 So I am happy to introduce Jens Ebler from Anwerp University, who is going to talk about the opposites of fashions on monoids as generalized topological spaces. So it's up to you. Thank you. Thank you for all the organizers for putting together a wonderful conference. And for the invitation. So I will talk about toposas of pre-shifts on monoids as generalized topological spaces. So we will really try to see how far we can push the analogy between toposas in this very special case and topological spaces. The talk will be based on joint work in progress with Morgan Rogers, and mostly the second part will be also based on joint work in progress with Adelian Sagny. So the first part will be about ethyl geometric morphisms and their dual, the complete spreads. So we will look at this in a special case of pre-shifts on monoids, which is a joint work with Morgan Rogers. And we will build on really a very large theory that is worked out by Bona and Funk. So this is something we already saw earlier in the school already and in the conference, is that the sheave on a topological space can be equivalently described by a local homeomorphism to this space. So here is some reminder of the notion of local homeomorphism. Say that F is a local homeomorphism if you have an open covering of the source to domain space, such that the restriction of this map to the open sets are homeomorphisms onto open sets for all these open sets. So here the condition that it is onto an open set is a bit tricky, but it is as necessary to include this that it is onto an open set. And we will use instead of local homeomorphism, we will use the terminology ethyl. And E will be called the ethyl space associated to the sheave. So here are some examples of local homeomorphisms to the circle. So these can be thought of as sheaves. So you can take an open interval of the circle that's the inclusion map is local homeomorphism. You can take the circle with double points. That's like a non-houser space similar to the line with double points. I mean, there's a circle and you take the projection to the circle that's also a local homeomorphism. And very well known local homeomorphism is the covering map from R to the circle here. So this gives a geometric interpretation of an object to an object of the topos. So this is a little bit further. And think of the sheave as not as an ethyl map of topological spaces but as an ethyl geometric morphism between the associated topos. And here we can say that an ethyl geometric morphism to this topos is precisely the ones to geometric morphisms that are induced by the local homeomorphisms of spaces. So this is an alternative characterization that can be extended to other topos. So the fundamental theorem of topos theory says that if E is a topos and X is an object in E, so X is a bit like a sheave. Then the comma category of E over X is a topos as well. So this category has as objects the maps from Y to X. They are for X to fixed object. And the morphisms are, as you might expect, they are the morphisms from Y to Y prime, making the diagram commute. So it's a bit, for me it was a bit strange to call this the fundamental theorem of topos theory at first. I don't construct new toposes but it was not clear to me what was so fundamental about it. But we will see later why it is important. So here is again the definition of geometric morphism is ethyl if and only if there is some object X in E. The domain topos F is equivalent to the slice topos over X. And well the map F is just agrees with the projection map by X here. So this is an isomorphism. So it really says that ethyl geometric morphisms are exactly the projection maps from the slice topos. And this is a description of what is a projection looks like. You can describe it by describing the inverse image functor. And this is exactly taking the product with X. And the projection map to X is the projection on the first component. So why are these slice topos as important. In addition there is a projection between objects of the topos and ethyl geometric morphisms from with the same code domain E. So these ethyl geometric morphisms are taken up to natural isomorphism. And this extends to an equivalence of categories. And the category of maps to the topos E, but the category of ethyl maps to the topos E. So I wanted to be sure about his results. So I looked it up in the literature with the help of Morgan. And I think at the moment, the only result that I found where only place where that I found where it is stated is in the literature on infinity topos. So there's a proof by David Garcetti. Using the result from infinity toposes and then truncating it to a result about one toposes. But I think now from the talk by Ricardo Sanza about the joint work with Olivia Caramello. It's not clear that this doesn't use infinity toposes and you also have a proof really just in one toposphere. So what's important is that here the category on the right hand side. It should be a two category, but it turns out that there are no natural transformations or no non trivial natural transformations. So it is it can be thought of as a one category and it is exactly the topos E. So how does this work for a monowit? We now look at the topos of pre-sheafs on a monowit N. And this topos of pre-sheafs is exactly the category of rights and sets. So if N is commutative, you don't have to mention that it's right and sets, but for non commutative monowits it's important. And so I use the letter M because it will be thought of as a co domain topos so we can keep the letter M for the domain. And also because a natural example is the natural numbers under addition. And then the topos that you get is the topos of different sets that Ivan Tomasic talked about earlier this week. So you can keep in mind, for example, and the natural numbers under addition. So at all topos is over this topos correspond to right and sets. And now how can we draw this? So we want to do geometry so we make pictures. And we take this special example of the natural numbers under addition and the base space we will write as this. It's one element and the action of the monowits is x by the successor and it acts trivially on this point. And this we can think of this as the base space. And then an adult topos over it is another set with an action. And for example, we can take two elements and one element is sense to this element is sense to the other one. And this other one is sent to itself. So I really want to stress that these pictures should be taken seriously and really they they give a lot of information about of the geometry of the topos. For example, if you will look at the double cover of the original space, then what you do is you just take the same base space twice. So this is a trivial double covering. And if you want to take a non trivial double covering, then you also take two elements, but they swap. So these are two covering spaces of index two. So, and now maybe it's a good time to start making this more precise. So more formally if you have a moment with n and access the right answer. Then we define the category of elements of x with respect to n as a category with as objects the elements of of x. So that's why it's called category of elements. And as morphisms, the elements of the monowit such that here this equation knows and is equal to be. And so the morphisms are of this form here. And further there is a natural projection from the category of elements to the monowits, sending every element to the unique object in the monowits and every morphism to the element of the monowit that labels this morphism. And we see that for every element of the monowits and every element of, of x, there is a unique lift to to a map in the category of elements that goes to a And categories with this property are called discrete vibrations, or more precisely it's the property of the functor and every discrete vibration is of this form. Okay. So, and it then turns out that the ethyl geometric morphism, given by slicing over the object x is exactly the same one as the geometric morphism that is induced by the projection functor from the category of elements to n. So we really can draw this category of elements, and toposes are then just the categories of pre-sheafs on it. So this is really how you would draw the Kaley graph of the of the n set. So now what happens if we take x to be the monowit and itself with right and action given by multiplication. For example, in our example of the natural numbers in our addition you get this picture here. And notice that the arrows go in the other direction. So that's part of the more formal definition. Okay, so that's what it looks like. And this is a general construction. It's called the root topos and it was introduced by Conan Konzani in 2019. And the root topos associated to a monowit n is the topos root n which is the pre-sheafs on this category of elements of n over itself. So if we take the cancelative monowit, then the root topos is equivalent to topos of sheaves on topological space. For example, in this case, you have that the topological space associated to the root topos will be the natural numbers together with pointed infinity. And some sets will be the sets that are upwards closed and contain at least one natural number or are not are empty. So and we can also look at this for the arithmetic side. If n is not the monowit of nonzero natural numbers in our addition, then you said addition. So it's under multiplication. Then this topos. This topos is underlying topos of the arithmetic side by Conan Konzani. And what happens in this case is that it's the category of nonzero natural numbers. And the morphisms are you have exactly a morphism from M to M, whenever M divides M. And another word for this is it's also called Conway's big cell, which appears in the paper covers of the arithmetic side by my PhD supervisor. So this is also how I got involved in topos theory in the first place. And so here is a picture of this big cell. So the topology on the big cell is non-house serve, but here we take the closest house of approximation. And the red dots together give them a topological space with a subspace topology of the plane. And the blue lines give the division relations, which are also important to recover the original topology. So for example, here you see that the prime numbers they converge to one for this house of topology. So now ethyl geometric morphisms to the base topos, they correspond to right and sets. So what happens now with left and sets. So the question is, is there a dual to the notion of ethyl geometric morphism. And the answer is yes and this is studied to in many papers by Bung Bung and funk. And to get to this notion we first have to look at the notion of spreads. So what is a spread. Well, we have a growth in the topos is F and E. And the spread is geometric morphism is a spread. If the complemented sub objects of objects of the form objects are in the inverse image of an object of E. Give an generating family of F. So this is a generalization of inclusions, because for inclusions we have that every every object in F is of the form, the inverse image of an element of an object in E. So we don't even need to take complemented sub objects. So that's why every inclusion is a spread. And also, if we would change the definition and we would remove the assumption that that you take only complement sub objects, if you would take all sub objects, then you would exactly get the definition of a localec geometric morphism. So all spreads are localec. And so another example is if you have the space, one, a half a third, and so on, converging to zero. Then this space as subset of the real line is a spread. The inclusion map is a spread. And this is a spread for both the subspace topology and for the discrete topology on why. So it's a more general. So we can have a look at which of the examples of it down maps or spreads. So the open interval is a spread because it's an inclusion map. The circle with double point will not be a spread because you can take here an open interval. So suppose that you take the open set here that only contains the upper point at the origin. Then, if you want to write it as a sub objects of the inverse image of this. And what happens is that it's not a complemented sub object because the complement is not open. So it doesn't work. It's a localec map but it is not the spread. And here, if you look at real line as a covering map to the circle, then it is a spread because, well, it has a basis consisting of these things here. They are precisely a component of the inverse image of a of an open set in the base space. So there, this R is a spread. And then the notion of complete geometric morphism is a bit more complicated. So here we take F and E to be growth in the toposis with F locally connected. And if, and we start with a geometric morphism. And then we construct a site from from a site of E. And it's given by objects in E together with or objects in the site for E together with the connect components. And I won't say very much about this definition. Maybe only the intuition behind it. So the intuition is that informally covering of the inverse image always comes from a covering on the original open set in the base space. And we stand to covering upstairs to covering in the base space for for any open set in the base space. So it's a rather complicated definition maybe, but it's already a special case of the definition by Boone and funk, because they give a more general definition for toposis over an arbitrary base topos. They are just working with growth in the toposis over the topos of sets as base topos. So if you want to have a very general definition then you have to look at the work of Boone and funk. So for example, their book sing singular coverings of the toposis. And then you have to be more careful with the definition. So the covering upstairs comes from a covering on the base space. So now we can see that that another one of our down maps will not be complete. So it was a spread, the open interval, the inclusion of the open interval was a spread but it won't be a complete spread. The reason is, suppose you take open set here and open interval, and then the pre image will be something like this, which isn't also open. And we can cover this by open sets that do not contain in some sense the point that this removed. So we have to extend this covering to covering on the base space, because on the base space. So here we all always have an open set containing this point that is missing upstairs. And so we will have here a half open interval happening here. So that's a bit intuition behind it. So the circle with double point was already not the spread so it won't be complete spread either. And the real line the covering map will be a complete spread. So and there is a process of completing spreads. So every geometric morphism with a locally connected domain can be factorized as a pure geometric morphism, followed by complete spread. And if you apply this to spread stand in particular a spread factorizes as a pure spread followed by a complete spread. So we have here. This was a spread to open interval. And then if you look at the completion. Then you will get the closed interval. So we will add two points there. And you can do the same with the circle with just one point removed. But then you will see that you get a point added on on both sides. So on the left and on the right. So you get again a closed interval. And there are two points lying above the same point of the circle. So the This notion is also inspired by notion in topology of a cut. And therefore, for example, you can look at the piece of paper and model this as a compact house or space in the clean space in our free. So this is a compact house or space, but then you do it in two. Then you still should get two compact house or spaces. So in some sense we have imagined adding here a boundary points on both pieces of paper. So that's the same thing that that happens in this situation. And so what if if ital geometric morphisms correspond to objects of the topos then what do complete spreads correspond to. And they correspond to some notion by love here. And the definition of love here is that the distribution is a co limit preserving functor from the topos to the topos of sets. And one thing that you can notice here is that an object of the topos is the same as a co limit preserving functor from the topos of sets to the topos. So distributions are already your dual to objects. And what is called by bone and funk is that there is a correspondence between distributions on the topos E and complete press that have E as a code domain. So what does this look like in our case. If we have the topos of pre chiefs and distributions are the same thing as functors from C to the category of sets. If he is pre chiefs on the moment with then distributions correspond to left M sets. And now we should describe what the associated complete spread is of this distribution, or in other words of this left M sets. We arrive at the category of elements but there are two categories of elements there is one for covariant functors and one for contra variant functors. And they are different in some sense. For N monowitz and why left and sets. We define the category of elements of why, as the category with as objects elements of why. And again as morphisms, the elements and such that and a is equal to be. And a bit different than the other category of elements and to distinguish them. We put here and on top of the integral side, and for the other category of elements we take it below the integral side. So, so the morphisms all look like this. And there is again some natural projection from this category of elements to N. And for every element of the monowitz and every a element of the set. There's a unique lift to a map starting in a in going somewhere else. And this factor is called the discrete of vibration, and every discrete of vibration is of this form. So distributions correspond to discrete of vibrations, while pre chiefs correspond to discrete vibrations. A complete spread associate to the left and set why is then this geometric morphism induced by this projection map or projection functor from the category of elements to N. And one thing we can do now is also define a dual to the root topos, which Morgan and I call the code root topos. And the definition is the same, but you take the category of elements of N as a left and set rather than as a, as a right and set. And so here I call roots, the core root topos of N, the map to the base topos will be a complete spreads. Whereas for the root topos the map to the base topos is a towel. And what are now the points of the, of the pre chief topos on this category of elements. So you can use the pure complete spread factorization by bone and funk. So if you have a point of the, of the base topos, the pre chiefs on N. So the points by the Oconasios theorem are exactly the flat left and sets. So we call this a flat left and sets a. And this point factorizes to the associated complete spread so the completion of this point so to speak, which is this topos, the complete spread associated to a, and then. We need to factorization to topos here because we are looking at the points of the topos there. And here this dashed arrow is exactly given by a map from a to y, which is a map of left and sets. So in this way you can show that the points of these topos on the category of elements for the left and sets y are precisely the flat left and sets together with the morphism from this flat left and set to y to y. So this connects back to the talk by Axel Osmo. So about overtoposis at the model. So what's a boom and fun show is that the complete spread associated to a to a point. So a flat distribution. Give exactly the category of points. They are exactly the overtoposis this model. So this is only with models in with values in the category of sets. If you want to look at models in in other toposes, then I, I don't know whether this still works. I don't know if there is something similar that you can do. Okay, so the points are left and sets a together with the map from a to y. And here why doesn't have to be a flat left and set it works for a general left and sets. Okay, so the first two examples were not complete spreads. Covering map by the real line was a complete spread. So it's both et al and a complete spread. And this is not really a coincidence. Because in our situation, if a geometric morphism is both a complete spread and et al, then it is a covering map. So what is covering map mean here is that you have a it is a morphism from F to E, and you have the fundamental group of the topos. And you can take a subgroup of the fundamental group and then take this pullback diagram. And then F is exactly the pullback of this morphism here that you get from the inclusion of H into the fundamental group. So this is exactly it works in exactly the same way as for topological spaces you have the correspondence between subgroups of the subgroups of the fundamental group and connected covering spaces. And it works here exactly in the same way in some special cases, namely, if, if the base topos E is a pre sheaves on a moment. And this was shown by Bonhe and funk, or more generally for connected pre sheaf toposis. And it also works for for either topos of sheaves on a house door of second countable connected topological manifold. And this was shown by funk and team shot in. It's not been very precise here. So the, the ones corresponding to subgroups will be exactly the connected covering maps, but you also have a non connected covering maps, and they correspond to sets with an action of the fundamental group. But, so this is corresponding to the set. There's a quotient set. So, how does this with the fundamental group work for pre sheaf toposes on a monolith. Well, to get the topos of pre sheaves on the fundamental group. You need to look at the locally constant objects. Or more generally, if you are in more general situations you may have to look at sums of locally constant objects but that's not necessary here. So you can look at the locally constant objects and these are precisely the right and sets on which and x by bijections. And the full subcategory on the locally constant objects is again a topos, and it's equivalent to the pre sheafs on the group G, which is a groupification of and so the closest group to the moderate and if x is such a locally constant right and set, then we can also define a left and action by just defining it as multiplication on the right but first taking the inverse of the element. So multiplication by the inverse elements. And in this case we get that for this left and action, the two. The categories of elements agree. So we will also get the same geometric morphism, which will be both a complete spread and et al. Okay, so let the end now be a monolith and extra right and sets. And we want to look at the geometric morphism associated to this right and sets. We want to classify et al geometric morphisms in the sense that we want to end up again with a topos of pre sheaves on the monolith. So when is it again of the form pre sheaves on the moment. And this is precisely when in the category of elements there is an element such that every other element is a retract of this one. And you can write this out algebraically. As follows, there is an element X subject for every other other element Y in X. There is some element you such that X you is equal to Y. And here you is an element that is a has a right inverse. So we use this notation to say that it is only a right inverse and not necessarily a left inverse. And then in this case if such element exists such that all other elements are retracts of it. Then, for this element X we get that the topos of pre sheaves on this category is the same as a topos of pre sheaves on the monolith and X, where, and X is the monolith and the morphisms of this element, which is exactly an element of M that fix X. So now let N be a monolith and why a left and set, and we can do exactly the same thing for complete spreads rather than a town maps. And the calculation is exactly the same. So this time, there must be an element Y in Y such that other elements are retracts of it. And in this kind of category of elements. This means that there is some element V with a left inverse such that Y is equal to V X. And again, we can then describe this pre sheaf topos as a pre topos of pre sheaves on the monolith. And the monolith is again the elements that fix Y. This doesn't really give a method of saying for a monolith map. Phi, whether it is at all or not because it starts from the base topos and then you construct a topos is over it. But say that you start with the monolith map from M to M, and you want to know whether it induces and it out geometric morphism. So I found some criteria to determine whether it is a tau. So Phi must be injective. If you have elements A and AB that are in the image then B must also be in the image. And for every element of N, there must be some right invertible element U such that N times U is in the image. And for example, you can check that these conditions are all satisfied. If you take the map from the natural numbers to the biotic numbers that sends N to P to the N. So this induces an ethyl geometric morphism. And the picture for complete spreads is exactly dual. You can work with elements that have left inverse, and here you can show that A is in the image if B and A times B are in the image. So now we will give some examples arising from arithmetic topos is for maximal orders that try to mimic the arithmetic side, but for certain rings that are called maximal orders. And this is joint work with Oriens Agni, but of course we focus here on the results that have to do with complete spreads in the town maps. So some background in their approach to the Riemann hypothesis, a Konen Konzani introduced in 2014 the arithmetic side. So it's a topos of pre-sheafs on the non-zero natural numbers under multiplication, equipped with a certain sheave of semi-rings. The sheave of semi-rings is very important, but still the topos of pre-sheafs is already very interesting. For example, it was computed by Konen Konzani that the points of the topos are classified up to isomorphism by the double quotient featuring the finite ideals. So here Z with a hat is the product of all over P, over all p-addict integers. It's called the profanity integers. And then if you take this profanity integers and you dancer with the rational numbers, then you get the finite ideals. So it's profanity integers, but you allow finite integers in the denominator. So to define maximal orders, we take R, a datacent domain, which is a field of fractions, a global field k. So global field is, for example, number field, which is an extension of the rational numbers. Or you can take the fraction field of a curve, a smooth curve in characteristic p, or extensions of that. So a maximal order over R is now an R algebra lambda, such that lambda is finely generated torsion free over R. And if you take the base change to the field of fractions, then you get a central simple algebra sigma over k. So lambda is a subring of the central simple algebra sigma. If suppose that R is the integers, then all finitely generated torsion free modules are free. So you really get the free module with the multiplication on it. So examples, the integers themselves are a maximal order. You can take n times n matrices over the integers. You can take polynomials in one variable over finite fields. You can take again n times n matrices over this polynomial ring. You can also look at the Gaussian integers. This is a case that was already studied by R&S Agni very early on. So as a general to generalize arithmetic side to by replacing the integers by the Gaussian integers. So we can also look at another example of maximal orders is the Hurwitz quaternions. And I gave a definition here. So they are the quaternions with such that the coefficients are either integers, or they are integers plus one half. So in the central maximal order we write the non-singular elements are exactly the elements that become invertible in the central simple algebra. And can we, some questions are can we associate an arithmetic side to maximal orders. For example, as an underlying topos is it a good idea to take the pre-sheafs on the non-singular elements of the monowitz, or do we need a different topos. We write structure sheave of semi-rings on this topos to be and maybe do we get a spectral interpretation of the zeta function of the maximal order in topos theoretic terms. So the spectral interpretation is already done. In relation to the arithmetic side by Konin Konzani, but it would be nice to have a spectral interpretation also for general maximal orders, but this this topic is needs work and will not be. I will not talk about this already in this talk. So if we are being very precise then the arithmetic side will not be a special case of this process of looking at maximal orders, because the closest approximation to the arithmetic side is a pre-sheafs on the non-singular integers. So how can we recover this, the arithmetic side from this. So the answer is we look at the following set with an action of CNS. So minus one acts, so you have two elements and minus one switches the two elements. And if you have a positive natural number, then it acts trivially on both elements. So this is a set where every element acts by bijections. So we are in a good situation where we have both a compete spread and an entailment at the same time. So and we can compute that this category of elements is equivalent to the category of the monowitz of non-zero natural numbers. So yeah, this is because minus one is an automorphism between the two objects. And in this way we get that the arithmetic side or rather the underlying topos is a twofold covering space of this topos that is more generally as a seed to the maximal order C. So we get this idea for more covering spaces. We can look at the maximal order lambda and look at the profanity elements. So we look at the pediatric completion at each eight one prime ideal and take the product over all the primes. And these are the profanity elements and then to define the lambda adels you again tensor with a fraction field. And then we can look again at the non-single elements which are elements that become invertible in the adel ring. And this the moment with is more complicated the moment with the profanity integers in some sense, but the topos is nicely behaved. So the root topos for example is a spectral space so it means that this coherent topos, and this is not necessarily true for the topos of pre-sheafs on the moment with lambda itself. For example for lambda equal to this ring it already fails because it does not have unique factorization and then you there is a problem and it is not a coherent topos anymore. So to go from this completion to the original maximal order, we can look at the right set, the set of the finite adels. So the adels depend on lambda to take the units and you quotient out by the units in sigma. So I use notation adels with sigma below it, but it does depend on lambda. So I mean the same thing. And the action is given by multiplication on the right. And again this is an action by bijection so we have a covering space. And then we are in a situation where we have a covering map so both a complete spread and a town map, and the associated category is this category of elements. So they both coincide. And can we give a more concrete description. Well, in the category of elements, two objects are isomorphic if they are related by a unit. A unit in the profanate profanate elements of the maximum order. And that's exactly because multiplication in this monolith in the category of elements is multiplication in the monolith. So the isomorphisms in the category of elements will exactly be the units in the monolith. And so the objects of the category of elements are classified up to isomorphism by the double quotient. Like this. And this is exactly the ideal clause group. So you get that. If you look at the objects of this category of elements up to isomorphism, then the number of objects is exactly the number of elements in the ideal clause group. And there's also a special elements to the clause of one. And the anamorphism monolith of this element is exactly the original maximum order lambda, the non-singular elements. And other elements you can also look at anamorphism monoliths of other elements, and they can have anamorphism monolith that is a non-singular elements of another maximum order lambda prime. So this gives one application of this covering maps. And so here is another application. So we will restrict to the case where lambda is integers to keep it a bit more simple. And the points of this, this topos is classified up to isomorphism by the double quotient involving the finite adels. And the question is now is there an alternative topos where the finite adels get replaced by the full ring of idea adels. So full ring of adels to get that you have to multiply the finite adels with real numbers. And the answer is yes. So this was already done in 2015 by Konen Konzani by looking at the scaling site. So I won't talk about the scaling site here, but I will try to describe a scaling site where we take the discrete topology on the real numbers. So in the scaling site, you really have topology on the real numbers, the usual Euclidean topology, which is relevant to define the scaling site. And then we try to forget the topology on the real numbers to get a much more simplified picture. And so how we do this is by looking at real numbers as a set on the mill under with an action of a ZNS to no singular integers. And then we then look at the associated covering map. So why is this a covering map exactly because the action is again by bijection so both a complete spread and an entire map. And the points of this topos. So I as I mentioned earlier it is an overtopos. So the points of this topos are given by flat sets. A flat left ZNS sets A together with a map from A to R. And this must be a map of left ZNS sets. And you can use this description as an overtopos at the model. To show that this the points are then classified to isomorphism by this double quotient, where you have the full ring of adels rather than just finite adels. And another thing we can do is, well, if we have just the real numbers with this action, then we can decompose this, this ZNS sets into as a co product of smaller ZNS sets. So you can, you have the element zero, which is a stable and their multiplication with element of ZNS. And you have the non zero real numbers which are also a sub ZNS set. And zero here the action of zero is trivial. So we really get an a copy. So this is really the terminal object. And this means also that this pre chief topos contains a copy of of the original pre chiefs on ZNS. So it's a covering map which is really a disjoint union of smaller toposes. And one of the smaller toposes is the original base topos pre chiefs on ZNS. And this corresponds to the decomposition of the double quotients, which is also considered in the article on the scaling side by Conan Konzani, which is that you first take the finite adels, and the, the component at the real zero, and then you look at the case where the component at the, or the entry at the real numbers is non zero. And more precisely there is a decomposition of discovering space as a really very much more smaller parts. So it's really disjoint union of toposes. So we already said that that there is a copy of the original base topos and then you have also coverings corresponding to subsets, given by the non zero rationals. And hopefully this gives a bit simplified picture of what is happening with the scaling site. So, thank you very much for your attention. So, thank you very much for your talk.