think he's referencing the fact that, well, there's a way to describe this pro-finite infinity groupoid that's pretty efficient. And this is sort of a glimmer of the future here. So if we take this pro-finite infinity groupoid, how could we describe it? Well, one concrete way of describing it is as follows.
11:02
We could take this category of lease sheaves, that is in 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,
11:23
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
11:42
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, oops.
12:00
Let me emphasize that I'm getting this without its topological information. So I take the materialization of that pro-finite stratified space. Or sorry, pro-finite space. We'll get to stratifications in a minute. Okay.
12:21
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 et al sheaves,
12:42
we're actually just taking the co-limit of the category of the p-constructible sheaves over all finite posets p. Okay, so we're just gonna take that co-limit and see what we get. And now, so what's the theorem? The basic theorem here,
13:00
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,
13:23
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,
13:41
along with the conservative functor to a poset. So I'm gonna 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 hat over p, that's an infinity category. And it has the property that, well,
14:02
it has a bunch of objects as 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. Okay, so the fibers are all infinity groupoids.
14:21
And what's the rule gonna be? The rule is gonna 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 for the etology topology. So that's the way this profinite stratified space
14:41
is gonna 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?
15:00
So let me tell you. Basically what we're gonna be doing, oh, so Sean asks, can I explain how to get from x-zar to p to x-et over p? To x-et over p.
15:21
Did I define x-et over p? I don't think I did. So x-et over p right now, this is one, let me emphasize, this is one piece of notation here. I didn't define something called x-et over p. I'm defining the entire notation, pi infinity of x-et relative to p.
15:42
So this is a relative notion of the Atal homotopy type. So before we had the Atal 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
16:02
to a finite coset p. So this is a relative form of the thing that already existed. I haven't defined anything that's by itself called x-et over p yet. And I'm not going to. There is something that you can define, but I'm not going to. Instead, what I'm gonna do is I'm gonna say, well, I'm trying to capture the p-constructible sheaves.
16:22
And so I'm gonna 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.
16:42
I'm trying to define the single object. And now the observation about Tsar 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 gonna give you this,
17:01
excuse me, I'm gonna give you this pro-finite stratified space for each p. And what I'm gonna try to do is I'm gonna 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 pro-finite stratified space
17:21
that classifies all constructible sheaves. That's the program. So first I have to work with fixed poset, and then I'm gonna let my poset vary. So first things first, I need to define this gadget so that this equivalence holds, this auxiliary equivalence holds.
17:41
Okay, 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. Those pieces are the strata and the links.
18:01
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 pro-finite homotopy type. And so I need to tell you what the strata are and I need to tell you what the links are.
18:21
And 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, right? And so that's exactly what I do. I'm just gonna 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?
18:41
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 what we need to understand how to pass from one stratum to the next in a way that respects all of the information that the atoll topology is providing us with.
19:01
And the answer to this question goes back to a really remarkable construction. And I guess it was originally written down by Giraud but it was really implemented for the first time by Deline and it's the oriented fiber product of Topoi.
19:25
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
19:41
in the atoll topology with sort of schemes that you already have or formal schemes in some cases 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
20:01
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.
20:20
It's gonna have a closed piece and a complimentary open piece. And this is the situation that's sometimes called a... In the world of the atoll sheaves, this is sometimes called a recoulement. But the question that we need to answer here is how do we define, how do we take these pieces Z and U
20:43
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 Tanakian kind of argument, like showing that XP constor at is of some kind
21:07
hence there is a profinite space such that the equivalence. I'm so sorry, I don't know if I understand.
21:23
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 Tanakian kind of argument. What I'm saying is that you can break up this sort of hard question of how to construct a P stratified space into the questions about how to construct
21:41
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 atoll homotopy type. And this is, well, it's the homotopy type of a topos. That topos isn't just the atoll topos of the scheme in general, it's something more exotic, but we can say precisely what that exotic thing is.
22:02
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, its strata and links,
22:21
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.
22:46
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 sub-scheme of X and a complimentary open sub-scheme.
23:01
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 ataltopos of X. I don't really care if I construct X or just it's ataltopos. I'm not gonna notice the difference for the purposes of this homotopy type.
23:21
I wanna reconstruct this piece, but from something back here. And while that 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
23:40
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.
24:04
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 exogamy is in analyzing these oriented fiber products.
24:22
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
24:45
for oriented fiber products. And this, it turns out, is absolutely critical for the entire development that we produce. Question, would taking a formal thickening of Z
25:02
and then taking a more classical fiber product not work? That's right, it would not work. I could show you precisely why some other 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
25:22
and then remove Z from that thickening that works in all cases. So one example of this kind of phenomenon, well, so there is something you can do in the case of curves.
25:40
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 can take the compliment of the FP, spec FP in that. And that will give you the right answer
26:01
for this to be distributed. But such moves are not always possible. Do U and Z have to be compliments of each other? Or is there some more abstract formulation? Like, do you require that the homotopy pullback square also be some sort of push out? So, right. So for a gluing square of the type that I'm saying,
26:23
I am talking about the situation in which they're compliments. 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.
26:45
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,
27:00
that there really is a definition of an oriented pullback for any two maps of infinity topology. That's right. So the stratification is precisely sort of controlling that open closed compliment data. And the whole sort of yoga of the sort of decolage type perspective, this functor from the subdivision op
27:21
to whatever your target is, spaces, I guess. That whole yoga is that you can 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 posets that look like that. And so that's our whole goal here
27:41
is that if we can understand well enough and in functorial enough in a way, 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?
28:02
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-invertible, not orientable, then yes, I'm not aware of such a thing,
28:20
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.
28:41
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, oh, Harry Guindy says, it's not a lax limit, it's a weighted limit.
29:03
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
29:23
that category theorists go back and forth about what, yeah, the 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
29:44
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.
30:02
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 post set,
30:22
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 atoll homotopy type of the corresponding stratum of the scheme. And the links are gonna be understood in terms of this oriented fiber product,
30:40
which is 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.
31:01
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 post set P, so then what happens if I want to sort of take the limit
31:21
over all of these post sets 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 stratification P.
31:40
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 post set.
32:03
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 profinite stratified space.
32:27
And I'll just call this pi infinity x at x is over. So at the moment I've solved the problem. I've found a profinite stratified space that definitely classified as my constructible sheaves.
32:42
But if you remember from the first lecture, I gave 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 profinite stratified space.
33:02
So I regard it over the profinite post at x are, 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.
33:21
Oh, there's a question. Pi infinity x at p was already profinite. Yes, it was already profinite, but I'm taking the limit in the pro category. So I can have, if you have a pro object of pro objects and I take the limit of the whole thing, I could just have a single pro object.
33:41
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 x are. So I'm gonna take a point of x are. So it's a point of the underlying
34:00
topological 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.
34:21
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.
34:42
These oriented fiber products they're trying to, so the situation is that sort of x nought is an element to the closure of y nought. 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.
35:00
And this turns out to just be the space of specializations of geometric points, x and y that cover your fixed x nought and y nought. And so the link here is really the sort of category of or not category, but a group void of specializations
35:22
of geometric points that cover your favorite Zariski points x nought and y nought. And that's what happens. So what's the upshot of this? The upshot of this is that this sort of very complicated looking profinite topological space or profinite stratified space becomes reduced
35:41
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,
36:02
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.
36:22
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.
36:42
But the second, I think example is a little more indicative of what's going on here. So let's look at the situation in which A is a DVR. And I might be interested in a picture of taking of gal of spec A. So A being a DVR,
37:02
that means that when I look at a spec A czar, this is an exceptionally boring profinite stratified space. It's really just a finite stratified space. It consists of two different points, the residue field and the fraction field.
37:21
And the residue field is an element of the closure of the fraction field. So that's it. It's just your Pinsky 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.
37:41
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 res of the absolute gala group of the residue field. And on this side here, I have the classifying space of the absolute gala group
38:01
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,
38:24
on this side, I'll have the absolute gala group of Q. On this side, I'll have the absolute gala group of FP. And in between I have something. So what should that something be?
38:41
That something should be the absolute gala 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.
39:01
Now notice that the decomposition group itself, as you might recall, is only defined up to choices. It's only defined up to conjugacy inside the absolute gala group GK. But the map from BDA to BGK, that's a completely canonical map.
39:20
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 gala group of the residue field.
39:45
So Stephen McCain asks, for knots and primes, the tubular neighborhood of the knot is the link from the knot stratum to the knot compliment 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
40:01
that's being described here. So that's exactly right. So we have the absolute gala group of Q. So this is the sort of, this is the situation of the single knot, right? I've got the absolute gala 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.
40:20
And the deleted tubular neighborhood is exactly this guy up here, is 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.
40:43
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 constructible sheaves for the atoll topology can be described as these continuous functors from Gal X, which now has a pretty explicit description
41:00
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 constructible sheaves valued in perfect complexes on lambda. And those are gonna be the same things as continuous functors from Gal X to perf lambda.
41:24
So remember the story that we were telling ourselves is that we wanted to have this kind of exotropy, 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,
41:40
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 a talk on ology, 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 Betti numbers of your favorite algebraic variety,
42:02
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,
42:23
I'm constantly using the fact that I have access to a category. And I'm talking about the continuity from gal X 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
42:41
needs to interact with the topology or the profinite structure if you like on gal X. And I need to be able to speak about continuous functions from gal X 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
43:01
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 Schulze
43:24
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
43:42
if you're not prepared. So let me just say what it is though. So a picnotic object of an infinity category C is a sheaf
44:02
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 compactor.
44:20
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,
44:41
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 in 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.
45:03
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 from the ones that you might be used to. You might be used to sort of building up
45:20
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
45:42
out of co-limits of compacta. 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,
46:08
then if I take maps from blank into Y, that gives me a functor in the correct direction from comp op into sats. And this turns out to be a picnotic set.
46:23
So that's all well and good. It's perfectly, and this works for any topological space at 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.
46:45
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.
47:01
And so that's why this becomes a fully faithful functor into picnotic sets. So this category of sheaves or 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.
47:21
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.
47:40
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 picnotic objects. So let's suppose that I have some random infinity category C. Well, the first thing I can do
48:01
is that if I have a picnotic object, 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.
48:27
Sean asks, is there a theorem that says that pic set is a co-completion of these old fashioned convenient categories of spaces?
48:41
I mean, it is true that every picnotic set 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,
49:01
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
49:21
when we think of a picnotic object in infinity category C? Yeah, yeah, absolutely. I mean, 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. Or something like Daguerre's universal homotopy theory's result. Pic-set is gotten by adjoining
49:41
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 steering you wrong here.
50:01
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, and actually 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.
50:22
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 still open. So these are very, very unusual topological spaces. So the stone check compactification of any set
50:41
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?
51:01
The story here is that we're gonna look at the projective compacta, and we're gonna look at functors from there into C. 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 picnotic objects of our C are the same things as functors from prog op to C
51:26
that carry finite coproducts of these projective compacta to finite products.
51:41
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 picnotic spaces now, you can think of this as the non-abelian derived category, derived infinity category of prog.
52:03
So if you like, it's really that I'm taking prog, 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
52:20
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 picnotic objects of the infinity category,
52:41
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,
53:01
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 that might take me out of the discrete category. So let me show you an example of that.
53:21
But first, let's see, is there a way to define pixie directly as the category of beta modules or algebras in C, where beta is the stone check compactification? So yeah, so roughly speaking, this is the category of algebras for an infinitary Laverre theory. I'm not sure I'm using that phrase correctly,
53:42
but yes, yeah. It's not a Laverre theory in the usual sense. It's like a Laverre 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.
54:05
But let me show you how I want to use this for this particular purpose, for the particular purposes of extraordinary 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.
54:21
Another way to build ZL is I could have just remembered that ZL is a topological space and then made it into a Picnotic set. And well, it's a topological group. It's even a topological ring. So I can turn all those things into Picnotic groups and Picnotic rings. That's all formal. As such, it's actually the limit of the discrete things.
54:43
That's good because that's the way it is in topology. The same thing happens in this context. Similarly QL, when I want to regard QL as a Picnotic 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.
55:03
So now what we're able to do because we're in this Picnotic 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 the topology or anything else because I'm working in the setting
55:21
where everything is just a sheaf. So since I'm only just working with sheaves, I can talk about this thing, the category of perfect complexes in ZL as a Picnotic category or a Picnotic infinity category as a limit of a bunch of Picnotic infinity categories. And which ones are they? Well, I'm gonna take the discrete ones
55:41
on perf of Z mod L. So this is now an object in Picnotic 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.
56:00
It somehow remembers the information that built up the topological structure on the 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 Picnotic categories, perf ZL, perf ZL, perf ZL with multiplication by L everywhere
56:21
and that's gonna provide for me another Picnotic infinity category. And again, this is just happening because I'm speaking about sheaves of categories on compactor. And so I can just say precisely what I mean by that. Okay, so now I've actually given this thing definition. I now know what I mean by perf QL,
56:41
I know what I mean by perf ZL. And so I need to now tell you how I'm gonna regard gal X as a Picnotic category. And well, remember what kind of object is gal X? Well, I constructed gal X as a profinite stratified space. So it's a sequence of pi finite stratified spaces
57:03
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 Picnotic world. So if I have an inverse system of pi finite stratified spaces, then the profinite space that I get by taking the limit of this,
57:21
I can do that in the Picnotic world. And one of the things that we prove is that doing that in the Picnotic world gives you the same structure. In other words, profinite stratified spaces in the sort of formal sense, embed fully faithfully into Picnotic stratified spaces. So that gives me permission to think of gal now
57:42
as an object of Picnotic stratified spaces. So remember what does that means? That means that I'm talking about a Picnotic category with a Picnotic functor down to a Picnotic poset. And the Picnotic poset in question is this X czar and the Picnotic stratified space is this gal X.
58:03
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
58:22
from the category of sheaves on K into the ataltopos 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.
58:41
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 in Picnotic rings? Yes, yes, absolutely. You're also asking about solid things. I'm not using any solidification here right now.
59:00
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 Picnotic category whose underlying infinity category really is just perfect complexes in ZL,
59:21
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, more precisely a Picnotic structure.
59:41
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 Picnotic functors.
01:00:00
from Gal X, which I just gave a pycnotic structure to, to perf lambda and now this is a at least a sensible theorem and the way in which you prove it is you're going to take the theorem that you had for finite rings and you're going to 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
01:00:23
here. What is the version of the earlier observation that Gal X is a finite one category in the pycnotic setting? It's just that it can be expressed as an inverse limit of finite one categories with the discrete pycnotic structure. Oh, and Mark Levine asks, in my description of Gal XK, what
01:00:46
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
01:01:03
got myself just this pure pro-finite 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
01:01:23
surprising to you, it surprised 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 monoidal category, will PICC have a closed symmetric monoidal structure too? Yes, absolutely, absolutely, definitely. And it's just the usual one where you
01:01:44
take the tensor product and use sheaf of phi, tensor product object-wise and then use sheaf of phi. What is the K-twiddle? K-twiddle here is the category of sheaves on K. So I'm going to think of K, K is a compact CalSTRS space, and I'm going to take sheaves of spaces on it in the good
01:02:04
old-fashioned sense. Federico asks, back to the theme, sheaves versus homotopy types, can one pin down a more specific class of analytic sheaves? Suppose that X is defined over a finite field and I'd like to study
01:02:21
lease 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
01:02:51
question. So you can cut out some kinds of interesting homotopy types here. Oh I see, you're thinking about these skeleton sheaves. I think that's a
01:03:06
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. So I think maybe
01:03:20
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? I have some information about that, but I don't have complete information about that. For example, I pretty well understand how the six functors work with respect to operations on Gal, but I may not
01:03:47
have access to 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
01:04:07
happens when you replace Picnotic perfect complexes with solid ones? Do they happen to be the same for QL? Those are the same in that case, right? I mean, if you just think about vector spaces for a
01:04:24
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
01:04:43
don't need the solidification story here. But in general, I mean to do something fancier 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
01:05:02
that the Zariski topology trivializes all higher et al homotopy groups beyond Pi 1? 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 this sort of approximation
01:05:22
to Gal. I finally understood what you're 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
01:05:44
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
01:06:00
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
01:06:26
personally, no. Although the person that I would ask about that is Dustin Clausen, who probably understands a strict superset of what I know about class field theory in the Picnotic setting. Joshua asks, well he says first
01:06:43
some very nice things. Thank you Joshua. Are there any limits to what profinite stratified homotopy types can be represented homotopically as Gal X for a schema stack? Do I understand this question? Any limits to what
01:07:08
profinite 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
01:07:23
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 that there are, you know, this is some kind of stratification of what you might call Galwajian duality. One of the
01:07:42
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
01:08:04
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
01:08:23
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
01:08:42
sorry, the oriented fiber product is 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
01:09:04
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
01:09:24
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 going to 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
01:09:42
some Z 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 this square. This is
01:10:02
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
01:10:20
that you're taking the oriented fiber product of Z with U over X is the same thing as the product of, okay actually I'm not going to write this down, it's this sort of orientation. I don't want to get things confused, but it's just this sort of orientation. If we consider the
01:10:44
oriented fiber product of all strata out of the least sheaves of all strata. Yeah, so if I take the oriented fiber product, ah, 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
01:11:09
three strata. So suppose I've got Z and W and U. And I try and take the oriented fiber product of all three of these. So Z oriented fiber product over W with, or sorry, over X with W and oriented fiber product over X with
01:11:31
U. And this does break down into Z oriented fiber product over X with W crossed over X. This is just the ordinary fiber product now of W oriented fiber
01:11:45
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 Galois problem. I'm not sure
01:12:04
what that is a reference to, Sean. I'm sorry. Michael says, back to knots and primes. Is there any hope that this point of view might actually recover any classical invariants that we weren't able to see very, very before, like a crossing number of polynomial invariants? Yes, that's my hope too.
01:12:25
Yeah, I absolutely hope so. I don't have anything. I don't have any questions. Does the picnotic formalism give a simpler construction of the QL
01:12:41
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 going to give you a
01:13:02
stratified version of the usual QL homotopy type. Absolutely. Yeah, given the analogy between motives over a base X and constructible sheaves, do you think one should be able to get a motivic Galois category so that functors out
01:13:21
of it give back motives over X? Wow, you guys ask such good questions. In the 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 L-adic story
01:13:40
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 constructible isocrystals of Lestum and constructing a kind of Galois category and it won't just be a category anymore,
01:14:02
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. I feel like the L-adic story by itself isn't going to be enough to show us where we need to point. So let's see, there are questions
01:14:22
on this other side too. I should have been reading this, sorry. Already what fundamental groups can, this is a question for Remi, what fundamental groups can occur for varieties is a widely studied question with positive and negative results. Oh, sorry, sorry, sorry. I think this is a reference
01:14:46
to a previous question, I misunderstood. Okay, all right, sorry. I'm doing my best to keep track of all the back-and-forth. Are there other
01:15:01
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 their columnar of G spectra along a family or Mackie functors out of a sieve? What does
01:15:22
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 poset that stratifies the infinity category of G spaces is the poset of
01:15:44
conjugacy classes of subgroups, you know, where subconjugation is the relationship. For G spectra, so I haven't thought of, I haven't, I don't think I
01:16:03
have anything concrete to say about what happens stably. I don't have anything to say. Oh, David Corwin says, sorry, at the beginning I was just wondering, did Art and Mazur talk about homotopy sheaves valued in spaces? I was
01:16:20
wondering why they were included in that. Well, they're included in that because they didn't talk about homotopy sheaves as spaces, but they're included in that because they constructed the atoll homotopy type for the first time. And they did construct what was amounting, you know, what they constructed classified simplicial sheaves, and so therefore the,
01:16:42
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
01:17:01
asking us a lecture. I think we're all very happy with you.