 Oui, peut-être que c'était un peu imprudent comme on dirait, je... En anglais, on dit, comment ? C'est incroyable ! Attention ! Incroyable ! À choisir cela, parce que j'ai pas eu le temps d'écouter avant cette conférence, et j'ai eu ce cours à donner. Donc je ne pouvais pas vérifier les choses que j'ai voulu prouver. Donc... Donc... Oui, le travail en progrès. Et je serai heureux d'avoir votre opinion sur peut-être un peu l'appel ou les conjectures que je fais. Peut-être que vous connaissez déjà comment prouver. Oui. Donc... Ce matin, la dernière parole par Mike Prest a été développée en analogie entre la théorie de Topos et les catégories d'Aberia. Donc cette analogie entre les catégories d'Aberia et la théorie de Topos est très dans le monde de la main de beaucoup de gens. Et je voudrais dire quelques mots. Je n'ai pas de... parce que j'ai beaucoup de choses à dire. Je suis présentement en faisant un travail avec Pierre Cartier sur ce. Et... Je suis un peu intrigué parce que je pensais que j'aurais conçu le sujet. et le plus je pense à cela, je réalise que il y a beaucoup de choses que je ne sais pas. Peut-être qu'une Bible pour les categories abéliennes est la thèse de Pierre-Gabriel, des categories abéliennes. Oui, oui, c'est vraiment une Bible parce que, oui, c'était un grand impact. Donc, je ne prétends pas comprendre tout ce qui est dans ce paper. Oh, oui, bien sûr. Tokyo était avant. Tokyo, 1957, ou quelque chose. Ah ah, 1955, 1958. La thèse de Gabrielle, probablement, était aussi écrite avant la publication, la publication formale. Ok, 1961, oui. Et là, il y a beaucoup d'interessants que je n'avais jamais observé avant. Et l'un d'entre eux est la construction d'une caractéristique d'une caractéristique narrative. Je vais vous donner la construction. So suppose that A is an additive category. By this, I mean that you do have direct sums, right? Of course, zero and finite sums. Otherwise, I would call that a pre-additive category. And in there, there is a construction of, maybe it could be called, well, I'm going to call it like this, a lex of the left exact completion of an additive category. There are many ways to do the left exact completions of by left exact I mean that the additive category now has kernel. You don't ask for an abelian category but just left exact category. I mean an additive category where every map has a kernel, ok? Otherwise, a right exact additive category would be one in which every map has a kernel, ok? And the construction is very simple because what you do is that you take the category of arrows of A and you mod out by an ideal. The notions of ideal make sense in any additive category or even pre-additive category. It's simply for each ARM AB, you need to have GAB subgroup which is closed on the compositions on the left and on the right. So it's a two-sided idea. Yes, right. That's a good way to say it, right? Yes. The ARM functor is additive and we know what is an additive sub functor of a functor so J is just a sub functor of the ARM functor. So it's a little bit like saying that a two-sided ideal is a sub-bimodule of the, yeah. Ok, so what is this construction? You can describe it like this. Yeah, so you have A0, A1. These are an object of the category of arrows. So a map in the category of arrows is a commutative square like this. So I'm describing ARM AB. ARM AB, ARM between two arrows, this is the set of all commutative square. Let's say F0, F1. And now the congruence, or J, JAB, is the set of, let's say, F0, F1 is in GAB. If there exists a map H from A1 to B0 such that F1 equals BH. So what you're asking is for a diagonal, but not exactly a diagonal, it's weaker, you just want this triangle to be commutative. A and B. And Gabriel studied this category and actually he explains where it is coming from. It is actually coming from the constructions of Cartier which is constructions of the derived category of an Aberian category. You see this is the beginning of a resolution or maybe of chain complexes. So Cartier would describe the derived category of an Aberian category using chain complexes of injective objects and take multiple classes of maps between injective objects. And for the homotopy relation, I hope I got it right. You, that's correct, huh? The homotopy is decreasing the degree because the differential is increasing. Yes, yes. Then if you tronquate this construction you get that. So but this construction seems to be due to Gabriel but it was inspired. Gabriel says explicitly in his paper that it is inspired by these constructions of Cartier. I was surprised. This is what Pierre told me also. He told that the idea maybe of a derived category maybe was the beginning of the theory of derived category was you have an important contribution to this. Trangulate the categories. Right, yes. So the nice thing about this is that this additive category, that's an additive category, you take the quotient, the definition is that you take the quotient by J, by this two-sided idea. And now this thing has as kernel. It's an additive category with kernel. And it's the solutions of a universal problem. In other words, if you have, there is an embedding into, let's call it why, because it's a bit like the unit I'm embedding. So y of a is this zero map. So this is how you embed a into alex. And this embedding has a universal property, which is that suppose that e is additive and left exact, which means that it has kernel, then for every additive function here, additive, there exists an extension, let's say f prime, which is left exact though, and essentially unique. Oh yes, right, right, right. My notation has been changing, yes. Okay. There are other ways to construct alex. And Mike pressed this morning was giving another construction of the dual, maybe, which is right exact. But if you want to add co-kernel, you can take what press would call little-mod A-ab, well, op. So you could take abelian pre-sheaves on A of finite presentation. So that's what I mean here. So abelian pre-sheaves of finite presentation. That will give you kernels and co-kernels. Oh, that will be the dual. Okay. But it's very much, okay. So that's another way of describing this construction. Okay. Now, there is a little miracle which is that, suppose that you start with a narrative category which has co-kernel. Suppose A as co-kernel. A as co-kernel. Then you do this construction, not knowing it as co-kernel. You treat it as an additive category, okay. So you have this completion. You add something, you add kernel to something having co-kernel. Well, the little miracle is that first, this has keep having co-kernel. And it's abelian. It's abelian. So that's the miracle. Because to be abelian means that there are relations between kernel and co-kernel, etc. And here you just do the next completion not taking care of the fact that A as co-kernel and not imposing abelian relation, but you do get something which is abelian. So in other words, this is abelian. And not only that, I mean it's universal in the sense that now if you take E as abelian category and you take here a right exact function because you need to take care of that, right exact. Then the left exact extension would be exact. So it would be an exact factor between abelian category. So a consequence of that is that if you want to construct the abelian envelope of an additive category. So A is additive now. So you want to construct the abelian envelope. You first do, let's say, the right exact completion which is just the dual of this thing. And then you do the left exact completions of the right exact completion. And this is abelian as the solution of the universal problem. I'm sure that you know this thing in one way or another, right? I think it was clear in your talk. Yeah, yeah, yeah. So this is the objects here so you call them imaginary objects. Okay, yeah. Yeah. Because they are not modules over A. Yeah, but you call them imaginary. So here you start with a function which is additive. No exact net, just additive. And here this, let me call it f double prime exact. It's an exact function between abelian category. Of course you could do the constructions the other way. I mean, you could take, now if you do it this way you go to A rex lex or you could do the lex and then A lex rex and you get the same thing because of the universal, the same thing, in other words there is an asomorphism between the two things. Therefore there is a commutation operator between completing on the left and on the right or a distributivity law or whatever. It's something commutes. Now, what about AB5? So if there is an analogy between topos theory and abelian category so AB5, abelian, AB5, so-called AB5, abelian categories. Okay, I will call it. It's more important. Yeah. That's why I mentioned the whole thing. Now, if you work with girls in the abelian categories they are really like toposes and in this case these results can be sort of improved or maybe there is a version which if you start with A, a left exact, maybe a right left exact category and you take the unedited embedding so pre-sheaves on it. That's the unedited embedding. The additive version. Yeah, that's the additive version of the unedited embedding. Now, we know that this category is abelian because AB is abelian, okay? But in addition it is also abelian, grottendic abelian category but it's the solutions of a universal problem. So, in other words, if E is a grottendic abelian category and if you have a left exact function F left exact then you will have essentially a unique co-continuous extension which is exact and co-continuous means that it preserve all co-products, all sums, right? So, if you look at this picture what is the left exact? From A up to AB. Here it's a native category with kernel. And you consider a native function which preserve kernel. You don't ask it to be, okay? And then you can extend it into an exact function which is co-continuous, right? Right. So, because of that you can easily construct the free grottendic abelian category generated by a native category, right? And now there is this theorem of Gabriel and Popescu, right? Which is going to say that any grottendic abelian category is a left exact localizations of that and of course the left exact localizations are given by some serre subcategory and the serre subcategory they play the roles of the topology in the abelian context and everything just repeat The whole theory of Topos theory is completely I mean very, very, very almost statement by statement you can find the analog statement people should write a book or maybe I should write one where you make this parallel very, very strongly because, yeah. Okay. But what else there is about abelian categories? Well, abelian categories or actually additive category can be tensed. You see, there are notions of a function additive in each variable. So, you can create a monoidal category of presentable additive category if you want and there is a nice tensor product in there so where should I write this maybe so let's say presentable additive categories presentable really just mean that you have collection of generator essentially and there is a tensor product between presentable additive categories tensor B it's the solution of exhibiting a functor which is from A cross B let's say let's call it theta whatever you want a functor which is additive in each variable and that's it I mean that defines you and examples of a functor additive in each variable is precisely the home functor there are plenty of examples of functors additive in each variable ok and there is a home home A B is just the category of co-continuous additive functors I mean co-continuous I mean you want co limits to be preserved because we are working with presentable category where essentially everything is generated by co limits ok but there is also a sub category of that I mean which consists of abelian categories and it turns out that if A B are abelian then their tensor product as an additive category is abelian so in this way you can construct a closed monoidal two category where you consider a co-continuous functor which are also exact ok monoidal category like this symmetric monoidal category it is very tempting to look at a dualizable object so when is it true that a growth and dick abelian category has a dual in a sense that you have a pairing which will go into ab and unit ab to satisfying the usual identity for duality and examples of that are given by the free one the free free over a narrative category so any of these if A is small maybe I should write A is small additive then if you take the let me take abelian pre-sheaves pre-sheave on A then you can show this is not difficult that the dual that this thing has a dual which is abelian pre-sheave on A op so we have plenty of example of abelian categories that can be dualized growth and dick abelian categories that can be dualized but the question is what are they in general I mean so after hearing my press this morning I'm ready to make a conjecture is that they will be if you take A a small abelian category this time here A was just additive and you put on the topology that would say that every pymorphism is a cover so that's going to be like looking for functors which from A op to ab which are left exact I believe so take any small abelian category and look at left exact functors from A op to ab so I would conjecture or a problem, I don't know it's a bit like a guess so it could be wrong it's a problem I mean that these are exactly the dualizable grossensik abelian category and the dual is what you think it will be just the same description but with A instead of A op you have an opinion? well it's true that one direction is yes which direction is yes? such categories will be dualizable oh that will be dualizable ok ok so it's an open problem ok ok now I want to go to another situation if you take topos you can generalize them in many ways maybe here you could put what could be called two rings so these are a monoidal category with a tensor which is co-consumers in each variable and with a unit object so here I should write set and the unit object is I mean a co-consumers function from set to E it's enough to know it on the value on the point so that's just an object of E so but these are let's say monoid in the category of presentable categories in a symmetric monoidal in a symmetric monoidal no well not for two rings they are just ok but the interesting thing would be the two symmetric symmetric ring so in this case the tensor product is symmetric and then you can specialize a bit further something I have been calling it a paratopos because they are close to toposes ok in this case the tensor product is just a Cartesian product so you have a a Cartesian product well you will have a Cartesian product in these categories too but there is nothing that guarantees that the Cartesian product will have a right adjoint an exponentiation the exponentiation here do exist but it's with respect to the tensor product ok so here the condition is that this thing should preserve so in one variables enough because the Cartesian product is commutative or symmetric so so a paratopo a small co-limit arbitrary yeah no ok no why why is this interesting well first c'est que parce que de cette condition un fonctionnel entre les catégories il y a toujours un adjoint donc ces catégories sont Cartesian si c'est un paratopos c'est que c'est Cartesian et donc il y a une exponentiation qui est le right adjoint à prendre un produit c'est pas local non c'est Cartesian maintenant c'est pas difficile à montrer que un category présentable c'est un paratopos si et non si c'est Cartesian c'est un paratopos pour être Cartesian c'est nécessaire, je suis désolé ce n'est pas ce que j'ai envie de dire parce que ce que j'ai dit c'est trivial parce ce que j'ai envie de dire c'est oui disons, on le fait mais c'est pas difficile à montrer ok ok ok présentable c'est un paratopos si et non si si et non si si et non si c'est présentable c'est présentable si et non si c'est un localisation de pré chif category on dirait un set c set c offset un localisation de pré-chif c'est un localisation préserve dans lequel le localisation préserve un paratopos donc on a un set c up et on a le localisation d'intlusion et il y a E et on a un paratopos donc paratopos exactement comme c'est d'excepter que tu répliques les conditions de l'exact localisation juste par localisation préserve un paratopos et parmi les touristes travaillent totalement paroles pour avoir ce genre de localisation tu dois expérir un class de morphisme que tu veux inverter et tu as que ce class devrait être fermé sur le product n'exprimer pas mais juste pour le product et tout le monde fonctionne Donc, il y a des notions de géométrique ou de l'algebraique morphe, où le foncteur doit être consigné et préserve un produit final, plutôt qu'il n'y a pas d'exact. Mais vous avez juste... Oui. Est-ce qu'il y a un produit entretenu ? Oui, bien sûr, oui. Donc, il y a toujours un objectif terminale et il y a un objectif terminale qui sera préservé par le morphe. Oui. Et tout le grand théorème de Topol Studio, par exemple, vous pouvez décomposer un morphe géométrique comme une localisation préservée par parac, suivre quelque chose qui est conservatif, vous avez un morphe géométrique, tout ça se répète. D'exemple que les caractéristiques que vous obtenez sont beaucoup moins. Donc, qu'est-ce qu'un exemple de ça ? Et c'est pourquoi j'ai été intéressé par ce sujet. Qu'est-ce qu'un exemple d'un morphe géométrique ? Le morphe géométrique. Le morphe géométrique de petites caractéristiques. Vous savez que ce morphe géométrique est fermé. C'est facile de vérifier que c'est localement présentable. Donc, c'est un exemple d'un morphe géométrique. Je peux vous montrer pourquoi c'est une localisation exacte de la caractéristique appréciée. Je peux l'exhabiller. C'est parce que, si vous regardez la caractéristique de la caractéristique, c'est la caractéristique appréciée à la caractéristique Delta. Maintenant, chaque set simple a une caractéristique fondamentale. Vous voyez, il y a une intrusion ici, qui est le nerf. Et ce nerf a une haute haute haute, qui est la caractéristique fondamentale de la caractéristique. Et c'est une théorème, qui n'est pas difficile de prouver que la caractéristique fondamentale est la caractéristique fondamentale. Et le pi1 n'est pas le groupoïde. Le pi1 est le pi1. Le pi1 est le groupoïde fondamentale. Si je l'utilise, dans le set simple, je verrai que le groupoïde est aussi un paratopos. Donc, c'est un autre exemple. Donc, cat est un exemple groupé avec posette. Vous voyez, posette. Posette est une subcategorie reflexive de cat. Parce que la partie de votre set est en fait une catégorie. Il y a un fonctionnement de réflexion. À chaque catégorie, vous pouvez associer un posette dans une façon assez visuelle. Vous dites que l'objectif A est plus élevé que l'objectif B si il y a un héros de A à B. Qu'est-ce que le différence entre un paratopos et un quasi-topos? Oh. Un quasi-topos est quelque chose de différent. Qu'est-ce que vous disiez par posette? Pré-audit set. Partially, partially-audit set. Partially-audit set. I'm sorry? When you are hoars in the two direction. It's free audio. Here by partially PO set. Partially-audit set, right? But... It says in the reflexive some catégorie. No, but this is... Question is whether when you have x more than y, and y is x equal to y. Oh yeah, yeah, yeah, sure. Oui, bien sûr, mais je peux partie, donc vous voulez un set préordinaire, ok ? Oui, c'est vrai. C'est vrai aussi, vous pouvez mettre l'équivalence, la catégorie de l'équivalence et les relations. Et il y a presque cette liste qui peut être étendue dans plusieurs directions, parce que vous avez deux cat, la catégorie de deux catégories, trois cat, ok ? Deux cat, deux catégories, ok, je suis désolé. Tout ce qui est considéré comme une catégorie ? Non, non, non, c'est deux cat. Oh, oui, bien sûr, nous sommes dans une catégorie. Oui, je ne suis pas en train de regarder les deux cellules entre deux catégories. C'est vrai, je suis en train de prendre deux catégories, et deux catégories, etc. Et ces catégories, je veux dire, elles sont probablement reconnues. On comprend bien ce que l'équivalence est, les propriétés. Nous aimerions avoir une compréhension de la catégorie abstractement. Il y a des axioms qui peuvent être mettés là-bas. Le premier, bien sûr, c'est que la catégorie n'est pas fermée. Il y a d'autres axioms qui peuvent être mettés. Mais que sont les propriétés de l'équivalence de la catégorie ? Nous savons, par exemple, que dans les topos, vous pouvez facturer l'immorphisme, avoir de bonnes surjections et suivre l'équivalence, etc. Nous avons toutes les types. Donc, c'est une grande question. Et je ne sais pas. Mais, laissez-moi observer que dans beaucoup, beaucoup de ces situations, il y a une factorisation qui est un couple of factorisation that are interesting. So, let me just discuss one is that if you have a function between two categories, then you can factor it as, for example, as a function surjective on objects. Let's say, E. Follow it by a fully faithful monomorphism. That's one possible factorisation. In other words, the inclusion of a full subcategory. Any factor can be factorised in this way and in a unique way. Essentially, what you do is that you take the image of the objects of A and you take the full subcategory generated by the image. But it looks like a bad factorisation because this map looks bad. I don't know. But there is something maybe interesting which was observed by... You have a objective on objects. There is another factorisation. Right. There is this Gabriel factorisation where you ask for this map to be objective on objects and this one to be just fully faithful and you drop mono. But I'm choosing this one because... Well, because I want to... There is a famous result by Dana Lach with another auto. I forgot to the moment. I'm sorry. Apologise. Which says that if you push out the push out of a fully faithful factor, the push out in cat. So if you have some fully faithful factor, it could be mono on object or it doesn't matter but it is also fully faithful. Now, what it seems to say is that a cat the opposite of cat is actually an exact category. In an exact category you expect that the pullback of a surjection is a surjection. This is not true at all in cat but the opposite of cat has this nice property where the surjections now are played by fully faithful embedding. So I'm giving this as an example because I think that this kind of result should be true also in higher categories. In other words, the latch theorem should be true also if you were to take infinity categories where it's probably much harder to prove. So infinity one categories that should be true also. So my understanding of cat is very poor. I only know that this is essentially cat is enclosed that it has this kind of property but it would be very nice if you could extend topostheory to this kind of category possibly by adding a few axioms I think it's very much needed given the importance of cat of two cat and their weak versions infinity categories infinity one and also for two etc. we should be able to extend topostheory to these categories. Ok My how much time? 10 minutes what about ok sub-object classifiers oh they are not there that seems not that I know to make difference topostheory yeah yeah ok 10 minutes 10 minutes yeah I would like to discuss with you pointed set this is not a topos because the initial objects and the terminal object coincide so the question is if it's not a topos what is it because especially with pointed spaces see this plays a big role in topology pointed spaces and so abstractly what is it so I would like to make first an observation that there is given a category any category C you can describe the category of families of objects of C a family of objects of C is what you think it is, it's a family of objects of C and a map between two families is also what you think it should be because what can you do to map this into this if you don't have any other information first you need to have a map from I to J and and for each I you need to have a map from A I to B of Phi I or all I in I so a map between these two consists of a pair F and no how to call it F bad notation yeah anyway it's a pair maybe capital okay it's a pair consisting of a map from I to J downstairs plus maybe a family Phi I okay of maps like this yeah okay so that's given a category C you can construct the category of families of objects of C now this is the co-product completions of C times C as co-product arbitrary co-product and the idea is actually that this family is somehow a formal co-product of the object A I you should think of it as the co-product of the A I the co-product are purely formal so they may not exist in C so I'm going to make sure that we know where we are so I'm going to write a bracket here just for the inclusion it's really like the unit of function and actually I'm looking this is actually some category of the appreciative category and I'm looking at co-product for the presentables okay so and this is free this is if you were to map C into a category with co-product you will be able to extend it along the unit I'm betting to a co-product preserving function from the family of C to okay so observation if I take family of let's say set of pointed set a family of pointed set what is a family of pointed set this is the same this is the totals this is the same as co-variant on the category which I'm going to describe like this section retraction and here identity so you take a category with two objects and which I call zero and one and SR is the identity with two arrows generating arrows and you just ask this equation and if you have a function from this to set you will have two sets X0 and X1 you will have a retraction and a section okay and of course if you take the fiber of the retraction at a given point then you get a pointed set because the section is providing the base point of the fiber so you get a pointed set and in this way you get a family of pointed set and the converse can be done so there is an equivalence of category between families of pointed set and pairs of sets equipped with two operators like this satisfying this equation and since this is a category of diagrams and sets this is a tuples so that's the moral of the story is that we achieve at least two elements that are going in fact it's a clue how does it work is one element just projector oh yes you could even have it with a projector yeah that's true so definition I have a name but I'm not sure a locus is a pointed locally presentable category such that if you take family a locus E take family such that is a tuples I'm sorry a pointed category it means that the initial object and the terminal object coincide what do you mean by pointed category it means that the initial for a presentable is a locally presentable point point category oh which name would you like me to use what's a pointed category it's a category where there are two notions it means that first that the arm sets are pointed sets that's one way or you could ask for a base point I mean for an object terminal and there is a very close to the equivalence between the two right ok ok well I would like to give you examples of locus and this notions becomes interesting when we look at infinity tuples there is a notions of infinity locus which is just the same but families here by a family an object here i maybe 5 I don't know normally you would describe a family as as a functor from the discrete category i to phi but in the context of infinity tuples i should not be discreet i should be an infinity group that is a nomotopy type right this is for but it makes sense if you have an infinity category you can consider families of object in it indexed by an infinity group actually by any same pieces set for example it's a very very standard notion in infinity category theory so here maybe I should may underline this because we are no longer for example I could be given by a group then that would it's a category with one object but where the automorphism are non trivial let's give a group and that would produce like an object in e equipped with an actions of that group so ok and then if you take the category of spaces but in that case it's a tuples I'm sorry in that case it's a tuples you can do the action it's a tuples this is for e an arbitrary infinity one category right I'm not supposing that tuples right so now you could consider pointed spaces and families of pointed spaces ok families in this sense and this is again the same relation as before so this relation shows that a family of pointed spaces family in some extended sense is actually the same thing as a map of spaces equipped with a section ok yeah I will 100 seconds ok what's interesting here is that there is a notions of a spectrum of pre-spectrum a pre-spectrum in topology is a sequence of spaces x0, x1 etc together with map from the suspensions of xn to xn plus 1 for every n and there is a category of pre-spectrum and it's very easy to verify that this is a locus well actually it's pointed spaces yeah that this is a locus in other words you can augment this description with a variable base space and this is suspension over B so you have base space with sections and I'm taking the point y suspension it's a pre-spectrum over B that you describe and then the classical theory of pre-spectrum is that pre-spect is that there is a function to spectra spectra a spectrum pre-spectrum is a spectrum the adjoint map from xn to omega of xn plus 1 is an equivalence that's called sometimes an omega spectrum or etc that's the idea of a spectra it means that you have a sequence of space with a sequence of equivalence normally what you have is a pre-spectrum and then there is an associated pre an associated spectrum non non now the function that takes a pre-spectrum to the associated spectra there is an explicit construction and this is a left exact and just by construction it's a cool limit of iterated loop space and it is a left exact pardon on est en train de travailler dans l'infini-world ou dans les années que vous êtes en train de travailler on est dans l'infini-world ok mais ces deux catégories sont pointées et vous pouvez faire cette construction de famille donc vous faites construction de famille sur les deux côtés alors vous considérez les familles de spectra et ces spectra sont sur bases où les bases peuvent varier ok vous voyez que la catégorie des familles de spectra est la plus exacte localisation de cette catégorie mais cette catégorie est une totale ok c'est une totale parce que de cette sorte de construction ici parce que de cette sorte de dictionary vous pouvez c'est pas difficile de montrer que les catégories des familles de spectra sont les totales et donc les catégories des familles de spectra sont les totales ok, maintenant c'est quelque chose qui a été observé par plusieurs personnes j'ai déjà appris de Georg Biedermann en 2007 et Charles Rex observé ça aussi en même temps et c'était venu d'un bon calcul mais la preuve que je fais en faisant ici est complètement élémentaire en un sens parce que ça ne utilise pas le calcul c'est complètement standard almost standard construction et la chose fantastique de cette regardez la catégorie de spectra c'est très loin d'être dans les totales très loin peut-être que des personnes vont dire que c'est très loin parce que c'est comme une catégorie ambilienne c'est comme une totale aussi mais si vous justifiez les familles de spectra les bundles de spectra ok, il y a un bundle où le morphisme c'est ce que vous pensez que ça devrait être ok, le morphisme c'est une map entre les bases pour que si vous reposez E' il y a une map sur le repose donc F' c'est cette catégorie de spectra par Peter May il y a un livre sur ça mais ils n'ont pas observé que c'est une totale bien, à ce moment les totales n'existent pas vraiment mais il s'imagine que l'un peut embêter la théorie des catégories stabiles dans la théorie totale que je ne sais pas si l'embêtissement est utile mais je dirais que si vous prenez une catégorie infinitive stable que si vous prenez des familles, exactement les mêmes constructions des familles d'objectif il doit être complet donc vous devez avoir un co-productif je veux dire la catégorie de spectra, de catégorie stabiles que vous regardez si vous faites cette construction que vous prenez des totales chaque fois incluant les cases de complexe de chaine je suis regardé un maximum peut-être un conjecteur et c'est un maximum peut-être juste poli si j'avais du temps je ferais un argument de soutenir mais je n'ai pas de temps merci