Extending the topological presheaf-bundle adjunction to sites and toposes
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Toposes online, 20211 / 31
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Musical ensemblePressureLogicMeeting/InterviewComputer animation
00:20
TopologyTopostheoriePhysical lawFiber bundleJunction (traffic)Slide ruleProjective planeMereology
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SpacetimeTopologyGrothendieck topologyGeometryLogicTopostheorieTheoryFiber bundleSheaf (mathematics)Local ringFunktorPoint (geometry)Gamma functionAbelian categoryMorphismusGrothendieck topologyCodomainHomöomorphismusJunction (traffic)TopostheoriePoint (geometry)Category of beingSheaf (mathematics)FunktorEnergy levelLocal ringEquivalence relationFiber bundleNetwork topologyDivisorSpacetimeGamma functionShift operatorResultantDescriptive statisticsFamilyArithmetic meanGeometryTerm (mathematics)Open setSet theoryElement (mathematics)Topologischer RaumNatural numberSlide ruleContinuous functionSocial classGoodness of fitPrincipal idealGroup actionBlind spot (vehicle)Free groupSpecial unitary groupRight angleRange (statistics)Discounts and allowancesGame theoryCovering space1 (number)Uniform boundedness principleMusical ensembleAnalytic continuationTime domainPhysicalismHydraulic motorMany-sorted logicMaxima and minimaLine (geometry)Proper mapLogicMathematical singularityProjective planeTrailSelectivity (electronic)Process (computing)ComputabilityProduct (business)Computer animation
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TopostheorieHomöomorphismusLocal ringEquivalence relationAbelian categoryRadiusCone penetration testElement (mathematics)Sheaf (mathematics)Fiber bundleFunktorGeometryMorphismusHurewicz-FaserungGrothendieck topologyCategory of beingElement (mathematics)Object (grammar)FunktorSocial classTopostheorieEnergy levelGamma functionLocal ringVibrationMorphismusCondition numberResultantArrow of timeFiber bundleSheaf (mathematics)Lemma (mathematics)Maß <Mathematik>Equivalence relationNetwork topologyShift operatorDivisorDescriptive statisticsTheorySlide ruleIterationReduction of orderDirection (geometry)HomöomorphismusSpacetimeFamilyGeometryIdeal (ethics)Point (geometry)WeightMultiplication signTerm (mathematics)Presentation of a groupModulo (jargon)Equaliser (mathematics)Inclusion mapSummierbarkeitHomomorphismusSet theoryTopologischer RaumTransformation (genetics)Natural numberModulformMoment (mathematics)Different (Kate Ryan album)Product (business)Hill differential equationOrder (biology)Closed setSeries (mathematics)PreorderPhysicalismSinc functionAbelian categoryMonster groupJunction (traffic)Körper <Algebra>Metric systemPrice indexLimit (category theory)Film editingRootCapillary actionMatching (graph theory)Open setTrailGroup actionHeegaard splittingSpecial unitary groupAdditionBeta functionEmpirical distribution functionTube (container)AlgebraBeat (acoustics)Thomas BayesCausalityGreen's functionForcing (mathematics)Line (geometry)Division (mathematics)Musical ensembleComputability2 (number)Physical lawScherbeanspruchungContent (media)Analytic continuationCanonical ensembleDependent and independent variablesProjective planeRational numberLatent heatMorley's categoricity theoremBuildingState of matterComputer animation
Transcript: English(auto-generated)
00:15
So the next talk will be by Ricardo Zampa, and he will explain extending the topological pressure
00:22
bundle junction to size and toposis. And this is joint work with Olivia Carmelo. Thank you. I hope everybody can see the slides. So here we'll be presenting to you a part of the joint project we've been working on,
00:41
Olivia and I. And so basically, what I will present to you today is this. So we have this known result in topology. When we have a topological space X, we know that there is an adjunction between the pre-ships over the space X and the bundles over X. That is, topological space is over it.
01:02
We wonder if we can do the same for an arbitrary small site. And in fact, we do. If there is a site, Cj, a small site, we can build an adjunction in the same way where instead of topological spaces, we'll have toposes of a certain kind over our base topos. And the properties of these adjunctions
01:22
do reflect what happens in the topological case. So as I was telling you, this is an extract of my joint work with Professor Carmelo, which is relative topos theory via stacks, which will be available in a few days. And in particular, I will recall briefly
01:43
the topological pre-shift bundle adjunction. Those of you who are interested and want to see it again, you can find it, for instance, in shifts in geometry and logic, though the result is quite ancient in topology, let's say. And I will also cite the localic version
02:00
of the same adjunction. This is somehow harder to find in literature. You can find it, though, still in shifts in geometry and logic, it's kind of implicit in some of the exercises. So I've written it here. So for those of you who haven't seen Olivia's lectures last week, I will recall how the topological
02:20
pre-shift bundle adjunction works. So we know that we have this adjunction where X is a topological space, any topological space. The functor lambda is called the bundle of germs. And basically it takes a pre-shift P. It computes this set. These are the stocks of germs of the pre-shift.
02:43
We won't need to see the definition again, but basically you take the germs of the elements of P in a point X, you glue all these sets together into this co-product, you have a canonical projection, and you can topologize the set so that this map becomes a continuous map.
03:01
And this is called the bundle of germs of the pre-shift P. On the opposite direction, the right joint, the gamma, is called the local sections functor. It takes a bundle over X. So we have our continuous map P from E to X, and we call local sections, the continuous maps from an open U inside X
03:23
such that they work as a section of our map P. So they are the local sections of P at the open U. This defines a pre-shift and the properties of this junction are the follow, the basic, the relevant properties of this junction are the following.
03:41
First of all, we can calculate the fixed points of the junction. The fixed points in the pre-shifts for our topological space are the sheets for the topology while the fixed points in the bundles are what are called the etal bundles over X, which are basically the local homeomorphisms
04:01
with co-domain X. And they are the fixed points of the junction. So in particular, we get an equivalence with this level. The second important property of the pre-shift bundle junction is that when we start from a pre-shift, we compute its bundle of germs and then go back to the local sections.
04:21
We end up getting the shiftification or the associated shift of our original pre-shift. So the composite gamma lambda is the shiftification factor. So you see that the basic points in this result are that pre-shifts and most importantly, shifts over X can be thought as a category of spaces
04:42
because we are really seeing them as spaces over our base space X. And also that the shiftification factor gets a geometric interpretation in terms of local sections. And this is interesting because when you compute shiftification for a general Grothendieck topos,
05:04
in the literature, you almost invariably find just the one description, which goes back to Grothendieck with a double plus construction, which is, I mean, it's a very nice construction, but it's very algebraic and very hard to use because when you compute that, you take the matching families for a pre-shift,
05:22
you quotient them with respect to an equivalence relation and then you do the same process again. So it's kind of hard to manipulate it in a concrete case to do computations with it. While for instance, when you have topological spaces, you have this very nice description. So you know that shiftifying means taking
05:41
the local sections of a particular bundle over your base topos. And at the beginning, I told you that there is also a localic version. This basically happens because even though when you build the bundle of germs, you use the points in the set, sorry, in the base space,
06:02
you don't really need points. So you can do the same things, point three, and you get to the pre-shift bundle junction in the formalism of locales. So if L is a base local, you can consider the pre-shifts over L, it's shifts, the locales over L and the etal, locales over L.
06:20
Well, etal, the map of locales is the natural generalization of the etal map of topological spaces. And you get all the same properties. So a junction, a joint equivalence at the level of shifts and you can recover the shiftification. So it is natural to- Ricardo, we don't see your video. Maybe you can switch it on.
06:42
You don't see my video? Is it only my problem or someone else? I can see it, I see it myself, I mean. Ah, okay, so maybe it's just me. Okay, sorry for the interruption. Okay. So we wonder if we can do the same for a generic site.
07:04
So we have our topological space and we want to substitute it with the site. So as the title of the slide suggests, we think of toposys as a generalization of spaces. And so the generalization of continuous maps
07:21
over a base topological space will generalize to geometric morphisms from any topos to our topos of shifts. This is the most natural generalization one can think of. There is however, a size issue here because when we define the local sections of a topological space, we are considering home sets.
07:44
If we want to do the same here, of course we cannot speak of home sets because the class of geometric morphisms between two topos is in general, not just the set. So we have to restrict that in a certain way and we propose this restriction.
08:02
We call a topos E over our base topos smaller relatively to the base. If for each X in the site, and in a moment I will tell you why we choose this, this category of geometric morphisms. So you see, these are the geometric morphisms from this slice topos over the representable given by X
08:23
to our topos E over the base topos, shifts Cj. We ask for this class of geometric morphisms to be a set after equivalence of geometric morphisms, of course. So why do we choose the elements in the site? Of course, we want a pre-ship over the site. That's obvious, but the reasoning is
08:42
in the topological case, the elements in the site are the open. So in this case, we are really saying we are taking something from an open in the topological space to the space we are considering to calculate its local sections. So given the notion of topos is smaller
09:01
relatively to the base, which mind is referred, I say topos but the object of interest is actually the geometric morphism because of course we may have the same topos with different geometric morphisms over the same base. And we call topos with this superscript S
09:20
topos over shifts Cj, the full subcategory of the topos is over our base that are small, that are relatively small. And this will allow us to define local sections and have them being a set, so a pre-ship. In particular, we also want to generalize the notion of etal topological space. And of course we already have a notion here of etal topos.
09:46
A topos over a base is said to be etal or equivalent to a local homeomorphism. The terminologies are equivalent. If it is the topos is equivalent to one of this form. So it is equivalent of a slice of our base topos.
10:01
And also this geometric morphism is the canonical geometric morphism to the base. So it's the one who's adjoins from the writer joint to the essential image out of the dependent product going this way, the pullback factor going this way and the dependent sum in this way.
10:21
We denote the category of etal topos over a base with etal over shifts. And in particular, all etal topos are small relatively to the base. So we have this inclusion. And in a moment, I will tell you why. So this completes the picture because we have our topological space becomes a site,
10:42
bundles over the space becomes relatively small topos. etal bundles become etal topos. Now to define the adjunction, the local sections factor at this point is straightforward with the generalization of what we have in the topological case.
11:00
Given an essentially small topos E over shifts Cj, we consider its image via gamma to be this pre-shift. So you see it's the local sections of E at the elements of C. And we know that these are sets because E is small relatively to the base topos. So we chose E to be sure that this lands in set
11:21
and not somewhere else. The definition of the bundle of germs, the left adjoint requires a bit more of work, not much, but basically we start with a pre-shift P. We consider it's Grothendieck construction, which is a vibration over C, discrete vibration in this case.
11:41
And since C has a topology J, those of you who have followed all these lectures last week already know this, we can endow this category with the smallest topology, which we call the JP, making this functoric homorphism of sites. We call this topology Giraud's topology for the pre-shift P. We call the site Giraud site,
12:02
and obviously topos is Giraud topos for P. Since this is a homorphism of sites with this topology, it induces covariantly a geometric morphism. So we have always this geometric morphism, which is furthermore in this case, it's also an etal geometric morphism, but in general we have it and it's always essential.
12:23
So this is the bundle of germs functor. And to prove the adjunction at this point, you will see it's quite easy to prove it because it's a couple of lemmas and the rest is pretty straightforward. The first lemma shows that the Giraud topos for a pre-shift P is a conical co-limit
12:42
in the category of topos over the base. So basically we have our category of elements here. You see every object in here is mapped to an etal topos, which is gonna be shifts over the representable indexed by that element. And so what happens?
13:00
Why do we want it to be a co-limit? Because we want lambda to be the left adjoint of a home functor, of a contravariant home functor. So it's going to behave like a co-limit. And this is in fact true. You can prove this, it's a computation. So you see, if you start from an arrow, for a geometric morphism from lambda P to E over our base topos,
13:22
since this is a co-limit, having an arrow here is the same as having a cocon. The cocon has legs starting from topos of this kind, this etal topos to E. Each of these legs is indexed by elements in P. So for instance, you will have a geometric morphism,
13:42
which is indexed by an element A in P of X. So a cocon of this form is the same thing as a natural transformation from P to this set of geometric morphisms. And so lambda is the left adjoint of gamma. So we have the adjunction at the level of pre-shifts.
14:03
We want to see what happens when we restrict to shifts. And for that, we can use these two lemmas on etal toposes. The first one tells you that if you work over a base topos, arrows between etal toposes can be presented directly with arrows inside of the base topos.
14:20
So any geometric morphism here above E is presented by an arrow in the base topos. Second, the second lemma tells you that the zero topos of a pre-shift P, which we described in this way, is actually equivalent to an etal topos, which is the etal topos over the shiftification of P.
14:40
And with these two lemma, you can prove all the nice consequences about the adjunction, the pre-shift bundle adjunction. Because, see, if you start with a etal topos, shifts Cj over f, and you compute its gamma, its image, the local sections are defined in this way. But using the first line, you can reduce these two arrows
15:02
from this representable to f. You go from the etal toposes over the base to arrows inside the topos. And this is just f by unit lemma. This equivalence allows you to conclude that fixed points in small toposes over the base are in fact the etal toposes,
15:21
the fixed points for our adjunction. In the opposite direction, if you start with a pre-shift P, and you consider its shiftification, again, we do the same thing in reverse. So we start from the shiftification, we use unit lemma, we move to etal toposes, arrows from here to here are the same
15:41
as arrows from this etal topos to this etal topos, but this is just gamma of lambda of P, because we said here that lambda of P is precisely this slice topos. So this, again, this allows you to conclude that fixed points of the adjunction are the J shifts, and also that's the composite of gamma and lambda
16:01
is the shiftification factor. So we end up with the, let's say the pre-shift bundle adjunction for general sites, which you see is really the same thing as what happens in the topological case. Pre-shifts is a relatively small toposis, restricts to an equivalence between shifts and etal toposes, no longer spaces,
16:21
and you can recover the shiftification by composing the two adjoints. So- Can you say, can you say at this point, how you overcome this iteration twice that, you know, for the shiftification in the topos case? Yes, I will tell you later how.
16:42
In a certain sense, in a couple of slides. So for the, let's say, what is interesting about this adjunction is that you can move in different directions. You can go up, let's say go up and see it as a truncation or a reduction
17:01
of a wider phenomenon to categorical phenomenon. You can also go down and describe the shiftification using sites and their morphisms or comorphisms, or you can cross the bridge, let's say. So in some cases, for instance, I will show you in pre-order sites, you can actually forget the topos theoretic information.
17:23
So you end up with a description of the shiftification that is really topos-less, let's say. So for the two categorical adjunction, I won't tell you much. Oliva has already told you about this on Saturday, but basically there is two categorical adjunction,
17:41
which leaves at the level of pseudo functors from our category to ket and the topos is over the base topos. We have these two adjunctions. The right adjoint is still a local sections functor. The bundle of germs left adjoint behaves in the same ways in the discrete case.
18:02
So it takes a syntax category and builds its zero topos, which is the topos over its associated fibrillation. And when you take this adjunction and restrict it to the one dimensional case, you get, again, our discrete adjunction. So for those of you who are interested,
18:20
of course you will find plenty more content on this on the forthcoming paper. Going down, what is interesting is that this provides a geometric interpretation of the shiftification as local sections, because you see the value of the shiftification of P attacks is set up to prevalence of geometric morphisms
18:42
from these topos, which is the one indexed by our object to these topos. And of course, I mean, speaking about classes of geometric morphisms is not that simplifying, let's say, but at this point, since you have geometric morphisms, you can go to sites, you can reduce the sites in various ways.
19:01
So one way is the most standard one is using flat j-continuous functors. So a geometric morphism from here to here will correspond to a flat j-continuous functor from this site to this topos or equivalently to a morphism of sites from this site to the canonical site of the topos.
19:24
Another way of seeing this, which is harder to see it first, but basically you can present this topos with this site. It's the slice category of X. This is the zero topology for this script vibration.
19:41
And all the functors in here can be presented by comorphisms of sites from this site, which presents the first topos, so to this site, which presents the second topos. Mind, it's not that all morphisms of sites or all comorphisms of sites will give you topos, sorry, geometric morphisms in here.
20:03
Of course, they have to satisfy some conditions to give you something which leaves over the base. Of course, you have that. The third point, which somehow is connected to the question is the following. It's hard to give you really a technical idea of why this happens,
20:21
but if you think of elements in the shiftification, an element in the shiftification of P at X is something which may not exist in P, but is given by gluing local information in P. In the same way, it happens that a geometric morphism in here may not be presented at the level of sites
20:42
by a comorphism, but it is locally presented by a comorphism of sites and even better by morphisms of vibrations. So what happens is that you can find for each geometric morphism in here, you can find the J covering the family of X and the family of morphisms of vibrations
21:02
from C over YI to the vibration of P. And this is really telling you, I'm taking elements of P, I'm seeing them locally. These are all comorphisms of sites. And these, let's say, jointly induce a geometric morphism over here. So we are saying, you may not have a presentation at the level of sites
21:21
in terms of comorphisms from the site presenting these topos to the site presenting these topos, but you can restrict to a J covering family so that you have this presentation. And this somehow connects with the double plus construction because when you do the double plus, what you have is at first you have matching families
21:41
up to let's say the local equality, and then you do a matching family of locally matching families. So what you get, sorry, a matching family of matching families modulo local equality. So what you get is the information of a, it's hard to explain, but you see it's,
22:01
you get basically a local information for which when you restrict nicely enough, you get the same geometric morphism. So the point in the plus construction is that you have your local information, but the local information doesn't induce something already at the level of the site. So nothing, not something from CJ to int P,
22:23
but at the level of topos, it does. That's where the double plus kind of strikes in. I know it's a bit foggy. I will try to be maybe clearer later. I'll finish now. And for the last point, as I was telling you, there are some cases in which you can
22:41
move away from topos altogether. You can forget the topos theoretic information as it happens for pre-order sites. So I recall you this result of locales. So basically locales over a base include into topos over the shifts for that local. And this inclusion goes into essentially small topos,
23:00
sorry, relatively small topos. And also I recall you, I remind you that for a pre-order site, this happens, the shifts over that pre-order site by pre-order I mean category in which between two elements, there is at most one arrow. It's a localist, a localic topos, and it is represented by the locales of ideals
23:20
of J-ideals of the category. And you can do the same for every pre-shift over your category. The topos you get is localic, and you can see it as a presented by the ideals of the vibration. Using this, if you consider in particular a pre-shift over a local, shiftification is described in this way. And this is what we said before,
23:41
this is gamma of lambda. But then since we are in topos is over a localic topos, this is localic and this is localic, we reduce to locales. So this is the same as maps of locales from this locale, which presents this is a sliced topos to this locale. And you can plug this in,
24:02
in the adjunction, and basically you get this for pre-orders. You get that your adjunction at the topos theoretic level already exists at the level of pre-shifts over your pre-order and locales over the ideals. And again, this induces the equivalences we expect.
24:20
In particular, we get the notion of, there is a notion of etal net of pre-orders. So you get the pre-shifts over a pre-order are the same thing as etal pre-orders over a pre-order and shifts of our J etal shifts, generalizing a result by this article by Hameler. And with this, I think I've used all my time.
24:44
So I thank you for your attention. Okay, thank you, Ricardo.