 from the University of Düsseldorf is going to talk to us about ranges of functions and elementary classes via topos theory. All right, then thanks for inviting me to speak here. I'm happy about this whole conference, it's always good to see topos theory in action. And here's my little contribution to the conference. So I'm going to talk about ranges of functions and geometry classes via topos theory, that's what I announced, that's what is the content. And I'll just dive in and tell you what's the rough kind of problem here that I want to address. So suppose you're given two first order theories, S and T, and a functor between the categories of models, and then many classical questions are of the following form. So you could ask, okay, is this functor essentially subjective, is every b of the form f of a for something in the category a? Or you could ask the logical version of this, is every b elementarily equivalent to some f of a? Or then there's a more general one, a more complicated question maybe. So what is the elementary class generated by the essential image of f? So do I mean by this? I mean, okay, I take all the things appearing in the image of f, each of them satisfies some sentences in my language. So here I'm in the category of models of s, so it should take the signature of s. And then I just take all the sentences satisfied by things in the image, and I look at the category of models of those. So sometimes that's the thing you want to ask. And there's really many classical questions of this kind. So yeah, rephrasing of three, as I wrote it here, you could ask, is there some first order statement that is true for everything in the image, but not necessarily for all the b? Something that would help me to separate a general model of s from the models that come from a. Okay, that's very theoretical, so let's dive in with a pretty relevant example here. Famous example from lattice theory was this year, Dilworth's congruent lattice problem. It was asked in the 40s by Dilworth, consider the functor from lattices to algebraic distributive lattices. So lattices are an algebraic theory. So you can make sense of the notion of congruence there, the thing that by which you can quotient there. And the congruences of a lattice form a lattice themselves, and it's an algebraic distributive lattice. So finally, locally presentable category, which is opposite. And the question was, is every algebraic distributive lattice of this form? It's a typical representation problem. Can every such lattice be represented as congruences on something, some other lattice? So this was a long open question and it was a driving force of much of lattice theory, while lattice theory was still on vogue. The solution relied on a slight formulation here. So you have to ask, is every distributive semi-lattice the form compact congruences on some lattice? It's equivalent. You can go from one thing to the other, but this latter formulation, I can say a few things about the eventual solution here now. So Dilworth himself proved that every finite distributive semi-lattice is of this form. That's how he got to the question. And here are some results then, Hoon in 1985 proved that actually every distributive semi-lattice of cardinality at most, RLF1, is of this form. So presentability solved that far and it looked good. The conjecture remained open many years longer until Wehrung in 2007 showed, OK, but it's in general not true. There are distributive semi-lattices, cardinality, well, for Wehrung it was cardinality bigger than RLF omega, and then Ruzhichka a year later showed, OK, RLF2 is actually enough in our count examples here. Right. So long story. It's an interesting story, extremely interesting. But here's what this tells us about our kind of problem from the last slide. So we can conclude from this that there is no first order sentence that holds for all compact congruence semi-lattices, but not for general distributive semi-lattices, because if there was such a thing, then if we had a model, so then we would have a model of its negation, distributive semi-lattice that does not satisfy it, it's not in the image here, but then this model would also admit a countable sub-model. So there would be some by Leupham-Skolem, download Leupham-Skolem, some countable model of this, but that countable model being countable is in the image by whom. So that cannot happen. And so this kind of thing tells us, yeah, sometimes we cannot distinguish the image of such a functor by first order sentences. And that's one, it's a very, very nice example, but there's many such questions and, well, possibly many such phenomena that counter-examples kick in at late cardinalities. All right, of course, I'm going to address this kind of thing through Topos theory. So let's, I had my own motivating examples how I got to this problem of quadratic form theory, but let's skip that. So yeah, I'll gather some notions. And I said loosely just that I would be talking about first order theories. So here's the more precise formula here. I'm talking about certain infinitary first order theories, namely kappa geometric theories. So these are fragments of L infinity kappa, if you know, two inter-languages, here's the definition. So a kappa geometric formula is a formula built from atomic formulas top and bottom, allowing arbitrary disjunctions, well, set size disjunctions, and conjunctions of cardinalities smaller than kappa. So if you know geometric formulas, then kappa is all of zero. It's finite conjunctions. And yeah, of the same size, I allow also existential quantification. So what's the kappa geometric theory? It's one that's axiomatized by formulas of this form. Well, it comes from kappa geometric sequence, right? So it's implications, maybe with a for all in front of it, between kappa geometric formulas. I'll also be using, so yeah, I'll be using, so given some class of sigma structures of some first order signature, the theory of the members of this class, it's just the dissection of all the theories of the members there. So all sequence valid in every member of C. And I'll be using this negative kappa geometric theory of C. So that's just the set of negations of kappa geometric formulas. So it's those sequence where I have a bottom instead in the place of psi here. All right. So it's this kind of first order theory that I can treat. And that's, of course, because I want to use Topos theory. Here's another piece of ingredient. So accessible categories for a regular cardinal kappa. Well, you probably know what accessible categories are. So it's a category kappa accessible. If there are kappa directed co-limits, and there is a set of kappa presentable objects, such that every object in this category is a kappa directed co-limit of these. So examples, of course, well, for example, an algebra and a variety is even, finally, locally presentable. So all of zero accessible in particular. Fields are all of zero accessible without being locally presentable because they're not complete. Models of geometric theories, as you know them, are accessible. And actually, so kappa accessible categories are exactly those of the form mod t for a kappa geometric theory t. So that's sort of what is the scope of what I'm going to say. I can talk about kappa accessible categories with this and then just assume they come from a kappa geometric theory. All right. So let me state the theorem. And it will take a while maybe. So it has three parts. And don't be put off by the slightly convoluted statements here. I think the proof will make everything clear. Proof is nicer than the statements. So suppose, as in the beginning, that we have two such kappa accessible categories, a and b, and a functor between them, preserving kappa filtered co-limits, and preserving kappa presentable objects. So that's what I, the notation I use for these subcategories here, a kappa, b kappa of the subcategories of kappa presentable objects. Well, then a bunch of things are true. So I was asking whether I can distinguish the image of f by some kappa geometric formulas. Well, if f is essentially subjective, if this restricted f kappa is essentially subjective, then I cannot. So this is sort of really a negative result about my question. If I find formulas distinguishing the two things, no, the theories are the same. Let's look at this example from the beginning, what it says here. So there was functor here from lattices to distributive semilattices is such a functor. It preserves lambda one filtered co-limits, lambda one presentable objects, alif, sorry. And well, we know by whom that it is essentially subjective, right? On this subcategory of alif one presentable objects. Well, corollary, there is no alif one geometric sequence that holds for all the things in the image, but not for general distributive semilattices. So we knew that there was no first order formula, finitary first formula by Louis-Vlemskoulem. But now from this, you can also say, okay, there's no alif one geometric sequence. And that's just a different class of statements where things could work or not. The very last slide, we will see an example where the things work out in geometric sequence, but not in classical. Okay, so I don't want to go too deeply into this here, but the theorem, as I warned you, is slightly convoluted. So I have more general statements than I'm giving you because they get hard and hard to process. So here, for example, would be a more general statement. If I have two functors from other accessible categories to the same B, then I can ask, okay, how do the theories, the couple geometric theories of their images compare? Both are extensions of the Bayes theory, S here, whose models are B. Well, they're equal if and only if these images of the sub-categories of copper presentable objects have equivalent idempotent completions. And that's still not the most general thing. That's already hard as process. Yeah, as I said, I can hardly remember these statements myself. I just remember the proof, and then that's what I apply usually in practice. Okay, so let's go to two more parts here. So, yeah, we can ask whether there are these distinguishing sentences or whether there are not such sentences. So here's, you can also ask whether the image of F is actually axiomatizable, whether we can really capture precisely what's in there or not by couple geometric formulas. And here's a result in that direction. So, if my functor restricted to copper presentable objects is fully faithful, yeah, then yes, I can precisely capture this by some extension of my target theory. Yeah, so this B is mod S. I can just add some accents and say, okay, that's everything that arrives here is can completely be captured by geometric sequence. And finally, so here's another maybe not so easy to process statement. If one has that every B and B in the target category, but I only have to look at copper presentable once here again, I admit some morphism to something in the image here. Again, the image of the copper presentable parts, then the negative copper geometric theories coincide. So, well, it's a bunch of criteria of sorting out the theories of the image in comparison to the theory of the ambient category in terms of the functor here, restrict to the copper presentable objects. Let me give you, well, let me jump over that toy problem. Let me give you one of my motivating examples here. So, there is a functor to what is called special groups, not the most fortunate name. You take a field and you send it to its square class group. Yeah, take the units module of the squares. You have the, so this is a quotient of monoids. So, we have induced one at multiplication and you have the graph of addition. Just a relation now, it's no longer an operation there, but okay. And this captures quadratic form theory on our field. It's a thing people consider. And it's a long open question since the 70s, whether these special groups really, so you write down some axioms for this kind of structure here. And it's a long open question whether you could write some more axioms. If you really captured everything you could by geometric axioms describing this thing here, we know also since the 70s, it's parallel to the Billworth problem that every finite special groups as a morphic of the special group are fields. And we know that's more from the 90s here, late 90s, that every special group can be embedded into the special group of a limit of fields, at least. So, what can we conclude then knowing these things from the theorem? So, at first, we have to look. So, this function preserves actually all of zero directed co-limits and all of one directed. Therefore, both categories are all of zero accessible. As far as I know, I actually wait a moment for the second one, I'm not so sure. But it definitely doesn't preserve all of zero presentable objects. I want to make a point here about how we can fall into the scope of the theorem. So, we want this accessible functor here between accessible categories, and we want it to preserve kappa presentable objects. So, this is often not true for too low kappa, but we can raise the kappa always and always achieve it. That's a theorem. So, this is a point about the scope of the theorem here. So, look at this here. A finitely presentable field tends to have a large square class group. So, an algebraically closed field, that's not finitely presentable, because I have to add infinitely many roots there to get to the algebraic closure. But it will have, of course, every element there is a square root of something. So, this will be trivial. So, but the smaller my field, the bigger this special group here gets. So, the finitely presentable special groups are the finite ones. And so, this completely inverts the presentability relation here. On the other hand, if I'm talking just about all of one presentable objects, well, that means I have a countable presentation of my field. Well, that definitely goes to countably presented special groups as well. So, and that's a standard phenomenon. If you raise the kappa, then the particularities of this construction get blurred, and everything happens below some big enough cardinal. So, you pretty often get into this framework here. So, consequences, well, you can conclude from Part C of the theorem that general special groups satisfy the same negations of kappa genetic formulas as those coming from fields. So, there's no negation, but that's pretty cheap and pretty easy to see by hand as well. And we can conclude from Part A that, yeah, if we look for, yeah, so how the two questions fit together somehow. So, there are the two questions. Can we distinguish general special groups from those fields by some geometric formula, and are there actually special groups up to isomorphism even that are different from those of fields? So, these two things are linked as we can now conclude from Part A. So, if there is a formula distinguishing these two, then there will also be some counter example special group, and we even know the cardinality I'll have one. So, that's sort of a, I don't know, a hint where to look at least. I didn't solve the problem, but at least that. Okay, time is flying. Let's look into the proof, and the proof uses toposes. Yeah, so everybody knows here that toposes can arise from geometric theories and from topological spaces. From geometric theories, they arise by taking the classifying toposes. From topological spaces, they arise by taking the topos of sheaves on that space. So, the map from spaces to toposes is actually very good and preserves a lot about the spaces. So, solar spaces can be reconstructed, for example. And you can also, you know, transform continuous maps into geometric morphisms. And you can actually, for good enough spaces, so it depends a bit on which property you're looking at, what good enough means, but really many. You can recognize what kind of map your geometric morphism came from. For example, sojective means if upper stars faithful, if ever there's an embedding, that means inclusion with a subspace topology, then if lower stars fully faithful, you can see denseness here, and you can see when something is a closed inclusion, by certain properties of these functors, really. And that's the definition of the corresponding notions for geometric morphisms in general. So, that's how you define surjective, geometric morphisms, inclusions, dominant morphisms, and closed inclusions. Now, in topological spaces, there is a factorization of any continuous map. You can just map it first, subjectively, to the image, f of big X. Then you can have a dense inclusion into the closure of that image, and you can have a closed inclusion into the target space. That's completely trivial. And the same happens for topos, less trivial. Now, every topos is a classifying topos of some geometric theory, and so we can ask what are the theories here in the middle? And the thing is, all the inclusions here correspond to axiomatic extensions of the target theory over the same signature. And you can say more, and you can find this in Olivier's book. So, these axiomatic extensions can be described a bit better. So, the S prime, the first one occurring here, is the theory of consisting of all geometric syphilis satisfied by the generic model transported to this topos here. And there's this S double prime, well, that's obtained from S, from the target theory by adding the negations of geometric formulas that are satisfied by that model. So, that's how you see some elements of the theorem popping up here. All right, so that's a classical piece of topos theory. Now, we use this for kappa geometric morphisms and kappa topos. That's maybe not the standard law. So, what's a kappa topos? Well, it's a localization of a pre-shave category. Such that the reflection functor, so the reflective subcategories, such that the reflection functor preserves kappa limits. So, limits of diagrams of cardinality smaller than kappa for some regular cardinal, again. And kappa geometric morphism, again, it's the obvious generalization of geometric morphisms. You have to demand that the f-upper star preserves kappa small limits now. Well, what's the role of this in logic? As in the usual geometric logic, the role of the preservation of small limits is the preservation of conjunctions, really, a final conjunctions in the usual case and now of conjunctions of size kappa in this kappa geometric case. So, and with this, kappa geometric morphism preserves all kappa geometric sequence. Now, this is not my invention. This has been elaborated by Christian Spindler. And I have to thank Olivia at this point for setting up all of these topos meetings here because I learned about this at the toposis in Como and I'm not sure I would have learned about it otherwise. So, it was great to see Christian talk about this the last event. So, he developed this theory and found lots of great generalizations of results and of model theory or geometric logic. Okay, but I want to use this now for this factorization here. I want to look at kappa geometric morphisms between kappa toposes. And well, kappa toposes are just also toposes. And we can use the usual factorization that we know from topos theory. So, actually, how, for example, does it work here? You take that geometric morphism, it's an adjunction. So, it uses a co-monad on the domain here. And the category of co-algebras, that's the first thing popping up here, for example. You can just do the same thing and the outcome, well, the factorization will go through kappa geometric morphisms again. That's a little thing to check, which I did, it's not hard. But co-algebras are something pretty abstract. So, how do you compute the factorization? Well, it's easy under the assumptions of the theorem because we can restrict to essential morphisms between pre-chief toposes here. So, for pre-chief toposes and essential morphisms, factorization actually yields, again, in the middle, a bunch of pre-chief toposes. So, that's really nice. And it's not a priori clear, really. It's also not hard to see. And so, all of these intermediate morphisms as well are essential. So, they are induced by a functor between the little index categories C to D. And, namely, it's the following, it's completely explicit, and that's what makes everything applicable in practice. So, if I have such an essential functor coming from a functor little f from C to D, well, I have factorization system on categories. Any functor factorizes as a functor that's subjective on objects, followed by a fully faithful functor. I just add the arrows here that are missing. And I can still have an intermediate factorization here through this full subcategory of things that admit a map to something in the image here. So, here you see some more elements from the theorem. And that's what induces the factorization here that we needed. Okay, so what's how to get the theorem from that? Well, we are talking about kappa-accessible categories. And kappa-accessible category always is the category of models of some kappa-geometric theory. And it can be taken to be a pre-chief category. Just it's a complete triviality, or a long known at least, that the models of this theory here, well, it's a pre-chief theory. That's the thing that wouldn't happen so easily for usual toposus. So, the extra flexibility of having this kappa here lets us fall into the setting of pre-chief categories. And that's really nice because then computations get very easy. And well, then the hypothesis of the theorem ensured that our given factor here from A to B really is induced by a map of classifying toposus by morphisms there. And it's an essential morphism. And so we can compute the factorization of this whole thing as we wanted to. So, these are the kappa-classifying toposus of our kappa-geometric theory. And well, now the conditions in this case A of the theorem ensure that the second maps here are equivalences so that the whole thing was subjective. So, they can't distinguish the geometric theories of things in the image with the other one. And well, that's how I roughly go there. Let me finish with an example. I know I'm over time, but this would be a nice one here. And this is really an example about not of anything that I wanted to formulate so strictly because it would just be off-putting. It's an example of this whole line of reasoning by factorizing the geometric morphism here. So, let's take the theory of groups for our target thing. And let's take the theory of the free group of n-generators as the thing that we include. So, really we include actually a very tiny thing. Now it was a question by Tarski asked in the 50s, I think, and long, long open where the first order logic was able to distinguish the theories of free groups on different numbers of generators, finite numbers. And it was finally, with a gigantic effort, resolved in 2006, independently by Chalapovich-Miasnikov and Zeller, that this is not possible. The first order theories, now first order really classical, finitary, full first order logic, of all finitely generated free groups coincide. You cannot distinguish the number of generators. Now, this is not true in finitary logic. There's a clear sentence here that tells you how to distinguish the number of generators, just this one. There are x1 to xn such that every other element y is one of the possible words. I need a countable junction here. It's one of the possible words that you can build from the x1 to xn. Now, that's not a geometric formula here because we have this existence before the fall. And there's no obvious equivalent geometric formula here. But now with the theorem, or with the setup here of this proof style with geometric opposites, we can say something about it here. So that there is actually a geometric formula, now a usual geometric formula, that allows to distinguish these different numbers of generators. And I haven't been able to find it explicitly and I have also asked a bunch of people who everybody is shrugging and they don't know. But the theorem spits this out. Why? Well, what we have to include here is just the free group on n generators into the finitely presentable group. So that's two sides here. This one is one defining side for the category of the theory of the free group on n generators, the geometric theory now. Okay, but the question is now whether this induces an equivalence of classifying opposites of kappa, well in this case, reducible classifying opposites. Well, it doesn't because this is the case if and only if the idopotant completions become equivalent here. But they don't because the idopotant completion of this one object category here, that's just consists of the three groups of it most n generators. Because it consists of subgroups, of course, that idopotant completion gives us subgroups and subgroups of free groups are free again. That's all we can get here. And this gives us different categories even for different numbers of generators. So we can distinguish them. That's a non-constructive proof. I don't know a formula doing this, but so that's one of the nice applications here. All right, I think I have to stop. There's stuff to think about further stuff, but don't let me bother you with this. Thanks. Thank you very much.