 So I'm pleased to introduce Ivan de Liberti, who will talk about the words, higher topology, please. OK, so hello, everyone. I hope you can hear me well. And yeah, now I'm going to talk to you about the words higher topology. So this talk is a summary or presentation of a paper, which was recently accepted by the journal by the algebra, which has the same title of the talk. But it is also, I mean, it's really not the full story. What I'm going to tell you, it's just a little part of the big picture. And the reason is quite simple. We don't have enough time to go through the whole story. So you find here a list of the other papers. And in particular, one could say that there are three parts of the story. There is a more geometric part, a more categorical part, and a more logical part. And today we will only discuss the geometric side of the story. To be very honest with you, my favorite part is the logical one. And this is where my motivation comes from. But there is no way to present only the logical one without telling the geometric part. And so for time reasons, I can only present the geometric one. And for the logicians, I recommend to read the paper number three in the list. And the paper number four is just my PhD thesis. So essentially it puts everything together. OK, I think we are ready to start. So what is this talk about? It's too early to tell. And we have to start from something simpler. And let's start from general topology and plain geometry, which is something that I'm sure everybody understood. So in this diagram, you see three categories. And top is the category of topological spaces and continuous maps. And pos omega is the category of posets with directed suprema and functions preserving directed suprema. And log is the category of locales. And a locale is just a frame. So this is just the opposite category of frames. Maybe I should tell you what's a frame, or maybe not. It really depends on your background. But let's recall it together. A frame is just a poset with infinitary joints and finite needs. And these two distribute with the infinitary distributivity rule. The canonical example of frame is the poset of open sets of topological space. And I'm sure since all of you have attended the whole conference, I'm sure you are quite used to the notion of frame. So why am I telling the story of these three categories? Why do we care? The reason is that these three categories are different approaches to geometry. Well, there is not much to say about topological spaces. I think we all agree that a topological space is a geometric gadget or entity. A locale is the point-free approach or constructive approach to spaces. And posets with directed suprema are not precisely a geometric entity per se, but they were approached by Scott with a very geometric perspective. And they are very relevant in domain theory and lambda calculus. So essentially these are three very interesting things in mathematics and geometric study of mathematics. And these three approaches have been related in the past in quite well-known ways that I will just recall. So we have some factors that relate these categories. So one of these is the factor of points. And as you see, there is an abuse of notation. Both these two factors are named points and you will see in a moment the reason for this. So points takes a locale and it maps it to the set of formal points. And the formal point of a locale is a morphism from the truth value locale into L. And this sets amidst both a topology and a partial order in a quite natural way. So if you put the partial order, then you land in posets with directed suprema. And if you put the topology, then you land in topological spaces. So S is a little bit more tricky to discuss and it doesn't appear in the literature. Well, it kind of does, but it doesn't appear explicitly in the literature. And it takes a poset P and it maps it to the home space into truth values, so zero and one. And one can check this is, I mean, you can find it in some works of beakers, but it's an exercise, it's not a super hard theorem. That this is always a locale. And so this spills a locale out of any poset with directed suprema. And then ST equips a poset with the Scott topology and people in domain theory and people in general topology are very much used to this notion. And essentially the Scott's topology over P was defined in the previous, by the previous sponsor, because we can see that this set is a subset of the power set of P. And so we can look at it as a family of open sets. And one can check that this define a topology and this is precisely the Scott's topology, which was defined by Scott in the, I think in the 80s or in the 70s while it was working on domain theory. Then I should tell you who is the functor O. This is the only one missing. And O is just the functor that takes the open set of topological space. So all these functors are quite well known, I would say with the very exception of ST, sorry, of S. And so the first one is called by some people the Isbel adjunction, even though it was not shown by Isbel by for the first time, I would say it was popularized by Isbel to the general audience. Then the second one has no name because it doesn't appear explicitly in the literature. So let's call it the Scott adjunction. And another information is that the solid diagram commutes. So if I take the Scott's topology and then I take its open sets, this is the same of taking directly the functor S. And this is more or less evident from the presentation I gave you of this functor S. And all these stories extremely classical. And so a good question is, okay, what did I do? What was my contribution to this? And what I did essentially was to categorify this. So I should tell you what means to categorify. And this is the project of categorification. So we have the same diagram, just all the notes have changed. And that, I mean, essentially during the talk, I will describe you all the two categories and all the functors involved in these constructions. And maybe I will also tell you why this is important from a logical point of view, just in a nutshell. So if we go back to this diagram here, we can see that the adjunction between topological spaces and locales, I mean logicians know that the adjunction between topological spaces and locales can be used to prove completeness for propositional logic. And this reconnects my talk with the talk of, with the first talk of the afternoon. So okay, this adjunction is important when you want to prove completeness like theorem for propositional logic. So the adjunction between this adjunction that we will prove here will be important to discuss completeness theorems for geometric theories. And we will see that, we will not see it, but if you read the papers, you can see that also the adjunction on the right is linked to completeness theorem. And in general, some people call them reconstruction theorems for theories, or you might just call them syntax semantics dualities. Okay, so in this picture, let me describe who are the main characters. So topoi is the two category of groten dictopoi, and a topos is a co-complete category with lexco limits. Lexco limits is a way to phrase descent. And if you look at this definition, you see the analogy with locales. So topoi are just categorified locales in the sense that they are complete posets where limits and finite limits and co-limits behave nicely. So this is the infinitary distributivity rule, which then becomes the descent property of the topos. And then there is the generating set assumption, which can be seen just as a tameness assumption. It is very important from a logical point of view because it allows to cut the internal logic to just a set, which then can be seen as the axioms. But it doesn't have a conceptual meaning. It is just important from the point of view of the presentation of the topos. And of course, the presentation of the topos is technically very important, but the whole internal logic does not depend on the fact that we have a generating set. And so I would say in this sense, the topos is really the natural categorification of a poset because again, it is a co-complete category together with a good interchange between co-limits and finite limits, which is precisely what a locale is. And acc-omega should be the analog of posets with directed co-limits, right? So in acc-omega, we find accessible categories with directed co-limits and factors preserving directed co-limits. So again, someone might raise his hand or her hand and say, look, but what is this accessibility assumption there? Well, it's like in the case of the generator of the topos. The point of an accessible category with directed co-limits is more the fact that we have directed co-limits than the accessibility assumption is important in order to have a decent generator, which is useful when we want to do computations. But as a fact, the internal logic of the category that we care about is just to have directed co-limits. And we see that this is a natural categorification of the notion of posets with directed co-limits because it's essentially the same definition. And then I should tell you about this LBIO. So this should be the replacement for the notion of topological space. And in a sense, it's the most important contribution of this talk. And this will be the notion of ENAT. So ENATs were introduced by Richard Gardner. And the idea of ENAT is the following. So what's a topological space? It's a set together with an interior operator or a closure operator, but we will focus on interior operators. So, and what's an interior operator is just a, if you think about it, it's just a common on its power set, which is either important, but this is trivial because every common on opposite will be either important. And also it must preserve finite limits, which is the fact that if I take finite intersection of opens, this is still open. This is the property. So Gardner went very naively in a sense and just said, look, a ENAT on a set is a set X together with a common over its co-press ship category, which preserves finite limits. Again, in the analogy between power sets and pre-shift categories, which category theorists are very used, this is the most naive categorification of the notion you can think about. And yet it's surprising how perfectly this notion works. And the first way that comes to my mind to convince you that this is a good notion is to look at this theorem by Gardner. So in general, if you take the category, sorry, if you take the co-algebras for an interior operator in a space, this is precisely the category of open sets. And the reason is stupid because since the interior operator is either important, the co-algebras are precisely the fixed points and the fixed points for the interior operator are precisely open sets. So the co-algebras for the interior operator are is precisely the local of open sets. In the same way, if you take the co-algebra for the interior operator in a ENAT, you get in general a co-complete elementary topos. So it is lacking a generator, but it's very close to be ungrothed with topos. And Gardner says that a ENAT is bounded if when I take the co-algebras, this has a generator. And so in particular, it's not just an elementary topos but it is ungrothed with topos. So if you forget for a moment about the size story, the size issues of the story, you get a natural factor from ENAT into topoi. And of course, this would be from ENAT into elementary topoi, but if you choose bounded ENATs, then you get growth and dectopo. And this is precisely the categorification of the factor which goes from topological spaces to locales and takes the open sets. And so the analogy is extremely tight. So essentially what I told you up to this point is that there are these three categories, topoi, ACC omega and Bayon. And then there is this factor O, which was defined by Gardner. Also the factor PT in this diagram is known to the literature and people have essentially presented it during this conference because it's precisely the factor that takes points of a topos. And one can show this is proven in many references in the literature that the category of points of a topos is always accessible and has directed coordinates. So we do lend inaccessible categories with directed coordinates. So essentially my contribution was to provide all the other factors, show that they are a junction and show properties of these factors. So let's start. How much time do we have? Aha, interesting. Okay. So the first one is the SCOTA junction which was found in collaboration with Simone Ari. So this is the junction between accessible categories with directed coordinates and topoi. And so I will not present again the categories. And I already defined you the factor PT. So I should tell you who is the factor S. So S is defined by taking accessible factors preserving directed coordinates from A to set. Now, if you remember the construction of S in the Posetal case, it was defined by taking factors preserving directed coordinates into truth values. So as always, we replace truth values with the category of sets and the construction remains the same. And one can show this is not trivial now. I mean, it's not a trivial verification that this category is always a topos. And the reason is that you can check all the zero axioms on the spot because essentially since directed co-limits commute with finite limits, things will work very well. I mean, I cannot say more than this, unfortunately. But you can show that this is always a topos. And how does it work with morphism? Well, you take a factor preserving directed co-limits and the F upper star clearly maps a factor preserving directed co-limits in a factor preserving directed co-limits. And then you can show that this is co-continuous and it preserve finite limits because of the structure of the category. And so this will induce a geometric morphism in the same direction because remember geometric morphism going the direction of the right a joint and this is gonna be the left a joint. Okay, so this is the Scott construction. Now let's move to the right a joint to O. Right a joint to O. Now, if you look at Garner definition, you really cannot provide a right a joint for O. The definition of Garner is not the right one in order to finish the proof. So we had to introduce a generalized notion of ENAT which I'm gonna present now. It will look scary at first sight just bear with me for a second. What's a generalized ENAT? So a generalized ENAT is a category, a locally small category, which is pre-finitely co-complete equipped with a lexical monad over this category. So I should say many things but maybe let's start from a question. Okay, I need to generalize ENAT and it makes sense that I want to put a locally small category instead of a set. But why isn't ENAT just the data of a lexical monad over set to the X? And the answer to this question is very delicate. And so in order to get this answer, I first make precise my definition. So this guy is a full subcategory of set to the X which is made of small copper sheets. Namely, those factors that are small co-limits of co-representables. These categories locally small has opposed to set to the X which might be locally large and it's not a category that I like. Of course, if X is small then every pre-sheet is small so we are closer to Garner definition when everything is small. And remember Garner was using just a set. So of course a set is small. And then P of X has a universal property and it is the free completion of X of under co-limits. And in general, P of X is a little bit pathological. You cannot prove many things about it. And so this additional assumption of being pre-finitely co-complete is needed in order to speak about Lex co-monance. And the point is that in general, this category will not have finite limits. So I need to put an assumption in such a way that this category has finite limits to talk about Lex co-monance. I could speak about flat co-monance but this would be a technical disaster. So I prefer to use this notion. And I will not tell you what means to be pre-finitely co-complete. I will just give you two sufficient condition. If X is small or accessible, then P of X is complete. So in particular, if X is small or accessible, it is finally pre-co-complete. And so P of X has finite limits. In fact, it has all limits. And so I can make sense of a Lex co-monant. So all in all, what I will say all the time is that a unit will be an accessible category X together with an interior operator over its category of small copper sheeps, which is Lex. And this is the most general, sorry, the most natural generalization of Gardner definition given the fact that we have to take into account size issues. So why do I care about size issues? And the reason is that when I have this proposition, the point is that when G is a decent category. So here I chose total but you can choose any reasonable assumption. And P of X is my small pre-sheeps. Then any co-continuous factor has a right adjoint. This is not true if P of X is locally large. And so I need a category which is locally small in order to apply any version of the joint factor theorem. So here size issues are extremely important in order to make anything works. And why do I care about this result? Well, the reason is very simple. The reason is that I want to induce naturally more co-monad over this. And when I have such a thing, then I can take the composition and this induces a co-monad over P of X. Now, I have very little time. So I will close it up very, very quickly. Maybe one thing I would like to say is that accessible ultra-categories Alamakai admit unnatural structure of Enad. So this means that Enads are not just geometric objects but they really are categories of models equipped with a special logical glue that puts everything together and keeps track of the logical structure of the category of models. And so for me, higher topology is really the correct place in which to study formal model theory. So Enad is formally a category of models of a theory. Okay, so I presented you the factor O and now I should introduce you the factor PT which is the right adjoint. No, sorry, I introduced you the factor. Sorry, I haven't, sorry, I should introduce you the factor PT which is the right adjoint for all. Now, unfortunately, I see that there really is no time. So if someone will ask me, I prepared the slide. So I will go directly to the results. So what I showed is that, so what I showed is that we have a two-adjunction between bounded Enads and Topoi. And the most nice result, I would say unexpected one is that this adjunction is idempotent, inducing an equivalence of two categories between Topoi with enough points and sober Enads. Now, of course, the notion of Topos with enough points was in the literature since forever, but this making the analogy with spec, with, sorry, with locales with enough points and sober spaces was not in the literature at all. And so this totally makes sense of the original notion of Topos with enough points because you really get the usual isbo duality between Topoi, sorry, between locales and spaces. Oh, sorry, and this is the end of the talk. So there are many open questions. For example, general topology is completely missing for Enads. For example, the notion of closed set is definitely missing. So what is a closed set for a Enad? Honestly, I don't know. And this would be important to characterize sober Enads because we don't have the same characterization we have for sober spaces. So in general, it would be nice to develop general topology for Enads because then it has logical implication on the category of models. And now I will stop. Thank you for your attention. Thank you, even.