Toposes generated by compact projectives, and the example of condensed sets
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Toposes online, 202123 / 31
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Set theoryComputer animation
Transcript: English(auto-generated)
00:16
So Dustin, this is yours, okay. Thank you very much, I'm very pleased to be talking here at
00:23
this TOPOS conference. Well, I don't know, like many people, my first time meeting TOPOS was when I was studying et al. cohomology as a student, and like many people I was given the advice that you shouldn't delve into the general theory because all you need to know is just looking at the example of et al. sites and doing things explicitly there, and like
00:44
thankfully enough people I've ignored that advice and studied the general theory anyway. So, and this has been good for me, so I'm very happy to be here talking about this in a setting where I don't feel like I have to shy away from the general nonsense. So, yes, so this is in some sense part one of two talk, I've got my collaborator Peter
01:04
Schultze here, we're going to be explaining some aspects of the work we've been doing over the past few years. Let me start by sharing my screen, let's see if it works. Okay, so there's the title, and let me draw just a rough schematic of the situation I want to
01:22
sketch out here. So we've got the, well, the concept of TOPOS, I guess. And then there's another very interesting, very general class of categories, we think of a TOPOS as a kind of category, a class of categories called the categories generated by compact
01:42
projectives, or the sifted end categories. So I'll just write S-end, or categories generated by compact projectives. Part of my goal is to explain a little more about what that means. And then there's some intersection here with things which are both a TOPOS and are generated by compact projectives, have very favorable properties, and specific examples are given
02:03
by pre-sheaf TOPOSes. And then there's one example, well, there are lots of examples, of course, outside that, but there's one example in particular I want to talk about, which is the example of condensed sets. So that's a rough schematic of the world we're going to be talking about here. And I'll take it for granted that, well,
02:26
the concept of TOPOS is at least somewhat familiar to everybody who's attending this this conference, but I want to spend a little bit of time reminding the definitions over in this half here and how they came up and what they do for you. So here's the basic definition.
02:45
So suppose C is a category with all co-limits, and let X be an object in C. So the first part is we say X is compact if, and this is the standard definition, if homs out of X
03:11
commutes with filtered co-limits. The second is we say X is projective if
03:33
homs out of X commutes with a different class of co-limits, namely
03:43
reflexive co-equalizers. So let me, well, what does that mean? So a co-equalizer is when you take a, well, when you co-equalize two maps and it's a reflexive co-equalizer, if there's a common
04:02
a common retraction for these, or what do they call it? Splitting, or it's a splitting, yeah, a common splitting for these two maps. So that seems, it might seem a little bit weird at first sight, but there's a method to the madness. So, well, so then X is compact and
04:22
projective, or compact projective, so well, compact projective if and only if it's compact and projective, but then you can also characterize it in terms of calming out commuting with certain co-limits. Well, it's whatever co-limits you can build by
04:42
combining filtered co-limits with reflexive co-equalizers, but there's a nice characterization of such co-limits. This should commute with sifted co-limits where a diagram is said to be sifted if, you know, if whenever you have two objects, then the category of common objects
05:08
they map to is connected. So it's a weakening of the notion of a filtered co-limit where you require sort of asymptotic equality of all, you know, morphisms originating from any two.
05:24
Maybe I should have a condition, and it should also be non-empty, I guess. I mean, really I should have a condition for any finite set of objects this analogous category is connected, and when that finite set is the empty set, I would just be saying my category is non-empty. So I probably have to throw that in. We finally talk about two objects.
05:41
So it's a weakening of the notion of filtered co-limit. And the sifted co-limits, the significance of them is that in sets finite products commute with sifted co-limits.
06:11
So you may know the fact about filtered co-limits that all finite limits commute with filtered co-limits, but if you only care about finite products and not general, say, fiber
06:21
products, then you get, well, it'll commute with more things, and it's exactly this extra reflexive co-equalizers that you get. So this is kind of, I mean, it's kind of a little bit obscure at first sight. So it's not true that a co-equalizer without the reflexive part commutes with finite products, and it might seem a little bit obscure why this little extra
06:41
bit of data guarantees you this property, but I don't know. If you've studied algebraic topology, then you may know this fact that geometric realization of simplicial sets commutes with products, and it's very closely related to that. So this is actually, I think, the right way to look at this reflexive co-equalizer category. It's some truncation of the
07:02
simplex category. It's the category of non-empty finite ordered sets with one or two elements. And you can kind of, I don't know, think about how when you take the product of two simplicial intervals, you get a simplicial square. That's kind of, you know, you need the
07:23
non-degenerate simplices to, in order to pop out and form the, you need the degenerate simplices on your interval to pop out and form the non-degenerate simplices of higher dimension you know must exist when you take their cartesian products. I don't know, there's some, there's something funny going on here. Well, it's not funny. There's something fundamental going
07:41
on. It's not obvious at a naive glance, but so I thought I'd, right, so let me continue the definition. So we say C is generated by compact projectives if the smallest co-complete
08:08
subcategory containing all the compact projectives is C. So if you start with
08:21
compact projectives and take co-limits and you're allowed to iterate if you like, then and if you reach C, any object of C by that procedure, then we say that C is generated by compact projectives. But you don't need to iterate. You can just take co-limits once,
08:41
and in fact you can add co-limits, spices only add co-limits of a very controlled fashion. So the first sort of proposition about this is that, well, if C is generated by compact projectives,
09:02
then every x in C is directly a sifted co-limit of compact projective objects,
09:20
and in fact you can say something more precise. So C is equal to, identifies with a certain categorical construction called the sifted in construction on the full subcategory of compact projective objects. So this is a category obtained by formally add-in sifted co-limits,
09:47
just like if you're used to the in category where you formally add in filtered co-limits. So an object in here can always be represented by some sifted co-limit of, whoops, sifted co-limit of x i's, and if you want to know how to hum from this to that, well,
10:06
you can pull out the first co-limit, that's no surprise. So then you get x i co-limit j y j, but you can also pull out the second co-limit because these are supposed to behave like compact projective objects. So you get co-limit over j, limit over i, x i y j, and that explains how
10:30
calculations in this category are formally reduced to calculations among the compact projective objects. So there's an analogy between generated by compact projectives and
10:40
generated by compact objects and between sifted co-limits and filtered co-limits, and this s ind construction and the usual ind construction. Okay, now let me tell you what the basic class of examples is.
11:08
So, well, any algebraic theory, and so, well, let's, so, I mean, if you have anything where
11:20
you have like an underlying set and then you're supposed to specify certain operations on that set from x to the n to x, and those maybe satisfy certain axioms for like associativity or commutativity or I don't know, so anything where you kind of, you have an underlying set and some operations specified like this with some relations also specified purely in terms of
11:45
Cartesian products, then the category of things satisfying those, that algebraic property forms a category generated by compact projectives. So let me give an example here. So, well, let's say abelian groups, so here you give a set and you give a map from x squared to x
12:06
and it satisfies some certain axioms that you can write down, associativity and so on, and essentially because everything is expressed on the level of finite products here and this sifted co-limits are the things that commute with finite products,
12:21
you get that the abelian groups is generated by compact projective objects and what are these compact projective objects going to be? Well, they're going to be the, well, a priori, the retracts of free abelian groups of finite
12:40
rank, so z to the n's. So about this business of retracts, I mean, so this s ind construction, it applies to say any category with a finite co-products, so maybe I should make a remark that
13:03
you know these compact projective objects is always closed under finite co-products and retracts. The retracts are not so important in essence because if this s ind construction
13:23
doesn't, I mean, it does the same thing for CCP, you know, for a category closed under finite co-products and it's idempotent completion, so if you close under retracts, it doesn't change this construction. So if you want to know what your category is, it's enough to know that it's generated by a class of compact projectors, which is
13:41
finite co-limits, you don't have to know that you have all the compact projectors there. All of them would be gotten by taking retracts, but it's often, often you don't have to understand explicitly what the retracts are. Right, so another example would be, say, commutative rings.
14:00
So they're the compact projectors, well, retracts again, of the polynomial rings. So the free object, you always do the free objects on finitely many generators in whatever algebraic context you're working in. So here it's free rebuilding groups of finite ring,
14:20
here it's polynomial rings, you know, non-commutative rings, you'd have the free free algebra and n generators over the integers. Another example, you can mix the two of course, and you can take the category of r modules, and then the compact projectors would be the finitely generated projective r modules. But these examples, algebraic theories,
14:44
algebraic theories have this specific property that it's generated by, well, it's in fact generated by a single compact predictive. So in this case, it's z, if there was the
15:02
free object on one generator, here would be z bracket x, here it would be r as an r module, then you'd just take all finite co-products of those to get a sufficient class of compact projectors. But the example that we're going to talk about, condensed sets, is somehow a different form than this. So it'll be generated not by a single compact predictive, but by a whole class
15:26
of compact projectors. And this means when you're generated by a single compact predictive, it makes sense to say that you have an underlying set that tells you a lot about your object, but in examples where it's not generated by a single compact predictive, well, it's not, you don't just have an underlying set, or you have an underlying set,
15:41
but it doesn't tell you all that you need to know. Okay, so now let's talk about again, this intersection of topos and s-in. So how can we characterize which topoi are generated by compact projectors? So again, there's a little proposition.
16:06
So in a topos, what is this concept of compact projective?
16:20
So it's compact projective if and only if, well, it's quasi-compact and projective. Again, so you always have to be careful. There are these two different notions of compactness in a topos, being a compact object and being quasi-compact, so every cover has a finite sub-cover
16:43
basically, do not agree in general, but when you add the projectivity condition, they become the same. So it's the same as saying if x is covered by a collection of maps, you know, f i, so if you have an epimorphism from some disjoint union of x i's to f,
17:01
then there exists a finite subset, j subset i, and a splitting of disjoint union i and j x i going to x. So it is kind of a mixture of what
17:32
you would think of as being quasi-compact, so every cover has a finite sub-cover, and what you would naively think of as being projective, so every projection has a splitting in essence.
17:42
So that's one thing, you can kind of make very explicit what the compact projectives are, and another thing is, so we'd like to understand what the topos is generated by compact projectives are, and in general, if you have a generating set for a topos, then you can describe
18:03
your topos as xi's with respect to the induced growing deep topology on that subcategory. So you might wonder what kind of growing deep topologies arise on the collection of compact projectives, in the case where compact projectives generate the topos. So suppose
18:22
c is a category with finite coproducts. So we'd like to know, so the compact
18:43
projective objects in a topos are basically the ones for which, so you know that when you have a splitting, then the question of the sheaf condition being satisfied over x for this covering here is equivalent to the same question for this disjoint union. So in other words, every covering is in some sense refined by a finite disjoint union covering. So if we
19:03
want to get our topos generated by compact projectives, we should start with a category of finite coproducts, and just try to make the growing deep topology which says that, you know, finite coproducts should be covers. So when is that a growing deep topology? Well, we need to make sure that the, in some sense, the axioms of a topos are satisfied,
19:21
or the ones that we have access to by virtue of the fact that c has finite coproducts. So suppose the finite coproducts are disjoint and universal. So what does this mean? This
19:40
means that here, this means that if you take x fiber product over x disjoint union y with y, then you get the empty set, or the initial object. And universal means that if you have x disjoint union y, and you have to say x mapping in, if you have x disjoint union y isomorphic to z, and then you have some z prime mapping to z,
20:04
then when you pull back x prime and y prime, you get also a disjoint union decomposition of z. So pullbacks of finite coproducts are still finite coproducts. Then you get a growing deep topology,
20:23
where the covering sieves are those generated by a finite disjoint union decomposition.
20:44
And this is a, this gives a topos generated by compact projectors. And also all topoi generated by compact projectors are of this form, with c is the compact projective objects in
21:13
the topos that you're considering. So it's just, I mean, it's kind of something, you know,
21:21
fairly obvious here, I guess, from topos perspective, you just, you know, if you're an s-in category, then all co-limits except for the finite coproducts are sort of formally built. So you just start with a category of finite coproducts, add the versions of the topos axiom that you can articulate with finite coproducts.
21:41
And then that's, yeah, then that will generate the topos of the required form. Okay, well, so this is a whole bunch of abstract nonsense. Let's get to the example that we care about. But maybe, well, maybe, yeah, maybe I should make a philosophical point here.
22:00
So, you know, this concept of topos is a generalization of the concept of topological space. This concept of s-in category is a generalization of the concept of algebraic theory. And this business of condensed sets was designed in order to be able to mix algebra with topology.
22:24
And it wasn't by thinking about this that we came to it, but somehow it fits, I don't know, you want a topos living in the intersection here. I mean, it was very concrete calculations that led me personally to go with this precise notion. So I wasn't thinking about these abstract concepts, but still it makes sense from a purely abstract perspective.
22:44
So what is the definition? So it's a sheaf on the site of profinite sets.
23:05
So these are, if you like, topological spaces, homeomorphic to a inverse, filtered inverse limit of finite discrete sets. This category is also equivalent to
23:23
the pro category of finite sets. So the name really does fit. With the topology generated by finite disjoint unions and surjective maps. There is a little set theoretic subtlety that one
23:53
perhaps ought to mention. This category of pro-finite sets is not a small category. So
24:01
the notion of a sheaf on a large category is not exactly well defined as a category, the category of sheaves at least. So one should fix maybe a cutoff cardinal kappa, which should be a strong limit cardinal. And then you'll see why it should be a strong limit cardinal in just a second. And then just consider only the pro-finite sets bounded by kappa.
24:21
This is a purely a technicality and it's not very important, but I feel like I should mention it. And I should also mention that the same concept had been developed by Barwick and Hayne. So they had also started studying this around the same time as us. They chose a different
24:41
name called Picnotic set, and it was a slightly different way of resolving the set theoretic technicalities, but it's really the same thing. Okay. So, well, so what, so if you're a condensed set, then if you're a condensed set X,
25:01
then by definition, you have a value X of T for any T in pro-finite sets. And you want to think of this as a continuous maps from T to X. So you think you're,
25:24
so the idea behind condensed sets is it's a replacement for topological spaces. And the way that it works is that you're sort of only specifying the topology by specifying what the continuous maps are from a pro-finite set. So, or the other way of thinking about it,
25:44
in terms of the fact that in a topos, every object is generated under program that's by objects in the image of the Oneida embedding, is that you're only looking at topological spaces, which are somewhat built from pro-finite sets. And well, and it's not really equivalent to
26:01
topological spaces, but it's something that's formally built from pro-finite sets subject only to the relations that are sort of implicit in the definition of the site here. So everything you're supposed to, when you do condensed sets, you're supposed to think everything is built out of pro-finite sets. And you're not supposed to care so much about open subsets anymore, but just the manner in which you build things from these basic pieces.
26:25
Okay. So the idea should be that this is a very nice topos. So it behaves very much like sets, a lot like sets.
26:46
Like the topos of sets and mixes with algebra well. But on the other hand, it contains lots of topological spaces of interest, or basically all topological,
27:05
that's maybe an exaggeration, contains most topological spaces people study as a full subcategory. And moreover, it doesn't do weird things to them.
27:22
Calculations with them, for example, work out nicely. And I'm going to give one example of that towards the end of the talk. So, but maybe I'll say right now what, make this more precise, contains most topological
27:43
spaces as a full subcategory. So there's a functor from topological spaces to condensed sets, which makes kind of this intuitive description of the t value points into a definition. So it's called x goes to x underline. So x underline of t is just the set of continuous
28:02
maps from t to x. This gives a fully faithful embedding from compactly generated
28:21
weak house store spaces. Okay. And maybe you want to put the kappa into there and like kappa compactly generated. Let me ignore that. Two condensed sets. And this is a very large class
28:42
of topological spaces, which includes, for example, the CW complexes people use in algebraic topology. It includes any metrizable space. It includes any locally compact space. And so really it's a, for a very large class of things, you can just put them in this world,
29:02
which again, since it's a nice topos, has very good general features. But on the other hand, yeah, it contains all of the reasonable examples that you'd be looking at. Okay. So it is supposed to be some general thing. If you're ever thinking of mixing topology
29:25
with some algebraic structure, it's a good idea to, we think, use condensed sets instead of using topological spaces. And so Peter is going to go into a specific example of this in terms of real functional analysis. So that's one place where you mix topology and algebra,
29:45
of course, to tame infinite dimensional vector spaces, the standard approaches to put a topology on it. But Peter's going to explain, well, that perhaps it's more appropriate to tame them by condensed structures instead. But also, but I want to say that the main thing we do with this
30:02
is the, so what do we do with these? Do we do with condensed sets? Well, the main thing
30:21
is it lets us make a definition, a notion of an analytic ring. So, and what is an analytic ring? So an analytic ring is a pair consisting of
30:46
a condensed string R, or maybe I should write R, I never know exactly what notation I should use. Well, I'll call the analytic ring R as well without maybe some kind of potential for confusion. Or no, I'll call it script, I'll call the analytic ring
31:03
script R, is normal R, and then something you call mod R, which is supposed to be a full subcategory of just modules over this condensed ring. So condensed abelian groups with an action by this condensed ring R. And this is supposed to satisfy some strong axioms.
31:25
And you're supposed to think of this as a of complete objects. So, you know, when you're doing a functional analysis or
31:46
analytic geometry, anytime you're dealing with, you know, modules over a topological ring, especially if they're infinite dimensional modules, and you kind of want these big things in there to make a good theory, then you run into this problem that the notion of
32:01
you want to have is a completed tensor product. But there's nothing a priori in the definition of the notion of an R module over a topological or condensed ring R that tells you what the completion functor should be. So you actually have to add it in some sense as part of the data defining the theory that you specify not just a ring, so basic objects in analytic geometry
32:23
as opposed to algebraic geometry is not just a ring, but a ring together with a notion of which you could say defines the notion of module over the analytic ring that you're interested in. It's the complete R modules in some sense, which is part of the definition.
32:42
And so, yeah, so this, and then we can globalize this analytic ring in the style of algebraic geometry. You can globalize analytic rings to analytic spaces and you get a theory which encompasses all sorts of classical
33:02
analytic or algebraic space. So schemes give examples, complex analytic spaces, rigid analytic spaces. I don't know. You can even do things like topological manifolds
33:20
or real manifolds. Sorry, not maybe not topological manifolds, not so obviously real manifolds, at least smooth manifolds. I mean, even the topological ones. Okay. That's not obvious to me, but I'll trust you on that. I mean, it's not nuclear.
33:47
Yeah, at least it gives me pause. Yeah. Yeah, no, but yeah. Anyway, it works. Okay. Overconversion continuous functions, they behave as they should. Oh, really? Okay. All right. Yeah, fine. And someone follows automatically from them
34:03
being a module over whatever they have to behave correctly. Anyway, you just have to check some. It follows. What? It follows automatically. It does. Okay. Well, I certainly trust you on that. Yeah. All right. So, yeah. Okay. So now,
34:33
oh, so there's something about this situation that I want to point out.
34:41
So, well, essentially because of the fact that, well, actually I haven't yet explained why condensed sets is generated by compact projectives. So why is condensed sets generated by compact projectives? Well, certainly this defining site of propionate sets does not consist in
35:04
general of compact projectives. I mean, we need both. We have the topology generated by finite disjoint unions and surjective maps. And these surjective maps really are playing an important role here. You can't just leave them out. But so nonetheless, there is a subcategory
35:20
which still generates the topos. And this comes from a remark of Gleason's, I guess. So the category of propionate sets has enough compact objects. So for all propionate sets T,
35:41
we're actually, it's enough, I mean, it's enough to talk about a compact house store space T. There exists a T prime, a propionate set, with a surjective map, T prime maps to T, such that, you know, any surjection to T prime splits. So you can cover any object by
36:13
a projective object. And since the topology is finitary, that will mean it will be a compact projective object.
36:24
Okay, so that's actually quite easy to see why this, well, once you have a certain construction, it's fairly easy. So in fact, you can take T prime to be the stone check compactification of the underlying discrete set, the underlying set of T. So certainly the discrete set underlying T has a continuous map to T.
36:48
And by the universal, surjective map, and by the universal property of stone check compactification, that extends to this thing here. And on the other hand, every surjection here splits because it suffices to split, again, the universal property,
37:01
it suffices to split the restriction to T discrete, but by the action of choice, any surjection to a discrete set admits a splitting. So that's that. So there is a different defining site, which consists of the projective profinite sets,
37:20
or you could just also just take the profinite sets of this form, stone check compactification of the discrete set, which defines this. Okay, now I want to point something out here that this automatically implies that, you know, that when R, if ever R is a condensed string,
37:42
then mod R is an abelian category. Well, that's, that's general topos theory, but it's an abelian category, which also is generated by compact projectives. And this implies that it has the exact same, has the, all the exactness properties,
38:06
the same exactness properties as, you know, the category of abelian groups, or the category of modules over an ordinary ring. Because if you have a compact projective object
38:20
in an abelian category, then homs out from your abelian category A to abelian groups, it commutes with all co-limits and all limits. Well, all limits is obvious. All co-limits, well, it commutes with all sifted co-limits because this is compact. But then the only thing, difference between a sifted co-limit and an arbitrary co-limit is a finite co-product, but
38:43
in an additive category, finite co-products and finite products are the same. So that, that's automatic as well. So this means, and now if you have compact projective generators, then you can test everything on maps from compact projective objects. And that means that any question of commuting a limit with a co-limit or a limit with another limit or
39:01
whatever, everything reduces to abelian groups. So for example, filtered co-limits are exact. Again, this is a general feature of a topos, but something that's definitely not a general feature of a topos, filtered inverse limits are also exact. And that's very, that's actually quite crucial for some of the calculations that we do. And yeah, it's a nice property. So,
39:26
or filtered or infinite products are exact. Right. So yeah, I meant just infinite products are exact, not filtered inverse limits. I apologize. So right. And then the axiomatics
39:45
that I didn't spell out, but they also imply that modules over a script are, is also a category generated by compact projectives. And this means that although it is encoding some kind of functional analysis or something,
40:03
where you have some more or less topological ring and some notion of complete modules over, over with some complete tensor product from a purely categorical algebraic perspective, these things are not that much different from working with modules over an ordinary ring. The only difference is that instead of having one generator, you have the whole family of
40:21
generators parameterized by your pro finite sets, say. So, but from a formal perspective, this lets you import many notions from pure algebra into this setting, but then it has, it gives you implications in analytic geometry or in functional analysis. Just through this, this route here and there, and there, it is really important that we,
40:44
not just that we live in the world of topos, although of course that gives you a lot, but you also live in this world here. This, these, these compact projectors are very important for us. Now, but you also might think, okay, well maybe now that you have,
41:00
how important is the topos concept really? So now that you have your an isobelian categories generated by compact projectors, you really need to know that they came from a topos? Well, maybe not, but to produce examples of this axiomatics that we have, it's extremely important that you, you come from a topos. And I want to explain a mechanism,
41:23
um, so I want to explain why, essentially. So, also it's extremely useful in practice to have it come from topos, because if you want to say how, what it means for like a group to act on such a thing, and this only really
41:43
makes sense if there's this ambient topos formalism around. Yeah, that's a very good point. Yeah, I would just be repeating what Peter said, but just in case, um, yeah, if you want to, you know, if you want to know what a representation of a group, you know, one thing you'd like to do
42:02
with abelian categories is, for example, look at, I don't know, uh, yeah, well basically just what Peter said, you might want to talk about g-modules for a condensed group g, and to make sense of that, you have to know that it came from a topos basically. And again, well yeah, I'm going to make a similar point in just a second at the more primitive levels of condensed
42:21
sets, but having viewing condensed sets in here is actually kind of quite crucial. Okay,
42:41
yes, but first I want to say something again quite general. So I was talking about topos generated by compact projectors, and it was the same thing as saying topos generated by quasi-compact projectors. But in this example, we have something, um, something even better.
43:03
These generating quasi-compact, quasi-compact objects are also quasi-compact quasi-separated. So if you have a topos generated by quasi-compact objects, then an object is said to be
43:26
quasi-separated. If whenever you have two quasi-compact objects mapping to it, then the fiber product is also quasi-compact. And it follows then that if you look at the collection of quasi-compact quasi-separated objects, so if you're generated by quasi-compact
43:43
quasi-separated objects, you get very good closure properties. So it's closed under finite limits, and one of the fundamental examples of a co-limit, namely and quotients by equivalence relations, meaning if you have a quasi-compact quasi-separated object and you quotient by
44:06
quasi-compact quasi-separated equivalence relation, you still get a quasi-compact quasi-separated object. So this QCQS is the most, maybe the most convenient finiteness property you can have in a topos. And it's convenient because it has all these good closure properties
44:27
which involve the usual operations one considers in topos theory. Well, not the infinitary ones, but to consider this as finitary and it's about the finitary closure properties that you usually use. In contrast to the projective objects, which don't have very good closure properties at all,
44:44
the only thing they're closed under is finite coproducts and retracts. So it follows also that the collection of quasi-compact quasi-separated objects will also give a different defining
45:00
site for our topos. And in our example of condensed sets, quasi-compact quasi-separated is exactly the same thing as compact Hausdorff. So this purely topos theoretic finiteness property
45:20
recovers this standard kind of finiteness property in point set topology. So that's kind of interesting. And it also tells us that we have another possible defining site for condensed sets. We can make the same definition and replace profinite sets by compact Hausdorff spaces and we'd get the same topos. Right. Now I want to say, so how do we produce examples of these
45:56
strong axioms? I haven't said what the strong axioms are, but to verify these things,
46:01
you essentially need to do X calculations. So to produce analytic rings, you need to be able to calculate X. And for this, we use a topos theoretic result. So even though it's X,
46:29
we're interested in calculating and appealing categories, we have to reduce ourselves to topos theory to make these calculations. And this is a very interesting theorem of Green-Deligne.
46:46
So there exists a functorial resolution of any abelian group terms of the form.
47:08
So you take a finite direct sum of the free abelian group on some product of copies.
47:25
We take a finite direct sum over the free abelian group on some number of copies of the free abelian group on the underlying set of the enfold Cartesian product of A with itself, where this is less than where the exponent is also a finite number.
47:45
So, well, it's easy to get started, right? I mean, you can always surject from Z bracket A, the free abelian group on A to A by just, you know, in the standard way, take a formal A and A here and map it to the actual A. And it's not too hard to see that the kernel of this
48:02
is generated by expressions of the form A plus B minus A minus B. So this lets you continue to a second term here. But then after that, it's actually not so obvious that the relations between relations here can also be written in this planetary form.
48:21
And that you can continue infinitely is actually a very serious theorem. And I want to give a hint of how it's proved, both because I think it's interesting and because it again generates, tells you a bit about the power of this notion of category generated
48:45
by compact projectives. So, right, so, well, the category of abelian groups is one of these simplicial in categories. And so the category itself is formally determined under sifted
49:00
co-limits by the free abelian groups. And that tells you, the finite free abelian groups, and that tells you it's enough to handle the case where A is a finite free abelian group, provided again, you ensure the resolution is functorial, then you'll be able to pass to arbitrary abelian groups. And then, so this is the first step. The second step is that you
49:26
recognize such a functorial resolution, this as a resolution of the identity functor. Well, not really identity functor, but so let's call this full subcategory of Z to the N's
49:44
so not the identity functor, but the inclusion functor by compact projective objects in this functor category. So additive functors from lattices to abelian groups.
50:08
So the claim is that this Breen-Deline theorem is exactly equivalent to saying that the inclusion functor from lattices to abelian groups admits a resolution by compact projective objects in this abelian category. And this is a very fluid notion, admitting a resolution by
50:24
compact projective objects. It's known as being pseudo-coherent. And it has a lot of permanent properties, which let's you make sort of a devizage style arguments with it. And that's very convenient. And the third main idea in the proof is quite remarkable, I think. Well, at least
50:41
in one way of presenting the proof is that if you do the analog over the sphere spectrum, then this is easy, easy and explicit. So what do I mean? I mean that you can make a resolution,
51:03
well, you can't say of the Eilenberg-McLean spectrum on an abelian group by finite direct sums of copies of the free spectrum on some finite product of copies of the underlying set of your abelian group A. And actually, well, this is easy and explicit if you
51:23
just know that the Eilenberg-McLean spectrum is constructed by taking an iterated bar construction of your abelian group. And in the iterated bar construction, the only things that it's a simplicial construction and the only terms that appear are finite products of copies of A. You just iterate that over and over again and you again only get finite products of copies of A. And this turns into exactly the desired statement. Now, and then the fourth main
51:45
point is Sarah's finiteness. So that the homotopic groups of spheres are finite abelian groups for i bigger than zero. And this tells you that the sphere version is only a little bit,
52:02
only finite much off from the desired version with abelian groups. And if you combine with the fact that this notion of pseudo-coherence is suitably flexible, a suitably flexible kind of finiteness property, then you see that in fact the sphere version for somewhat in explicit but not difficult to explain reasons implies the abelian group version.
52:24
But it's totally non-obvious to me how to calculate it or whether it's even possible to write an algorithm to calculate like a resolution up to 100 terms or something like this. I don't say homology of Alain Bremerton spaces to homotopic groups of spheres this problem lies.
52:43
So it's really quite an explicit and for that reason I think it's all the more remarkable that this in explicit resolution is our main calculational tool. So how can you use an in explicit resolution to do calculations? And I want to end the lecture
53:02
giving just one example of this. Well, a state of results which I think is the first theorem, this first real theorem in this study of condensed sets. And it was the thing that,
53:20
well from my perspective, I knew I needed the theorem to be satisfied but I didn't know what the correct definition necessarily of condensed sets was. And I ended up studying this one because it was the only one for which the theorem had a proof. So let me state the theorem. So well I already said that locally compact house to our spaces all give examples of condensed
53:43
sets and in particular locally compact abelian groups is a full subcategory of condensed abelian groups. Well that's not too difficult but then you also want to know that x's between them are reasonable. So if you have a and b in there then the claim is that if you take the x i
54:06
in condensed abelian groups from a underline to b underline you get just the continuous homomorphisms from a to b in degree zero. Well that's just what I said about the full faithfulness. You get the extensions of a by b when i equals one. So meaning the locally
54:28
compact abelian groups which sit in a short exact sequence, they sit as the middle term of a short exact sequence with b on the end and a on the other end. And then you get zero for i bigger than one. So this is an example of calculations in condensed abelian groups. So
54:43
doing what you want or giving something completely reasonable so that this abstract nonsense stuff actually matches with something that looks like it makes sense in the real world. So you use a well a lot of reductions reduced to two key calculations.
55:08
One is that x i from the real numbers to z say is zero in all degrees.
55:20
And the second thing is it's kind of I guess I know the Pontryagin tool to that that it that x i from r mod z to r not yeah r mod z to r is zero for all i bigger than zero. And I want to explain how to use the Breen-Dellin resolution to prove something like this
55:45
to finish up the talk. So the Breen-Dellin resolution, so I said it gives you a
56:01
functorial resolution of any abelian group by free abelian groups on the underlying set. Now since it's functorial it automatically passes to any pre-sheaf category. But then you can sheafify and sheafification is exact. So in fact the statement exactly the same statement holds in any topos. So any abelian group object in a topos
56:22
admits a resolution by free abelian group objects on the underlying object of a finite product. So we can in particular apply it to this topos of condensed sets. So then this shows you that x to i r mod z to r is calculated by complex
56:47
with terms some finite direct sum of continuous maps from some product of copies again finite product of copies of r mod z to r.
57:05
And this is actually a complex of Banach spaces. So with just the sup norm but that's actually also just the Banach space that if you take the internal hom and condensed abelian groups it is the same as the condensed abelian group associated
57:22
to the Banach space with sup norm. But we need one more observation. So the fact that the Breen-Dellin resolution is functorial and it's a projective resolution, so it's unique up to chain homotopy, tells you a certain interesting scaling
57:41
property of the Breen-Dellin resolution. So if you take for example the natural number two then on the Breen-Dellin resolution you can consider two different versions of the multiplication by two map. You can have multiplication by two on the level of abelian groups or on the level of coefficients of Breen-Dellin. Or you can consider like what
58:07
we call square brackets two which is given induced by functoriality by multiplication by two on a on the abelian group a. So those will do two different things to this term here. The
58:24
first map multiplication by two just multiplies all your continuous maps by two just using multiplication in the reals whereas this one is induced by multiplication by two on the inside here. The fact that the Breen-Dellin resolution is a projective resolution tells you that these two
58:43
are these are canonically and functorially chain homotopic by some chain homotopy so h. And this also tells you then by just composing by just iterating these things and composing the chain homotopy so multiplication by two to the n is a chain
59:04
homotopic via some h n to multiplication by two to the n on the inside. And now you just have you just so here's a then you just have a small lemma here.
59:20
Suppose you have a complex of Banach spaces with a self-map let's call it let's call it multiplication by two because that's what it will be in the example of bounded of norm less
59:43
than or equal to one which is homotopic to multiplication by two then the complex is acyclic.
01:00:00
So the kernel of D is equal to the image of D. There's no homology. And the proof is kind of simple, I guess. So, well, if you have any sort of cycle, so if you have DX equals zero, then you can write,
01:00:21
well, you can write two to the N X equals two to the N X plus D to the N X of this N homotopy HN of X. And this tells you that X is equal to one over two to the N two to the N of X plus D one over two to the N HN of X.
01:00:46
And then you just check that by the explicit formula for composing homotopies, that this forms a Cauchy sequence. So that you'll be able to control the norms of the terms rather easily. And then you can, so then you can take the limit
01:01:02
and then you'll find that, well, when you take the limit here, because this has norm less than or equal to one, and then we're dividing by one over two to the N, this goes to zero, and you'll find that X is equal to D of something, namely the limit of that Cauchy sequence right there. How do you know that the image of D is closed? It follows from the proof.
01:01:20
Yeah, so you prove that it's equal to the kernel of D by exactly this argument. So if you knew in advance that it was closed, you would reduce to a much simpler fact that if you have a Banach space on which multiplication by two has a norm less than or equal to one, then the Banach space is zero.
01:01:41
But by the way, that gives an argument in degree zero. It comes from R on Z to R has to be zero. And that's kind of a higher homotopy generalization of that, but yeah, the proof works perfectly fine without assuming anything about the image being closed. Okay, so that was just a little tour
01:02:00
of some general category theory nonsense related to Gedim's set. Thank you very much all for listening. Okay, thank you. Thanks a lot, Bastien.