 Merci, thank you. So I'd like to start by thanking all the organizers for the opportunity to speak here and also to say that I'm very very happy to be speaking not only at the IHES but in a conference that is pretty much focusing on exactly the kind of thing that I do. For me it's like a one-of-a-kind experience. So okay, so I'll start by defining what a local topos is and this is the definition that you have in SGA4 by Grossendijk. So Grossendijk topos is local if the unique geometric morphism into set has an extra right adjoint which I refer that to as C okay. What happens then is that well here's your original geometric morphism and you get another geometric morphism going in the other direction okay. An example of this is sheaves under the risky spectrum of a local ring. This is where the name comes from but you can also generalize this to any topological space having a point whose only neighborhood is the whole space. Another example very simple example that if you get bored of the talk you can work this out for yourself is take C a category of a terminal object and then the direct image functor there is just evaluation on the terminal object so you're pre-composing with the functor that picks out that terminal object and then you get your left and right con extensions there which are going to be the left and right adjoints okay. That's another example and as Joijel pointed out in the workshop when you have a property of a topos you can think of what would be the property of a geometric morphism right. Generalize it to a geometric morphism right and so we can you can define what a local geometric morphism is and I'm not going to explain exactly where the definition comes from you can see that it's very very similar right as in well if if your if your direct image functor has extra right adjoint the full and faithfulness comes out of the fact that you're now working on a slice okay so now your local geometric morphism isn't just going between two topos is it's actually commuting with a bunch of other diagrams and that's where you get that some things have to be the identity okay and so the situation that you have is well again that C is full and faithful because of the string of adjoints the the inverse image functor is also going to be full and faithful okay so we have this situation here now where C is on the right and this guy is on the left right and you have a geometric morphism going in this direction and another geometric morphism going in the other direction okay so you have an inclusion the top one is an inclusion because C is full and faithful and the bottom one is a connected geometric morphism slightly stronger than a suggestion right because f uh f upper star there is full and faithful you can also compose these these two geometric morphisms and get a further adjoint a lot of the times I'll be a bit sketchy and I will mix these two guys up and you'll see that there is no real danger there it facilitates things a bit okay now when you work with growth and ectopuses this becomes even simpler now to think of because now again I have the same string of adjoints here I've given them different names the names that I'll be using uh from now on so I is just the inclusion here and A is an associated chief functor and conveniently my left adjoint is L okay and what you have in this situation since you have an inclusion is that you can fix the the base category and like C here is fixed and just think of a stronger growth in ectopology on the other topos okay and we know when this information comes out of the inclusion that we have here right that we have an inclusion and therefore you just you can just fix the fix the the base category of the site and think of a stronger growth in ectopology right so my the question the problem of my my whole phd was well what information is l telling us about k right what what extra information can we get out of l which is a left adjoint which is preserving finite limits okay and what the approach is to try to explore the symmetry that we have here of this triple adjoint not so much in the fact that we have a geometric morphism going one way and a geometric morphism going in the other direction but first try to explore the fact that this is reflective here and this is core if wants to be co-reflective right and so this approach involves defining what a discrete object is which is pretty much a dual notion to what a sheaf is because the sheafs sort of belong to the inclusion and now we're going to look at the core reflection and see what's going on there so a discrete object well you you have a local operator already right you have a category of sheaves right and it's defined as the objects which are left orthogonal to the morphisms inverted by the associated sheaf function it's exactly the dual notion of of a sheaf right and it's so dual that you can actually even just take appease here you know as in and a sheaf usually just take it's only necessary to take monos you know but here we just take appease but it can it works for any morphism that's inverted by the associated sheaf function and so that is if you remember what the diagram for diagram for the sheaf condition is this is exactly the dual thing right and you get a full subcategory of discrete objects okay now there's a theorem by Kelly and Lovier that if the category of discrete objects is co-reflective then it's it's equivalent to the category of sheaves then pretty much you get a symmetrical uh totally symmetrical situation here you basically join those two together and this and this arrow here which was the co-reflector joins with the reflector here right and you get your triple adjoins right and there's another theorem by Kelly and Lovier which pretty much characterizes what's happening with respect to the site and i think for a month or maybe a bit less i thought this theorem was mine but then i found out that it had already been proved well more than 20 years ago so yes and anyways no yeah and so basically if you take i took the theorem as it stated in in uh in the paper it's in much more generality than than i'm putting down here i'm just putting in a simpler version so they can have a have an understanding of what's going on so you take a subcanonical site you don't need to take a subcanonical site but i've taken it subcanonical to make things easier and then inclusion of growth and ectopuses has an extra left adjoint if and only if each representable has a smallest k dense sub-object okay so let's break down what exactly this is saying we have a smallest k dense sub-object there which i baptize as sigma okay and it's not necessarily a covering sieve okay it doesn't live in k because the covering sieves they don't actually live in this category here right the sigma a is living in this category the sieves lives in the pre-sheaf category what is happening is that a sieve belongs to k if and only if its closure contains sigma a right because sigma a is a member of of this collection here of j closed sieves okay there's a slight subtlety there there's a there are rays around a ways around this subtlety which i discuss in my thesis but i won't get into them right now okay so some of the properties of of of this smallest covering sieve it has some very interesting properties so some of them is first that it's functorial also in well in the other variable right and so it's it turns into a profunctor so you can you can the a can vary and it varies functorially with respect to the a okay so basically we know that sieves are closed under pre-composition but this one's also closed under post-composition right you post-compose with an f you land in another sieve okay and one of the most interesting facts is that you can now define an interior operator on your toppers okay you have an interior operator on the whole toppers which restricts to interior operator on on the the sub-object vibration there which is left adjoint to the closure right this is the formula for it okay the explanation for this formula is also in my thesis the important fact is that basically the sigma a's are the interior of the representables you know and so yes and that is you can also see that's a lot of interesting structure is arising here and you may also want to think well just take the transpose of the the profunctor and you know that you're landing in sieves right because it's closed and so you may also want to think of this as a functor going from sea into sieves into j okay now here's a theorem of kelly and lovier which gives an interesting interesting duality theorem that we have here that there's an order preserving bijection between essential localizations of the pre-shift category and idempotent ideals of sea this is a theorem that i did quite a few generalizations of it in my thesis as well only talk about one here what i mean by an ideal well it's just a collection of morphisms that is closed not only on the pre-composition but post-composition exactly like uh smallest covering sieve and it's idempotent well this is the formula here but pretty much it says that you can take an element of your ideal and you can split it into two different elements there right okay basically if you took the the product of the ideal of itself uh thinking of the composition as a product okay so now here's an interesting result that came out in 99 by last bergdorf was working with realizability toposys but what he did in his phd thesis was he axiomatized the notion of a local map of toposys and this can come in quite handy as well so basically basically saying that you're going to get a local geometric morphism okay between a topos and a category of sheaves here when you pick out a local operator if the following things hold the closure operation has a left adjoint like i said that always does right and he calls this a principal local operator i prefer to call it an essential local operator because if you remember the theorem theorem by kelly and levier you can realize that this is actually going to give you it's going to give you the essential inclusion part of the of the local map okay this is just saying that the local geometric morphism is going to have is going to be bounded which is not very useful in our case because when working with growth in the toposys all local geometric morphisms are automatically bounded this axiom here is saying that basically your bound has that this you can calculate the discrete a discrete object associated to your bound okay which also comes for free when you're working with growth in the toposys and you assume axiom one right it because you're already going to get by the fact they have a generating set that you're going to get your left adjoint and then this comes for free the the more interesting axiom for me is this one which translates to the the fact that if you take an open sub-object of a discrete object then it's discrete this is for me the most interesting one because it's it's slightly non-trivial okay and um this was saying oh by open i mean it's the interior of it is itself okay and the last axiom for me is slightly cheating because basically he's putting in these axioms to make sure that your left adjoint when it exists is going to preserve finite limits that's what he wants for it to be a local geometric morphism and a result of topos theory says that if your functor preserves products monos and pushouts it will preserve finite limits this is one of those topos theory magic things because uh pullbacks turn into pushouts and things like that so he's pretty much just putting in what he needs to get products you know this this one implies that things are monos that that it preserves monos but other things as well okay so now let's look at what's happening when we take our topos to be uh sheaves on a locale a localic topos right in this case since one is always discrete um you can easily check that it satisfies the the the diagram that i put there for an object being discrete okay when e is localic we have that well the interior of sigma a is sigma a again because sigma a is the interior of of of y of a okay and so and it's a sub object of y of a uh the representable which is a sub object of one so if the axiom four holds right which says that an open sub object of a discrete object is discrete then you have that your smallest covering sieve is discrete and then from then it becomes very easy to construct your left adjoint l your left adjoint l is just going to be tensoring with sigma okay it's going to be the left con extension along your nadir of sigma i'll give a short proof of this okay well since since since sigma a is discrete and that guy is dense you can draw this diagram and you can see that the unique morphism there exists if you turn your head around or switch the diagram around you can see that it's is a definition of discreteness but it's also saying that sigma a satisfies the universal property that it needs to satisfy for to to be in the image of the co-reflection right so we found that la of y is sigma a and then it just follows from adjointness here this is this the yonade okay and this follows from i of a being a left adjoint to l of a you get this right and from this you can see that i've represented right this this functor here and if and basically what you have looking at the tensor homo junction right we have that tensoring with sigma is a left adjoint to this functor here thus let me just go back a slide right thus if this guy is this one and this one has a left adjoint is this is the left adjoint thus l must be okay i've cheated a bit there i've cancelled out an a but you can do that by precomposing with the i this is just a mere merely a sketch of of the proof now the generalization of the theorem by lovier that i pointed out what i did in the case for a locale is that there's a bijective correspondence between local geometric morphisms not not essential inclusions but local geometric morphisms and sub functors of the yonade embedding which is just well my rephrasing what an ideal is right which are cartesian and such that they are now idempotent with respect to tensoring to respect to composition of pro functors i think this is also saying that well basically you have an idempotent pro functor commoded here but that that's just lots of names you can just it is what it is right okay now let's look at the case where where your topos is a is a pre-shift topos which is also a fairly well behaved case in this situation we have the associated chief functor the most unpopular functor that we probably have in all of topos theory because this description is usually horrible right you have to iterate this plus construction so the associated chief functor apply to a pre-shift and then apply to an object is basically where you iterate the plus construction twice and the plus construction once is a co-limit okay but since topology k had a smallest object okay this co-limit trivializes and it turns into this right and then it makes the plus construction very simple you don't need to take a co-limit okay and then again we have the tensor homo junction again okay and the plus construction thus becomes this which then by the tensor homo junction turns into this right and look at the variables here right we can see that the plus construction also becomes representable okay therefore again another string of adjoints here you have l of a here is just going to be equivalent to this guy here which is the plus construction which is i composed with a which again by the adjunction turns into this and so l is pretty much just tensoring twice or tensoring with this tensor here right and now the question of whether l preserves finite limits or preserves finite products reduces to flatness of this guy here or sifted flatness of this functor here okay so that's a well behaved case because you have a tensor homo junction right what about the general case when you're looking at sheaves right on a on a site well then you don't have necessarily a tensor homo junction you would like to use the same technique but you don't right i mean and so what are the barriers well what what's stopping us from doing this right and so one of the things is first is sigma j continuous right because we want to get closer and closer to to diakonescu's theorem right and the answer is is yes it is j continuous here's a short proof you take a jcf s and you i pick a bunch of morphisms in there right here here's the cf represented i apply my sigma to it okay and then i take the the the co-product of of the guys in the in the domain right and i want to ask myself is this is this an epimorphism and one fact is that if you take the co-unit of your adjunction and you do the cover image factorization you get your sigma there okay this is one of the many different characterizations of sigma and so since this functor is j continuous well this is definitely j continuous for the associated chief functor with respect to j and l preserves co-limits and epis therefore the composite is j continuous looking at this diagram okay you can see that this is an epi because la of yoneda is j continuous this is an epi because of the factoring that we have here right and since all these guys are epis this one must be an epi as well okay so we have a tensor homojunction right we also have jackonescu's theorem that says that j continuous flat functors correspond to geometric morphisms and we can subtract the the the j continuity right and get just that flat functors geometric morphism but instead of sheaves here you have just just a pre-shift category and what i would like to do is like theorem algebra i would like to subtract this theorem sorry this theorem from that theorem and add it to this one but that doesn't work right you don't have that j continuous functors are equivalent to to adjoints here right that just doesn't work and the the what you need is that for this guy to factor through sheaves you need that its left adjoints send covering sieves to isos okay so that when you tensor with sigma you get this right and for all for all your sieves in j but b and j continuous is saying that it's an epi right and now we just need we just need this morphism to also be a mono okay we just need that no tensoring with with with monos will give us back a mono and how much time do i have ten ten minutes okay i can go into there are many okay okay i think i can go into again a sketch of the proof of one one of the i have many well not many three conditions for preservation of monos of this functor i'll go into one of them okay so if you assume that c has pullbacks and you take you take a situation like this where the g's are in sorry your g's are in your in your sieve and and your f's belong to your your k sieve sigma for it's not a k sieve but your your your smallest covering your smallest dense sub-object sigma you take the pullback okay this is the pullback of of g and g prime and then okay looking at this diagram again you construct this other diagram where i've applied sigma as a functor to this area here right and i want the sigma to sort of preserve pullback up to the site okay so basically instead of a represent if you want to preserve pullbacks you get a representable functor here but instead you can take another just just a covering sieve of j right and so condition is that if there exists an r such that this commutes then then your sigma will be preserving monos okay but you can just take a much stronger assumption which is that the sigma preserves pullbacks and then your r is just going to be the maximum sieve okay which is what happens in well it's not what happens in the locale case but it's it's equivalent in the locale case to it being flat so the theorem that i have is that you have a small category and you start with a triple adjunction there right um an essential inclusion right uh and you take its minimal sieves functor right this this time it is a minimal sieves functor because we don't have a topology here then l preserves finite limits or products if this functor here is flat or sifted flat and you also get the slightly more well behaved version for locale's where you don't need to tensor it again it's going to be it's left it's cartesian if and only if sigma is cartesian because well since l has has finite limits cartesian and flat are the same thing and so okay and so the general case and how to control the general case um i've explored in more detail in my in my phd thesis but i won't go into it now because i don't have the time but yes it's i don't think it is available online yet because i haven't had my viva but but it should be around after january or so and that is all i have to say thank you have your questions have you tried to have a look at the notion of local geometric morphism from the logical point of view no i haven't and that that is one thing that i would love to do you know i mean what what is this saying about you know in terms of of the the geometric theory yeah yeah yeah yeah that is that was one thing that i think is missing right because we've gone all the way over to the site right but we need to go all the way to the theory right yeah yeah yeah i have to remark on that i mean there's was trial who's not here right now it looks into cohesive type theory so this extra adjunction gives rise to some modernities and he's been studying this a lot so for the higher order logic there are some oh okay okay i'll look into that yeah yeah