 So, I am happy to introduce Francesco Genovese. The title of your talk is derived Gabrielle Popescu-Teoreng for T-Structures via Derived Injectives. Okay, thanks for giving me the opportunity to speak in this conference. Just to say this work is joined with Julia Ramos Gonzalez. We are both in the University of Antwerp. I mean, I'm going to leave soon, but we're still in the University of Antwerp. So, okay, what I want to talk about here to today is not really toposteery, but maybe something which looks more like linear toposteery. So, first, let me just tell you how can we understand the concept of linear topoey? What should they be? Well, first, for ordinary topoey, we have categories and pre-shifts. So, if we look for linear analogs, then what should they be? Well, not really surprisingly, categories should be replaced by linear categories. I mean, over the integers, or if you want over any commutative ring, it's still okay. And instead of pre-shifts, you'll have modules. Modules on what? Modules over linear categories. So, indeed, if you have a linear category, let's say this small a, what you can define is the abelian category of modules. Right modules, if you want, which are just linear functors from the opposite to the category of abelian groups. Here, I wrote modules over the integers. If you want to replace the integers with any other base ring, it's okay. It's perfectly okay. So, what should a linear topo speed then? Well, this is kind of well understood. And the answer, this should be a grotenic abelian category. So, grotenic abelian category means just that it's an abelian category that it is co-complete, filtered, co-limits are exact, and it has a small set of generators. And the key result on grotenic, a key result on grotenic abelian categories, which, if you want, tells you that they are really something like linear topo is the famous Gabriel Popescu theorem. So, let me just recall it. So, here you are, here you have it. So, you start with a grotenic abelian category. You take the small subcategory spun by the generators, and then you have the restricted unit of functor. Just take an object with the modules over the subcategory of generators obtained just by restriction. And this functor is fully faithful. And moreover, it has an exact left of joint. And also, in particular, let me tell you just a little detail. A way you can prove this theorem is contained in a very short paper by Mitchell from 1981, I think, and one key argument, I mean, this is based on a trick. And one key argument is that exactness of the left of the left of joint is the same as G preserving injectives. Okay. Okay, fine. So, here if you want you have a you have a slogan. Now, then grotenic abelian categories are exact localizations of module categories. So categories of the form module modules over some small linear category if you want. If you want another analogy, then you can view discoverable Pesco theorem if you want is somehow a linear version of the G code theorem. So, then, I mean this was the like the preamble if you want, then what do we want, what do we want to do. The goal is like go derived. So, the goal is to try and understand the notion of linear higher top way. So higher top way we we we now know we know know something about them. Thanks to the course we had this conference but I want to concentrate on how do you how do you make this linear. So, first, then you see that what we want is indeed some higher counterpart of grotenic abelian categories. And in particular, higher counterpart of the category modules. And what what would be the idea to go to go derived I mean that there are actually many approaches, and the one I'm going to follow here is like using this this if you want this this criteria. Every time you you encounter an abelian group, you replace it with a complex of abelian groups. I mean if not a billion group of course it works on modules over any commutative ring of course. Also, we want to work up to weak isomorphism so quasi isomorphism is isomorphism means some maps with some map which induces an isomorphism co homology. And if you then linear categories are replaced by what by small the g categories I mean smaller, or big of course. The g categories doesn't mean if you linear categories is a is a category enriched in like a billion groups, then a dg category is a category enriched in complexes. And nice thing is that if you have a dg category then you get back a linear category by taking the zero to come out. And then maybe more interestingly what is the replacement then of the category of modules. Well, what we actually have is the so called derived category actually work with the g categories, or implicitly. So what, how, how do you define the derived category I'm going to cheat a little bit, but these are essentially the g fun tours. Maybe this should be understood quite, quite better but let's let's say that we cheat a little bit here, the g fun to from the opposite to the category of to the g category of complexes of a billion groups or vector spaces or whatever base you want. So there's there's there's an interesting observation to be made. This drive categories has some nice properties. For example, if you take an object in the derived category, then you can build a shift. In a complex you just shift the complex. And then if you have a map inside this derived category if you if you want this is a chain map of complexes, maybe like a little more complicated but that says a chain map of complexes sense essentially, then you can form a distinguish triangle. And in general you say that any digi category a you say that it's pre triangulated if essentially you you have these shifts, and you have these distinguished triangles inside it's inside it. So, yeah, so far so good. So we have, we have a replacement of like linear categories of the module categories but then how we understand, then the, the replacement of growth in the cabillian categories then something more has to be said. So just let me let me recall that there is a that there is a government for this pre triangulated the g categories if you want, by part in 2010, you need some suitable co completeness, and also suitable generators, but still this is not. This is not the, like, the real the complete story if you want. Namely, the problem is that if you just to just take pre triangulated the g categories by themselves. They don't look really like the higher counterpart of a billion categories, for example, I can ask this question, what is an injective object. We have an ocean of injective object. If we decide on a billion category then we maybe would like an ocean of injective object in this higher framework. We need something more. So, we make another assumption here. We assume, namely that that that that that we take a small digi category, which is now, which is then concentrated in non positive degrees. If you want, this is enriched in complexes which are just concentrated in non positive degrees. Okay, then something we can do is the following so take any object in the divided category. Then you can truncate it's just a complex if you want of a billion groups. Sorry, off of modules over this teaching category you can truncate, and you land in some subcategory of truncated objects. And then it's nice if you have truncation and then you can then you can get back homology. How do you define the homology of a module, you can you can just truncate from one side and then to another. And this thing, maybe have to put some. Okay, and this guy here is actually a module. It's actually a module over the zeroed homology of this digi category. Also, if you want to get the end homology you just shift just take the zeroed homology of a suitable shift. Okay, and then you see that that if you take that if you take the intersection of the dis truncated objects on like on the left if you want. If you take the truncated objects on the right, then it's indeed the category of modules over the H zero over the zeroed homology. So, can we generalize this maybe and the answer is yes. You have the notion of these structures. It's a crucial one here. So, what is a T structure. So you start, you start with a pre triangulated digi category a well actually you actually need just it's a multiple category. And then you have some additional data which amounts to like if you want a replacement formal replacements of this subcategory of truncated objects. Subject to some axioms of course and you and you end up with some with the possibility of making of truncating objects. So you can truncate objects and like, as I said in this subcategory of truncated objects and truncate in like on the left or on the right. And then, perhaps more crucially you can take homology of objects. You can define this age zero and also of course the H and by just shifting. And the point is here that this age zero is going to be a functor. And this functor is going to land into a specific subcategory you take the intersection of these truncated objects. So it's the heart of the T structure. And the important fact here is that this this heart is going to be is always going to be an abelian category. As like here modules over this age zero is clearly an abelian category in this case is also a growth in the category so maybe we are in the right direction. What can we do here. Yes, so just let me say going to abuse notation a little bit somehow like I make no difference between a and it's a multiple category so forgive me but just for the sake of simplicity. This this looks like to be the the right setting indeed. And let's make that let me make an example and try to convince you. I don't have much time but at least try to convince you that having a T structure enables you to, for example, speak of injective objects which we will now call derived injective objects. And the guy the injective objects what is going to be is going to be some object E in which is like lights in the like this this subcategory of truncated objects which we want it's concentrated. It's concentrated in in positive degrees if you want. Then we require that the quality of this object with respect to the T structure is injective in the heart. And also we require this compatibility if you want to take the home from any object to the injective object. And you apply the H zero factor, you land into the heart, you get this, and this has to be an isomorphism objects. I mean, and these, these injective objects actually behave quite nicely. For example, you can do some derived injective resolutions and this is exploited by by by in a work with myself with Wendy and Michelle. Or maybe if you recall, I said that we had a proof of the Gabriel Popescu theorem that the classical one for a billion categories. And a trick and involving and improving exactness of the left adjoint involved like preservation of injective objects. And then and then and then we thought well maybe we have now the derived injective objects we have the T structures which look like a nice setting where to do like higher. If you want to generalize a billion categories. And maybe so can we achieve a proof of some Gabriel Popescu theorem in the setting of these structures using these derived injectives and the answer is yes. As you know, as I'm going to explain you briefly. So, what, what what is a slogan, if you want now. If you take pre triangulated the DJ categories. And you endow them with some suitable growth in the structures. Finally, you would like to require that the heart is growth index somehow, plus some notion of generators. And these, these things now we are in a setting where we can understand this, this notion of growth index structure, even if I won't tell you because it's a little bit technical but at least if you want to have just an idea, that is going to be growth and they can have to be a notion of generators, and they're going to be some more nice compatibilities but well whatever. These things are going to be our linear higher top way. Once we have a Gabriel Popescu theorem, of course, which I'm not, which I'm now going to, to, to, to explain. So, what would be a Gabriel Popescu theorem for these structures then. So, you start with some pre triangulated DJ category, and you endow them and you endow it with a growth index structure so as I said this is going to be a T structure which is nicely behaved. And then the heart is going to be growth index, you'll have some notion of generators, which, which I'm now going to explain you. You take a set of generators. These generators are going to be objects which are going to lie in the this truncated part. So they're going to be in, if you want negative degrees, non positive degrees, and also, if you take any object, also in the left part. This is going to admit a morphism from some direct sum of objects in the two category of generators. Such that if you take the H zero, so the zeroed comology with respect to the T structure, you'll get an epimorphism in the heart. And I mean for that to be well behaved, you want also that this H zero is going to be is going to commute with arbitrary direct sums. Okay. And then what you do is you take a small you, you truncate this cut the subcategory of generators, you remember I said that perhaps I didn't say it well but before I told you about the, the derived category. The first derived categories have to be our main example of growth in the T structures, but for some reason, I mean, technically, this works. If you, if your DG category, the small a is concentrated in non positive degrees. You, you don't have a natural T structure, taking the usual truncation of complexes and the usual comology of complexes. So you need to truncate, but still, it's fine. You can truncate this DG subcategory of generators. And of course you have a functor from this, this, this thing to your bigger category this structure and then you'll also have the unit that restricted you need a functor. I mean this is not really a DG functor. I'm, as I said, I'm a little bit cheating here. Yeah, but let's say it's a DG fun to even if it's not really, it's not really a DG fun. But in any case, it works like this. And the conclusion is that, that this is fully faithful in a weak sense, of course, this has to be interpreted in a weak sense, but still, and has what the exact left adjoint. The exact means a functor, the G fun to if you want, which preserves the T structures. Here you put the natural T structure I've explained before, and the structure here is given. So if you want here, the slogan. And yeah, sorry, let me just explain just just briefly, the T exactness amounts to G preserving derived injectives. So the slogan if you want that every pre triangulated DG category with this growth and the structure nice to structure is a T exact localization of a derived category of a DG category, negative non positive degrees. Okay, so it seems really that these guys are the, at least a nice way to interpret the concept of linear derived or higher topics. So as some final remarks now times almost up. This is not the only possible approach. Indeed, if you look at the spectrum algebra geometry, and you, and you go to appendix C, then you'll find the theory of pre stable infinity categories and also pre stable grotenic pre stable infinity categories. And he also has a double Pesco theorem which is more or less the same as, as the one I've told you right now. The whole thing, the thing which is, which is kind of new in our approach is that the proof is, is actually new, and is more based on the short proof of the classical Gabriels Popescu by Mitchell. And it uses it crucially uses these derived injectives. And by the way, if you just to say, just another thing to say this right injectives. You can also prove a kind of a bio criterion if you want so then really nice, there is a nice, a nice generalization of injective objects and to do to do to get this, you need these structures, or if you want, if you want to work with pre stable infinity you need something like this. So what the next then you can use of course Gabriels Popescu theorem is quite powerful you can use it to understand tensor product of this kind of categories with these structures. And in general, but this may be more far reaching then this is going to be used to understand what is a linear derived site, and also topologies, but we think that this is going to be perhaps a little more a little more difficult. So yeah, this is it and so thanks. Okay, so thank you very much for your nice token.