Toposes of presheaves on monoids as generalized topological spaces
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Hemelaer, Jens
Various ideas from topology have been generalized to toposes, for example surjections and inclusions, local homeomorphisms, or the fundamental group. Another interesting concept, that is less well-known, is the notion of a complete spread, that was brought from topology to topos theory by Bunge and Funk. We will discuss these concepts in the special case of toposes of presheaves on monoids. The aim is to gain geometric intuition about things that are usually thought of as algebraic. Special attention will go to the underlying topos of the Arithmetic Site by Connes and Consani, corresponding to the monoid of nonzero natural numbers under multiplication. The topological concepts mentioned earlier will be illustrated using this topos and some of its generalizations corresponding to maximal orders. The talk will be based on joint work with Morgan Rogers and joint work with Aurélien Sagnier.
 So I am happy to introduce Jens Ebler from Anwerp University, who is going to talk about toposies of fish shaves on Monoiza as generalized topological spaces. Thank you for the organizers for putting together a wonderful conference and for the invitation. So I will talk about toposies of fish shaves on Monoiza as generalized topological spaces. So we will really try to see how far we can push the analogy between toposies in this very special case and topological spaces. So 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 Sagni. 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-Sheafs on Monowitz, which is a joint work with Morgan Rogers. And we will build on really a very large theory that is worked out by Boone and Funk. So this is something we already saw earlier in the school already and in the conference, is that the sheave on the topological space can be equivalently described by a local homeomorphism to this space. So here is a some reminder of the notion of local homeomorphism. Say that F is 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 necessary to include this that it is onto an open set. And we will use instead of local homeomorphism, we will use the terminology et al. And it will be called the etal 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. But this time it is a circle and you take the projection to the circle. There's also a local homeomorphism. And a 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. And we can take this a little bit further and think of the sheave as not as an etal map of topological spaces but as an etal geometric morphism between the associated topos. And here we can say that an etal geometric morphism to this topos is precisely the ones the geometric morphisms that are induced by the local homeomorphisms of spaces. But there 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 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. It gives a method to 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, a geometric morphism is etal if and only if there is some object x in e and this 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 etal geometric morphisms are exactly the projection maps from the slice topos. And here 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? Well by definition there is a projection between objects of the topos and etal geometric morphisms from with the same co-domain e. So these etal geometric morphisms are taken up to natural isomorphism. And this extends to an equivalence of categories between e and the category of maps to the topos e, or the category of etal maps to the topos e. So I wanted to be sure about this result. 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 that I found where it is stated is in the literature on infinity toposis. So there's a proof by David Carchetti using the result from infinity toposis and then truncating it to a result about one toposis. But I think now from the talk by Riccardo Zanfa about the joint work with Olivia Caramello, it's now clear that this doesn't use infinity toposis 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 is 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 right n sets. So if n is commutative, you don't have to mention that it's right n 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 in their 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 m the natural numbers in their addition. So at all topos is over this topos correspond to right n 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 in their addition and the base space we will write as this. So it's one element and the action of the monowits is x by the successor and it's 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 n action. For example we can take two elements and one element is sent to this element is sent 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 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 monowit n and x is the right n set then we define the category of elements of x with respect to n as a category which as objects the elements 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 of n is equal to p. 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 x there is a unique lift 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-sheaves on it. So this is really how you would draw the k-graph of the of the n-set. So now what happens if we take x to be the monowit n itself with right n action given by multiplication? Well in our example of the natural numbers under 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-sheaves on this category of elements of n over itself. And if n is left cancelative monowit then the root topos is equivalent to a 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 open sets will be the sets that are upwards closed and contain at least one natural number or are empty. So and we can also look at this for the arithmetic side. If n is now the monowit of nonzero natural numbers under addition then under multiplication you say that addition. So it's under multiplication then this topos is the 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 morphisms are you have exactly a morphism from m to n whenever m divides m. And another word for this is it's also called Konway's big cell which appears in the paper covers of the arithmetic side by my PhD supervisor Yvonne Le Brun. So this is also how I got involved in toposphere in the first place. And so here is a picture of this big cell. So the topology on the big cell is non-house dwarf but here we take the closest house dwarf approximation and the red dots together give then 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 dwarf topology. So now ethyl geometric morphisms to the base topos they correspond to right endsets. So what happens now with left endsets? So the question is is there a dual to the notion of ethyl geometric morphism and the answer is yes and this is studied in many papers by Bung Bung and Fung. And to get to this notion we first have to look at the notion of spreads. So what is a spread? Well we have growth in the toposis F and E and the spread is a 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 a generative 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 if we would change the definition and we would remove the assumption that the that you take only complement sub-objects if you would take all sub-objects then you would exactly get the definition of a locale geometric morphism. So all spreads are locale. 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. So the inclusion map is a spread and this is a spread for both the subspace topology and for the discrete topology on Y. So it's a more general. So we can have a look at which of the examples of etymaps are spreads. So the open interval is a spread because it's an inclusion map. But the circle with double point will not be a spread because you can take here an open interval. So suppose that you take that only contains the upper point at the origin. Then if you want to write it as a sub-object of the inverse image of this then 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 locale map but it is not a spread. And here if you look at the 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. And they are precisely a component of the inverse image of an open set in the base space. So 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 and dectoposys with f locally connected. And we start with a geometric morphism. And then we construct a site from a site of e. And it's given by objects in e together with or objects in a site for e together with the connected components. So 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. So you can extend the covering upstairs to covering in the base space 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 Fung because they give a more general definition for toposys over an arbitrary base topos. Whereas we are just working with growth and dectoposys 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 Fung. So for example, their book singular coverings of the toposys. 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. So the reason is, suppose you take an open set here, an open interval. And then the pre-image will be something like this, which is also open. And we can cover this by open sets that do not contain in some sense the point that is removed. But we cannot extend this covering to a covering on the base space because on the base space. So here we 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 of intuition behind it. And so the circle with double point was already not the spread. So it won't be a 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 a complete spread. And if you apply this to spreads, then in particular a spread factorizes as a pure spread followed by a complete spread. So we have here, this was a spread, the 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 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 notion is also inspired by a notion in topology of a cut. And for example, you can look at the piece of paper and model this as a compact house door space in the euclidean space in R3. So this is a compact house door space, but then if you do it in two, then you still should get two compact house door 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 happens in this situation. So what if ethyl geometric morphisms correspond to objects of the topos, then what do complete spreads correspond to? And they correspond to some notion by law of year. And the definition of law of year is that 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 e. So distributions are already here dual to objects. And the result by Bono and Funk is that there is a correspondence between distributions on the topos e and complete spreads that have e as a co-domain. So what does this look like in our case? If we have a topos of pre-shefs, then distributions are the same thing as functors from C to the category of sets. And in particular, if e is pre-shefs on the monowheats, 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 set. And again, we arrive at the category of elements, but there are two categories of elements. There is one for covariant functors and one for contravariant functors. And they are different in some sense. So for n, a monowheats, and y left n sets, we define the category of elements of y as the category with as objects elements of y. And again, as morphisms, the elements n such that n a is equal to b. So this is a bit different than the other category of elements. And to distinguish them, we put here n on top of the integral side. And for the other category of elements, we take it below the integral side. 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 monowheats and every a element of the set, there's a unique lift to a map starting in a in a going somewhere else, lifting the element n. 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-shefs correspond to discrete vibrations. So a complete spread associated to the left n set y 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 core root topos. And the definition is the same. But you take the category of elements of n as a left n set rather than as a as a right n set. And so here I call root the core root topos of n. The map to the base topos will be a complete spread. Whereas for the root topos, the map to the base topos is a tau. So what are now the points of the of the pre-shef topos on this category of elements? So you can use the pure complete spread factorization by Bono and Funk. So if you have a point of the of the base topos, the pre-shefs on n, then the points by the Okonassius theorem are exactly the flat left n sets. So we call this a flat left n set 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 a factorization to the 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 n sets. So in this way, you can show that the points of these topos on the category of elements for the left n sets y are precisely the flat left n sets together with the morphism from this flat left n sets to y. So this connects back to the talk by Axel Osmum about overtoposis at the model. So what's a boong and func show is that the complete spread associated to a point, so a flat distribution, they give exactly the category of points. They are exactly the over topos at this model. So this is only with models, with values in the category of sets. If you want to look at models in other toposes, then I don't know whether this still works. Or if there is something similar that you can do. Okay, so the points are left n sets a together with the map from a to y. And here y doesn't have to be a flat left n set. It works for a general left n sets. Okay, so the first two examples were not complete spreads. And this covering map by the real line was a complete spread. So it's both a tile 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 a tile, then it is a covering map. So what does covering map mean here is that you have it is a morphism from F to E. And you have the fundamental group of the topos E. 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 the base topos E is a pre-sheaf on a monwit. And this was shown by Bonhe and Funk, or more generally for connected pre-sheaf toposes. And it also works for for E, the topos of sheaves on a house door, second countable connected topological manifold. And this was shown by Funk and Tim Shatin. So I have not been very precise here. So the the ones corresponding to subgroups will be exactly the connected covering maps, but you also have non-connected covering maps and they correspond to sets with an action of the fundamental group. But so this is corresponding to the set, this quotient set. Okay, so how does this with the fundamental group work here for pre-sheaf toposes on a monwit? Well, to get the topos of pre-sheafs 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 n sets on which nx 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 n, so the closest group to the monwit n. And if x is such a locally constant right n set, then we can also define a left n action by just defining it as multiplication on the right, but first taking the inverse of the element. So multiplication by the inverse element. And in this case, we get that for this left n action, the two categories of elements agree. So we will also get the same geometric morphism, which will be both the complete spread and et al. And okay, so let n now be a monowit and x-right n sets. And we want to look at a geometric morphism associated to this right n sets. And we want to classify et al geometric morphisms in the sense that we want to end up again with a topos of pre-sheafs on the monowit. So when is it again of the form pre-sheafs on the monowit? 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 such that for every other element y in x, there is some element u such that x u is equal to y. And here u is an element that 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 in this case, if such an element exists such that all other elements are retracts of it, then for this element x, we get that the topos of pre-sheafs on this category is the same as the topos of pre-sheafs on the monowit nx, where nx is the monowit of n-demorphisms of this element, which is exactly the elements of n that fix x. So now let n be a monowit and y a left n set, and we can do exactly the same thing for complete spreads rather than a tau maps. And the calculation is exactly the same. So this time there must be an element y in y such that all 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 vx. And again, we can then describe this pre-sheaf topos as a pre-chief topos of pre-sheafs on a monowit, and the monowit is again the elements that fix y. And this doesn't really give a method of saying for a monowit map a phi, whether it is a tau or not, because it starts from the base topos and then you construct the topos over it. But say that you start with a monowit map from m to n, and you want to know whether it induces an etal geometric morphism. Then we also found some criteria to determine whether it is etal. So phi must be injective. If you have elements a and a, b 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. For example, you can check that these conditions are all satisfied if you take the map from the natural numbers to the paedic numbers that sends n to p to the n. So this induces an etal geometric morphism. And the picture for complete spreads is exactly dual. So then you 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 for maximal orders. They 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 and etal maps. So some background in their approach to the Riemann hypothesis, Konning-Konzani, introduced in 2014 the arithmetic site. So it's a topos of pre-sheeps on the nonzero natural numbers under multiplication equipped with a certain sheave of semi-rings. And this sheave of semi-rings is very important, but still the topos of pre-sheeps is already very interesting. For example, it was computed by Konning-Konzani that the points of the topos are classified up to isomorphism by the double quotients 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 dance 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, it's, for example, number field, which is an extension of the rational numbers. Or it's, 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 towards and 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. And if, suppose that r is the integers, then all finely generated towards and 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. And this is a case that was already studied by Arjen Sagny very early on. So to generalize the arithmetic side by replacing the integers by the Gaussian integers. Or we can also look at another example of maximal orders is the Herwitz 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. And for general maximal order, we write the non-singular elements are exactly the elements that become invertible in the 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-shefs on the non-singular elements of the monowheats? Or do we need a different topos? Or what is the right structure sheave of semi-rings on this topos? Do we and maybe do we get a spectral interpretation of the zeta function of the maximal order in topospheric 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 a general maximal orders. But this topic needs work and I 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-shefs on the non-singular integers. So how can we recover this arithmetic side from this? So the answer is we look at the following set with an action of c and s. 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 the good situation where we have both a compete spread and a net town map at the same time. So and we can compute that this category of elements is equivalent to the category of the monowit 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 two-fold covering space of this topos that is more generally associated to the maximal order c. So can we get this idea for more covering spaces? We can look at the maximal order lambda and look at the provenance elements. So we look at the periodic completion at each 8-1 prime ideal and take the product over all the primes. And these are the provenance elements. And then to define the lambda ideals you again tensor with a fraction field. And we can look again at the non-single elements which are elements that become invertible in the ideal ring. And the monowit is more complicated, the monowit of provenance integers in some sense, but the topos is nicely behaved. If you take the completion, 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 monowit 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 of the finite ideals. So the ideals depend on lambda. You take the units and you quotient out by the units in sigma. So I use notation the ideals 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 we are in a situation where we have a covering map, so both a complete spread and a down 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. So a unit in the profanate elements of the maximal order. And that's exactly because multiplication in the category of elements is multiplication in the monowheat. So the isomorphisms in the category of elements will exactly be the units in the monowheat. 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 element, so the clause of one. And the anamorphism monowheat of this element is exactly the original maximal order lambda, the non-singular elements. And other elements you can also look at anamorphism monowheats of other elements, and they can have anamorphism monowheat that is the non-singular elements of another maximal 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 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 adels? So the 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. But here 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 with an action of ZNS, the non-singular integers. And we then look at the associated covering map. So why is this a covering map? Exactly because the action is again by bijections. So both a complete spread and an ethyl map. And the points of this topos, so as I mentioned earlier, it is an over topos. 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 a left ZNS sets. And you can use this description as an over topos at the model to show that these, the points are then classified up to isomorphism by these double quotients, where you have the full ring of adales rather than just finite adales. And another thing we can do is, well, if we have just the real numbers with this action, then we can decompose this ZNS set into, as a co-product of smaller ZNS sets. For example, you can, you have the element zero, which is stable under multiplication with an element of ZNS. And you have the non-zero real numbers, which are also a sub ZNS set. And zero here, the action on zero is trivial. So we really get a copy, this is really the terminal object. And this means also that this pre-sheaf topos contains a copy of the original pre-sheafs 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-sheafs on ZNS. So, and this corresponds to the composition of the double quotients, which is also considered in the article on the scaling side by Konning Konzani, which is a, that you first take the finite ideals and the component at the real numbers is 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 a discovering space as a really very much more smaller parts. So it's really a disjoint union of toposes. And so we already said 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 side. So thank you very much for your attention. Okay, so thank you very much for your talk.
 So I am happy to introduce Jens Ebler from Anwerp University, who is going to talk about  toposies of fish shaves on Monoiza as generalized topological spaces.  Thank you for the organizers for putting together a wonderful conference and for the invitation.  So I will talk about toposies of fish shaves on Monoiza as generalized topological spaces.  So we will really try to see how far we can push the analogy between toposies in this very  special case and topological spaces. So 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 Sagni. 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-Sheafs on Monowitz,  which is a joint work with Morgan Rogers. And we will build on really a very large  theory that is worked out by Boone and Funk.  So this is something we already saw earlier in the school already and in the conference,  is that the sheave on the topological space can be equivalently described by a local homeomorphism  to this space. So here is a some reminder of the notion of local homeomorphism. Say that F is  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  necessary to include this that it is onto an open set. And we will use instead of local homeomorphism,  we will use the terminology et al. And it will be called the etal 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. But this time it is a circle and you take the projection  to the circle. There's also a local homeomorphism. And a 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. And we can take this a little bit further and think of the sheave as not  as an etal map of topological spaces but as an etal geometric morphism between the associated  topos. And here we can say that an etal geometric morphism to this topos is precisely the ones  the geometric morphisms that are induced by the local homeomorphisms of spaces. But there 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 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.  It gives a method to 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, a geometric  morphism is etal if and only if there is some object x in e and this 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 etal geometric morphisms are exactly the  projection maps from the slice topos. And here 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? Well by definition there is a projection between  objects of the topos and etal geometric morphisms from with the same co-domain e.  So these etal geometric morphisms are taken up to natural isomorphism. And this extends to an  equivalence of categories between e and the category of maps to the topos e,  or the category of etal maps to the topos e. So I wanted to be sure about this result. 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 that I found where it is stated is in the literature  on infinity toposis. So there's a proof by David Carchetti using the result from infinity toposis  and then truncating it to a result about one toposis. But I think now from the talk by Riccardo  Zanfa about the joint work with Olivia Caramello, it's now clear that this doesn't use infinity  toposis 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 is 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 right n sets. So if n is commutative, you don't have to mention that it's right n 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 in their 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 m the natural numbers in their addition.  So at all topos is over this topos correspond to right n 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 in their addition and the base space we will write as this.  So it's one element and the action of the monowits is x by the successor and it's  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 n action. For example we can take two elements and one element is  sent to this element is sent 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 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 monowit n and x is the right n set then we define the category of  elements of x with respect to n as a category which as objects the elements 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 of n is equal to p. 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 x there is a unique lift 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-sheaves on it. So this is really how you  would draw the k-graph of the of the n-set. So now what happens if we take x to be the monowit  n itself with right n action given by multiplication? Well in our example of the natural numbers  under 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-sheaves on this category of elements of n over itself.  And if n is left cancelative monowit then the root topos is equivalent to a 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 open sets will  be the sets that are upwards closed and contain at least one natural number or are empty.  So and we can also look at this for the arithmetic side. If n is now the monowit of nonzero natural  numbers under addition then under multiplication you say that addition. So it's under multiplication  then this topos is the 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 morphisms are you have exactly a morphism from m to n whenever m divides m.  And another word for this is it's also called Konway's big cell which appears in the paper  covers of the arithmetic side by my PhD supervisor Yvonne Le Brun. So this is also how I got  involved in toposphere in the first place.  And so here is a picture of this big cell. So the topology on the big cell is non-house  dwarf but here we take the closest house dwarf approximation and the red dots together give  then 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 dwarf topology.  So now ethyl geometric morphisms to the base topos they correspond to right endsets.  So what happens now with left endsets? So the question is is there a dual to the notion of  ethyl geometric morphism and the answer is yes and this is studied in many papers by  Bung Bung and Fung. And to get to this notion we first have to look at the notion of spreads.  So what is a spread? Well we have growth in the toposis F and E and the spread is a 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 a generative 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  if we would change the definition and we would remove the assumption that the  that you take only complement sub-objects if you would take all sub-objects then you would exactly  get the definition of a locale geometric morphism. So all spreads are locale.  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. So the inclusion map is a spread  and this is a spread for both the subspace topology and for the discrete topology on Y.  So it's a more general. So we can have a look at which of the examples of etymaps are spreads.  So the open interval is a spread because it's an inclusion map. But the circle with double  point will not be a spread because you can take here an open interval. So suppose that you take  that only contains the upper point at the origin. Then if you want to write it as a sub-object of  the inverse image of this then 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 locale map but it is not a spread. And  here if you look at the 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. And they are precisely a component of the  inverse image of an open set in the base space. So 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 and dectoposys with f locally connected. And we start with a geometric morphism.  And then we construct a site from a site of e. And it's given by objects in e together with  or objects in a site for e together with the connected components.  So 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. So you can extend the covering upstairs to covering in the base space  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 Fung because they give a more general  definition for toposys over an arbitrary base topos. Whereas we are just working with growth  and dectoposys 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 Fung. So for example, their book  singular coverings of the toposys. 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.  So the reason is, suppose you take an open set here, an open interval.  And then the pre-image will be something like this,  which is also open. And we can cover this by open sets that do not contain in some sense  the point that is removed. But we cannot extend this covering to a covering on the base space  because on the base space. So here we 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 of intuition  behind it. And so the circle with double point was already not the spread. So it won't be a  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 a complete spread. And if you  apply this to spreads, then in particular a spread factorizes as a pure spread followed by a complete  spread. So we have here, this was a spread, the 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  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 notion is also inspired by a notion in topology  of a cut. And for example, you can look at the piece of paper and model this as a  compact house door space in the euclidean space in R3. So this is a compact house door space,  but then if you do it in two, then you still should get two compact house door 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 happens in this situation.  So what if ethyl geometric morphisms correspond to objects of the topos,  then what do complete spreads correspond to? And they correspond to some notion by law of  year. And the definition of law of year is that 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 e. So distributions are already here dual to objects. And the result by Bono and  Funk is that there is a correspondence between distributions on the topos e and complete spreads  that have e as a co-domain. So what does this look like in our case? If we have a topos of  pre-shefs, then distributions are the same thing as functors from C to the category of sets.  And in particular, if e is pre-shefs on the monowheats, 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 set. And again, we arrive at the category of  elements, but there are two categories of elements. There is one for covariant functors and one for  contravariant functors. And they are different in some sense. So for n, a monowheats, and y left  n sets, we define the category of elements of y as the category with as objects elements of y.  And again, as morphisms, the elements n such that n a is equal to b. So this is a bit different  than the other category of elements. And to distinguish them, we put here n on top of the  integral side. And for the other category of elements, we take it below the integral side.  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 monowheats and every a element of the set, there's  a unique lift to a map starting in a in a going somewhere else, lifting the element n. 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-shefs correspond to  discrete vibrations. So a complete spread associated to the left n set y 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 core root topos. And the definition is the same. But you take the category of elements  of n as a left n set rather than as a as a right n set. And so here I call root the core root topos  of n. The map to the base topos will be a complete spread. Whereas for the root topos, the map to  the base topos is a tau. So what are now the points of the of the pre-shef topos on this  category of elements? So you can use the pure complete spread factorization by Bono and Funk.  So if you have a point of the of the base topos, the pre-shefs on n, then the points by the  Okonassius theorem are exactly the flat left n sets. So we call this a flat left n set 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 a factorization to the 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 n sets. So in this way, you can show that the points of these  topos on the category of elements for the left n sets y are precisely the flat left n sets  together with the morphism from this flat left n sets to y. So this connects back to the  talk by Axel Osmum about overtoposis at the model. So what's a  boong and func show is that the complete spread associated to a point, so a flat distribution,  they give exactly the category of points. They are exactly the over topos at this model.  So this is only with models, with values in the category of sets. If you want to look at  models in other toposes, then I don't know whether this still works.  Or if there is something similar that you can do.  Okay, so the points are left n sets a together with the map from a to y. And here y doesn't  have to be a flat left n set. It works for a general left n sets.  Okay, so the first two examples were not complete spreads. And this covering map by the real line  was a complete spread. So it's both a tile 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 a tile,  then it is a covering map. So what does covering map mean here is that you have  it is a morphism from F to E. And you have the fundamental group of the topos E.  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 the base topos E is a pre-sheaf on a monwit. And this was  shown by Bonhe and Funk, or more generally for connected pre-sheaf toposes. And it also works for  for E, the topos of sheaves on a house door, second countable connected topological manifold.  And this was shown by Funk and Tim Shatin. So I have not been very precise here. So the  the ones corresponding to subgroups will be exactly the connected covering maps,  but you also have non-connected covering maps and they correspond to sets with an action of the  fundamental group. But so this is corresponding to the set, this quotient set.  Okay, so how does this with the fundamental group work here for pre-sheaf toposes on a monwit?  Well, to get the topos of pre-sheafs 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 n sets on which nx 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 n, so the  closest group to the monwit n. And if x is such a locally constant right n set,  then we can also define a left n action by just defining it as multiplication on the right,  but first taking the inverse of the element. So multiplication by the inverse element.  And in this case, we get that for this left n action, the two categories of elements agree.  So we will also get the same geometric morphism, which will be both the complete spread and et al.  And okay, so let n now be a monowit and x-right n sets. And we want to look at  a geometric morphism associated to this right n sets. And we want to classify et al geometric  morphisms in the sense that we want to end up again with a topos of pre-sheafs on the monowit.  So when is it again of the form pre-sheafs on the monowit? 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 such that for  every other element y in x, there is some element u such that x u is equal to y. And here u  is an element that 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 in this case, if such an element exists such that  all other elements are retracts of it, then for this element x, we get that the topos of  pre-sheafs on this category is the same as the topos of pre-sheafs on the monowit nx, where  nx is the monowit of n-demorphisms of this element, which is exactly the elements of n that fix x.  So now let n be a monowit and y a left n set, and we can do exactly the same thing for complete  spreads rather than a tau maps. And the calculation is exactly the same. So this time there must be  an element y in y such that all 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 vx.  And again, we can then describe this pre-sheaf topos as a pre-chief topos of pre-sheafs on a  monowit, and the monowit is again the elements that fix y. And this doesn't really give a method of  saying for a monowit map a phi, whether it is a tau or not, because it starts from the base  topos and then you construct the topos over it. But say that you start with a monowit map from  m to n, and you want to know whether it induces an etal geometric morphism. Then we also found  some criteria to determine whether it is etal. So phi must be injective. If you have elements a  and a, b 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.  For example, you can check that these conditions are all satisfied if you take the map from the  natural numbers to the paedic numbers that sends n to p to the n. So this induces an etal  geometric morphism. And the picture for complete spreads is exactly dual. So then you 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 for maximal orders.  They 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 and etal maps. So some background in their approach to the Riemann  hypothesis, Konning-Konzani, introduced in 2014 the arithmetic site. So it's a topos of pre-sheeps  on the nonzero natural numbers under multiplication equipped with a certain sheave of semi-rings.  And this sheave of semi-rings is very important, but still the topos of pre-sheeps is already very  interesting. For example, it was computed by Konning-Konzani that the points of the topos  are classified up to isomorphism by the double quotients 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 dance 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, it's, for example, number field,  which is an extension of the rational numbers. Or it's, 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 towards and 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. And if,  suppose that r is the integers, then all finely generated towards and 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.  And this is a case that was already studied by Arjen Sagny very early on.  So to generalize the arithmetic side by replacing the integers by the Gaussian integers.  Or we can also look at another example of maximal orders is the Herwitz 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. And for general maximal order, we write  the non-singular elements are exactly the elements that become invertible in the  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-shefs on the non-singular elements of the monowheats? Or do we need a different topos?  Or what is the right structure sheave of semi-rings on this topos? Do we and maybe do we get a  spectral interpretation of the zeta function of the maximal order in topospheric 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  a general maximal orders. But this topic needs work and I 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-shefs on the non-singular integers. So how can we recover this arithmetic  side from this? So the answer is we look at the following set with an action of c and s.  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 the good situation where we have both a compete  spread and a net town map at the same time. So and we can compute that this category of elements  is equivalent to the category of the monowit 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 two-fold covering space of this topos  that is more generally associated to the maximal order c. So can we get this idea for more covering  spaces? We can look at the maximal order lambda and look at the provenance elements. So we look  at the periodic completion at each 8-1 prime ideal and take the product over all the primes.  And these are the provenance elements. And then to define the lambda ideals you again tensor with  a fraction field. And we can look again at the non-single elements which are elements that become  invertible in the ideal ring. And the monowit is more complicated, the monowit of provenance  integers in some sense, but the topos is nicely behaved. If you take the completion, 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 monowit 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 of the finite ideals. So the ideals depend on  lambda. You take the units and you quotient out by the units in sigma. So I use notation the  ideals 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 we are in a situation where we have a covering map,  so both a complete spread and a down 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.  So a unit in the profanate elements of the maximal order. And that's exactly because  multiplication in the category of elements is multiplication in the monowheat. So the isomorphisms  in the category of elements will exactly be the units in the monowheat. 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 element, so the clause of one. And the  anamorphism monowheat of this element is exactly the original maximal order lambda, the non-singular  elements. And other elements you can also look at anamorphism monowheats of other elements,  and they can have anamorphism monowheat that is the non-singular elements of another  maximal 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 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 adels? So the 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. But here 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 with an action of ZNS,  the non-singular integers. And we then look at the associated covering map.  So why is this a covering map? Exactly because the action is again by bijections. So both a  complete spread and an ethyl map. And the points of this topos, so as I mentioned earlier, it is  an over topos. 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 a left ZNS sets.  And you can use this description as an over topos at the model to show that these, the  points are then classified up to isomorphism by these double quotients, where you have the full  ring of adales rather than just finite adales. And another thing we can do is, well, if we have  just the real numbers with this action, then we can decompose this ZNS set into, as a co-product  of smaller ZNS sets. For example, you can, you have the element zero, which is stable under  multiplication with an element of ZNS. And you have the non-zero real numbers, which are also a sub  ZNS set. And zero here, the action on zero is trivial. So we really get a copy,  this is really the terminal object. And this means also that this pre-sheaf topos contains a copy  of the original pre-sheafs 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-sheafs on ZNS.  So, and this corresponds to the composition of the double quotients, which is also considered in  the article on the scaling side by Konning Konzani, which is  a, that you first take the finite ideals and the component at the real numbers is 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 a discovering space as a really  very much more smaller parts. So it's really a disjoint union of toposes.  And so we already said 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 side. So thank you very much for  your attention. Okay, so thank you very much for your talk.
Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_0_47.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_48_119.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_120_395.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_396_1163.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_1164_1247.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_1248_1763.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_1764_2111.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_2112_2831.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_2832_4619.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_4620_5651.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_5652_6707.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_6708_8111.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_8112_9959.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_9960_13007.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_13008_14771.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_14772_16775.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_16776_16883.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_16884_16931.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_16932_17495.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_17496_19703.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_19704_19763.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_19764_20771.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_20772_21551.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_21552_21623.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_21624_22499.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_22500_23219.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_23220_23843.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_23844_25043.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_25044_25883.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_25884_26051.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_26052_26099.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_26100_26147.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_26148_26255.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_26256_27083.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_27084_28775.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_28776_28823.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_28824_30083.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_30084_32267.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_32268_35507.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_35508_37463.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_37464_37523.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_37524_37955.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_37956_41027.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_41028_42503.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_42504_43307.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_43308_45887.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_45888_47435.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_47436_51827.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_51828_52307.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_52308_55679.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_55680_57755.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_57756_58091.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_58092_60671.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_60672_61511.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_61512_62087.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_62088_63851.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_63852_64091.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_64092_64331.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_64332_65063.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_65064_67007.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_67008_70259.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_70260_71819.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_71820_73139.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_73140_73355.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_73356_74855.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_74856_75239.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_75240_75419.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_75420_75815.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_75816_77819.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_77820_79619.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_79620_81767.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_81768_82211.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_82212_84335.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_84336_84383.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_84384_85619.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_85620_86243.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_86244_87167.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_87168_88259.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_88260_88991.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_88992_90179.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_90180_90227.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_90228_90299.png Jens_Hemelaer_-_Toposes_of_presheaves_on_monoids_a_1GI7HLX_ed0_90300_90324.png