 So it is my pleasure to introduce Morgan Rogers from the University of Insubriya in Como and he will talk about the opposites of topological manoid actions. Thank you very much. So thank you very much to the organisers for accepting my talk 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 toposes of group actions. So we'll start with this group 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 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-sheeps. So I can think of my group G as a category with a single object. And then an action specifically a right action determines a contra variant functor from G into sets. Now, because groups are self-dual, it doesn't matter whether I treat that so much whether I treat this as a fun to define on G 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, 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 doesn't this topology doesn't have to make G into a topological group. Because all I'm going to use this apology for is for a continuity condition on G actions. So if I take my G action, I can put the discrete apology on 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 full and faithful function. And this function has a lot of nice properties. So it's left exact. It's closed on the sub objects, which is to say any subset of a continuous, any sort of sorry sub action of a continuous G action is still continuous. 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. 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, 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 subgroups. 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 box. So I've shown that V is left exact. I don't know why the end is here. But because it also has a right adjoint, that's more than enough to make sure it's common attic. And it's quite a classical result that a category of cow algebras for a common ad on a top boss is still a top boss. But that's elementary top of theory. We obviously want more than that. So we can also check that it's a grid indeed top boss. And since I'll be mirroring this argument later, I'll spell this out. So obviously, we've already shown that it's a top, but an elementary top boss. And so we can think of V and it's right adjoint R as a geometric morphism. So now what we do is we take the representables in our pre chiefs on G, and we consider that quotients. And the reason we consider that 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. But remember that this subcategory is closed on the sub objects, which means that these S is this SNS primed will actually be contained in the subcategory of continuous actions. But because any top boss is well co powered, there's only a set of quotients of each representable up to isomorphism. Then this gives me an indexing set. And so basically I'm taking a subset of the quotients of representables, and I can use those automatically generating. We also have the top boss is has all of the required properties for zeroed characterization of grid and deep top bosses inherited via this commanding functor. And so everything works out as a great. We can go further. We can look at special properties of these top bosses. So remember that an atomic grid and deep top boss is a top boss in which has enough atoms. So an object of a top boss is called an atom. If it has no non trivial sub objects, so it's only sub objects or itself and the initial object. And if I look at the quotients of representables in my pre chief top boss, 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 elements at all. And because they're still atoms in this subcategory. We have a separating collection of atoms for that subcategory and hence for that. So the final property that I want to mention is that that this top boss has a rather special point. So if I look at my category of actions for the discrete group, then it has a forgetful function to sets. 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. But which is quite well known. I'm not laying any claim to that. But I mean, it's a property of this top boss that it has a point, the factors in this way. And then we get around to more minds this will be important, which we're going to do now. So, once again, if I take a moment, I can consider it as a category with a single object. And it's right actions for my pre chief of us just as before. And these topics 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 and sets is a mapping between the underlying sets. So if I have an essential surjective point of a top boss, 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 objective sets under this monad here. 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're preparing with a topology. Once again, we don't require the result to be a top logical monoid. And we can once again consider the subcategory of our pre chief top boss on these continuous actions. Now you'll know that I wrote the arrow both early and now in this direction. So because this has a right adjoint, this is not an inclusion of top bosses. The geometric morphism that we end up with is a geometric morphism 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 hyper connected 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. Left exactness is quite straightforward. The issue is showing is constructing this right adjunct because we can no longer just take stabilizer submonoids because for a lot of monoids. Sub stabilizing submoids are going to be trivial. So what we do is we consider what I'm going to call necessary clothes. So if I have a right answer x and an element x in that answer. And then the action is continuous about element. If and only if these sets which partition the mode M are all open. So I need to not only look at what happens for the stabilizer submoids so to speak when I miss the identity element. I also need to look at all images of X under the action. So I can equivalently express that as the requirement that this congruence this right congruence of pairs which act in the same way on at this element X is open in the product apology of towel itself. So finally, I need in order for X to be in the sub action which I'm going to define. So I need the action to be continuous not only at X, but at the image of X under the action event. And so if I define this, I can check that the adjunct properties are satisfied. And so I have two different ways both in terms of necessary clothes and in terms of these open congruences to describe this right adjunct. But I mean this is the result 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 in the group case, I need to tell you what the thing corresponding to atomicity is. So we saw that groups give us atomic toposes. 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. If whenever I have a joining epic family over that object, then one of the morphisms in that family is epic. And in particular if I have a pre-chief topos, then the quotients of representables are exactly the super compact objects. So representables are super compact, but also the quotients are. This is one of the generalized compactness properties that Olivia mentioned in passing in one of her lectures. So a 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 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 gridly topos, and it once again has a point of this 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 topologies as the basis of bridges in the style of Olivia Caramella. So we want to be able to identify when a topos can be expressed as the topos of continuous actions for some monoid, according to the topology. So I'm just going to present some questions that we might ask on the way to getting there. The first is suppose I have this setup, but I don't know what tau is. I only know that it's a topos of a category of continuous actions for some 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 elements of the set. And so if I equip R with its 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 indiscreet topology in that case. Or I can't tell which topology I started with anywhere between the ordinary topology on the reels and the indiscreet 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 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 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 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 exist as elements in here. Naively, one might hope that we can recover a topology from this object somehow. 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 co-monad induced by that hyperconnective morphism that we saw earlier. And so this turns out to be a base of cloak and sets for the courses 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 if I look at the necessary tokens 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 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 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 a topological monoid in the first place, then you didn't have worried because any monoid equipped with a topology is canonically marital equivalent to a topological monoid. And we can do even a bit better than that. So if I take the comma comma gar of quotient of my monoid, which is to say I identify any elements which can't be distinguished by this policy. And then I still have a valid topological monoid, and it produces the same actions. So if I call a one of these canonical topologies and action topology, and that makes sense as definition because the construction is by an important by nature. Then I can replace my original monoid with monoid with this action topology and then take the comma gar of quotient and I get a zero dimensional house dwarf monoid. So if I get a very nice, or at least I get a monoid which falls into quite a specific class, and I call these powder monoids, and they're not entirely characterized by the fact that they are house dwarf and zero dimensional, but those are some nice properties of them. So, you know, in a way of saying that is that any topological monoid or any monoid with a topology is discrete action marital equivalent to a powder monoid. And here I'm saying discrete action marital equivalent because it's more difficult to consider topological monoid acting on topological spaces. And so you need to be careful about the specific marital equivalents that you're talking about when you're discussing this with people. For examples, I'll skip straight to pro discreet monointing groups being examples. Those are quite nice. If you look in sketches of an elephant, the, the process that I've just described amounts to the reduction of a group to what Johnson calls a nearly discrete group where the intersection of all the opens contain of all the open subgroups gives the identity, and plus there's a second condition. And here is the kind of classical example that gives us the chanel topos with the automorphisms of the natural numbers, but where the topology is taken in terms of the stabilizers upon its upsets. I'm going to speed on because I have a 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 emits a point of this form without loss of generality I can put this monoid in the middle here. Because, like I said, the existence of a point, an essential surjective point characterizes these. But the question is, if I have a hyper connected more geometric morphism from a topos of discrete monoid actions. What can I say about this topos. I mean, is it the topos of actions of and was on 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, because they're the quotients of the representables in the pre chief topos, they are exactly the principal m sets with a single generator. And specifically they're the ones which are continuous with respect to town. Therefore, we get a site of those objects. We can index those objects by the open right congruences are all m with respect to this topology. So the right congruences, which are open with respect to town. So what we're going to try and do is reconstruct a representing monoid for funny from the endomorphisms of this canonical point right 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. 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 projective limit with respect to. 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. I mean that's 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 discreet apology that comes from the expression of L as this limit. Then we get a topological monoid, which represents this top loss. So here it's a pro discreet apology, but this is not a pro discreet monoid in general. Which is quite an important distinction to make. Because I, we have this representation theorem. And we know that any top of continuous actions of a monoid has a point of this form. We arrive at this final result that a top is equivalent to the top of actions of topological monoid on sets. If and only if it has a point of this form factoring as an essential suggestion, followed by a hyper connected geometric morphism. So there is a final point here to be made, which is L isn't the same as M in general. 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 a nice morphism. And then we get a nice characterization of pattern lines as those for which this comparison monoid homomorphism is injected. So obviously I haven't presented that many specific examples. So here is a counter example, which Olivia gave me at the time when I thought the power the monoid should coincide with complete monoids. If we take the integers with addition, and I equip them with the topology having these prime power subgroups as a base of opens, then they form a nearly discreet group, which is a special case of a power monoid which I mentioned earlier. But it's not complete. Indeed, it's completion is the group of pediatric integers. 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. Thank you very much. Here are my references. Okay, thank you for your talk.