 So thank you very much. I'd like to thank the organizers for inviting me. Is this working? Yeah. So it's a pleasure to be here, particularly to talk about toposes, because this is the very place where toposes were created. So I'm quite happy to be there. So I don't know yet if we coordinate well with André, because when I gave my title and decided on my talk, I didn't know André Joillard was going to do the same thing. Well, I guess it's for the best, because now I don't have to justify the analogy between toposes and commutative algebra. So this is why I changed the title. And so instead of explaining that toposes or commutative ring, oh, yes. That's an idiosyncrasy of mine. I write topos in the plural as topos. So this has been done. So rings, yes. No, no, this one. OK. Excellent. This one I usually put an S, yes. So I want to try to describe a research program, which is the development of the analog. If we take this analogy with commutative ring seriously, then there should be such a thing as commutative algebra for toposes. See, I almost wrote. OK. Maybe I should please everybody. So this is a joint research program with André. This is why we decided to talk on the same matter. Georg Biedermann, Eric Finster, and Damien Leger. So some of the things I'm going to talk about today are not proved. They are conjectural, the kind of trying to develop an understanding of what these commutative algebra for toposes could be. Most of the naive questions about that are still without answers. But hopefully I'm going to give pieces of the theory that should convince that there should exist such a thing. OK. So I'm going to work in the setting of infinity categories and infinity toposes. So let me briefly say that I'm not going to define infinity category. I'm going to use the notion of infinity category as if they were ordinary category, because actually the syntax of the two theories are exactly the same. But if you want to have something in your mind, an infinity category is probably best simply defined as a category enriched over spaces, as André proposed. And what I mean by spaces, it's a bit ambiguous. So by space, I mean infinity group weights, which can be defined. So we can be formally defined as topological spaces up to a monopie or can complexes. So today I'm going to do some synthetic higher category theory. I'm not going to use explicit models for the objects that I'm going to talk about. So I'm going to start with the definition of an infinity topos. So André gave already the definition, so let me recall what it was. So I'm going to use the letter S. There's been news a lot. I'm going to use it to refer to the category of spaces. So it's not sets anymore, nor the Sierpinski space anymore. So the category will be called presentable if it is an accessible localization of a pre-shift category, where C is a small category. So I'm using categories, but I really meant infinity category. By presentable, do you mean what's usually called local? Exactly, yes. I'm using the same convention as André did in his talk. So a category E will be an infinity topos if it is presentable. And for any diagram from a small category I, we have the so-called rest condition that the category of objects over the co-limit of the diagram is the same thing as the limit of objects over each component of the diagram. So I'm going to detail that a bit. So actually, if I define those two categories, there's two obvious fractals between them. So the limit of this category can be described as I start with the category of I diagram in E. And I look at all my diagram over my diagram X. And then I look only at this full subcategory of morphisms of diagram that are Cartesian, which means that I'm looking at a diagram Yi over Xi, such that every square like this for any arrow in I is Cartesian. So I can take the co-limit of such a diagram and that produce an object over the co-limit of X. And here, given an object, I can just define a kind of constant diagram by just pulling back my object here over each of the Xi. So the fact that this is an equivalence, so this is an adjunction. And the fact that this is an equivalence can be stated by two conditions. It's equivalent to say that the function of constant diagram is fully phase full and the co-limit function is fully phase full. So if we unravel what this condition means, this is equivalent to the universality of co-limits. And this condition, which is new, is a very strong descent condition. It's called effectivity of co-limits. So in a usual topos, only equivalence relations are effective. In an infinity topos, there's a variation, there's a set of axiomalagiro for infinity topos, where every infinity groupoid is effective instead of every equivalence relation. And with this formulation, it's even stronger. It's effectivity of any co-limit. So this is true in what I will call one topos. These are ordinary toposes. So this condition is true, but this one is very false, as André explained, in ordinary toposes. So let me give you some examples of so. The left one, you understand that you are working with kind of things like pseudo-facult, that is diagonal, that is commuter, really up to some moment of, I mean, it's not so strict sense. So this is different between the treatment to usual categories. Exactly, exactly. You have to, you didn't define what you mean, that is, you said that you have category and reach to other spaces, but are these either such activities strictly or not? There are many choices. Yes, so this is the part I wanted to avoid. So probably a complete formal approach to this would use model categories. And we'll have to use usual techniques for categories of diagrams and define them using models and the right functions and everything. But I'm kind of skipping this part and I'm kind of presenting the theory as if it would work without the formalism of model categories and without presenting the theory formally within set theory, somehow. So I'm assuming that there exists a theory of infinity categories, that the notion of diagram makes sense, and that I can construct a category of diagram and et cetera. So I'm not describing the basic setting, but I'm assuming it exists. And for the set theory, so you indicated the home, home categories are small and the seven objects could be large, do you mean? No, that's not what I meant. So in usual categories, the objects form a class and the homes are a set. So what do you do? So here, I'm not assuming this, actually. So the definition of the presentable category will be the case that the home space between two objects will be small. But we can even not use that if we want. OK, so I come back to my examples. So Andre explained how to construct the free topos. So if we start with a small category C, we can add finite limits to C and then take the pre-shift over this. More generally, the pre-shift over any small category is already a topos. It's not free, but it's OK. As a particular case of a pre-shift, I have the category of spaces. It's a topos. The category of group action in spaces. So this is a group up to a monopie and this is just a space with an action of G. I can do the usual thing and start with a theory, a logical theory. And it will have an enveloping infinity topos. Let me give you an example of a very silly theory. So I can consider the free topos on one variable. So this is the topos classifying objects. So it's the theory associated to the theory with one type. And that's it. And action, yes. I can consider the slice topos of S of X over X. So I will call it like this. And this one, it classifies objects with a global section. So I'm going to call that pointed objects. Finally, let me mention a source of example. Any accessible localization, which is left exact, over topos is again a topos. So this was expected. So as an example of such thing, and I hope I don't think I will have the time to develop it further, but let me mention that in the work of Goodwillie, Goodwillie construct left exact localization of this topos. So this topos, concretely, it's funtals from finite pointed spaces. And the theory of Goodwillie is essentially the study of a tower of left exact localization of this topos that are called an excisive localization. So these are the embedding of this in two S of X is the so-called an excisive funtals. When you say funtor, just to make clear, do you mean funtals from finite pointed set to set? Yes. I'm seeing this as a subcategory, as a full subcategory of this one. I'm sorry. So now you are in usual topos, right? No, no, no. This is a higher topos. Yeah, this is the category of spaces, yeah. Not pointed spaces? No, this one is pointed. This one is not pointed, yes. And finite space, you mean finite CW complex Yeah, exactly, yeah, finite. Wait, exactly. What time did we start? OK. In fact, the clock is just earlier now. Ah, well. OK, good to know. So let me say a word about the difference between infinity toposes and one toposes. So there's many differences. I'm just going to mention three. So the first one is one that Andres mentioned, that the sub-object classifier is enhanced into an object classifier. So I'm not going to elaborate on this one. Let me just say that this is useful, for example, to write in the, sorry, I'm going to use something like the Mitchell-Bennabou language. This is useful to write formula like this. Now this makes sense. Now this is an object of the topos, modulo of the size issues. OK, so now the second point, which is probably the main difference is that left-exact localization are no longer controlled by Goten-Dick topologies. So notion of Goten-Dick topology still makes sense. But not every left-exact localization is associated to that. So let me give you an example. If X is a topological space, it's possible to build. An infinity topos of shifts on X. And these are frank-talls from the open subset of X into spaces. And that satisfies some condition. And the condition is the usual shift condition for coverings of any open of X. So maybe the notation is a bit ambiguous because you write shift on X. Everybody is thinking of their shift. So I mean infinity shifts. And now there's another topos that I can construct associated to X. It's the so-called hypercompletion of this. It's that I impose descent not only on coverings, but on hypercoverings. And it's possible to prove that a second one is a left-exact localization of the first one. But this is not a localization which is given by Goten-Dick topology. So this is the most difficult technical thing about higher topos. That all the technology of Goten-Dick topology is not enough to control left-exact localization. So every time we have a proof of classical topo theory that uses a Goten-Dick topology, it may not generalize easily as a statement for infinity topos. So it is not like in usual algebraic geometry that somehow to prove descent is enough to do it with coverings. So the check. Yeah, now there's a difference between the two descent conditions. And this is related to some unboundedness or something. What's that? Yes, yes. For truncated objects, if I look only at shifts with value in truncated spaces, then the two notions are the same. So it's only because we look at shifts of possibly unbounded homotopy types that the two things are different. Yes. OK, let me mention a third point that in infinity toposes, we can impose some fun equations like something like this. So this is an equation that has no non-trivial solution in S, but it has many, many models in infinity topos. Yeah, sorry, sigma is suspension. So I'm assuming x is a pointed object. And this is the suspension of the loop space of x. Yes, sorry. Could be worthwhile to just say briefly what suspension is in the context of topos. OK, so the loop space is defined as the fiber product of this diagram, the homotopy fiber product of this diagram. And the suspension of an object is defined as the push out, the homotopy push out of this diagram. So if I compute the suspension of the loop of x, it's easy to see that there's a canonical map like this. And we ask that this map is an equivalent. So this is, for example, we can consider a left exact localization that would be generated by this. And I don't think it'll be generated by a grotenic topology. This is also the kind of condition that is not seen by grotenic topologies. Let me mention why. The idea about grotenic topology is just that grotenic topology are inverting monomorphisms only. And the question is, can we control any map by inverting a bunch of monomorphism? And the answer is yes, when the category is truncated. But in infinity category, it's no longer true. OK, so a lot of time to elaborate. Sorry. OK, I'm half my time. So let me ask a few questions about this analogy with topos and commutative rings. So if infinity toposes are commutative rings, then we can ask the following question. Do they compare to ordinary rings to other kind of categorical rings? Is there a corresponding linear algebra? If I'm thinking my rings as functions, is there such a thing as a distribution theory, measure theory, morphisms of toposes? I mean, when you said that infinity toposes are commutative rings, what are the morphisms and in which direction? Oh, sorry. Yeah, I'm assuming that people attended the talk of Andre, actually. So that probably makes sense. What's the direction? I mean, when you say OK. OK, that's a good. OK, let me make a remark over there. Sorry, I should have said that. So it's convenient to introduce two categories for toposes. So I'm going to introduce a category that I will call shoes and another category that I will call topos. So this will be the algebraic side and this will be the geometric side. And this category will be opposite one to another. So here, the objects, they are the topos as I defined them. But it's convenient to have two names for when we look at the topos geometrically and algebraically, a bit like locales. We have locales and frames. And so I'm going to call them categories of shoes. And I will keep the name topos for the geometric side. Maybe certainly you could call them infinity frames. Oh, infinity frames? No, I don't like it. So the morphisms are the so-called algebraic morphisms that André defined. So between two categories of shoes, I'm looking at functors that are co-continues and left exact. And here, the morphisms are the geometric morphisms, which are defined just by defining this category as the opposite of this one. I don't want to talk about the two cells. No, no, I don't have time. I don't have time, all right? Maybe you can ask again the question at the end. OK, so last time, so I come back to my questions. Yes, so the comparison of topos with commutative rings is on the algebraic side, yes. So the co-limits are seen as sums, and the finite limits are seen as some kind of products. And so the morphisms are the functors preserving sums and products. OK, last question. If this is indeed a theory of rings, is there a notion of polynomial and is there a notion of differential calculus? OK, so I want to have time to answer all the questions. And actually, I don't know how to answer all the questions. But I'm going to sketch pieces of answers. So let me say a brief word about the first point. It's I'm going to use this part, actually. So in my head, I have the following analogy that is helping me, so I guess it's a good thing that I share it. So I'm going to draw a table. So I start with the Sierpinski space. And associated to that, we have the notion of locales. Then we have sets. And associated to that, we have one topos. And then we have spaces. And associated to this, we have infinity topos. Do you want to draw an horizontal line between the second line? Here? OK, if you want. Because of what you're having in mind. OK. So I'm going to compare this classification with a classification in ordinary commutative algebra. So it's a bit rough and I don't have time to be more precise. But essentially, we can do this. So I'm going to write it and explain it. It's possible to think this way. So a Sierpinski space, it functions with coefficients in 0 and 1. So it's like polynomial with coefficients in 0 and 1. In topos, we look at functions with value in sets. And so by using the cardinality, we can think of them as functions with value in integers. And if we look at functions with value in spaces, we can think of x as something as the cell of dimension one. And essentially, the way to go from the left to the right is taking some Euler characteristic. And then we see that a topos is a kind of z algebra. And then infinity topos would be a z of x algebra. So this is ordinary topology. So ordinary topology is to the power of topos topology. So what algebraic geometry over z2 is to algebraic geometry over z or over more complex objects. x as degree one, yes. x, I'm thinking that x is an interval without a boundary. No, x plus 1 would be the circle. Is x square equal to 0? It's super kind of 2. No, no, no. So this is ordinary algebra. So it's a commutative grading. If you want, I can assign a space here to its characteristic series. And x is just the variable for that. OK, so let me add something which is not a topos. But it's useful to have in mind. To have it in mind, it's the category of spectra. And the category of spectra in this language is correspond to this ring. So we have inverted x minus 1. x. No, x plus 1 is the circle. And we invert x for the smash product. So the anti-circle will be x minus 1 plus 1. It's a 0 cell and a 1 minus 1 cell. OK, 20 minutes. So in this comparison, so the free topos, compares to the free ring. Andrei explained that. And a topos without points, it's the same thing as a ring with no rational point. And now if I have a first order theory or a site, this is just data to generate a ring. So it's a presentation. I'm sorry, but if you say that, then there should be an analog of taking the algebraic closure in the ring case. I have no idea. Well, OK, the question is, what is an algebraic equation? It's not even clear. Well, you have written in one of there. Just an example. Sigma over. Well, this would have to be an example of an algebraic equation. Yes. Because sigma is a sum. Omega is a product. Kind of. But maybe we can discuss that after. Yes? You can associate to a topos a commutative ring, but you just made an analogy. No, it's just an analogy. It's just an analogy. I don't pretend that this is a, I don't know how to formalize this. And if we take the theory of the commutative rings and the theory of the infinite topos, and we take the classifying topos of the two theory, that's there, more than thank you all. I'm not sure I understood the question. So maybe we can discuss that later. No questions. OK. OK, so that was for the. What is the analog of an ideal? Yeah, that's a very good question. I don't know. I guess this goes with differential calculus, but it's an ideal. Is there such a thing as an ideal? It's not obvious because this is highly nonlinear. I mean, we have a kind of sum, but we don't have a subtraction. So it's not clear what an ideal could be. Yeah, but what I can explain this morning is that you don't need ideas. What you need is a monoidal categorical. It's a closed monoidal categorical. Then you can dispense with ideas. OK, maybe. So where was I? 20 minutes. So let me say a word about linear algebra. So essentially, the thing that corresponds to linear algebra for toposes is the theory of presentable categories. So this is the category of presentable categories. I have an obvious forgetful factor. And a modular size issue that I don't want to discuss now, there's the construction of a free topos from a presentable category. It's an analog of taking the symmetric algebra. I cannot go a bit more. It's a presentable category with collimates. Exactly. So yeah, it will work, yes. And we can even generalize the notion of topos by removing the presentability assumption. OK, so the category of presentable categories is a very nice category. It's a monoidal, it's a symmetric monoidal closed category. All right? Yeah, that's exactly what I was suggesting before. That you could apply back here, it was this. Yeah, yeah, well, that's a good question, yeah. OK, so let me describe the tensor product in the particular case of two pre-shifts category, the tensor product is just the pre-shift over the Cartesian product of the categories. And then the general tensor product is computed from this formula by composing with localization on the left and on the side. I don't have time to give the precise formula for the tensor product, but it's exactly the same as the one computing tensor product of vector spaces. We take something freely generated by pairs, and then we impose linearity on one side and linearity on the other side, and this gives the formula. So the analogy of presentable categories with vector spaces is perfect. It works all the formulas and all the results that one can expect are exactly the same. Is it a theorem that that's independent of the present date of the choice? Oh, yes, yes. Sure, yeah, yeah, yeah, yeah, it is, yes. It will be, yeah. It will be. No, no, no, it is, it is. So this is done, for example, this is done in Nuri with higher tuples. OK, where was I? What are the monoregues in this? There would be symmetric monoregues and all presentable categories. No, no, I know, but what are the monoregues in this category? In this category? Symmetric monoregues and all presentable categories. You said the category itself is a monoregual tensor category. Close monoregues. Close. So my question is what are the monoregues in this category? In the category of presentable categories? No, no, there will be presentable categories with a symmetric monoregual structure. Which is closed, what are the morphisms? Which will be closed because it will have to be, OK. So I didn't mention what kind of morphisms I want here, but I want co-continuous morphisms. What are the morphisms between? Yeah, the co-continuous morphisms between presentable categories. Co-continuous morphisms. Co-continuous factors. Sorry, I couldn't hear. Everything is infinity category, but it works the same as classical in the classical setting. OK, where am I? So example of monoregues here are, so S is the unit for the category of spaces is the unit for the distance of product. If I look at the category of truncated spaces, it is a monoid. And it's actually quite important. And it's the same thing for spectra. You have not said what a truncated space is. Oh, OK, I'm assuming. OK, so sorry. Truncated space is a space whose homotopy groups vanish after all the n. In this case. So if I tensor the category of spectra with itself, I get spectra. And this is related to this formula here. If I tensor this over z of x, I get itself. Is there a universal property of the tensor product? Of the tensor product? Yes, yes. It's the universal object that classifies B co-continuous factors. OK, so this distinguish, the module over this guy will be presentable n categories. The module over spectra will be the stable presentable categories. Presentable n comma 1 categories. Sorry? Presentable n comma 1 categories. Yes, yes. So categories with no higher morphisms on degree higher than n. OK, now I'm late. So I want to say more about the linear algebra part. Let me say a word about measure theory. So if A is a commutative ring, I can define a distribution on A as a linear map. So I can do exactly the same for. Sorry? I have no topology on my rings. It's commutative rings, yes. I'm all in a commutative setting. So if E is a topos, I can define a distribution as a co-continuous factor from E to spaces. Actually, there's not a lot of example of such things. The point of E will be examples, but it's difficult to produce other examples. So it's useful to generalize a bit this definition by saying that instead of looking at a distribution with value in S, we are going to look at a distribution with value in any category. So I'm going to restrict to the case where the category is spectra. So such factors, co-continuous factors, they are called co-shifts also. So we can define a kind of dual of E by looking at co-continuous factor from E to spectra. And we can also define by a kind of base change the topos of stable sheaves. So this is the category of sheaves of spectra. So this tensor product is very useful to produce this. It's not the usual definition, but it's a very nice definition. Is it the same as the presentable tensor product? Yes, this is the tensor product of presentable categories. So you're saying that if you take two topois, two toposes, and you do the presentable tensor product, then it's also a topos? Yes, it's a ferre bin l'urie. Sheaves on E, on the topos E. So there's actually a formula that says that this category is the same thing as the category of co-variant factor, sorry, André, from E up to SP that are continuous. That's a general formula for the tensor product. Of two things, that's kind of the value. Yeah, I should have said it earlier. This has nothing to do with the fact that E is the topos. How much time? 10 minutes? Eight, nine. Nine, huh? OK, so let me try to use this. So I'd like to explain how this is related to vertier duality. So vertier duality, classically, it's a setting of operations on the category of sheaves with value in chain complexes or in spectra. But there's a nice result. I don't know who came up with this idea, actually. It's but there's a statement in the high algebra book of l'urie that vertier duality should be seen as a duality between a category and its dual in the sense of presentable categories. So vertier duality with l'urie is to say that if X is a compact outdoor space, then I can define the category of infinity sheaves on X. And I can take sheaves of spectra. I can also compute the dual in the previous sense. And the statement of l'urie is that vertier duality can be understood as an equivalence of categories between those two things. So this can be also written as infinitely co-sheaves with value in spectra on X. And so we have an equivalence between a category and its dual. Compact or locally compact? What? The space? Compact outdoor. OK, maybe you're right. Maybe locally compact should be enough. You're right. On the left-hand side, you have the sheaves of spectra, co-sheaves. Here it's pre-sheaves. Here it's co-sheaves. This is sheaves with value in spectra. Yeah, so that sheaves with value in spectra on X is equivalent to co-sheaves. With value in spectra. Yes. And sheaves are the same. Yeah, exactly. So this is a duality between those categories of functions. OK, so with Damien Loge, we are working on a statement that generalizes this and that explain a bit what is going on for arbitrary toposes. So let me explain what we could do. So the question is, can we replace X by a topos? And what kind of hypothesis do we need on a topos? Such that such a statement is true. Oh, sorry. Let me mention that the way to transform a sheave into a co-sheave is by taking a section with compact support. So it's essentially this factor that produce a co-sheave from a sheave and vice versa. So we need some local capacity condition. So question, what is a locally compact topos? It's actually, so there's several possible answers to the question. And here is one that is satisfying. A locally compact space is the same thing as an exponentiable object in the category of locales. So this is the well-known construction of the compact open topology on the space of function. And so we can use this to define locally compact toposes as exponentiable objects in the category of toposes. So a locally compact topos is an exponentiable topos. So, sorry? It's clear that if you have a locally compact space, you can exponentiate it, but do you mean the converses? Yes, it's proven in the book of Peter Johnstone, for example, and stone spaces. And actually, any exponentiable local is a space. So when it's a locally compact, there's some variations in the references. So is it locally compact out of the space, or just locally it is isomorphic to say? No, it's more locally quasi-compact. This characterization is actually for locally quasi-compact spaces, not house-off. OK, I have a minus one minute. Two minutes in including questions. Ah! OK, so I'm going to try to write a statement. So here is the first result. A topos is exponentiable if and only if its category of sheaves is what is called a continuous category. I'm going to write the definition immediately. Is there a tract of a category of an object by functors preserving filtered collimates? So this is exactly the same statement as in the paper of Joyal and Johnstone. The thing is, the proof is different because they use gotonic topology to prove the statement. And here, we cannot use gotonic topology. So we need another approach. And now, let me finish by a statement which is not fully written. So I'm going to call it a wannabe theorem. Maybe one difference is that what we had, Peter Johnstone, is a characterization of a punishable one topos. And you are extending it to infinity. Yes, I should maybe kind of drop the infinity because I'm used to forget about that. Oh, I forgot an n here. OK, so let me finish with a kind of verdié theorem. So if e is locally compact, meaning exponential, and with locally compact diagonal, so which is a kind of separation assumption, then there exists a factor that will play the role of a section with compact support. And this factor is the trace of a four-menu structure on the category of stable sheaves. So this means that I have a factor of a section with compact support with sheave of spectra with value in spectra. And this induces a perfect pairing between sheaves and so it produces an equivalence between the category and its dual. So now I will finish with one last remark. So this presentation of verdié de réalité is kind of nice because it explains the nature of the Schrick factors. We have sheaves and co-sheaves, and both categories are naturally factorial with respect to e. So here we have functorialities like this between sheaves. Here we have functorialities like this between co-sheaves. And using the equivalence, we can transport these functorialities into sheaves. And these are the Schrick factors. And now because it's a four-menu structure, the commutative algebra structure here, the symmetric monoidal structure here, produces a symmetric co-algebra structure on this side. So it appears that the operation of Gotendic has to be completed a bit because we have a commutification also to take care of. Is it compatible with the multiplication when you transport it? With what? Sorry? With the multiplication when you transport it. It's not a bi-algebra, right? No, it's not a bi-algebra. It's a four-menu algebra. So it's like the structure of matrix algebra. Where is the commutative application exactly? You said there is a commutative application? There's a symmetric tensor product of sheaves. And so by duality, because this is contravariant, I guess. So it produces a co-structure on the other side. OK, thank you. I'm going to stop here. Andre, you want to ask a question? The twosome. Because we had discussions about this yesterday, and we agree that it's time that this thing should be fixed. OK, OK, OK, OK. So let me add one comment. So there's two conventions for two cells in the category of two topos. SGA convention and the convention of everybody else. So the usual convention is to look at the inverse image and to take natural transformation between the inverse image. And this is not the SGA convention. This is a convention that is nice for application in logic because with this convention, the category of points of the free topos is the category S. Now on the geometric side, so by definition, the morphism from E to F is an algebraic morphism from F to E. And a two cell between F and G will be a two cell between the corresponding factors. And so I'm inverting the order also for two cells. So with this convention, the point of the topos S of X becomes S up. So both conventions are useful. This corresponds to the convention where the point of a space are ordered by specialization. This is the convention where they are ordered by generalization. So I guess it's a problem of religion. And yesterday you suggested that on the left hand side, you should not call that points, but models. Yeah, I like to call them models. And then everything is fine. So there's a category of models and a category of points. And the category of points is the opposite of the category of models. And then, yeah, it's convenient. Thank you, Andre. Thank you. Thank you. Thank you. Thank you.