 And here we take a presentable category. Everything just, the pattern just repeats itself. So maybe I should first give a definition of what Topos is. It would be very abstract. Suppose that E is a complete category. Oh, I'm sorry, co-complete category. I will say it. So maybe I should draw a picture here. Okay, so here are posets. And then in the theory of posets, you can talk about, for example, the enthema, the meat between A and B, or Suprema. And of course, you have also enthema, as well as Supre joint. So this is called meat. This is just the finite, the enthema. When it exists between two elements in a poset, it's called the meat. Just a terminology. This is called the joint. Now, if you replace poset by categories, then you have also the same thing. Here, this is called co-limit of a diagram. So you have a diagram, and you take its co-limit. Here you take the limit of the diagram. And here you could take the co-product, or more better, even the amalgamated co-product. And here you take the fabric product. Let's see. I said that a category is, I'm sorry, a lattice is called a Sup lattice. A Sup lattice means that Sup lattice is poset with Suprema, arbitrary Suprema. So that's the notions of a Sup lattice. But there is a notion of co-complete category. It's a category closed under co-limits. So a co-complete category is very much like a Sup lattice, if you like, in the world of categories. There is also a notion of a complete category. It's closed under limits. No, here everything is small limits. I mean, you have a small diagram. Okay, yeah. For finite limits, I like to call that Cartesian. You don't like. I can say it's finitely complete. Oh, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah. It's like saying that a poset is closed under Suprema. We could say with, yeah, yeah, yeah. Yeah, it's an abuse of language, right? Yeah, yeah, yeah. But in writing, I can make it straight, yeah. But Grotenzik had the idea that when you take a limit of a diagram, you take it in the category of pre-sheeps. And if, so the limit always exists in the category of pre-sheeps. And if this limit is actually representable, then you say that the category is closed under limit. So if we adopt the point of view of Grotenzik, we can use the word closed. Okay, so it's not completely wrong. Okay, yeah. Okay. So what is a presentable category? Presentable category is a bit of a technical notion. Roughly speaking, it's a co-complete category with generators. Let's see. A small set of generators. It's not too wild. It is generated in some nice way by a certain set, a small category inside. So I could take some time to explain to you what is a presentable category, but I'm afraid that it may use a lot of my time. So I will suppose that this notion is known, but I'm writing some notes and I will make it, give the definitions in the notes. So presentable. So an example of presentable category is that if you take A, a small category, then you take pre-sheeps on A, which by this I mean contravariant from A to set, which are, of course, also the same as co-variant from A up to set. This is an example of a presentable category. In this case, the generators, it is generated by A. So you have the unit of function in the category of pre-sheeps. And every object, every pre-sheep is a co-limit of representable. So that tells you somehow that the category of pre-sheep is generated by A, but not only it is generated by A, but it is freely generated by A. It is freely generated. That's very important for the category of pre-sheeps. Exactly. So what this means is that if you have a function here from E into a co-complete category, then there is essentially a unique way to extend it into a co-considuous function. I forgot to say what a co-considuous function is. A function is co-considuous if it preserves co-limits. It is continuous if it preserves limits. So given a function from A to E, there is essentially a unique function f, a lower freak, which is co-considuous and which extends f. This function can be described easily. It's called the left-can extensions of f along y. That's a technical name. But it has a concrete description. The reason is that if you take a pre-sheep, then every pre-sheep is a co-limit of representable, but in a canonical way, this means that x is actually the co-limit over the diagram A over x. Let me write it like this. A of yA. I'm sorry. I think it's Yuneda who first observed that. Any pre-sheep is a canonical co-limit of representable. This indexing category here is the category of elements. I'm using the Yuneda lemma to identify natural transformation from the representable from yA to x because I use the same notation as a map from A to x. I write arrow from A to x where A is an object of the category A. The corresponding natural transformations are like this. By Yuneda, this is the same as an element in xA. There is a category, the category of elements of x. The objects are pair AA where A is index A, and morphism, our commutative triangle of this kind should say A is in the category A. This is a notation for this category, which Groznyczyk used a lot. The category of elements of a functor is written by Groznyczyk's element of x. A slice is A over x. It's the category of representable objects mapping to x. This is the same thing. You have this category of elements of a pre-sheep, and there is a projection from A over x to A, and you can compose with the Yuneda functor, and this is a projection, and you get a diagram of representable indexed by the category of elements of A. The category of elements of A is here, of x is here, that's here. A typical object is like that, and the value of this functor at a given element is just Y of A. You forget the structure map, so every pre-sheep is a canonical collimate of representable. It's a miracle. I think that mathematics works on a certain number of miracles, but it happens that something becomes very, very simple, and we don't really know why it is so, but it's like the complex numbers. It's very, very simple, but why is it there? I don't know, but without these simple facts, mathematics would be very hard or impossible. It's one of the Yuneda lemma, and this fact that every pre-sheep is a collimate of representable in a canonical way makes life very easy for us magnetists. Because if you want to extend continuously, or co-continuously a functor, you use this formula here, because if I have a functor, which I want to extend, so I assume that the functor preserves a collimate, so I will have the formula that if F preserves a collimate, it will be the collimate of F of Y A with A in the same diagrams. If there is a co-construed extension, it will have to be that, and F of Y A is by definition F, you see? So we see that it's the collimate. So we have a formula for extending an arbitrary functor into a co-conservative functor, and we see there that it's unique. There's a unique way to do that. So the category of pre-sheep is the free co-completion of completions under a collimate of a category. And I would like to think of this as the linearization of the category, because I want to think of Suprema as sum, and I want to think of finite limits as product. So what I'm looking here is the linearizations of a category. This is very much like taking a set and embedding it into the linearizations of it into the free Abellion group generated by it. I'm sorry. Oh, right, right, right. Thank you. Here. Okay. Okay, so we have this linearization process. Now, do we have the other process which is adding multiplication? Let's see. Remember that a ring is made in two steps. From set, you take community monoid and then rings. So we need to understand what is the first step of building the free category with finite limits. Okay. How do we build up the free category with finite limits from a given category? So in this case, it would be like cat, cat with finite limits. And then here, actually, we go with the topos. Okay. So we are going to tell you how to construct the free topos generated by a category. So at least we still don't know what the topos is. I didn't define it, but we will be able to construct the free topos. So how do we add finite limits to a category? Well, it is easier to add finite co-limits because we know how to add all co-limits. It's the unit I'm betting. This add all co-limits. So in order to add finite co-limits, you just need to take the sub-category of finitely presented perceives. Some perceives are finite co-limits of representable. So a perceive is finitely presented. Essentially, if it can be described by a co-limit co-kernel of a finite co-product of representable. So you take a finite co-product here, a finite co-product there, and you take the co-equalizer, and that's what you get there is a finitely pre-sheaf of finite presentation. Now, the category of pre-sheaf of finite presentation is actually closed under finite co-limits. It's not difficult to check. Okay. And the unit I've talked to and use something here, which you could call the unit I've talked to again, but it lands into a smaller category. And this is the free co-completion under finite co-limits of A, which means that now, that if E is finitely co-complete, co-complete under finite co-product and co-equalizer, and if you have a functor here, you will be able to extend your functor in essentially a unique way, unique up to unique azimorphism to have a triangle which commute up to canonical azimorphism. So we know how to co-complete a small category, co-complete under finite co-limits. But what about completing other finite limits? Well, one of the marvelous things about categories is that every category has an opposite, right? It's a marvelous thing. I mean, before Allen-Burr and McLean, we had very few examples of duality, you see? But since every category has an opposite, we can always change it, and it's just an amazing thing. It's very trivial, but it's a very important process. So in order to co-complete under finite limits, so you take the opposite of A, I first take the opposite of B, and then I take its finite co-completion, and then I take the opposite. And from the unidad functor, you will get the functor here, which you could call the unidad again, but which is not exactly the unidad. And this category has finite limits, and it is the three completions under finite limits. Okay, so maybe we should give it a notation. I... Jean, do you have an notation to suggest me? So I will pretend that it's like completing under joint. You see, joints are finite and fema, okay? No, meat, completing under meat, okay? It could mean big completions, right? Okay, this is bad. Okay, FLA, no? Finite limit. Not very inspiring. Okay, and which direction? This direction? Yeah, but I will put the arrow in this direction. Okay, for the finite limit completions, okay? Okay, thank you. Yes, I agree. Yeah, notations are extremely difficult, and terminology also very difficult. Grufnzik was very good to find the right notations in the right terminology. I have a lot of admiration. Yeah, so, okay, now, what will be the three topos generated by a small category? Well, the three topos, I did not even tell you what is a topos and what is the category of it, but the three topos generated by A are actually the pre-shifts, so the core limit completion of the finite limit completion. I'm sorry? All pre-shifts. Yeah, there is no topology because it's free. Topology comes when you want to make a quotient. Remember that in the case of friends, the quotient friends were in projections with the nuclei. So if you want to make a quotient, you need the analog of nucleus in this context, and this is exactly what the Grufnzik topology are providing. Okay, now I should give you some axiom, the zero axioms or versions of zero axiom. Is this a monad? Yeah, with a small problem because you started with a small category and you land with a big one, and so a monad should be an endofonter, but otherwise it's a monad. Okay, yeah, I'm not finished. I did not, yeah, sure. Yeah, yeah, yeah. You see, remember that morphisms between friends are maps that preserve suprema and meet, and meet are finite and fema. So the same thing for topos, so I will define what a topos is by giving zero axiom. You could use zero definition to define what a topos is. Zero characterizes topos in a certain way, so you could define this axiom definition as a presentable category. E is a topos if the following conditions hold. So the first one would be that pullback functors are co-considutes. The second axiom is that the following square is a pullback for A and B. So if you take the co-product of two objects, A and B, and you look at the inclusion, and you take the pullback, you take the intersections of A and B into their co-product, then that's the initial object, that's zero. So this is a pullback. And three equivalence relations are effective, which means that if you have an equivalence relation, it's always the equivalence relation of a map. It's always obtained by pulling back a diagonal, if you like. Okay, that's a version of zero axiom. It's not exactly the zero axiom, but I want to make some remark. The first axiom here is distributivity. You see, the distributivity of pullback over co-limit. So, because what it says is that if you have a map from A to B and if you have a diagram now, Di mapping to B. So D is a diagram from a smaller category to E over B. So you have a family of objects over B and then you can take the co-limit of the pullback, co-limit, thank you, of the pullback. I'm sorry. Yeah, yeah, presentable category fortunately has limits. Presentable category have by definition co-limits, but they have also all limits, all limits. You see, it's the analog of this theorem, or results, elementary, that if you have a poset, which is a soup lattice, suprama, then it must have enfima. Yes, I mentioned this. And the exact analog of that is to say that the presentable category, which is actually defined as a category with co-limit, has limits also. Okay, so this is why I can write the here pullback and there will be a canonical map from this to A, the co-limit of the DI, and the condition is that this canonical map is an isomorphism. And you see that this is really a distributivity law, because for example, if the co-limit is a co-product, you see, I'm saying that this is the same thing as the co-product of A cross B over DI. Now, I'm defining topos this way. It's equivalent to the classical notion. It's a theorem, right? It's a theorem. It's not obvious completely. It's essentially a zero theorem. But that's one way of defining what a topos is. You may have a definition because there are many ways to characterize a topos. Co-consumers means preserving co-limits. No, no, no, no, no. I forget about the definitions of the geometry. This is not one. That's equivalent. That's equivalent. I cannot prove it in this talk, but this definition here is equivalent to the classical notion. That's a theorem. So we can use this one if you like. If you don't like, you can use another one. But the reason I'm putting it here is that it clearly shows that the distributivity of finite limits and co-limits plays an essential role in the notions of a topos. Now, what is a morphism, a geometric morphism? Actually, I'm working here not in the category of topos, but in the opposite category. It's a bit like working in the category of frames and in the category of locales. So I'm going to define the algebraic notions of a topos. So, or the algebra, okay. So a morphism and algebraic morphism, algebraic morphism of toposes, let's say from E to E prime. Okay. It's a functor which preserves co-limits and finite limits. That's it. No, no, no, no, no. This way, this way. An algebraic morphism, okay. So in this direction, an algebraic morphism of topos is a functor from E to E prime. It's a functor that takes... That's what you wanted to say. Yes, and this category, I call it a top. It's like the category of rings. Let's say a topos, okay. And finite limits, finite limits, and arbitrary co-limits. And then there is the category of topos, okay. The category is the opposite of that, okay. So this is like here, then when we say that frame opposite are locales, okay. Is the category of locales? Well, we do the same thing here. In other words, locales are kind of fiction with respect to frames because it's just obtained by reversing things upside down, okay. Well, topos are also fiction. They are just obtained by reversing algebraic topos upside down. And these things are rings, okay. And what is the proof of that? It's the fact that the constructions I just gave of the three topos works. In other words, if you take... If you take A category with finite limits. So this is like... And then we know that this is the enveloping topos of the category with finite limits. In the sense that it has this universal property. Well, actually this is a theorem. You take the UNEDA. You have the partial category. So you start with the... Now you take a topos satisfying this condition. Take a topos satisfying these axioms, okay. Take the topos, okay. Now the topos is here. And I take a fronter that preserve finite limits. Preserve finite limits, okay. And then I know that there is a linear extension of it which is what we saw before. It's the left-hand extension, the linear extension. We extend it linearly. Just with the formula. Extend it. Linear. Theorem. This fronter is not only linear. It is multiplicative. It preserve finite limits, okay. This is, in some sense, there should be or there could be the fundamental theorem of topos theory, but I don't want to say that because it's maybe an SGA4, actually. I should check. There are so many things in SGA4 and I never read it systematically. I may have missed this book. It could be there, but really what it says is that there is a distributivity law between arbitrary core limits and finite limits. And from this distributivity law, the notions of topos, which is ear, emerge. That's essentially the idea. And the theorem here is not... I think I will try to give a proof in the notes, but you may receive the note in a couple of months. I have a lot of things to do. But this theorem is kind of interesting to prove. Let me just prove you an instance of it. Let me show you why this preserve finite product, exact. It may say a function of preserving finite limits can be said to be left exact. Okay, so I'm starting with... I'm sorry? Yes, exactly. Okay, yeah. It's a morphisms of this type. It's an algebraic morphism of topos. It's a function that preserves... Okay, so let me try to give you a quick proof of that, but it will be not complete. I will not prove that it preserves all the back. I will just prove that it preserves product. And the proof is trivia. The difficulty is to show that it preserves all the back for product is very easy. To preserve product, you want to show that f of... if x cross y, that the canonical map to f of x cross f of y is an isomorphism. You want to show that this is an isomorphism for every x and y. Okay? So let's suppose that x and y are representable. Let's look at the case where f, x, and y are representable. So in this case, you have f of y, a cross y, b, and it goes into f of y, a cross f of y, b. That's the special case. But the product of two representable... Oh, oh, oh. Yeah. I'm finished. So you want to show that this map... Okay, so in this case where you have a representable, this is y of a cross b, because since the category has finite limits, it has product, and the unit has function to preserve. So in fact, you are looking at a map like this, and this is f of a, and this is f of b, because f, our shriek, extend. So you see this is f of a cross b going into f of a cross f of b, and we're posing that f is left exact. So this map is an azimorphism in the case of representable. Okay. Now, taking product, fixing one variable, preserve co-limits. Okay. By distributivity. You see, by distributivity. So there is a diagram here, and if you use the canonical co-limit, y is a co-limit of yb, and use the fact that this is an azimorphism, you fix one variable, you play with the other, and then you can't do it simultaneously. You have to do it one at a time. It's a function of two variables, which is co-consumers in each, but not simultaneously. You have to be careful about that. And the theorem is done. The difficulty is to prove the general case. It's, you have to slice, and there are something a bit more subtle. Here you see I have used very little of the actions. Now, why is it fundamental as a result? I will be finished with that. It's just, if you have topos in the sense that I have defined with the axioms, then since it is a locally presentable category, you can always find a subcategory a, which is large enough so that the corresponding functor, just because it's locally representable, a to e, is a localization, which means that the right adjoint is fully faithful. That's just because e is, that's a general result for presentable category. You can always do that. Okay, once you have done that, what about if e is a topos? Well, in this case, you could choose a to be closed under finite limits. Okay, and then the inclusion has preserved finite limits. So the extension here, let's call that f, the inclusion, the extension f star will preserve finite limits. It will be a geometric morphisms of topos where the right adjoint, with a right adjoint is fully faithful. Well, this is exactly the other definitions of what a topos is. It's a left exact localizations of a topos. So that prove, at the same time, zero still run and etc. I guess I should stop here. Yes, thank you.