 I thought this would be very, very sketchy, because this is a subject that is growing and with contributions, important contributions by United peoples. First, it was the idea of a higher purpose, and it was about Charles. Charles, right? He is an homotopy theorist. Actually, I had the idea, just for a minute, in 1985, that one could possibly do homotopy of exact localizations of a diagram continually, because I had a monostructure for simply such sheaves in a general growth of the topos, something that I have described in the letter to growth of zinc, which was never published, but the letter circulated. And then I asked a bit why there was a meeting in Bangor if he knew that exact localizations of the categories of spaces. He immediately told me there is not, except the trivial. The trivial one collapsed everything to a point. The non-trivial one, so to speak, is just the identity, and that's the only things you can do. And, oh, I said, oh my God, this, my idea is wrong. So I stopped thinking about it. Of course, Dwyer was right. There is no, I had, my question was wrong. The question I should have addressed to him, do you know any left exact localizations of a category of diagrams of space? This would have been the right question, okay? But I asked the wrong question, and so I stopped thinking about the subject. This idea. But Charles Rex, maybe in the late 90s, understood that the category of spaces and diagrams of spaces had very nice property. I mean, sometimes we think that the category of set is very nice. I mean, Hilbert said that no one could keep us out of the paradise that Cantor has created for us. And I still believe that the category of set is very nice. But from a matter of your point, sometimes the category of spaces are even nicer. What do you mean by spaces? Do you mean selfishness sets? Yes, that's a good question. Yeah, you could take the category of spaces, a couple of spaces with, but you have to somehow invert a weak motor vehicle. There's a model structure there. And every space is weakly equivalent to a CW complex, and therefore you could take the category of CW complexes. Except that the problem with the CW complexes is very nice, but it's not cataclysm-close. You take the hum between suits again. So you don't see what happens if you exponentiate. And we would like to have a category which is close under exponentiation, for example. And therefore CW complexes is not the best thing to take. CW complexes turns out to be very good, because they are close under exponentiation. And so a good approximation for the category of spaces is the category of CW complexes. Yeah. This exponentiation problem, by the way, was just a critical question for setting this disease about group spaces and so on. It's a typical case of exponentiation. Oh, yes, right. Do not use CW complexes or that. That's what you have to reinvent it as you feel. Aha. You use a cubical set in this case. Well, but don't use that. Okay. Yes, right. Yeah. Yeah. Yeah. So maybe just a small picture of, I mean, in my talk yesterday, I said that we study spaces by looking at maps into a ring, continuous map into, or maybe smooth map or algebraic map into a ring. And so we look at map R, and we formalize the properties of this object. It is also a ring. And in the case of locale, the ring, or in the case of locale, is 0, 1. I mean, with the topology where 1 is open, the Sierpensky space, that's the theory of locale. No. No. No. Because the topology is... Yes. No, no. There is no... The negation, the... Let's call that Sierpensky. No, no. What I mean is that, okay, the sum is the obvious sum, with 1 plus 1 is 1, and the product is the ordinary product. Yes. Oh. Yes. Okay. That's what I mean. Ah. Yes. Yes. The only thing is that you change the topology to make a semi-continuous function. Yes. But they are actually continuous. I know. But what I mean is that the topology that you take is really the topology of semi-continuous functions. Of course, semi-continuous functions. Because 1 is open, and 0 is not. It's a semi-interface. Well, the semi-continuous functions are continuous. For this, when you choose a certain Sierpensky topology. Well, that's what I choose. Ah. You know that you are laughing, but... Because the point is that, for instance, if you take values in the r-max and such things, okay, then the topology on r-max would be... the correct topology would be the semi-continuous topology. So, to imitate the Sierpensky topology. Well, I guess we are talking about the same thing. We need a slightly different language. This picture, I mean, somehow, there is a map to the Sierpensky space. What did they... somehow the text open some set. Actually, the video description should be open some set. But what happened in the case of the tropos, they maybe as a Cartier was saying that with multi-valued functions, you have a space X right there. And with the multi-value functions, you have some kind of covering of our etal space over X. And the functions itself is defined here. So, that's somehow the picture. And the idea of Grupensky was to say, well, let us study all the etal space over X. Okay? So, this is the category of sheaves over X. This is the tropos of sheaves. This is... And then, you can express this map from E to R by pulling back R to X and looking as a map from E to R X in... So, you put back X to R and take the product and you internalize R and you may study maps from E to R X if you like. Okay? Now, if you want to study homology and homology of sheaves, you need to replace E by chain complexes and I'll have again groups, for example. There could be two chain complexes over E over X. So, they are objects over X and look at the resolution, objective resolution technology, et cetera. But another way of doing this already in affinity is to look at senbichel objects senbichel objects in X to internalize the notion of senbichel object you get senbichel sheaves and do the homotopy theory of senbichel objects in X. So, you mean this is for the Dalkan correspondence? Yeah, but you can do it non-Abelian if you like and you get non-Abelian homology if you do that. There is a model structure here. When you say in ex-Amenian sheaves of X. Ah, yes. So, in other words, in this picture you replace instead of just the pure sets, you have arrows, for example because the senbichel set has arrows and maybe triangles you look at structure like this over over X. Okay. So, this looks very much like a combinatorial representations of a space over X, because at each point you will have a senbichel set which is equivalent to a space by I have to be question. I mean, if you take senbichel object in a sheaves of X, is that a topos? Yes, it is a topos. Right. It is a topos. Yeah, but the one thing it's maybe the first one of the first examples of an infinity topos because the notions of isomorphism between object here is not the notion of usual notions of isomorphism it's notions of equivalence or formative equivalence. So, if you use this notion you do get an example of an infinity topos. Okay. So, so topos theory in some sense is about replacing the Sierpensky space by the category of sets. So, let me draw it like this. In the theory of local or frames you contour maps from X to the Sierpensky space. That's the basic thing. In topos theory you contour maps from X to the category of sets because a family, a sheath can be regarded as a family of set, a sheath over X. You have a point and if you look at the father so for each point of X you have a set and so you look at families of sets parametrized by the beds and so you're actually looking at a map from the space to set. So you start to sort of considering that the category of set is a space itself because a sheath can be regarded as a continuous map from the space to this kind of new space. And this point of view is completely correct in the sense that in topos theory there is a copy of the category of set which is a classifying it's like the affine line it's a classifying object topos and a point of this topos is a set it's like if you look at it a point of this topos, a geometric point of this topos of the set. So now a geometric morphism is the same as choosing an object here because algebraic can be it goes the other way and you just need to interpret x into u so from this point of view an object of the topos is the same thing as a geometric morphism into the space of all sets this represents the space of all sets okay and higher topos theory we want to replace higher topos theory we want to replace the category of set by the category of spaces so we want to have a new form of topos where this would be true that an object in the topos would be regarded as a smooth map from x indicating your space but this is just a geometric picture algebraic can be would have the classifying object topos and I write p because it's about spaces and have the same picture algebraic can be so in other words instead of looking at families of set parametrized by the base point of the space we want to look at families of space parametrized by this point of the space that's what's happening with higher topos theory there's a new notion of space which is based on the idea that a sheaf here is no longer a family of sets parametrized by the points but it's a family of space typically for example bundled in bundled theory there's a topology if you have bundled bundles of geometric kinds of bundles the bundles normally is not discrete the gluing maps are not discrete and there's a problem in geometry you can go around the fact that this is not discrete by looking at gross topos sheaves and etc but you need to go to gross topos for that now we want to take a different point of view but the inventions of the higher topos by Charles Reck I don't know exactly what was his motivation but his observations is this this set so what is this set suppose that you have an object in a topos which is a coordinate of a family so suppose that you have an object let's say b which is a coordinate of some family or diagram so you have a diagram let me write the diagram here I have dr the object in the diagrams and b is a coordinate now if you have you would like to understand how to build things over b in terms of building things over di which mean that if you have some object over b and this is the projection pr you do pr star p and you get a new diagram over your base diagram and you would like to understand in which sense e is obtained from the pi and you would like to put some descent condition so that you have an equivalence category of category between e over b the topos and here I'm pulling back so I have the little pieces e over di and I have a diagram of toposes so I would like to take the projective limit of this in some sense and I would like to say that there is an equivalence of category the diagram is given the diagram is given and we would like to know for which diagram is this true that I will give you examples where this is true and examples where this is false suppose that you have a co-product of two of two things you have first inclusion and B conclusion and this co-product is actually the pull limit of the discrete diagram with two objects A and B and therefore it leads to a diagram like this and you would like to say that this is a product diagram so for each object here you get two objects and you get a filter here and we all know that this is an isomorphism because if you want to construct a bundle over the disjoint union of two objects you just construct something on the first and second and second so the descent condition is true trivially another example is playing with the cleavage relation and the usual descent condition I will not do that I will just give you an example where the general descent idea which is false we will look at the case of a push out and the push out will be really very simple and this is an example by Charles Rex we take two points you map to one point you map to another point so it's two two one so this is just the sum of the territorial object with itself and the push out of that is just one point so this is a push out diagram in any topos in fact this is a push out now let's look at the dual diagram so we have E over one which is E E over one E over one E over three and we would like to know if this is a comeback ok so what is the fabric rock of these two maps so I can write E instead of E over one and ok so I would like to take the fabric rock of these two maps what is this it takes an object X and it takes it to X plus two copies of X over one plus one in fact it's X plus two and here the same thing so we want to look at the pullback now the pullback should be of course in categories so we should have two objects here so that their image with an isomorphism between the two we are not going to ask for inequality so suppose that we have two objects and we make two copies of them we have a first copy for X and second copy and a copy for Y and then we take another model between X plus X and Y one ok another morphism that would be the descent data another morphism between these two things another morphism between these two things would be two asomorphism completely independent so what is this category that you put back it's two objects different and two asomorphism between well if there are two objects like this you could use the first asomorphism to identify and for the other asomorphism you get an automorphism so actually this is the category the pullback will be E to ok E to the loop I mean I'm just a little loop actually this is Z Z is as one invertible generator so it's the the category of Z actions of E which is of course very different from this therefore the descent conditions cannot go it's one of them it's one of them what we have yeah but I remember I had talked about this the descent conditions you wonder this kind of questions and in my early days I would say ok I think I even constructed this example and I say oh it's terrible the descent conditions cannot be true in general ok this was a wrong idea ok the idea that the descent conditions should be true is wrong but Charles Rex said it should be true and we need to change the notions of purpose and this is how higher purposes were invented so what happened with the idea of Charles Rex is that if you look at this diagram if you draw the arrows this diagram in the category of spaces in the category of spaces this diagram has an automorphical limit and the automorphical limit I don't know if you're familiar with that it's just the ordinary you constructed using cone I mean if you want to compute the automorphical limit of something you construct the cone or maybe the cylinder you construct the mapping cylinder of the first map ok so you have A becomes its inclusion in the mapping cylinder and then the mapping cylinder of the second map and so you have A mapping cylinder of the first map U it's included included in the mapping cylinder of the second map it now is included and now you take the push up mv union over A ok taking the automorphical limit was a kind of art for a long time I think the idea of the automorphical limit was first introduced by Graham Siegel the idea of Graham Siegel I guess was that if you have a diagram indexed by a category let's say even in set the usual the limit of the diagram is actually pi zero of the nerve of the category of elements of I so you take the category of elements of I you take its nerve it's a simple object and you take pi zero and that's equivalent and so the idea of Graham Siegel was why should we take pi pi one pi two actually zero in his book on very fatal non-community non-community right, right, right he has a big limit for diagrams because of the big limit and the big limit actually is the fundamental groupoid of this gadget so I guess that group and they can zero already had the idea of automorphical limits but they stopped at one maybe because they didn't have the formalism so the idea of Graham Siegel is that the automorphical limit should be the automorphical type of this so this is called H limit of D and then people I guess can and Bardsfield understood how to construct a motorpy limits, I'm sorry, this is H code limit, I'm sorry H they understood how to construct motorpy limits and things started to develop from that, people in motorpy theory were using motorpy limits motorpy code limits all the time and Charles Beck if you apply the constructions of the motorpy code limit the motorpy push out what you get here is this figure you see you have let's say this is the point A and B and now I have the A, B X, Y and you have yes X is X and Y you have this if you look at the diagram you get this thing you see a circle aha we have a circle therefore if we change the notions of motorpy code limit by the notions of motorpy code limit we will have a general descent theorem but for that we need to have new subjects the motorpy code limit of a diagram of proposes will be a celestial object for example but then you need to rework the whole theory of proposes using spaces or celestial objects this is how the subject started maybe just a word or two to finish so I mean I was fascinated when I was reading the stuff about the motorpy code limit by the fact the sizes of motorpy code limits were absolutely needed in order to have a good compatibility with fixed points and finite limits so I presume that so then this means that in certain circumstances working with ordinary code limits is totally wrong and one has to elevate the situation and work with motorpy code limits so and this is what people do in algebraic topology very currently I mean they use this technique all the time and it's very surprising indeed that then you have this amazing compatibility that for instance if you take if you have a finite compacting on the various terms of the code limits in order to take the fixed points after taking the motorpy code limit it's the same thing as taking the motorpy code limit we have discussions about that and we need to discuss that because I disagree is this the motivation somehow I mean you know the fact that one wants to replace and have a coherent salary the motivation for the amazing thing is that everything we have done so far with tuples so this can be generalized to spaces so you start with so called infinity one categories but or infinity categories we are just let's say categories in virtual work can complexes for example okay and then there is a notion of pre-sheaves in the sense that you need to harm these things into spaces spaces and then there is a notion of localization and of that exact localization and you do that and you get the general higher purposes that's how they are constructed and everything works this is is there another way to think about it because we were saying that if you take central objects in a tuples they form a tuples but then the new ingredient is localizations yes right so could you think of it as performing some kind of localization in a given tuples rather than higher tuples yes you can yeah there are variations right because some aspect of this theory was already present in the work of Roy Wojcicki also because introducing a mutipi metal with algebraic geometry looking at pre-sheaves which is a sebaceous value of pre-sheaf things of this sort so the theory of higher tuples is not coming from nowhere there are various roots but the fantastic thing maybe is that essentially a higher tuples is a characterization theorem of Dury which is that a presentable so-called infinity category don't worry about the terminology but this is essentially categories of over 70% or there are very small is a higher tuples and being higher tuples means that let's say it's a mutipi that exact localization of this kind of thing if and only if the descent condition is true so and the descent conditions means that e over a coelibit of a diagram can be equivalent to the limit of the it becomes completely general and it's even only if this is zero theorem but everything is captured by one descent condition which is general can one go between you say infinity but can one word insert infinity or something between like hand hand tuples how do you like to look at only infinity well yeah that's a good question I don't know what time I have now I still have 35 minutes oh good I almost said everything I wanted to say so at the moment everything is perhaps formal there is also an ocean of elementary we don't know what it should be elementary higher tuples we don't know exactly what this should be but it's based on the theorem of Lurie that if you take Kazna I think it should be regular but maybe it's not Kazna then in a higher tuples there is an ocean of small object so you look at k small objects and also k small maps so a map is k small if the object exp the slice tuples e over y is k small the notions of small map is somehow feather wise you just relativize the tuples over y so there are notions of k small k small is like yeah it's rather standard let me not okay and then you can look at the category of k small maps with a Cartesian square so k small you look at the category of Cartesian square where the vertical maps are k small but with squares which are Cartesian, they want to be Cartesian and this has a final object which in other words there is a universal I write omega just because I want to stress the analogy with a classifying sub-object tuples there is a terminal object the fact that there is a terminal object means essentially but there is a universal k small map as long as k is large enough you need to put some choose k okay and then it means that given a k small map there is a pullback square like this and the pullback square is the map here the pair of maps here is only to be unique so you may think of this as the universe classifying k small maps it's a direct generalizations of the omega where in the case of omega what it classified is the universal monomorphism Jesus, I'm not monomorphistic no, that's the point in fact if you allowed to a little bit of imagination you could remove the k and then you say in the topos there will be a universal map okay in practice there is because whatever you do you always work with k small map for some k so there's always a universal thing and this map here is special because this was maybe observed by one of us here not in the context of topos theory but in the context of mutipita theory this map is univalent so I would like to explain what univalent means I will consider univalent maps in the category of spaces so there are many approaches to univalence but suppose that you have a map in spaces space in the category of spaces so you may think of this map as describing a family of spaces index its equivalent to a family x i or maybe x y spaces indexed by y and univalence means that you don't have a repetition in other words univalence means that x y as a morphic to x y prime should implies that y so this is a universal case yes exactly it's a classification ok now the main thing would be to say y is equal to y prime but no or monotopy so that there is a pact between y and y prime and not only that there is a pact but actually there is an equivalence between the asomorphism here and the pact between y and y prime there is a complete an equation between I would say the space of asomorphism from y to y prime or I would say equivalence monotopy the space of monotopy equivalence between the two fibres and the space of pact between y and y prime that's the notions of univalence and these are classifying maps there are all univalent this is typical of a universal fibrations of any type if you have an anthropology let's say that is based on BG for instance yeah right it's construction exactly exactly one remark the isomorphism which you erased namely x over A plus B is the same the same and the notion of case mode maps yes which you are defining now I think there are notions which should be expressed not for the situation you are looking at but for fibrations fibred category just explain why if you have a vibration you can say for example if you see the vibration as the fiber over some i as families of object indexed by i object of then you can say families of small objects okay independently of the fact what you are looking at is the vibration such that the fiber over each A is A over A but this can be defined for any vibration the same goes for example let me give one example which doesn't fit no no which doesn't fit with what you say suppose you have an internal category internal you can define a vibration by looking at just map into this internal category this vibration will always fiber over B equal and many other vibrations that's a property of vibration let me comment it's a notion of fiber gain not just the vibration for which the fiber over A is E over A let me comment what you just said in your work you have constructed given a map an internal group associated to that given the map for example the objects are the base and given two points let's imagine you have elements the arrows between the two points are the azimorphism or you have even constructed an internal category and in some sense I'm not saying the special case is not important but what I'm saying is that the general situation should be stated in terms of vibrations and then look at the vibrations whose fibers are fiber over A is E over A this is a property of let me comment what you just said I want you to hear I think we can continue this discussion so you have constructed this group associated to a map you may think of this as a a situation of a fiber category you may think of the object over the base and then you have constructed this internal groupoid in the base that represent that and I like this construction and I use it as often as I can it's a very interesting constructions and somehow it is saying that this groupoid this is an azimorphism between two points of the base it's reflect an azimorphism between the fibers that's what you have done now I agree with this philosophy where there is a difference is that what you are comparing here are two objects let's say of a topos if these things are groupoids themselves in other words if there are two vibrations or whatever there is a problem here you will have to put two categories because here you will have to put equivalences so that's the problem in higher topos theory we have everything we don't need to filter things by one category, two categories etc I am not saying anything about higher topos I am saying something about some ideas you may have about families and these ideas are encompassed by the vibrations ok let me add also something about that that what is an elementary higher topos it's still a problem at the moment people using working in so-called homotopy category which is category with homotopy interpretation they want to develop a formal logical system that would fit higher topos theory you don't always say that but I don't know if Thierry would agree with that but Thierry I don't know enough ok but some people at least maybe Mike Schumann and other and so the problem is that these higher topos are hard to formalize using just category theory because we are talking about homotopy pullback homotopy push out everything is up to homotopy and this makes things difficult from the point of view of category theory so there are proposals precisely to develop a kind of model based on query models categories or something like that where fabrations plays an important role in other words a family of objects like P for example here I said there is a family of objects that this family should be expressed by a fabration so things are developing there is not yet a precise formal system of that I am personally thinking about the problem I don't have a complete solution I am sure that your work is relevant your work on fabrations is relevant because in the axiomatic of an element of the idea is that there is a notion of a fabration and then a universal one in addition so instead of having an object omega there is a universe which classify fabrations and this fabration is univalent that is very important that the discovery of Wojcicki Wojcicki was learning type theory and he saw that there are universes in type theory but then he saw that these universes should be universal in some unique way and this is where he had the idea of introducing the notions of univalence for the universe and it turns out that if you take this formal system for multiple type theory where you have a univalent universe you can prove things in type theory that you cannot prove without univalence but it is a very strong axiom it has interesting consequences and some people were able to for example to compute a few multiple groups of spheres using this axiom so it's interesting that possibly we will eventually maybe have a way to compute multiple groups of spheres purely by using logical reasoning type theory is really logical reasoning based on univalence but this is a big project it may take years before giving something really powerful but there is a potential so maybe first can you make a remark I think the situation that Wojcicki called univalence the statement that you have already he proved that because the kappa small object classifier densifies it really up to a multiple he proved that the group point of kappa small max over x is equivalent to the group point of max he doesn't have the name maybe he doesn't he doesn't have the sexy name univalence now and of course he doesn't bother about this rectification issue which is maybe the the most you cannot read a higher proposal I mean Wojcicki is a very interesting character once I had a discussion with him where a group of people and then he said maybe he was over enthusiast he said in his enthusiasm category theory is finish okay I don't know I think he would like to use for foundations of mathematics and it would be that formally is not category theory it's a dream who knows maybe this I don't know I have a different opinion but it's a matter of opinion so it could be that Luri already had the idea of univalence before he certainly has the formal statement for infinity he just doesn't consider it in this rectified statement but what do you have on the board maybe just to finish about univalence it's kind of fun to do the following you take a fabrication which is not univalent you take a fabrication which is not univalent so it means intuitively that there are fibers which are asomorphic and the paths between these two the base points don't match very well the asomorphism so you feel that maybe you could even have fibers which are asomorphic and so therefore you could reduce the family it means that you have many copies of the same thing and you could maybe compress and in fact there is a compression give a fabrication you can always compress compress it you can always find another fabrication peeperon univalent in a pullback square and how can you do that well you can do it directly or if you have the universal fabrication it's very simple you classify your fabrication in omega prime omega you classify it and you take the image and you take the reaction for the variable of water you see the incomprensible incomprensible incomprensible fabrications those that cannot be compressed further or they are univalent fabrications they are not necessarily universal you see but they are all somehow sub-objects of the universe so the universe is really a union of all univalent fabrication that maybe I'm not universal okay I think