Higher Sheaves
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Toposes online, 202121 / 31
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Transcript: English(auto-generated)
00:15
So the last talk this afternoon will be by Joerg Biderman, and I mean he will talk about
00:23
higher sheaves. So Joerg, it is yours now. You can go. Thank you very much. I would like to thank the organizers for this nice meeting and for inviting me. This is joint work with Mathieu Anel, Eric Finster and André Joyal, and the results that I'm going to talk about are in this preprint, higher sheaves
00:43
and left exact localizations in infinity topos. It's available on the archive. Some related results are in this other article, modalities in homotopy type theory by Reiche, Schulman and Spitters. Okay, so let me start by recalling some facts about one topoi. So if you have a
01:02
small category C and you take pre-sheaves with values and sets, that's a one topos. And every one topos is the left exact localization of such a pre-sheave topos. And then it's true that any left exact localization of a pre-sheave topos corresponds
01:20
bijectively to a Grotendig topology on C. So any left exact localization is given as the sheafification with respect to this Grotendig topology. And then there is also this set theoretic thing that in a one topos any left exact localization is accessible.
01:40
Okay, now I'm going to infinity topoi. So when I, so C is now an infinity one category, and usually I will just say category, and S is my notation for the category of infinity groupoids, and I will basically just say the category of spaces, and an object I will call a space. It's modeled by topological spaces up to weak equivalences or by simplicial sets,
02:06
for example, up to weak equivalences. And then again we have the basic fact in infinity topos theory that if we take a small category C and then we take pre-sheaves on that with values in the category of spaces, well that's an infinity topos.
02:25
Okay, and then it's still true that every infinity topos is the left exact localization of such a pre-sheaf topos. Okay, here I have to say accessible. There might be things that are inaccessible, but I let them be inaccessible. But then here comes the catch now. In higher topos
02:44
theory, not every left exact localization of such a pre-sheaf topos is the sheafification with respect to a Grotendig topology. So Grotendig topologies give us left exact localizations, but there are more. Now how about that? So we are faced with a few questions.
03:06
So given a small category, how do we describe all the left exact localizations of a pre-sheaf topos? Or more generally, if we are given an infinity topos E, how do we describe all the left exact localizations of E? Or slightly reformulated, if I'm given a set S
03:24
of maps in an infinity topos, how can I invert S in a left exact way? So I would like to describe the left exact localization generated by S. I know how, or we know for a long time how to localize with respect to S, but it's not clear how to left exact localize with respect to S,
03:43
at least. And then finally, we need to say what is a sheaf with respect to S. And we can answer all these questions. So the answer will come in a few steps. I will define a nested sequence of full subcategories of E. So given a set S of maps in an infinity
04:07
topos E, I will first define the local objects with respect to S, then the modal objects with respect to S, and finally the sheafs with respect to S. And I will try to explain how all works. So first, so we are given a set of maps S in an infinity topos E,
04:26
and then I call an object X in E, I call it S local, if for any map F in that set S, pre-composing with this map induces a weak equivalence of these mapping spaces. So recall that in an infinity category, an infinity category always comes enriched over
04:44
the category of spaces. So this map E is the mapping space, it's a space. And I'm pre-composing here with F, and if this is a weak equivalence for all the Fs in S, then I call X S local. This is an instance of orthogonality that Charles Resk mentioned in his talk. So we basically,
05:05
this is the condition saying that the map from X to the terminal object is right orthogonal to all the maps in S. Okay, and I will write log ES for the full subcategory of S local objects in E.
05:20
And it's a fact that this log ES is a presentable category, and it's reflective in E. So reflective means that the inclusion functor has a left adjoint, and this left adjoint is the reflector, but sometimes I will just call it the localization, slightly imprecisely. Okay, next I will tell you what are the modal objects. So an object X in an infinity topos E
05:46
is called S modal if it's local with respect to all the maps in S and all their base changes. So that's now a bigger class of maps, and so modal objects are local with respect
06:03
to, they are local with respect to a slightly bigger or bigger class of maps. And I will say, or I will write mod ES for the full subcategory of S modal objects. And now it's true again that these modal objects are a presentable category and they are reflective in E. So why is that? Well,
06:26
there is a trick. In the definition of an infinity topos, one always says that E is a presentable category. So presentable means that there is a choice of generators for that category, and if G is such a set of generators, so there's a choice of set of generators,
06:45
and if G is a set of generators for E, then in the above definition I don't need to take all the base changes of the maps in S, but only the base changes over the generators. So I'm taking my maps in S, and if the target of this map in S is B, then I take all the maps from the
07:05
objects in G to S and I pull back along them. That gives me now a new set, because S is a set and G is a set, so now I have a new set which is bigger but it's still a set, and then the modal objects are the local objects with respect to this bigger set. Okay, so I need to now add
07:27
base changes. Next, I need to define what is the diagonal of a map. So if F is a map from A to B, then I pull it back again along itself, and then I have a little square, and the diagonal,
07:43
there is some other co-cartesian, the cartesian gap map of this little square. And then I need to define iterates of the diagonal, so the convention is that delta zero of F is F, and then delta n of F is just the diagonal of the diagonal of the diagonal,
08:02
and so on. Example, if I take the map from A to the terminal object, then the diagonal of that map is just the diagonal of A, A to A times A. And the nth diagonal of that map is A mapping to A to the S n minus one. So for example, in spaces, this A to the S n minus one
08:24
is just the functions from the sphere to A, and among them you have the constant functions, and this A that maps there picks out exactly the constant functions. And then I can define the
08:41
diagonal closure of this set S. So I take all the maps in S and then I add all their all their higher diagonals, their iterated diagonals, and I call this set delta infinity S, and I will call it the diagonal closure. So now I can give you our definition of sheaves.
09:01
So S is a set of maps in an infinity topos, and now X in E is called an S-sheave if it is modal with respect to this diagonal closure of S. And the full subcategory of E given by the S-sheaves is denoted by sheaves ES. And our main theorem is the following. The category of sheaves ES
09:23
is presentable and reflective in E, and the reflector that goes from from E to the sheaves in E is a left exact localization. In particular, the sheaves themselves form again an infinity topos, and L has the correct universal property that we would like it to have,
09:45
its initial among all the left exact and co-continuous functions that invert invert the maps in S. So left exact means it preserves finite limits and co-continuous means it commutes with all with all co-limits. Okay, so in this way we this is the left exact localization
10:05
generated by my set S of maps. And now I can define what is a higher site. So if I'm back in the case of a pre-sheave topos, I take as generators the representable functors, and I call
10:21
RC the set of representable functors in this pre-sheave topos. And then on the infinity side, it's just a pair of a category C with a set S of maps in the pre-sheave topos. And then a sheave with respect to this infinity site is just a pre-sheave which is local with
10:42
respect to all the RC base changes of the diagonal closure of S. So this is the recipe to get to get sheaves. You take first your set S, you add all the higher diagonals, then you pull them back over your your generators, in this case the representable functors, and then you take the local
11:01
objects. Now this definition is not exactly the definition that Charles gave in his lecture. In his lecture, he defined a higher site, but S in his case was just a set of monomorphisms. But here we have a general set of maps, and we can define now the sheaves with respect
11:24
to a general set of maps. And then we have the corollary that any infinity topos is an infinity site. Okay, I would like to now relate this notion of sheaf to the classical notion of
11:46
sheaf. For that I need the notion of a monomorphism. So a monomorphism is a map such that its diagonal is an isomorphism. An isomorphism here is to interpret it in the higher categorical context, so you would probably say a weak equivalence. Now a monomorphism is the
12:07
same as a minus one truncated map, it's just another name for that. And in the category of spaces, a monomorphism is just the inclusion of a bunch of connected components into your space.
12:24
And in the category of sets, a monomorphism is just an injective map. And there is
12:41
set S, and this set S only consists of monomorphisms, then you call this localization topological. So topological localization is a left exact localization that is generated by some monomorphisms. And then Charles Rask explained, he sketched somehow why these localizations, these topological localizations of appreciative topos correspond exactly to the
13:04
topologies on C. So these are the ones that we already know somehow. And so what is now the classical sheaf condition is just somehow, if you take a set of maps in an infinity topos that
13:26
only consists of monomorphisms, then you don't, then somehow the first step that we took is to take the diagonal closure of your set S, but if you have monomorphisms, well then the diagram is an isomorphism, and the higher diagonals are also isomorphisms. So what you add is just isomorphisms,
13:45
you have your set S plus a bunch of isomorphisms. So those ones you don't need to invert anymore, they are already isomorphisms. So this means that in this case the sheaves are exactly the same as the modal objects. The step of adding diagonals was unnecessary, and then our sheaf
14:02
condition reduces exactly to the classical sheaf condition given by golden-dip topologies, because now you have a set of of monomorphisms, and when you take this step where you pull back over the generators, well if you pull back a monomorphism, you end up with a monomorphism.
14:21
Now you have monomorphisms which target the representable frontus. This is what people call a sieve, right? This thing is now stabled by base change because we just added all the base changes. So this is how you get a classical sieve. And the proof I just explained here, if you have a monomorphism, then the diagonal is an isomorphism. Now in one-topos theory, this is all there is.
14:47
Every Lex localization is topological, so why is that? Well if we take a map f from a to b in a one-topos and we would like to invert it in a left exact way, how do we do it? Well we factor it into a suggestion followed by a monomorphism, and then I start inverting the monomorphism
15:06
h here in the factorization and all its base changes. I need to add the base changes in order to make it left exact, the localization. But that's fine, so far we are only inverting monomorphisms. And then in the next step I need to invert the diagonal of f.
15:25
Well the image of the diagonal of f is the diagonal of the image because the localization induced by inverting h is left exact. So now I invert the diagonal, and this makes then the
15:41
map f also a mono, so now it's a suggestion and a mono, and therefore it's an isomorphism. So this is the recipe to invert a map in a left exact way. But now comes the point, in a one category the diagonal of any map is already a monomorphism. So if you
16:00
do the next step, the second diagonal, it's an isomorphism, there is nothing anymore. This is all there is, you get away in a one-topos to generate left exact localizations in a one-topos, you get away by inverting only monomorphisms. Therefore they are all topological. In higher topos theory that's not the case.
16:20
If you have a space and you take the map to the terminal object, then the diagonal of that map is the diagonal of a, so you go from a to a times a. And again in spaces, yeah you have, or in sets, you take, you can draw the square and then you can draw in the diagonal, it's a subset. But in the higher categorical context this is not a sub-object.
16:44
If it were a sub-object all the fibers would vanish. Let's compute the fibers of this map. Well, in the higher categorical context I need to compute, not the strict fibers, I need to compute the homotopy fibers. And how do I do that? I need to replace my map, my diagonal, with a vibration.
17:03
And one way of doing it, that is by replacing a by the path space. So now I have maps from the unit interval to a, and it has two loose ends, the unit interval, and I can evaluate at each one, and this gives me a map downstairs to a times a. This is a vibration, and this is the one that
17:22
replaces my diagonal, and then I calculate the fiber. So the general fiber at a point a comma b in the product is now the maps, oh sorry, the paths that start at a and end at b. Now if a is equal to b, so if I take a point in the diagonal, well then the fiber is a loop space.
17:44
The loop space of a at the point a, and in general in homotopy theory they don't vanish. That's just a fact of life. They get into the way, and so the diagonal of a space is not a monomorphism. You cannot do the trick I just did in one category here.
18:03
You just, you have to add higher diagonals. So let me just see how many, how much time. Ah, I'm very fast. I'm very fast. Okay, so what is left? What is left? These are the, Lurie recognized this, and he called them the cotopological localizations.
18:24
They are the ones, the left exact localizations, that invert no monomorphism at all. And he also proved the theorem that any left exact localization in a topos, in an infinity topos, can be factored into a topological localization,
18:42
followed by a cotopological localization. And well, the topological localizations are the ones we somehow we know. I mean, for a long time people know how how to deal and to handle Grothendieck topologies and sheaves and things like that. But what are these
19:01
cotopological localizations? I mean, maybe they are just an artifact of higher topos theory. What about them? Well, in fact, they are very important somehow. I'll give you an example. Let fin denote the category of finite spaces. So these are the spaces whose homotopy type is
19:24
equivalent to a finite CW complex. And then I take the category of functos, covariant functos, from finite spaces to spaces. So we denote them by s, a joint x, because it has the universal property of a polynomial ring, but now our ground ring is the category of spaces somehow.
19:44
It's that the infinity topos generated by one object. And in this infinity topos, I can take evaluation at the terminal space, or the one point space, and this gives me a left exact localization. Well, it also preserves all limits because evaluation preserves all limits, but
20:06
that's not so important at the moment. I take evaluation at the point and I call it p0. p0, why? Because it's the zeroth level of the Good Willy Tower. The Good Willy Tower is supposed to be a replacement of the Taylor series,
20:21
and the local objects with respect to p0 are the constant functos. So like for any Taylor series, for any good Taylor series, the zeroth level is given by the constant things. And this localization p0, it has a non-trivial topological part, but it also has a non-trivial co-topological part. And somehow this is somehow what forced us to consider
20:50
co-topological localizations because in our group, the four of us, we started because we wanted to understand the relation between Good Willy calculus and higher
21:01
topos theory. And then this observation, it forced us somehow to come to grips with co-topological localization, and then we started to understand how to generate left exact localizations and everything. And then somehow the point is, that's in a forthcoming paper, is that Good Willy calculus happens completely on the co-topological side somehow.
21:24
So the fact that these co-topological localizations exist gives rise to the Good Willy Tower, and the Good Willy Tower, even though you might not know what it is, doesn't matter, it's central to homotopy theory. So this is supposed to tell you that co-topological
21:41
localizations are not something pathological, they are in fact central. And let me just see, maybe I have time, maybe I don't explain this, well okay, this somehow, this slide
22:02
illustrates how our theorem works. So if we want to generate this localization p0 according to our theorem, we can do it in the following way. We take x, well this x, it can be identified with the functor, the inclusion of finite spaces to spaces, so it's
22:24
the identity functor. And then I consider the map from x to the terminal functor, the functor that is constant, the point. And this, so these two functors, they are representable, the identity is representable at the terminal object, and the terminal functor is representable
22:40
by the empty space. And now we can go through the list, what are the local, the modal objects and the sheaves? Well, the local objects are just the ones that, such that the map from the initial object to the terminal object are isomorphic. Okay, because you see, if you map this map from x to the point to f, you use the unita lemma, and then you get this
23:08
map from f to the, of the empty set to f to the point, and being local forces you that this map is a weak equivalence. Okay, well that's maybe not so interesting. What are the modal objects?
23:21
Well now you can add base changes. So a base change of the map upstairs looks like this, x times a representable functor to a representable functor. The representable functors are now covariant. And a modal object now is just a functor such that the value at k is the same as the value at k union at this joint point. Okay, that's maybe also not so
23:46
interesting. But now what are the sheaves? Well now recall that in fact the first step to generate sheaves is to add all the higher diagonals. Well the diagonals, they look like this x to x to the sn minus one, and x to the sn minus one again is a representable functor,
24:03
representable in a sphere. And now when you take base changes of those maps, you will realize that they give you, they are the unita image of maps from a sphere to k, and the co-fiber is gluing on a cell, it's gluing on an n cell, and any finite space
24:26
is obtained in a finite wave from gluing on these cells. So now a sheave, an x sheave, is a functor that sends, that doesn't distinguish between gluing on sheaves, so f of k
24:41
and f of k with the gluing on cells, f of k with f of k glued on that n cell is just the same. So this means that all the values are the same, so the f is now a constant functor. So this means that p0 is generated by this single map from x to the point. And now our theorem is that
25:03
the Goodwillie Tower is somehow just generated by taking higher joins of that object x somehow. What you do is you take somehow the kernel of your, we call it a congruence class, so that's the map that I inverted by your localization, in this case p0,
25:21
and then you take higher powers of this thing, which is an ideal, you should think of this like an ideal of a ring. So now you take higher powers of that ideal and you divide it out, and what you get is the Goodwillie Tower. In fact, thank you very much, that's all.
25:41
A lot, okay, I mean you have really explained quite well. So are there questions from the audience? I mean I guess I have a, you know, a general question, more philosophical, I mean I understand very well that you want to pass to spaces and so on and so forth, but
26:04
what is really the origin of infinity topos? I mean in that, I mean I mean in a very general broad philosophical sense. Well the fact is that somehow these functors from finite spaces to
26:30
spaces, it's just an infinity topos, right? You mean this is a basic example, you mean? I mean for me, I mean the basic example is maybe the category of spaces itself,
26:42
that's an infinity. Okay. And then let me say it like this, so... But I'm sorry, I mean when you talk about the category of spaces you mean, for instance, simplicial sets or what? I mean... Yeah, I mean the, well I mean the infinity category associated to, for example, simplicial sets. Okay, okay, okay, yeah, as an infinity category,
27:06
via Khan localization. Sure, okay, fine, okay. And so spaces and infinity, let's say co-complete infinity categories, you can think of them as the modules over spaces. So spaces are now thought
27:24
of as a ring. Yeah, okay, fine, that's perfectly good, yes, yeah. And the co-continuous, sorry, the co-complete infinity categories are now like the modules. Okay. And the infinity topos are like s algebras, they are like the algebras in the world. And then somehow...
27:46
But, yes, wait, wait a second, I mean if I take the analogy with gamma sets, where I don't have spaces but I have just sets, I mean, okay. Yeah, so gamma, yeah, gamma spaces, so there that the source category is a one category,
28:11
right, it's just pointed sets or the opposite of pointed sets. Yes, finite sets, let's say, yeah. Yes, and in this case you don't have
28:30
so there are still of course topological localizations somehow. But what we want is, we want a finite space. So if you take functus, covariant functus from finite pointed sets to
28:47
spaces, you can't extend them to all spaces. Exactly, yes, yes. And then you get functus that are, they are called, how do you call them, maybe strongly finitary, they commute with all sifted columns.
29:01
Yes. That's a nice class of functus, but it's a bit restrictive when you take finite spaces as your source category and you can't extend, these are now finitary functus, these are the ones that commute with filtered columns, not with all sifted columns, only with the filtered
29:21
columns. And this is somehow the place where good willy calculus happens. And then you can think of that really just as a ring. Yeah, yeah, okay, as the analog of a ring, yes, yeah. And here somehow we can give this interpretation. In fact, we have a more general
29:45
construction which we call the nilpotence tower, which we are still about, it's in the forthcoming paper, it's not yet on the archive. So you take any infinity topos and any left exact localization, and to this data you can associate a tower of left exact
30:00
localization. Which is like the good willy tower. Which is like the good willy tower, exactly. And the good willy tower is an example, as is the orthogonal tower by Weiss, and there's also a unitary tower by Neill, they are all examples of that. Well, that's very good, yeah. Okay, so thank you, and bye-bye. Thank you very much.