From Essential Inclusions to Local Geometric Morphisms

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From Essential Inclusions to Local Geometric Morphisms
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It is well known that, given a site of denition, a subtopos of Grothendieck topos can be obtained by strengthening the Grothendieck topology, thus obtaining an inclusion of toposes. An essential inclusion is one where the inverse image functor of this inclusion has a left adjoint. Kelly and Lawvere proved in [1] that an inclusion is essential if, and only if, the stronger topology is closed under arbitrary intersections. They also showed that such a topology generates idempotent ideals on the base category of the site, and this fact fully characterises the Grothendieck topologies which give rise to essential inclusions into presheaf toposes. In SGA 4, Grothendieck and Verdier dened a local topos as one where the canonical geometric morphism into Set has an extra right adjoint. One can generalise this denition by taking an arbitrary topos instead of Set, and thus dening a local geometric morphism between two toposes. Such a geometric morphism is always connected, i.e. that the extra adjoint is full and faithful, and this implies that the codomain is a subtopos of the domain. Thus one can view a local geometric morphism as an essential inclusion where the extra left adjoint preserves nite limits, in other words, a cartesian essential inclusion. Somewhat midway between local geometric morphisms and essential inclusions are nite-product-preserving essential inclusions, which are essential inclusions where the leftmost adjoint preserves nite products. The process of understanding the invariant of a topos in terms of its sites of denition is paramount in the \toposes as bridges" approach of Caramello, which is outlined in [2]. In this talk I shall explain how to obtain characterisations of both the sites that induce cartesian essential inclusions and the sites that induce nite-product-preserving essential inclusions, and also oer exten- sions of the theorem of Kelly and Lawvere, which states that there is a bijection between essential inclusions into a presheaf topos [Cop; Set] and two-sided idem-potent ideals on C.
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select to start by thanking all the organizers of the dividend to speak here and also to say that I'm very very happy to be speaking not only of the ITS 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 OK so far both felony a fine with local publishers and this definition they have MSG for liberals and the slow-growth unique topless is local if the unique genetic morphism into has an extra right adjoint which I referred to as the OK what happens then is that while he is your original genetic morphism and you get another geometric morphism going in the other direction give an example of this is achieved the risky spectrum of a local ring this is where the name comes from we also generalizes to any topological space having .period was only neighborhood is the whole space another example very simple example that if you get bored of the torture you can work this out for yourself is see a of eternal object and then the direct image front there is just evaluation With the Tamil object supra composing with the fund to the pedia picks out that Tamil objects in get left a mark on extensions there which are going to be left in my dad that's another example and enjoy L pointed out in the workshop when you have a property of a top was you can't think of what would be the poverty of geometric morphism by generalize it to a dramatic move for the nite and so the weekend you can define what a local dramatic morphism is and unknown explain exactly when the definition comes from you can see that it's been very similar writers and well if if fuel if you direct image from 2 has extra right joined the fallen faithfulness comes out of the fact that you are now working on a slice OK so now your local dramatic more them isn't just going between 2 publicists is actually commuting with a bunch of other diagrams and that's where get us some things have to be the identity of a themselves situation that you have is well again that's useful they've sold because of a string of bad joins below inverse image from 2 is also going to be full and faithful OK so we have this situation here now where C is on the right and in the skies on the left right and you have a geometric moves going the direction and another dramatic more them going the other direction OK so you have an inclusion in the top honors inclusion because the useful faithful and the bottom line is a connector geometric more slightly stronger than its rejection right because after up stars there is full and faithful because a composer these the student of malls and a further adjoint all other times albeit a bit sketchy and our mix of these 2 guys up and you see that there is no real danger there is facilitating the bit of now when you look at growth and offices this becomes even simpler to think of it because now again have the same string of about joints here I've given them different names the names abusing from now on so I just the inclusion here and EIS an associate chief functor and conveniently mine laughter Georgia's L. OK and what you have in this situation since have inclusion is that you can fix the the base category and like the is fixed and just think of a stronger growth than the topology on the other topples perfect and we know we know that this information comes out of the inclusion that we have here right there have an inclusion and therefore you just you can just fix the fixed the debates kind of oversight and think of us from the real thing topology so Mike the pressure and the problem of my my hope his duty was slow watch information is Els telling us about Kerry right what what extra information coming out of that which is a laughter joint which is preserving finite limits OK and what is the approach is to try to explore the symmetry that we have here of the legend not so much in that sector we have a dramatic move going 1 way and a dramatic move isn't going the other direction but 1st I'd explore the fact that this is refill active here and this is Coleridge wants to be called reflected right and so this approach involves
defining what a discrete object is which is pretty much a dual notions when she fears because the she's sort of belongs to the inclusion and now they're look at the coal reflection and see what's going on there so discrete objects well you you have a local operators already right to have a career achieves right as defined as the objects which are left orthogonal to the morphisms inverted by The Associated she it's exactly the dual notion of of a sheaf right and it's so they actually Evander stake at peace here you know as an amateur fusion it's only necessary to take models in but here we just happy but he can it works for any more for the Mets invented by The Associated chief of and that is he fumes from what the diagram for diving for the sheep condition is this exactly the dual thing right and you get a full subcategory of discrete objects now this is their by killing this that if the category of discrete objects is call reflective then it's it's equivalent to the category of she's them pretty much you get of symmetrical competently symmetrical situation here you basically joined those 2 together and this and this hour here which was the call reflected joins with the reflected here right and you get your triple adjoint right ends and there's another theory by cantilevered their which pretty much characterizes what's happening with respect to the site and I think for months or maybe a bit less so I thought this there was mine but then I found out that had already been proved while more than 20 years ago so yes anyways know it and so basically if you take a took the Ferron as it stated in In In their papers and much more generality there then and putting them here I'm just putting in a simpler version so we can have a have understanding what's going on to take some canonical cited only to take of subcritical side with figures of canonical to make things easier and then inclusion of growth and officers has an extra laughter joint if only if each representable had the smallest cadence of object OK so let's break down what exactly this is saying we have smaller scale than some of the exam which I baptize a signal OK and it's not necessarily covering said OK it doesn't live in K because the covering since they don't actually live in this category here right the same way as living in this category the saves lives and appreciate category what is happening is that citizen belongs to decay if and only if its closure of contains Sigma white because Sigma is a member of the of this collection here of J. closed seas regained there's a slight subtlety there knows raise around always around the subtlety which I discussed my thesis but I won't get into that right now and they sold some of the properties of of all of this of smallest covering the years it has some very interesting properties so some of them as 1st that it's functorial also in 1 and the other variables right and so its citizens profiling too so you can't you can't be a can vary in embarrassment orally with respect to the military so basically we know that sees closed under free composition but this was also closed down there post composition right post composer land in another city and 1 of the most interesting fact is they can now define an interior operator on topless getting you have an interior operate on the whole doubles which restricts Dr operator on the symbolic vibration there which is less damaging to the closure right this is the former French don't deflation for the former is also my thesis the important fact is that basically the signatories the interior of the representable you and so we assume that is it you can also see that's a lot of interesting structure is arising here thing and I also want to thank while adjusting the transpose although the Prefontaine in earlier landing and she's right because it's close to and may also want to think of this as as a functor going from the and achieve integer no His at
their home of cantilevered which it gives us an interesting Anderson duality theorem that we have here is an order presented by junctions between essential localization of depreciate category and i'd important ideals of Cedars the theorem that I did quite a few generalizations of it and my thesis as well we talk about 1 here's what I mean by an ideal was just a collection of morphism that is close not only on the precomposition but post-composition exactly like Smalls soon and it's item potent well this is the former here but very much says they can take an element of your Split into 2 different elements that right basic issue took the the product of the ideal of itself and the thinking of the composition as a part of the game so no here an
interesting result that came out in 1990 by Rosberg Dove was working with realizability topless is for what he did in his PhD thesis was the action ties the notion of a local map of officers in his income in quite handy as well so basically basically saying that you're going to get a local geometric morphism market between topless and Academy of of she's pick local operator if the following things hold the closure operation has left a joint like this said there's always those rights and he calls this an principle a local operator I prefer to call it an essential local operator because if remember the terror on Kelly love you can realize that this is actually gonna give you is going to give you the essentially inclusion part of the of the local men very this is just saying that the local dramatic move as it was then have risen rebounded which is not very useful our case because when workers Roseneath publicist all of which are much more than that automatically bounded this action here is saying that basically abounds hazardous K-III can calculate the discrete at discrete objects associated to your bound in chains which also comes for free when you worked with Wilson at offices and you assume action 1 it because you already going to get by the fire they have a generating said that you're going to get your left eye joints and then this comes for free the them the more interesting action for me is this 1 which translates to the fact that if you take an open subobject of discrete object than this great this is for me the most interesting 1 because it's it's slightly nontrivial holiday ended this was all by open I mean it's the interior of it is itself and the last exon for me is slightly cheating because basically he's putting in these actions to make sure that you left joints and when it exists is going to preserve finite limits that's what he wants for it to be a lower geometric morphism and the result of topless theory says that if your fund to preserves products models and push out to preserve finite limits this is 1 of those theory magic thing because pullback student to push out and things like that much splitting in what he needs to get get products in this this 1 implies that things Amano said that that preserves models but other things as well so now let's look at what's happening when we take top was to be a seasonal cow Ocala topples right in this case since 1 is always discreet some you can easily track that satisfies the the the diagram that put their fallen object being discreet OK and Wendy's locality Aqeel have that well the interior of the same way as cemented again because the Sigma interior of of of wiry OK and so it is a subject why wait we represent which is a subject of 1 so yes Jackson fall holds rights which says that an open some object of discrete objects discrete then you have that you'll all smallest covering service discreet and from then it becomes very easy to construct you after joint L left as L is just going to be 1017 OK so middle left on extensions Colombian leader of Sigmund a shockingly short proof of this losses since the fulfillment and that dance you can draw this diagram and you can see that the unique morphism there exists if you turn your head around us that nightmare around you can see that it is a definition of discrete but it's also saying that seem a satisfied the Universal property there needs to satisfy floated to be in the image of the call reflection right so we found that L a ablaze signal and then it just follows from a joint us here this is just beyond a dilemma OK In this falls from a high of aid being left out joint to elevate you get there right and from this you can see that I represented this this fund to hear anything and basically what you have a look at the as a homer junction we have that with save it is a left joint To dysfunction here thus let me just go back a slight rise thus if this guy is this 1 and this 1 has a left at Giants Is this left at Giants those L must be against cheated the better their life canceled out and they will you can do that sound decomposing the I. this is just a mere merely a sketch of the proof no
the generalization of the theorem by Levine and that I pointed out with that in in the case full of local the visit by GEC of correspondence between local demented morphisms not non-essential inclusions blockage of metamorphose into himself onto the being made in betting which is just too well my rephrasing what an ideal is right which are Cartesian and such that they are now I don't potent with respect to soaring to respect the composition of pro functions I think this is also saying that basically you have on denied potent from Tacoma loaded here but that's just what it means you can just do it is what is right OK let's look at the case work where you'll topless is is appreciative topples which is also fairly well-behaved case In this situation we have the Associated chief fronted the most unpopular functor that we probably have involved top of Serie B because his description is usually horrible right you have to iterate this construction so the associated she found to apply to appreciate and applied object is basically where you try oppose construction twice and the plant's construction was is a cold limit OK but since topology K had us from all those objects of carrying this column materializes and it turns into this right and then it makes the PLO's construction very simply deleted a colon and then again we have the tells home attention again Monday and the blast construction thus becomes this which down by the at times home attention turns into this write a look at the variables here right we can see that the polls construction also becomes representable OK therefore In another thing about until you have elevators is just going to be equivalent to this guy here which is the PLO's construction which is I compose of which again by the injunction turns into this and so on L is pretty much just transforming twice supporters removed with the transfer here right and on the question of whether L preserves a finite limits or preserves finite products reduces to a flatness of this guy guy here also sifted flatness of this functor here OK that's a well-behaved case could you have a 10th homer junction right of
what about the general case when you're looking at she right on on the site of the new don't have necessarily at his home attention you would like to use the same technique but you don't write I mean and also water of the barriers what what what's stopping us from doing this right now so 1 of things 1st is signal J. continues right because we will want to get closer and closer to 2 Diaconescu things right and that the answer is yes it is to continue come on his show proof from you take us Jaycees as saying that because a bunch of morphisms and their rights here the the represented I apply my signature to it OK and then I take the good the coproduct all all of the guys in the middle me right and when I asked myself is this Is this another morphism and 1 fact is that if you take the Co unit of your adult Junction and you do the cover image factorization you get a signal that the OK this is 1 of the many different characterizations of of stigma and so on since this fund is there continues Well this is deftly J. continues for The Associated she respected J & L preserves Co Ltd nappies therefore the composition is the continuous looking at this diagram OK you can see that this is unhappy because l alienated is day continuous this is unhappy because of the factory that we have here right and since all these guys Abbey's this 1 must be happy as well so we have a home junction right we also have Diaconescu around that Savage a continuous flat-front is crisp onto geometric morphisms and we can subtract will the J. continuity rights and get just that flat-front argument wasn't but instead of cheese here you have just just appreciate category and what I would like to do is like theorem Algebra I would like to subtracted disappearance so distinct that added to this 1 but that doesn't work right you don't have a J continuous punters are equivalent to 2 . 20 agitated when that doesn't work and the reason for that is what you need is that for this to factor through She's you need that it's left joints and covering serves to Isaac OK so that when utensil Sigma you get this right and for all for all users injured but the day continues as saying that it's an empty riot and now we just need to see we just need this office and to also be a Amano OK you just me that notice with with with models will give respect kimono and how much time do I have 10 10 minutes OK I can go into the mixture of I think I can go into a again a sketch of the proof off 1 by 1 of the concerns that I have many candidate will not meet 3 conditions fall preservation of models of this fund I'll go into 1 of them OK so efficiency has fullbacks and you take on a take a situation like this where the duties and started using your and then you ask belongs to you or your was a case of Sigmund Freud ,comma cases but fuel you the smallest covering His most there's some object signal you take the pullback OK this is the pullback of some of G and the G Brian and then OK look at this time against you constructors other diagram wherever applied Sigma as a front to this area here right and I want to the signatures sort of preserved pullback up to the site OK so basically instead of a representative if elected president back together representable fund to him but instead you can take another just just a covering civil J but it is so the condition is that if there exists an author and such that this commutes van and then your signal representing 1 0 OK but you can just take a much stronger assumption which is that the singer preserves fullback seminars of commuter Maxson said which is what happens in well and is not what happens in the localities but is equivalent in the localities retrieving flat so the that I have is that we have a small category and you start with a triple junction there right on an essential inclusion rights and you take it's minimal saves from 2 but this time it is a minimal since from 2 because we don't have a topology here than l president that limits for all products this In this to here is flat or sifted flour and you also get the slightly more well behaved version for locales where you don't need to cancel again it's going to be used leftist parties in Italy if significant because busses L has has finally meets countries and flat of the same thing and so OK and so the general case and how to control the general case I have 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 yeah it's I don't think it is available online at :colon had my revival verdicts but it should be around after January also and that is all I
have to say thank you home with him please write to you can have a look at the end of the season of the year and we haven't and that's that is 1 thing that I would love to do 91 1 is the saying about you know and in terms of the the dramatic clearing the area that is that is 1 thing that I think is missing right because not all the way over to the side right politically to the theory that it is the festival's hands do you that this during the rest of the season concerning his future will be on the altar all of the few