 So Dustin, this is yours. Okay. Thank you very much. I'm very pleased to be talking here at this Topos conference. Like many people, my first time meeting Topos was when I was studying Atalco homology 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 atal sites and doing things explicitly there. And like, thankfully enough people, I've ignored that advice and studied the general theory anyway. So 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 Schultz here working with me, 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. There's the title. And let me draw just a rough schematic of the situation I want to sketch out here. So we've got the concept of Topos, I guess. And then there's another very interesting, very general class of categories. We think about Topos as a kind of category. Class of categories called the categories generated by compact projectives or the sifted in categories. So I'll just write S in for 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 in Topos and are generated by compact projectives and very favorable properties. And specific examples are given by pre-sheaf Topos. 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, the concept of Topos is at least somewhat familiar to everybody who's attending 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. 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 homed out of X commutes with filtered co-limits. The second is we say X is projective if homed out of X commutes with a different class of co-limits, namely 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, a common retraction for these, or what do they call it, a splitting or it's a splitting, yeah, a common splitting for these two maps. So that seems, 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 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 homing out commuting with certain co-limits. Well, it's whatever co-limits you can build by 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, well, you know, if whenever you have two objects then the category of common objects 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. And 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. 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. 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 safe hyper-products then it's, you get, well, it'll commute with more things and it's exactly these extra, these 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 might seem a little bit obscure why there's a little extra bit of data guaranteed to 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 the, it's some truncation of the 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, there's a, 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 non-degenerate simplices to, in order to pop out and form the, that 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 product. I don't know. There's, there's, there's some, there's something funny going on here. Well, it's not funny. There's something fundamental going 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 subcategory containing all the compact projectives is C. So if you start with the compact projectives and take, take co-limits and you're allowed to iterate if you like, then 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, you can just take co-limits once. And in fact, you can add co, co, it, it 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, then every x in C is directly a sifted co-limit of compact projective objects. And in fact, you can say something more precise. So C is equal to, identifies with a certain categorical construction called the, I don't know, sifted in construction on the full subcategory of compact projective objects. Where this is some, so this is a category obtained by a formally add in sifted co-limits, 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, you know, sifted co-limit of, whoops, sifted co-limit of x i's. And if you want to know how to home from this to, to that, well, 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, you know, co-limit over j, limit over i, x i, y j. And that explains how 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 generated by compact objects and between sifted co-limits and filtered co-limits and this s in construction and the usual in construction. Okay, now let me tell you what the basic class of examples is. So, so, well, any algebraic theory. And so, well, it's, so I mean, if you have anything where 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 may be satisfy certain axioms like associativity or commutativity or I don't know. So anything where you kind of an, you have an underlying set and some operations specified like this with some relations also specified purely in terms of Cartesian products. Then, then the category of things satisfying those, that algebraic property forms a category generated by compact projectives. So let me give example here. So, well, let's say a billion groups. So here you give a set and you give a map from x squared to x and 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, 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 rank. So z to the n. So about this business of retracts, I mean, so this s-ind construction, it applies to say any category with finite co-products. So maybe I should make a remark that, you know, these compact projective objects is always closed under finite co-products and retracts. And the retracts are not so important in essence, because if this s-ind construction doesn't define it, it does the same thing for a category closed under finite co-products and its item potent 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 projectives, which is closed under finite co-limits. You don't have to know that you have all the compact projective there. All of them would be gotten by taking retracts. But it's often, you don't have to understand explicitly what the retracts are. Right. So another example would be, say, commutative rings. So they're the compact projectives are, well, retracts again, of the polynomial range. So the free object, you always do the free objects on finally many generators in whatever algebraic context you're working in. So here it's for your building groups of finite rank, there it's polynomial rings, non-commutative rings, you'd have a 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 projectives would be the finely generated projective r modules. So with these examples, algebraic theories have this specific property that the, it's generated by, well, it's in fact generated by a single compact projective. So, so in this case, it's z, if it was the free object on one generator, here would be z bracket x, here it would be r as an r module. Then you just take all finite co-products of those to get a sufficient class of compact projectives. But the example that we're going to talk about and then sets is somehow of a, yeah, a different form than this. So it'll be generated not by a single compact projective, but by a whole class of compact projectives. And this means when you're generated by a single compact projective, it makes sense to say that you have an underlying set that tells you a lot about your object, right? But in examples where it's not generated by a single compact projective, well, it's not, you don't just have an underlying set, or you have an underlying set, 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 s in, so how can we characterize which topoi are generated by compact projectives? So again, there's a little proposition. So in a topos, what is this concept of compact projective? 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, you know, these two different notions of compactness in a topos being a compact object and being quasi-compact, so every cover has a finite subtower basically, do not agree in general. But when you add the projectivity condition, they become the same. So and it's the same as saying if x is covered by a collection of maps, you know, fi, so if you have a epimorphism from some disjoint union of x i's to f, 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 you would think of as being quasi-compact, so every cover has a finite subcover, and what you would naively think of as being projective, so every surjection has a splitting in essence. 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 your topos as she's with respect to the induced groten-deac topology on that subcategory. So you might wonder what kind of groten-deac topologies arise on the collection of compact projectives in the case where compact projectives generate the topos. So suppose c is a category with finite co-products. So we'd like to know, so the compact 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 want to get our topos as generated by compact projectives, we should start with a category of finite co-products and just try to make the groten-deac topology which says that finite co-products should be covers. So when is that a groten-deac topology? Well, we need to make sure that the in some sense the axioms of a topos are set, the axioms of topos are satisfied, or the ones that we have access to by virtue of the fact that c has finite co-products. So suppose that finite co-products are disjoint and universal. So what does this mean? This 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, then when you pull back x prime and y prime, you get also a disjoint union decomposition of z. So the pullbacks of finite co-products are still finite co-products. Then you get a groten-deac topology where the covering c's are those generated by a finite disjoint union decomposition. And this gives a topos generated by compact projectives. And also all topoi generated by compact projectives are of this form. With c is the compact projective objects in the topos that you're considering. So it's just I mean it's kind of something fairly obvious here I guess from topos perspective. You just if you're an S in category, then all co-limits except for the finite co-products are sort of formally built. So you just start with a category of finite co-products, add the versions of the topos axiom that you can articulate with finite co-products. And 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 I should make a philosophical point here. So 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. And it wasn't by thinking about this that we came to it, but somehow it fits. I don't know. We want a topos living in the intersection here. I mean it was very concrete calculations that led me personally to 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. So what is the definition? So it's a sheaf on the site of pro-finite sets. So these are, if you like, topological spaces, homeomorphic to a filtered inverse limit of finite discrete sets. This category is also equivalent to 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, there is a little set theoretic subtlety that one perhaps ought to mention. This category of pro-finite sets is not a small category. So 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. But, and then just consider only the pro-finite sets bounded by kappa. This is 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 has been developed by Barwick and Heim. So they had also started studying this around the same time as us. They chose a different name called Picnotic set and they chose 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, 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, so the idea behind condensed sets is a, is it's a replacement for topological spaces. And the way that it works is that you, 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 in terms of, you know, the fact that in a topos, every object is generated under co-limits by objects in the image of the Oneida embedding, is that you're only looking at topological spaces, which are somehow built from pro-finite sets. And, well, it's not really equivalent to 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 that 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. Okay. So the idea should be that this is a very nice topos. So it behaves very much like sets, a lot like sets, 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, that's maybe exaggeration, things most topological spaces people study as a full subcategory. And moreover, it doesn't do weird things to them, and 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 spaces as a full subcategory. So there's a, 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 maps from T to x. This gives a fully faithful embedding compactly generated weak house dwarf spaces. Okay, and maybe you want to put the kappa into there, and like kappa compactly generated. Let me ignore that to condense sets. And this is a very large class of topological spaces, which includes, for example, the CW complexes, people use an 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, 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, you'd be looking at. Okay. So, so it is supposed to be some general thing. If you're ever thinking of mixing topology with some algebraic structure. It's a good idea to, we think used to condense 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, of course, to tame infinite dimensional vector spaces, the standard approach is 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 is the So what do we do with these? Do we do with condense sets? Well, the main thing is it lets us make a definition notion of an analytic ring. So and what is an analytic ring? So an analytic ring is a pair consisting of a condensed ring 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 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. And you're supposed to think of this as a complete objects. So, you know, when you're doing functional analysis or analytic geometry or any time 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 tensor product 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 where a topological or condensed ring R that tells you what the completion function 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 a basic objects in analytic geometry as opposed to algebraic geometry is not just a ring, but a ring together with a notion of completion, 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. 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 notions of analytic or algebraic space. So schemes give examples, complex analytics spaces, rigid analytics spaces. Well, I don't know how you can even do things like topological manifolds or real manifolds. Sorry, not maybe not topological manifolds, not so obviously real manifolds, at least smooth manifolds. I mean, no, even the topological ones. That's not obvious to me. But I'll trust you on that. I mean, it's not nuclear. Yeah, at least it gives me pause. Yeah. Yeah, no, but anyway, it works. Okay. Over convergent continuous functions, they behave as they should. Oh, really? Okay. All right. Yeah, fine. And someone follows automatically from them somewhere being a module over things, using whatever smooth functions. They have to behave correctly. Anyway, you have to check some it follows. What? It follows automatically. It does. Okay. Well, I certainly trust you on that. All right. So, yeah. Okay. So now, so there's something about this situation that I want to point out. So, well, essentially, because of the fact that, actually, I haven't yet explained why condense sets is generated by compact projectives. So why is condense sets generated by compact projectives? Well, certainly this defining site of propanite sets does not consist in general of compact projectives. I mean, we need both, we have the topology generated by finite distro and 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 which still generates the topos. And this comes from a remark of Gleason's, I guess. So the category of propanite sets has enough compact objects. So for all propanite sets T, we're actually, I mean, it's enough to talk about a compact house door space T. There exists a T prime, a propanite 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 a projective object. And since the topology is finitary, that will mean it will be a compact projective object. 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 and by the universe, a 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 the universal property. It suffices to split the restriction to T discrete, but by the axiom 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 propanite sets or you could just also just take the propanite sets of this form, stone check compactification of a 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, then mod R is an abelian category. Well, that's that's general topo 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 as the same exactness properties as, you know, category of abelian groups or the category of modules over an ordinary ring. Because if you have a compact projective object in an abelian category, then harms out from your abelian category a to a billion groups, it commits 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 arbitrary co limit is a finite co product. But in an additive category, finite co products and finite products are the same. So that that's automatic as well. So this means that 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 limit with another limit or whatever, everything reduces to abelian groups. So for example, if 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 or filtered or infinite products are exact. Right. So I meant just infinite products are exact, not filtered inverse limits. I apologize. So right. And the axiomatics that I didn't spell out, but they also imply that modules over a script are is also category generated by compact projectives. And this means that although it is encoding some kind of functional analysis or something, we 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 a whole family of generators parameterized by your pro bonite 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 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 projectives are very important for us. Now, but you also might think, okay, well, maybe now that you have how important is the topos concept really. So now that you have your anisobelian categories generated by compact projectives, 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. So I want to explain why essentially. So also it's extremely useful in practice to have it come from a topos because if you want to say how what it means for like a group to act on such a thing and make a logical contents group. So this only really makes sense if there's this ambient topos formalism around. Yeah, that's a very good point. Yeah, thank you. So I mean, I would just be repeating what Peter said, but just in case. Yeah, if you want to know what a representation of a group, you know, one thing you'd like to do with a billion categories is, for example, look at, I don't know. 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 sets, but having new and condensed sets in here is actually kind of quite crucial. Okay. 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 something even better. These generating quasi compact, quasi compact objects are also quasi compact, quasi separated. So so if you have a topos generated by quasi compact objects, then an object is said to be quasi separated. If whenever you have two quasi compact objects mapping to it, then the fiber product is also quasi compact. And it follows them that if you look at the collection of quasi compact, quasi separated objects. So if you're generated by quasi compact, 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 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, which involve the usual operations one considers in topos theory. Well, not the infinitary ones, but you know, if you consider this, this is finitary limits 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, 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 site for our topos. In our, in our example of condensed sets, quasi compact, quasi separated is exactly the same thing as compact house dwarf. So, so this purely topos theoretic finiteness property 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 house dwarf spaces and we'd get the same topos. Right. Now, I want to say, so how do we produce examples of these strong axioms? I haven't said what the strong axioms are, but to verify these things, you essentially need to do X calculations. So to produce analytic rings, you need to be able to calculate Xs. And for this, we use it, we use a use a topos theoretic result. So even though it's X, we're interested in calculating and abelian categories, we have to reduce ourselves to topos theory to make these calculations. And this is a very interesting theorem of Breen Deline. So there exists a functorial resolution of any abelian group terms of the form. You take a finite direct sum of the free abelian group on some product of copies. You 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. 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 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. 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 a power of this notion of category generated by compact projectives. So, all 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 pole 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 recognize such a functorial, 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 lattices. 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. So, the claim is that this Brindelene 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 of being a resolution by compact projective objects, it's known as being pseudo coherent. And it has a lot of permanence properties, which lets 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, and so at least 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, 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 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 appear, 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 can only get finite products of copies of A. And this turns into exactly the desired statement. Now, and then the fourth main point is Sarah's finiteness, so that the homotopy groups of spheres are finite abelian groups for a high degree of zero. And this tells you that the sphere version is only a little bit, 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 inexplicit but not difficult to explain reasons implies the abelian group version. 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 know where on the scale of complexity from, say, homology of A. and B. spaces to homotopy groups of spheres, this problem lies. So it's really quite inexplicit. And for that reason, I think it's all the more remarkable that this inexplicit resolution is our main calculational tool. So how can you use an inexplicit resolution to do calculations? And I want to end the lecture by giving just one example of this. Well, state of result, which I think is the first theorem in this first real theorem in the study of condensed sets. And it was the thing that, 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 door spaces all give examples of condensed sets. And in particular, locally compact a billion groups is a full subcategory of condensed a billion groups. Well, that's not too difficult. But then you also want to know that X between them are reasonable. So if you have A and B in there, then the claim is that if you take the X, the X I in condensed a billion groups from a underline to be 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 compact a billion 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 a billion groups. So 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. One is that X I from the real numbers to Z say is zero in all degrees. And the second thing is kind of, I guess, I don't know, but I'm sure I can do to that that X die from R mod Z to R, no, yeah, R mod Z to R is zero for all I bigger than zero. And I want to explain how to use the Breen Deline resolution to prove something like this to finish up the talk. So the Breen Deline resolution. So I said it gives you a fun for a resolution of any a billion group by three billion groups on the underlying set. Now, since it's funtorial, it automatically passes to any pre-sheaf category. But then you can chiefify and chiefification is exact. So in fact, the statement exactly the same statement holds in any topos. So any a billion group object in a topos admits a resolution by free a billion 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 i R mod Z to R is calculated by by complex with terms, some finite direct sum of continuous maps from some product of copies, again, finite product of copies of R Z to R. And it's actually, this is actually a complex of Bona spaces. So it's just the sup nor, but that's actually also just the Bona space that if you take the internal home and condensed a billion groups, it is the same as the condensed a billion group associated to the Bona space with super long. So, right, but we need one more observation. So the fact that the Brindelene resolution is functorial. And it's a projective resolution. So it's unique up to chain homotopy tells you a certain interesting scaling property of the Brindelene resolution. So if you take, for example, the natural number two, then on the Brindelene resolution, you can consider two different versions of the multiplication by two map, you can have multiplication by two on the level of a billion groups or on the level of coefficients of Brindelene. Or you can consider like what we call square brackets to, which is given induced by functoriality by multiplication by two on a on the building a. So those will do two different things to this term here, the first map multiplication by two just multiplies all your continuous maps by two, just using multiplication in the reels. Whereas this one is induced by multiplication by two on the inside here. And well, the fact that the Brindelene resolution is a projective resolution tells you that these two 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 homotopic via some hn 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. Suppose you have a complex of bonac 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 norm less than or equal to one, which is homotopic to multiplication by two, then the complex is a cyclic. So the kernel 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, well, you can write two to the nx equals two to the nx plus D 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. 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 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. 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 norm less than or equal to one, then the Banach space is zero. But by the way, that gives an argument in degree zero that comms from r hz 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 of some general category theory nonsense related to condensed set. Thank you very much all for listening. Okay, thank you. Thanks a lot, Bestien.