 perhaps I think you can start with a high octopi. Okay, I'll do that. All right, so I've told you enough about infinity categories to get going with, I hope. So let me recall a definition of Groton-Dictopos. This is my favorite definition because it's so compact. So a Groton-Dictopos, it's a category E, so there exists a bunch of stuff. So you have a small category C. These are one categories. I'm in one categories now. A fully faithful functor from E to the category of pre-sheaves of sets on C, which has a left adjoint. And furthermore, that left adjoint preserves finite limits, so it's left exact. So there are the left exact localizations of pre-sheave categories is how people often say this. So you can show, it's well known, that there's a correspondence between these left exact localizations of the pre-sheave category and the Groton-Dictopologies on C. So this actually requires so this actually recovers the very familiar, familiar to many people, characterization of Groton-Dictopoi as sheaves on a Groton-Dict site. Let me rant a little bit about this. This is well known. This whole theory, it's in the SGA, for instance, goes back to the beginning of the subject. It's not as well known as it should be, at least when I was young and learning about all these things, learning what a Topos is, learning what Groton-Dictopologies, I did not know this fact for a long time. I knew about a Groton-Dictopology and that seemed interesting, but kind of complicated and a little ad hoc. And I didn't understand that well, it actually has this very simple characterization. Of course, you need to know what Groton-Dictopologies are in order to work with us, but it's a very cute characterization. You know, some books on, you know, Topoi don't mention this at all. Many do mention it, but it's not in chapter one, so I never ran into it for a long time. So that's what a Groton-Dictopos is. It's convenient to not have to mention Groton-Dictopologies at this point. We'll see why later. But the one reason is that we can take this definition and automatically port it into infinity categories. Oh, before I do that, let's notice that some of the properties of Topoi actually just fall directly from that definition. For instance, pre-sheeps of sets are Cartesian closed by a standard argument. You know how to write down the internal function object. And if you take one of those things, and if you have one of these left exact localizations, let's say E is actually a subcategory. You have this left adjoint, which is called sheetification. Then if you have an object Y in the subcategory, then it's easy to check that the internal function object is also in the subcategory. And in fact, it's an internal function object there. This actually only uses the fact that L preserves products. That's the property you need. One fact about Topoi is that the slice of a Topos over an object is also a Topos. So if I have pre-sheeps unseen, and then I have one of these full subcategories E, and I pick an object in the full subcategory, I get functors on the slices. The inclusion restricts to a fully faithful functor on slices. And the left adjoint restricts, because X is LX or isomorphic, restricts to a functor on the slices, which is a left adjoint, and in fact is left exact. And this category of pre-sheeps sliced over X is actually equivalent to a pre-sheaf category of sheaves on some comma category. So from this definition, it's kind of easy to show that a slice is a Topos. And then you can also produce sub-object classifiers. It's easy. You can construct the sub-object classifier in pre-sheaves. And then the left adjoint gives you an item potent on that sub-object classifier. And you split off that item potent. That turns out to be a sheaf. And that's the guy. That's a theory that's probably familiar to some people. Okay, so this is a nice characterization. And it's the one that I'll generalize. So an infinity Topos. So I've put in red the things that change from the previous definition. It's an infinity category E. Since there exists a small infinity category C, an accessible and fully faithful functor from E to the infinity category of pre-sheaves on C valued in infinity group voids. I will use this notation a lot. This will always mean pre-sheaves of infinity group voids. I can't spell pre-sheaves. And it has to have a left adjoint. And that left adjoint has to preserve finite limits. That's the definition. I do need to explain accessible. So that's a technical condition that did not appear in the original definition I gave for a Groentink Topos. And I don't want to dwell on this. It's technical. And it comes from the infinity categorical analog of the theory of accessible categories, accessible one categories. So I've written out the definition here. If I have some co-complete C, I can talk about a functor preserving kappa-filtered co-limits. That's what an accessible functor is. Or kappa is a regular cardinal. And the functor is kappa-filtered. So a kappa-filtered co-limit is a co-limit on a kappa-filtered category. And there's a definition of kappa-filtered category in terms of extending cones of kappa-small infinity categories. So if you take out the infindies here, you get a very familiar notion. In fact, for omega, a caramel cardinal, in one category, that's the notion of a filtered one category. And there's these cardinal generalizations. And all that carries out in infinity categories. This is done by Jacob Lurie in his big book. In a Groentink Topos, the one categorical notion, accessibility of the inclusion follows from the other actions. So it's accessible too. I just didn't need to include it as an axiom. Here I'm going to want to. Presumably, I could leave it out, but I don't think that's been very well-studied. So we'll stick with this. All right. While I'm at it, let me introduce this other concept called a presentable infinity category. And this is the infinity categorical generalization of what's called locally presentable infinity categories, locally presentable categories, excuse me. If you don't know what that is, it's good to learn about it. I wish I'd known this much earlier than I learned it. A presentable infinity category, it's the same list of definitions, except that I'll drop the fourth one that the left adjoint be left exact. It doesn't have to preserve the finite limits. Almost anything of consequence ends up being a presentable infinity category, or in one category is a local presentable category. Categories of algebras like groups, rings, those are all locally presentable. So this is a very large class of things, which are always complete and co-complete and have various good properties. I don't want to dwell on this, but it's an interesting and useful concept. So one question you may ask is, okay, I didn't use Groton-Dictapology in my definition, so where are they going to be? And I will come back to this question later. The following is true though. If I have a Groton-Dictapology on a one category, a usual notion of a Groton-Dictapology, then I can form an infinity topos, which will be a full subcategory of the pre-sheaves of infinity categories on this one category C consisting of sheeps. For instance, in the special case when I have a topological space, then C is the poset of open sets on X, the Groton-Dictapology is the usual topology, and the sheaves are the ones I defined in the previous hour. It's literally that definition I gave. And I could give a similar definition in the case of a general Groton-Dict site. So these are examples of infinity topoi. Are they all of them? We'll find out. All right. I do want to make sure I don't go over on time. There's a lot of material here, some of which I can pass over quickly. So what I want to do in this hour is talk about a characterization of Groton-Dictapology that's more intrinsic. So what I'm going to talk about is analogous, although not identical to what's called the Gero theorem, which is a characterization of Groton-Dictapoi. And I've written it here. I'm not really going to talk about most of the elements, some of the elements of this definition, but I just wanted to put up here so you know what I'm talking about. One category is a Groton-Dictapose if it's locally presentable. So it's one of these nice categories. And then it has three more properties which have a more elementary character. Co-limits are universal, co-products are disjoint, and equivalence relations are effective. We're not going to meet these last two. This particular formulation is not the one that's going to generalize to infinity topoi. So I don't want to take a lot of time talking about it, but there are characterizations like this for Groton-Dictapoi. It's a locally presentable category with some additional properties. So here's what one of the possible characterizations of infinity topoi you can write down analogous to the one I just gave for one topoi, for Groton-Dictapoi. An infinity category E is an infinity topose if and only if it is presentable in the sense I described earlier. And then two more properties, which are the ones I want to focus on. Co-limits are universal. That appeared in my statement of Gero's theorem. And co-limits satisfy dissent. So what I'm going to do now is I'm going to talk about this equivalence and these properties. I should say here before I go on, people often group these two things as one property and just call it dissent. If you have property three, you usually want to have property two, two, so they go together. I'm following the luri here and keeping the separate concepts. What I want to do first before I go into the definitions of these, I want to talk about homotopy theory. These things actually have roots in homotopy theory. And I think it's good to sort of establish, see how that works out because these kinds of conditions don't come from nowhere. They actually come from something that, you know, it already existed and was there to be sort of generalized. Another reason is this particular property of dissent that I'm going to talk about isn't satisfied in any one category at all, except the trivial one category. So there isn't a good one categorical model for it. And for that reason, it's good to give you some intuitions to have a think about. All right. So I want to think a little bit about homotopy theory. So I've told you that the homotopy theory of spaces is the same as somehow the same as the infinity categories, infinity group points, whatever that means. But people did homotopy theory long before they knew what an infinity category was. In that context, we talked about something called homotopy limits and co limits. So going back very early into the subject, people recognize there are certain kinds of diagrams, often limit or co-limit diagrams in spaces that had a special role. For instance, when you write a space as a union of two open sets, and then you have the intersection, that's important. This behaves well for many reasons. For instance, it behaves well with respect to invariance like homology theories. That's the Myer-Vatoris theorem, which tells you how to compute roughly how to compute the homology of x from the homology of the pieces you've been doing with your sect V. Another example are fiber bundles. So if you have a fiber bundle of fiber, this homomorphic f, then this behaves well with respect to homotopy groups. You have the long exact sequence in homotopy. So these have a very important role, and there are special cases of what are called homotopy pushouts and homotopy pullbacks. So in homotopy theory, say in spaces, you can identify certain classes of commutative squares as being homotopy pushouts or homotopy pullbacks. One way you can do is you can say that a commutative square is a homotopy pushout. If you can connect it by natural transformations, which are weak equivalences at every corner to a square of a particular form. For instance, a square, which is an honest pushout, along a nice map called a cofibration. And analogously, there's a complementary theory where you take a pullback along a vibration, and that gives you the basic examples of homotopy pullbacks. So there's a recipe for computing homotopy pushouts or homotopy pullbacks. So if you have a homotopy pushout, very often you have some random map on one side of your square. And what you do is you factor it through a cofibration and a weak homotopy equivalence. And then you would take the actual pushout, where you replace, that's called B prime, you replace the original, let's call it B tilde, B prime on the page, you replace the original map by that cofibration. You take the pushout and that's a homotopy pushout. So if you have a general square that doesn't involve a cofibration, that's how you compute it. So there are recipes like this, the details aren't too important here. You can do this in spaces, you can do this in some special sets, which is also a model for the homotopy theory of spaces. Or you can do it in any Quillen model category. When I say, by the way, a homotopy theory has a model, I mean in a sense of Quillen model categories, which gives you these vibrations and cofibrations and such. In spaces, this actually leads to very geometric pictures. For instance, if I have a span like this, A maps to X and Y by some maps, which might be complicated, not inclusions in particular, I can replace it by something called the double mapping cylinder. I connect the images of those maps by a tube, A times the unit interval. And then I'm really forming a pushout along of A, including to some replacements for X and Y, which are X and Y together with some tubes with A on the end. That's an explicit construction of a homotopy pushout. So it's a good geometric picture to have. So the classical understanding of homotopy co-limits and limits was that they were derived functors. So they're in some sense, a precise sense, the best homotopy invariant approximation to the actual limit or co-limit in topological spaces or some partial sets, or in general, in some Quillen model category, depending on what kind of homotopy theory you want to study. That was the classical understanding that was made formal in the 70s. But now in the infinity categorical language, these just correspond to what I called limits and co-limits. So the infinity categorical limits and co-limits, which are meant to capture a universe characterized by some of the infinity universal property correspond to this older notion. Now, in homotopy theory, the homotopy theory of spaces play a special role. There are many other homotopies. There's chain complexes and chain homotopy equivalences. There's some official rings and all sorts of weird things you can construct. But spaces are, of course, special, and they're not just special because they're the first example. They're special. Well, they're special for various reasons. One thing I want to emphasize is that in spaces, homotopy limits and co-limits have some additional properties, which are not shared by general homotopy theoretic settings. So one question that you might ask is, what are the properties of these constructions like homotopy limits and co-limits that are characteristic to classical homotopy theory, or what we would now say the infinity category of infinity group points? And the notion of infinity topos actually arises from one answer to these questions. So you may have heard or know that you can think of a topos as some sort of generalization of a category of sets, sort of a universe of generalized sets in some sense. It has features like the category of sets. And there's a precise analogy. An infinity topos is a universe of things that are like spaces from the point of view of homotopy theory. So this analogy is what is going to lead us to this characterization of infinity topoy that I want to talk about. Okay. So still talking about homotopy theory, one thing I can talk about is what I'll call the universality of homotopy co-limits. And I'll just do this in this special case of pushouts, because I can draw the diagrams. I'll think about this primarily in the simplicial set model. The simplicial set model has a feature, this underlying category, simplicial sets, is a topos, pre-chieved sets, delta op, or delta rather. And I'm going to make reference to the fact that it is a topos. But I also care about the homotopy theoretic aspects. So we can think about homotopy pushouts and simplicial sets. And those, they're always weakly equivalent to pushout squares along monomorphisms. Now I could do the following. I could pick some map from y to x, call it p, and I can pull back the whole diagram, the whole square along that map p. So these are also, I feel like, the pre-images, which is actually y0, the pre-images of the x's along this p. And I get a new commutative square. Let's do it in this case. I've drawn this picture again. So there it is. When I pull back, so I form this by pullback, that the right hand square then maps the left hand square. Notice the monomorphic co-fibrations, by the way, I should have said here, in simplicial sets, it's very convenient in this case, co-fibrations are exactly the monomorphisms. So co-fibrations, prolactic co-fibrations. Furthermore, you have an interesting property. This pulled back square is also a pushout. That is the fact that pushouts are universal, as they say, in simplicial sets. In fact, they are universal in any topos. If you pull back a pushout along a map to the target, you get another pushout diagram. Therefore, this other square, this square I said was some kind of homotopy pushout, it's an actual pushout, that's along co-fibrations. This one's also a homotopy pushout. Let me put one more thing into the mix. Let's suppose that, oh, sorry, I can keep going. This is just a slide explaining what universality of co-limits means in a one topos or in a one category. So co-limits of universal just means that if I have any morphism, if I consider the base change functor, which I'll call f up or star from the slice over x to the slice over y, that preserves all co-limits. That's the definition of co-limits or universal in a category with pullbacks. In a one topos, it's even better. You actually have a right adjoint. Actually, you have two adjoints. You have a left adjoint always, but there's an interesting right adjoint. Because it preserves co-limits, you expect it to have a right adjoint. It does. Okay, so that's just a problem of topos. But now I want to put in the homotopy theory. I'll take my original, that diagram I had before, but now I'll actually draw it as a cube. So the top and the bottom squares are pushouts. And in fact, are examples of homotopy pushouts because they're along monomorphisms. And I'll suppose that this map here is a confibration, which is the correct notion of vibration in some partial sets. It doesn't matter what it is. The important thing is that if I form pullbacks, I get confibrations along all the sides. That's what these double arrows mean. And therefore, by my general theory of homotopy pullbacks, all four sides are homotopy pullbacks. So I have a commutative square where I took the bottom scorches, the homotopy pushout, and then took the homotopy pullback along p of everything. And I observe I get a homotopy pushout along the top. So we'll give this property a name. We could say that homotopy pushouts are universal in the homotopy theory of some partial sets, which is the same as the homotopy theory of spaces. That's if you like, the universality of homotopy pushouts. And you can do this generally for any kind of homotopy code. I won't try to draw or define arbitrary homotopy code diagrams. The same thing works. You actually only need two cases. You need homotopy pushouts, and you need coproducts. To those two cases, you can drive everything. All right. Now there's a more subtle property called descent. It's kind of called descent. It's kind of what happens if you do things in the opposite order. So the idea was that in this picture, I started with some kind of pushout, a homotopy pushout. I pulled it back along a map to over the co-limit itself, and then I got another pushout. I'm going to do this in the other order. I'm going to start with a commutative diagram like on the left here, where the squares are both homotopy pullback squares. And I'm being a little bit careful here. I'm not assuming they're honest pullback squares. A homotopy pullback doesn't have to be a pullback. It's just a constructed. That's the most convenient way to compute it. It just has to be weakly equivalent to a homotopy pullback square. But if I have a commutative diagram like this, I could take the homotopy co-limits horizontally, the homotopy co-limit construction to be constructed as an honest functor, so you'll get a map between homotopy co-limits. And then the descent condition says that if I form commutative squares, which I have for each i, which involve the inclusion of each of these xis into x and the corresponding ones of the ys into y, this resulting square is a homotopy pullback for all the values of i. So I start with a diagram with these pullbacks. I push out, and then I pull back again. And the descent says I get back to where I started, except they're homotopy pushouts and homotopy pullbacks. Let me draw pictures. So let's take a map and spaces. So I'm going to draw the picture on top, but it's supposed to be the same. I would draw exactly the same pictures, because honestly, as you can tell, I think of them as almost the same thing. Well, let's take a diagram like this. On the bottom, I'll just have the one-point space and the two-point space. And then on the top, I'll just have copies of x, x over the point, x over the point, and then two copies of x, one over each point. And I'll have this commutative diagram. I'll use the identity map in most places along the top. But here, I'll use f in one of the places. So if I form a homotopy pushout horizontally, on the bottom, so what I'm going to do is I'm going to form these double mapping cylinder constructions. On the bottom, I'll just get a circle. On the top, I'll get a construction, which is also called the mapping cylinder of f. So geometrically, you take x times an interval, here it is, and then you glue the ends together using the map f to identify the two ends or points in one end with the other end. If f is a homeomorphism, this ends up being a fiber bundle with fibers homeomorphic to x. And furthermore, you get maps from each of these things back into the whole thing, although you get some pullback squares. By the way, I should say here, when f is a homeomorphism, these are actual pullback squares. And so we're in a situation where we have actual pullback squares and we get a fiber bundle. And then we check, oh, if we pull back, we get back to the original things. The fibers of this P are actually x. My picture, by the way, looks kind of like a Klein bottle because I can't draw very well. But also, the Klein bottle is an example. You use the sort of the obvious inversion of the circle that produces the Klein bottle. But I could instead do the following. Maybe f isn't a homeomorphism, it's just a homotopy equivalence. Then you don't get a fiber bundle in general. But you can still say that the homotopy fibers, that is the homotopy pullbacks along any point, are homotopy equivalent, or weekly equivalent, to x. So you get a vibration up to homotopy, which has the correct fibers. If it's a homotopy equivalence, of course, if it's a homotopy equivalence, these aren't pullback squares anymore because you're using some weird map, which isn't the homeomorphism. So in particular, the right hand one could fail to be a homotopy, fail to be a pullback. But if it's a homotopy equivalence, there'll still be homotopy pullbacks. Okay. This is not true in sets, in the one category of sets, where I'll just take equivalence to mean isomorphism, and I don't have homotopy pullbacks, I just have pullbacks. Let's draw the same picture. I have a set x where, and I have some, you know, automorphism of my set. I'll take pushouts. The pushout of the bottom is just a single point, not a circle. The pushout of the top is, well, it's a quotient of x where I sort of identify any point with its image under f, the orbits of f. And then for like I, the one or two in my original diagram, I'll get squares like this. And then I can ask myself, is this a pullback square? And the answer is that this is a pullback only if, if was actually the identity map. But most of the time it's not a pullback. It just fails. Descent does not work in sets. Descent even for pushouts. I should clarify, and in fact, in a one category in general, this doesn't happen very often. It does work sometimes. For instance, if the maps along the bottom are monomorphisms, then you actually, it's okay. If you take a code, if you take this diagram with actual pullbacks and sets or a Roman diktopos, and the horizontal maps are monomorphisms, then it is true that if you push out and then pull back, you get to where you started. But in general, it is not true. Okay. So this page is for the homotopy series. I'll just pass by it. The summary is that I don't know who first understood that descent was a thing. I think it's Graham Siegel, who wrote a paper where he said it was well known. But then apparently that held up the paper for a long time, getting a proof of that fact. I have here a sketch of a proof. Shall I give this or shall I pass over it? I want to say one thing about this. So you actually have to prove descent and really the key case is pushouts. So you can do this in some official sets. You write down one of these diagrams where you have these pair of homotopy pullbacks. You can always set things up so that the horizontal maps are co-fibrations down here and then these vertical things are fibrations. You replace the squares by equivalent squares. That's a standard reduction. Then you just have to form the homotopy co-limit and that will also be the form the co-limit rather. And that will also be the homotopy co-limit. And now you're asking if the resulting squares are pullbacks. Now there's a special case which has to work. It's when the squares are homotopy pullbacks and actual pullbacks of some official sets. Pullbacks in the underlying category. They're actually pullbacks, which generally a homotopy pullback doesn't have to be. So all you got to do is you got to replace your diagram with one where the squares are pullbacks and the tricky part is you have to do it for both squares at the same time. So you're reduced to one particular problem. So I've expanded out one of these squares. So the original community square involved y0, y1, x0, x1. I have some co-fibrations, some confibrations. There's the honest pullback and it's a homotopy pullback means that the honest pullback along this confibration is weakly equivalent to the other states, the one that the other special set that was originally there. What you want to do is you want to find something that goes here so that the top square is also a pullback so that this map on the side is a weaker coincidence and this map is a confibration. If you can do that, then the whole rectangle is a pullback and it's a model for the homotopy pullback you wanted. Well, there's something you could try. If I have a map in some official sets, I have a pullback functor and that has two adjoints as I pointed out, topos, and I'm interested in this right adjoint. If j is a monomorphism, then in fact, if you form the right adjoint and then pull back again, that's the identity, that's a formal property. So if I put in here pi j of y0, j is this map here, I'll get a pullback square and so the miracle is that this actually works. That's actually the solution to the problem. So I've now constructed an actual pullback square which is also a homotopy pullback square. The hard part is showing these properties and that's the solution. I mentioned this because there's a long literature attempt at proofs of descent in various contexts and they're often quite technical. I'm even responsible for such a paper but now that nowadays there's a clean proof. This proof is due to Wojcicki. He introduced this argument not to prove descent per se, but to prove something which is called the fact that some official sets are a model for univalent type theory. I may mention something about that before the end of these lectures. So I just wanted to sketch out that there is a proof to this in the language of classical homotopy theory. I thought we were quite possibly. That's the end of this excursion to homotopy theory. So I'm going to return to the infinity categorical setting and give you some proper definitions. So if I have an infinity category that's complete and co-complete, co-limits, and finite limits it has, then I'll say it has universal co-limits if for all morphisms the induced pullback functor preserves finite limits is left exact. Oh, and of course it's wrong, preserves co-limits though. It is a finite limit. Okay, that's the university quality of co-limits very easy to define. Now I want to give descent. So this is a little bit more tricky. So, okay, some definitions. Suppose I have a natural transformation of functors to an infinity category from an infinity category. I'll say it's Cartesian if for every morphism in the domain category, every one morphism, every one cell, the resultant commutative square I get is a pullback in E. This definition makes sense for one categories. I'll call this a Cartesian natural transformation. Now let's consider the arrow category of my infinity category. So delta one is the walking one morphism. So I'll call this even an arrow. And I'll define a subcategory of the arrow category, which I'll call cart of the arrow category. This is the subcategory. It's not full, but it's wide the right term. It has all the objects, but it only has the morphisms that are Cartesian transformations. In other words, the objects are arrows, but the morphisms are just the pullbacks squares. And that actually does turn out to be an infinity category. And now I can define descent. I'll say that E has descent if this cart of the arrow category has all co limits. And when I say that, I mean all small co limits as one does. And if the evident functor, the inclusion functor back to the arrow category preserves co limits. That is descent for all co limits. You can also talk about descent for particular shapes of co limits. You just restrict the co limits of particular shape. I put this reference here because although in 1974, Puppa was not did not know what an infinity category was. He only had homotopy columns. He nonetheless actually wrote down a formulation of descent in the homotopy theory of spaces, which is almost exactly the one I've just told you. All right. So given this, I can now give you the big theorem. An infinity topos, sorry, a theorem is that infinity topoi have universal co limits and descent. I'll sketch the proof. It's in steps. First, you show it for infinity group voids. And that was the purpose of the previous discussion. This is this uses the simplicial set model of infinity groupers. Well, infinity groupers are con complex, so it's sort of was our definition. So those arguments I gave you before are the proof and simplicial set. I don't know any other way to do it. Then from that, you get that it's true for pre sheaves with values in infinity group voids, because limits and co limits in a pre sheaf category like this are computed point wise. So you can use the fact that you have these properties and s to prove them for pre sheaves. And then finally, you recall that a general infinity topos is a left exact localization of a pre sheaf category. And so you use the properties of left exact localization. L and I preserve finite limits. And all co small co limits. And those are the things that appear in these definitions pullbacks and co limits. And so, you know, I prove descent right down to your diagram and E from the pullback. You compute the limits in E, and you compute the co limits by going into pre sheaves computing it there and then applying the left adjoint. And that's also compatible with forming the limits because everybody preserves the finite limits pullbacks. It's it's just a the obvious argument. Every infinity topos has these properties, because infinity group voids do. We actually get a characterization. An infinity category is an infinity topos, if and only if it's presentable, co limits are universal and co limits have descent. I've already told you one direction. So I'll sketch why the other direction is true because it's interesting. And it is an illustration of descent. So this is the key property in some sense, it's the one that doesn't happen in one categories. So that's the one we want to sort of keep an eye on. All right. So I'm going to briefly sketch this. The first step is formal and is really sort of an application of this theory of accessible infinity categories. The analog works the same way in one categories. So if you have your infinity topos neat, you want to first find an essentially small subcategory C, which is closed under finite limits. And such that the restricted yoneta function and the functor, so you have a yoneta functor for E, but then you restrict the functors, the precheves to the subcategory, you want that to be fully faithful and have a left edge. So everything except except the the the left edge being left exact. Also, I guess I also want i to be accessible. So that comes out of this theory of accessible infinity categories. So you just pick C to be big enough. So you might say something like this, where this is some full subcategory of what in this theory are called kappa compact objects. In the theory of the one categorical theory, the usual term is kappa presentable, but we changed all the terms for some reason. So with something if you like a size condition. This is sort of standard, standard idea works just the same way, except that you need to do several hundred pages of work just to make sense of it. In the end, you get to where you wanted. So I have everything except L being left exact. And so of course, I need the following proposition. So if I have E is a co complete finite complete infinity category, which has these two properties of universal co limits and descent. I have a small finite complete infinity category. And then I have a co limit preserving functor L. Then L is left exact. If only if it's composite with the native functor is left exact. I hope it's clear. I always write, I know, I always write your row for the native functor for some reason. So that's what I need. There's a list works in one case or what's the works for topoy. So this works with this is also true. If he is a one topos. You can, you can characterize this meaning if I replace infinity group words with sets, and that's co limit preserving, then if CS finite limits, I can determine these sunkers by using from this property restricted on your data. If that was the whole story, I would just stop here and say it's like one case, but the proof is actually a little different because it actually gives you something a little bit stronger. This thing on this page is does not work the same way as it does in the classical setting. So let's suppose I have a co complete finite complete infinity category universal commas descent and see is small, but maybe it's not finite complete. So I'll take that away. Same conditions. L is co and preserving. I want to know when the L is left exact. It's left exact if only if two things are true. One is that L preserves the terminal object. And the second is that L preserves pullbacks of the form like this, their pullbacks of span or co spans of representable functions. So that's a particular class of pullbacks and pre sheaves. If L preserves these and it preserves the terminal object, then it's left exact. This does not work in the same way if he is a one topos. As we saw yesterday, there's a condition that you have to put called, I believe filtering, which is not this. It's a little bit more complex to state. But in infinity categories, you get a very slick looking condition. It's actually kind of a miraculous this works out this way. It's a difference between the infinity and one sex. It's really a difference between sets and infinity group points. That's that's what's different here. Not using sets. I'm using infinity group points. It doesn't work in a one topos. Oh, here's the illustration of why doesn't work in a one topos in case you're wondering. So let me take my category C to be a group. And my topos will be sets. My co preserving functor will be co limits. So I can form a pullback in the pre sheave category, which are just sets with a G action. Where I'll take the terminal object, and I'll take the representable function just G acting on itself. By the way, it's a hypothesis of that we're in this is this statement are true. Co limit of the terminology is the terminal object. And it preserves pullbacks of representables because well, because the group a group has a group has pullbacks. So pullback of representables are representable. And therefore it has to preserve it. But here's a pullback down. That's not a representables because the terminal object isn't. So the pullback is really the product. If I form the co limits of the G action, then I get it. I just get the point in three locations. This thing will be isomorphic to as a set to G. Although this was a pullback. When I take the co limits with respect to G, this is not a pullback. That's the illustration that this doesn't work in sets. One way people sometimes talk about as they say that pullbacks are not a sound doctrine. But that's an only one catch infinity categories pullbacks are a sound doctrine. All right, so I have a few more minutes. Let me serve. Okay, let me sort of sketch a proof I may go through this quickly. So this is my setting. Co complete finite complete infinity category universal co limits. I won't need dissent for this. Here's a sort of a special case, which is much easier to prove. C is small. I have a colon preserving L. I'd like to know when it preserves pairwise products. And here are the condition it preserves pairwise products if only if it preserves pairwise products of representatives. So I mentioned this special case because this works the same way in one category. The proof is the same as for infinity categories. This uses the fact that co limits distribute over products in a topos. When I say it works for one categories, by the way, I mean it works in a one topos also works if he is a one topos. So you do have universality of co limits in a one topos in particular of you have this property that if you take a product of two co limits over two different indexing categories, that's equivalent to the co limits of the product over the product of the indexing categories. And you can use that to prove that this implies this. It doesn't need dissent. So it works for classical topo. Now let's think about this general case of pullbacks. There's a special case I get immediately in my setting. So this is the special case where I have a pullback. This is in the category of pre sheaves. So it's a pullback of the diagram of pre sheaves where B is itself representable. So saying that L preserves this pullback. Well, this is this pullback is really a product in the slice category. And L carries the slice to the slice over LB. These are both infinity. These are both infinity. Well, they both have universal co limits and dissent. I'm not writing very clearly. Universal co limits and dissent they inherit it from pre sheaves and from E But of course, this is also a pre sheave category. It's equivalent to one. It's pre sheaves over the slice because that's representable. And I just told you the recipe for knowing that something preserves products, it just has to preserve products of representables and a product of representables. You know, you know, in here, well, it's really a diagram like this. It's one of these pullback squares where it's over a coastline of representables. So that's the lemma. So the lemma gives us the special case where it's a pullback where I have comma representable. I guess this part of the argument also works in the topos. I think the problem is the general case. So here I have a general pullback in pre sheaves. I'll write the bottom object as a code limit of representables. Then for each of those representables be I can pull back the whole pullback over that representable. So I'll get a for each object in my indexing category. Oh, sorry, I'm kind of screwing things up here. This is not a these are not pullbacks in I these are pullbacks in the functor category from some indexing category to pre sheaves. Sorry, forgot there's another indexing here. So for each I get a square like this and that's just a pullback square in pre sheaves. So I got this whole collection of pullbacks. I have this pullback of diagrams should have said what fun categories these were in. You get this pullback in pre sheaves for each eye. But when you pullback pullbacks, you get more pullbacks. So you get a bunch of cubes all whose sides are pullbacks. So for each morphism in the indexing category, you get pullbacks of pre sheaves. However, we also know that be by construction was the code limit of the representables. And then we have on this property that code limits are universal in pre sheaves. So if I take the code limits of these other functors in the P, P the P sub I X sub I and X prime sub I those also recover P X and X prime. Now, in all these pullbacks, the bottom right object is a representable. And we know that L preserves these pullbacks over a representable because that was the special case. Well, actually, not the left hand square, but there's a rectangle where they're both pullbacks and the rectangle and the right hand pullback are over one of these representables. And so you can patch pullbacks. So the left hand pullback will also be preserved by L. All right. So what I'll do is I'll apply L to all these. So I've done that shown this here for this picture, apply L, I get some more pullbacks. Oh, but L preserves code limits. So if I take the code limit of the P is the excise and so forth, I get L, P, L of X, L of B. Now, I have this diagram on the top. If you think about this as these natural transformations, these are actually Cartesian natural transformations. These things here, I'm sorry, I wrote these here. These represent Cartesian natural transformations of functors from I into precies. Each edge here. Then I have this property called descent. Descent says that this has colonists. Actually, these are in E, not in precies. This has colonists and the, if you like, the inclusion functor preserves colonists. So these are colonists in the Cartesian category, but they're in the Cartesian category. So that means that these are also pullbacks. I form the pullback. L, P is a pullback of these LPI's. Sorry, I've formed a co-limit L, P is the colon of the LPI's. Then I pull back again. I get pullbacks. That's descent. So what I care about is this map, L applied to the original pullback square. I'd like to know that's an isomorphism. What I've shown is that the pullback along each of these LPI's is an isomorphism. The pullback of this, so I didn't write the squares here, but the pullback of the original square of this square is the square LPI to LXI. LPI to LXI prime. And then universality of co-limits, in E, tells you that F is an isomorphism because LB is a co-limit of the LPI's. So if you pull back to all the pieces of the co-limit and you've gotten isomorphism, then you had to have an isomorphism. All right. That was, I hope, clear enough. This was the key step. I used descent here at this step. Otherwise, I could have done this in a one-topos, but this is the key step. So that's the proof of that property. So that's the story of a characterization of infinity topo. What I want to do next time, which I guess is tomorrow, is to develop some consequences of that. The most interesting one, which is the main one for the first hour next time, is to talk about the object classifier. So you may remember that topos has a sub-object classifier, and infinity topos has an object classifier, in some sense. You have something that's a little bit better than a sub-object classifier, and I'll say what that is next time. And I'll also talk about some other aspects, which are truncation and connectivity. As the first hour and the second hour do some more things. We'll leave that till tomorrow. Okay. Thank you. Okay. Thank you very much for your very nice lecture.