Toposes of Topological Monoid Actions
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Toposes online, 20219 / 31
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Formation <Mathematik>GrundraumOrtsoperatorMonoidGruppenoperationTopologieBesprechung/Interview
00:37
GruppenoperationTopostheorieWechselsprung
01:01
Mengentheoretische TopologieMonoidGruppenoperationTopostheorieGeometriePunktrechnungBeweistheorieMorphismusTopologieSurjektivitätHyperebeneEinfach zusammenhängender RaumPotenzmengeFunktorGibbs-VerteilungMengenlehreTheoremMultiplikationObjekt <Kategorie>Kompakter RaumStetige FunktionHauptidealGrothendieck-TopologieIndexberechnungMonoidGruppenoperationTopologieAnalytische FortsetzungTopostheorieKategorie <Mathematik>MengenlehreOrdnungsreduktionProzess <Physik>ÄquivalenzklasseMereologieZweiUntergruppeNichtunterscheidbarkeitElement <Gruppentheorie>MomentenproblemQuotientKlasse <Mathematik>Natürliche ZahlResultanteTransformation <Mathematik>Monade <Mathematik>Rechter WinkelEndomorphismenmonoidRadikal <Mathematik>KongruenzuntergruppeObjekt <Kategorie>EinsInverseRauschenGruppendarstellungZusammenhängender GraphArithmetisches MittelModulformOrdinalzahlFinitismusTeilmengeAutomorphismusKonditionszahlKlassische PhysikTeilbarkeitFunktionalEinfügungsdämpfungModallogikOrdnung <Mathematik>TabelleInverser LimesArithmetischer AusdruckRichtungInklusion <Mathematik>Projektive EbeneZeitrichtungTermNichtlinearer OperatorProdukt <Mathematik>Kompakter RaumPunktVollständigkeitGanze ZahlIndexberechnungMultiplikationFrequenzHauptidealGibbs-VerteilungMorphismusFamilie <Mathematik>Leistung <Physik>AdditionPrimidealMultiplikationsoperatorHomomorphismusPaarvergleichAnalogieschlussIsomorphieklasseExistenzsatzAussage <Mathematik>Grothendieck-TopologieFunktorNichtlineares GleichungssystemReelle ZahlStützpunkt <Mathematik>SurjektivitätUnrundheitGeradeDualitätstheorieParametersystemTheoremBeweistheoriesinc-FunktionKoalgebraPhysikalische TheorieRechenschieberVarietät <Mathematik>Offene MengeDiskrete GruppeTopologische GruppeTopologischer RaumGruppentheorieFormfaktorGegenbeispielQuotientenraumPotenzmengeEuklidischer RaumProduktraumUmwandlungsenthalpieExogene VariableHyperbelverfahrenBasis <Mathematik>Sortierte LogikComputeranimation
Transkript: English(automatisch erzeugt)
00:16
So it is my pleasure to introduce Morgan Rogers from University of Insobriar at in Como,
00:30
and he will talk about toposis of topological minority actions. Thank you very much. So thank you very much to the organizers for accepting my talk.
00:46
And thanks for a really great conference. I'm looking forward to the last few talks after this. So since we've seen plenty of topos theory during the school and in the talk so far, let's jump straight into it. So first, I'm going to tell you about toposis of group actions.
01:06
So we'll start with discrete groups. So if I have a discrete group G, I can consider actions of that group on sets. And so classically, we think of an action of a group as an operation
01:21
that goes from the product of my set A with G to A, which respects the group operations. Which is, you know, a nice alternative way to think of them once we're used to thinking of them as pre-sheaves. So I can think of my group G as a category with a single object.
01:45
And then an action, specifically a right action, determines a contra-variant functor from the G-intercepts. Now, because groups are self-dual, it doesn't matter whether I treat it so much, whether I treat this as a functor defined on G
02:02
or G-op. But the important thing is that this is a topos and the kind of topos that we're quite familiar with. So now suppose that we equip G with a topology tau. Note that I'm saying it this way around. I'm not thinking of G as necessarily a topological group,
02:21
which is to say a group in the category of topological spaces. I'm again working classically and equipping G with a topology. And in particular, this topology doesn't have to make G into a topological group, because all I'm going to use this topology for is for a continuity condition on G actions. So if I take my G action, I can put the discrete topology on
02:47
my set, I have my topology on G, and I'm just looking at whether the action is continuous with respect to the product of those topologies. So because that's just a condition, it defines some full subcategory of my category of right G actions, which therefore comes with a
03:08
functor. And this functor has a lot of nice properties. So it's left exact, it's closed on the sub object, which is to say, any subset of a continuous, any sort of, sorry, a sub action of a continuous G action is still continuous,
03:24
and it has a right adjoint. So the left exactness comes from the fact that when I take a product of continuous G sets, then when I need to look at an open map in the inverse image, I can take the opens corresponding to the G sets individually, and I can take their intersection.
03:45
And so I can verify the continuity condition for the product. And checking that sub actions of continuous actions are continuous is quite straightforward. So all I need to really spell out is the adjoint. And this is typically presented like this. So it sends for my G set X,
04:06
it sends it to the collection of elements whose stabilizer subgroup of G is open in the topology. So I can just express it as a conditional open subgroup.
04:20
So from the existence of this functor with all these properties, I can deduce a variety of properties of this category. So first and foremost, that it's a top-ups. So I've shown that V is left exact. I don't know why the and is here, but because it also has a right adjoint, that's more than enough to make sure it's kominadic. And it's quite a classical
04:44
result that a category of co-algebras for a kominad on a top-ups is still a top-ups. But elementary top-ups theory, we obviously want more than that. So we can also check that it's a great indeed top-ups. And since I'll be mirroring this argument later,
05:01
I'll spell this out. So obviously, we've already shown that it's a top-up, an elementary top-ups. So we can think of V and its right adjoint R as a geometric morphism. So now what we do is we take the representables in our pre-sheaves on G, and we consider their
05:23
quotients. And the reason we consider their quotients is because if I take any continuous G action X, then it's covered by representables. That is, there's a jointly epimorphic family from representables. And I can factor each morphism in that family via its epimono factorization.
05:43
But remember that this subcategory is closed under subobjects, which means that these S's, this S and S primed, will actually be contained in the subcategory of continuous actions. But because any top-ups is well co-powered, there's only a set of quotients of each
06:04
representable up to isomorphism. And then this gives me an indexing set. And so basically, I'm taking a subset of the quotients of representables. And those are automatically generating. We also have that the top-ups has all of the required properties for
06:24
Giraud's characterization of great and deep top-ups' inherited via this chromatic functor. And so everything works out. It's a great leap. We can go further. We can look at special properties of these top-ups. So remember that an atomic growth and leap top-ups is a top-ups
06:44
which has enough atoms. So an object of a top-ups is called an atom if it has no non-trivial sub-objects. So it's only sub-objects are itself and the initial object. And if I look at the quotients of representables in my pre-shift top-ups,
07:02
then those will exactly be the transitive g-actions, which in particular, I can't take any sub-action, which is not trivial because the action is transitive. And so I get all the elements back or no amounts at all. And because they're still atoms in this subcategory,
07:22
we have a separating collection of atoms for that subcategory and that's what happens. So the final property that I want to mention is that this top-ups has a rather special point. So if I look at my category of actions for the discrete group, then it has a forgetful
07:43
function to set. That's the inverse image of a geometric morphism, which is a point of this, but it's more of an essential point because this forgetful factor has a left adjoint. And then we have this hyper-connected morphism, which I've just constructed,
08:00
but which is quite well known. I'm not laying any claim to that, but I mean, it's a property of this top-ups that it has a point, the factors in this way. And when we get around to monoids, this will be important, which we're going to do now. So once again, if I take a monoid, I can consider it as a category with a single object
08:23
and its right actions form appreciative top-ups just as before. And these top-ups are actually characterized by the existence of that essential point. So it's also a surjective point because this inverse image functor is faithful. Any mapping between M sets is a mapping between the
08:45
underlying sets. So if I have an essential surjective point of a top-ups, then I can recover a monoid which represents it. And there are a couple of ways of doing that. The first is to look at what happens to the terminal object of sets under this monad here.
09:03
And the second is to look at the endomorphisms of this point. And that second way of constructing a monoid will be important later. So now we equip M with a topology. Once again, we do require the result to be a topological monoid. And we can once again consider the subcategory of our appreciative top-ups
09:25
on these continuous actions. Now you'll note that I wrote the arrow both earlier and now in this direction. So because this has a right adjoint, this is not an inclusion of top-ups. It's the geometric morphism that we end up with is a geometric morphism
09:42
going from left to right and not from right to left. So because I keep talking about it as a subcategory, that might be a point of confusion. But what we end up with is a connected, in fact, a hyperconnected morphism. So I've kind of jumped the gun a bit because I still need to prove that this actually has an adjoint. So the first part of the proof is identical.
10:04
Left exactness is quite straightforward. The issue is showing, is constructing this right adjoint because we can no longer just take stabilizer submonoids because for a lot of monoids, stabilizing submonoids are going to be trivial.
10:25
So what we do is we consider what I'm going to call necessary clopements. So if I have a right n set x and an element x in that n set, then the m action is continuous about element.
10:40
If and only if these sets which partition the monoid m are all open. So I need to not only look at what happens for the stabilizer submonoid, so to speak, when m is the identity element, I also need to look at all images of x under the action. So I can equivalently express that
11:06
as the requirement that this congruence, this right congruence of pairs which act in the same way at this element x is open in the product topology of tau itself. So finally, I need
11:22
in order for x to be in the sub action, which I'm going to define, I need the action to be continuous, not only at x, but at the image of x under the action of m. And so if I define this, I can check that the actual ness properties are satisfied. And so I have two different ways,
11:41
both in terms of necessary clopements and in terms of these open congruences to describe this right action. But I mean, it's the results of the proposition that's important as far as understanding the results that will come next. So before I re-express the analogs of the results
12:00
in the group case, I need to tell you what the thing corresponding to atomicity is. So we saw the groups give us atomic topos, but that's not true for monoids and we need to see what the replacement is. So an object in a topos is called super compact.
12:20
If whenever I have a jointly epic family over that object, then one of the morphisms in that family is epic. And in particular, if I have a pre-sheathed topos, then the quotients of representables are exactly the super compact objects. So representables are super compact, but also like questions are, this is one of the generalized compactness properties that Olivia
12:44
mentioned in passing in one of her lectures. So topos is said to be super compactly generated if it has enough of those. So every object is covered by super compact objects. And in particular, because these are closed under quotients, it's enough to talk about super
13:03
compact sub-objects. So skipping straight to the final result by exactly analogous arguments to what we saw in the group case, this topos of continuous actions for M with respect to topology tau is a super compactly generated grittendy topos. And it once again has a point of this
13:25
form. Okay. So having gotten that far, we've learned something about these categories in particular that they have all these properties. We start to want to use these monoids with
13:43
topologies as the basis of bridges in the style of Olivia Caramello. So we want to be able to identify when a topos can be expressed as the topos of continuous actions for some monoid equipped with a topology. So I'm just going to present some questions that we might ask
14:06
on the way to getting there. The first is suppose I have this set up, but I don't know what tau is. I only know that it's a topos of act of there's a category of continuous actions for some
14:21
topology. Can I recover that topology? And immediately I can say the answer is no. So, for example, if I take the real numbers here, the continuous actions of the real numbers on sets are rather boring because it being a connected monoid, it has to act trivially on all of the
14:42
elements of the set. And so if I equip R with this usual topology, then I just get sets here. The resulting category will be equivalent to the category of sets. And so the best I can hope to do is recover the indiscrete topology in that case. Or, you know, I can't tell which topology
15:02
I've started with anywhere between the ordinary topology on the reals and the indiscrete topology. But there is a best answer. And to understand what it is, we need to look at some other objects in the topos than the equations of representables. So I can consider
15:22
left actions of the monoid. We saw a bunch about those in the answers talk. And if I take any left action and I apply the power set functor, this is the ordinary power set functor, I take a set, I take its collection of subsets, then actually I get a right action on that power set. And specifically in this case, if I take my left action to be the action of M
15:47
on itself, I get a right action of M on the power set of M, which acts by inverse images. So if I take any subset and I take an element of M, I can ask which elements of my action
16:03
are mapped into that subset by M. So because I have a power set of M here, and obviously any topology on M is some collection of subsets of M, so they all have, they all exist as elements in here, naively one might hope that we can recover a topology from this object somehow.
16:24
And we can. So here's how we do it. We consider the object T, which is the image of this power set of M under the comonad induced by that hyper-connective morphism that we saw earlier.
16:44
And so this turns out to be a base of clopin sets for the coarsest topology, tau tilde on M, which gives us the same category of actions. And so I should, since I haven't put the proof in the slides, I should give a word to how this works. And basically it's the fact that
17:04
if I look at the necessary clopins that we saw earlier as elements of this power set monoid, they interact really well with the inverse image action. And so in particular, if I look at what, when an element A of the power set of M is continuous, I can decompose A into the union
17:26
over its elements of the image of A under the action of those, under the inverse image action of those elements. And so we get this nice interaction, which allows us to show that whenever something is continuous with respect to this topology, it has to be continuous with
17:43
respect to the original topology and vice versa. Moreover, what we end up with when we equip M with this new topology tau tilde is a topological monoid, even if the original data that we had wasn't. So if you were worried about the fact that I was working classically and not considering
18:03
a topological monoid in the first place, then you didn't have worried because with any monoid equipped with a topology is canonically more or equivalent to a topological monoid. And we can do even a bit better than that. So if I take the Kolmogorov quotient of my monoid,
18:23
which is to say I identify any elements which can't be distinguished by the topology, then I still have a valid topological monoid and it produces the same actions. So if I call one of
18:42
these canonical topologies an action topology, and that makes sense as a definition because the construction is item potent by nature, then I can replace my original monoid with the monoid with this action topology and then take the Kolmogorov quotient and I get a zero-dimensional Hausdorff monoid. So I get a very nice, or at least I get a monoid
19:10
which falls into quite a specific class. And I call these power monoids. They're not entirely characterized by the fact that they are Hausdorff and zero-dimensional, but those are some nice
19:21
properties of them. And so, you know, another way of saying that is that any topological monoid or any monoid with a topology is discrete action or equivalent to a power monoid. And here I'm saying discrete action or equivalent because it's more difficult to consider topological monoids acting on topological spaces. And so you need to be careful about the
19:43
specific more or equivalent that you're talking about when you're discussing this with people. So here are some examples. I'll skip straight to pro-discrete monoids and groups being examples. Those are quite nice. If you look in sketches of an elephant, the process that I've just
20:04
described amounts to the reduction of a group to what Johnson calls a nearly discrete group, where the intersection of all the open subgroups gives the identity, and plus there's a second condition. And here is the kind of classical example that
20:26
gives us the shannual topos with the automorphism of the natural numbers, but where the topology is taken in terms of the stabilizers of finite subsets. I'm just going to speed on because I have another question I want to cover, which is, suppose I have a topos of this form, that is, I have a topos and I know that it omits a point
20:44
of this form without loss of generality, I can put this monoid in the middle here, because like I said, the existence of an essential surjective point characterizes these. But the question is, if I have a hyper-connected geometric morphism from a topos of discrete
21:01
monoid actions, what can I say about this topos? I mean, is it the topos of actions of M with some topology? The answer to that turns out to be no, but we can find a topological monoid which represents it. So, remember the super-compact objects which generate this topos,
21:20
because they're the quotients of the representables in the pre-sheaf topos, they are exactly the principal M sets with a single generator, and specifically they're the ones which are continuous with respect to tau. So, therefore, we get a sight of those objects, and we can index those objects by
21:42
the open right congruences on M with respect to the topology, so the right congruences which are open with respect to tau. So, what we're going to try and do is reconstruct a representing monoid for E from the endomorphisms of
22:04
this canonical point, or this point that we're assuming exists. So, what we have to do is reduce the data of such an endomorphism to the data of the components of natural transformations defined on that generating subset,
22:24
and we can reconstruct a monoid by looking at the underlying sets of these principal actions indexed by the right congruences, and the reason we need to index by the right congruences is so that we have this ordering which we can take the
22:42
projective limit with respect to. So, actually, the multiplication on L is inherited from M just because the naturality conditions ensure that any endomorphism of this point has to interact well with the images under the quotient maps of elements of M.
23:05
So, I mean, that's where this expression comes from. It's essentially compressing the data of endomorphisms into a form that can be expressed in terms of these quotients, and if we equip this with the pro-discrete topology that comes from the expression of L as this limit, then we get
23:27
a topological monoid which represents this topos. So, a word of caution here, it's a pro-discrete topology, but this is not a pro-discrete monoid in general, which is quite an important distinction to make.
23:42
So, because we have this representation theorem, and we know that any topos of continuous actions of a monoid has a point of this form, we arrive at this final result that a topos is equivalent to the topos of actions of a topological monoid on sets if and only if it has a point
24:03
of this form factoring as an essential surjection followed by a hyperconnected geometric morphism. So, there is a final point here to be made, which is L isn't the same as M in general,
24:20
and so we get the idea that this L is a completion of M, and indeed, whatever M was, it emits a monoid homomorphism to L, and we can say that the monoid is complete if that's an isomorphism, and then we get a nice characterization of pallet monoids as those for which this comparison monoid homomorphism is injected. So, obviously, I haven't presented
24:47
that many specific examples. So, here is a counterexample which Olivia gave me at the time when I thought the pallet monoid should coincide with complete monoids. If we take the integers with addition, and I equip them with the topology having these
25:05
prime power subgroups as a base of opens, then they form a nearly discrete group, which is a special case of a pallet monoid, which I mentioned earlier, but it's not complete. Indeed, its completion is the group of periodic integers.
25:21
So, it's nice that this completion process does give an intuitive notion of completion, at least in this example. I have some further highlights, but I think I'm out of time. So, I'm just going to stop there, and thank you very much. Here are my references.
25:44
Okay, thank you for your talk.
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