 So I want to have a sort of a grab bag of topics in this last lecture. So I've talked about infinity topoi characterization in terms of descent, local classes and universal families and truncation connectivity. So I've noticed that the, I can look at the full subcategory of in truncated infinity group oids. And so these are what are called in group oids or sorry. Yes, these are called in group oids. S less than or equal to zero is equivalent to the one category of sets. S less than or equal to minus one is equivalent to the one category actually poset of propositions. S less than or equal to N is an example of an N plus one category or as I said it's really in plus one comma one category, but I'll use Larry's terminology it's easier. So in this setting and is an N plus one category of all of its mapping spaces are in points. That's the definition. So just as we can look at pre-sheaves of infinity group oids, we can look at pre-sheaves of N group oids, I'll say N minus one group oids. So C will be some infinity category probably small. And I can look at functors on C op with values in N minus one infinity group oids. This is actually the same or equivalent to the full subcategory of N minus one truncated objects in the pre-sheaf category pre-sheaves of pre-sheaves valued in infinity group oids. That's because truncation of pre-sheaves is computed point wise. So I'll write this as pre-sheaves C less than or equal to N minus one. One more fact about this, if you have an infinity category C, then there's a best approximation to an N category. So I'll call that thing the map from C to HNC. So HNC is an N category. And this map is initial among functors to N categories. So it turns out this functor category is equivalent to a functor, always equivalent to a functor category on some N category. So when I talk about pre-sheaves, a category of pre-sheaves of N minus one group oids, I might as well assume that the domain is an N category. Well, last journal, at least I can do that. So we can define analogously to infinity topos the notion of an N topos. So it's the same definition, except that I replaced the role of infinity group oids, pre-sheaves of infinity group oids with pre-sheaves of N minus one group oids. So it's an infinity category E, such there exists a small infinity category of C, an accessible fully faithful embedding of E into pre-sheaves of N minus one group oids, which has a left adjoint, which is left exact. And as I said before, I can actually, without loss of generality, assume that C is itself an N category, if I wish. So, for example, a one topos is exactly, well, the same in the sense that it's the infinity categorical, in an infinity categorical language, it's the same as a groten diktopos. Every one topos in this sense is equivalent to a groten diktopos in the classical sense, because when N equals one, that's pre-sheaves of C less than zero, that's the zero truncated pre-sheaves zero truncated infinity group oids, so really pre-sheaves of sets. A zero topos is exactly the same as a locale, really a frame or a complete hitting algebra. If you start with an infinity topos and you look at the N minus one truncated objects, that is an N topos. That's pretty easy to assume. Remember, let's suppose that our infinity topos, that's easy to prove. I suppose our infinity topos is, let's pick a left exact localization, so presentation is left exact localization of some pre-sheaves category. So I have a full subcategory of N minus one truncated objects in E, but I have a similar thing in pre-sheaves. And remember that truncation is characterized by a condition on the iterated diagonal, so it's a limit condition, a condition in terms of finite limits, and both L and I preserve this condition. And therefore they preserve truncated objects, therefore they restrict to functors on these subcategories, which are necessarily have the appropriate properties as you can prove. Certainly I is fully faithful, you can show it's accessible, and L is still left exact. In case for any infinity topos, the zero truncated objects are a groten-deak topos. And the minus one truncated objects, which is the same thing as sub-objects, so the terminal object is a local, opposite of a local. Furthermore there's a fact, I won't try to prove it now, but any N topos is an N minus one truncation of some infinity topos. I'll warn you here, so when you have an infinity topos E, you will call E less than or equal to zero the underlying one topos. But that is not a complete invariant, you can certainly have different infinity topos, which have the same underlying one topos. Easy example comes from infinity group boards, take your favorite infinity group board, then you can form the slice over that infinity group board and that is an infinity topos. Now I can take the zero truncation, that is equivalent to the functor category from the fundamental groupoid of your space into sets, that's the fundamental groupoid. So the zero truncation of this infinity topos only depends on the fundamental groupoid of X. However, the infinity topos itself actually depends on X itself. The fundamental groupoid is certainly not a complete homotopy invariant of spaces. Therefore for any infinity topos we get this chain of full subcategories of objects with various truncation levels, and we have left adjoints going back, which are called truncation functors. So at this point I'm going to return to this question that I introduced in the first lecture, which is where the second lecture I guess where the growth and deac topologies in this game. So I have to think generally about left exact localizations. Let's first establish some terminology. So by a localization and I think reflection is a better term here, but it's sort of standard to call localization. So it's a picture like this you have a fully faithful embedding with a left adjoint. I can identify because it's fully faithful I can identify D with its essential image and so in practice I will just do that I will assume that D is a subcategory. When you have a localization you have something that I'll call the kernel this may have some standard name but I couldn't figure out what it was I'll just call it the kernel. That's the class of morphisms and E that are inverted by the left adjoint. So it's the class. So it's the class of morphisms that it'll takes to an isomorphism. Any kernel has the property of being strongly saturated assuming he has co limits. Isomorphisms of course are in the kernel logically. It has the two out of three property. So, if I have two morphisms and a composition and any to. And T implies all of them are empty. And lastly this class this kernel is stable under code limits computed in the arrow category. So if I compute a column in the arrow category. It's actually also T. So every localization determines a strongly saturated class, you send the localization. It's a that's an injective correspondence in the sense that from the kernel you can recover the localization. The full subcategory is the class of what are called local objects. So it's the class of objects, such that the map induced by mapping by any T, going to access an isomorphism for every morphism in the class T. If he is a presentable infinity category, then you can classify the accessible localizations using the theory of presentable infinity categories. These correspond exactly those strongly saturated on classes that are generated by a set. So T equals s bar for some set of morphisms size restriction. So this s bar is the strong saturation of s. It's the smallest class of morphisms containing s and closed under those properties. Isomorphisms two out of three columns. So this is a correspondence for any set, you can build localization whose kernel is the strong saturation that set. If I want to think about left exact localizations. So those are going to this should probably be accessible. And these are strongly saturated classes that are the saturation of a set. But the left exact property corresponds to the classes where the kernel is in addition closed under base change it's closed under taking pullbacks along the map. And that's proved by an elementary argument which actually works just as well in the one category setting in general. Right. Okay, so the key question that one would like to answer and have a good answer for if you have an infinity topos, how do you classify the left exact localizations. There's some things to say about this I will not say everything there is to say. I think there's going to be a talk in the conference next week about this exact topic, but I'll tell you a little bit. So, let's suppose I have an infinity topos. Let's suppose I have a left exact localization, and I'll call its kernel T. So that T gives you a local class I think of the kernel as a thing of it as a full subcategory of the arrow category. So I'll take the intersection with the Cartesian arrow category. The intersection is a local class. That's actually very straightforward to prove using the left exactness of the localization certainly closed under base change. And it's closed under a co limits in the Cartesian category because in fact it's closed under co limits in the arrow category. As a consequence we have the following if I have a pullback square where I'm pulling back along a cover. F is in the kernel. And so is G. So that's a property of kernels because they are local classes. Now still assuming I have an infinity topos in the left exact localization with kernel T. I have the following interesting condition on elements in the kernel. If I take a morphism F, and I make its cover mono factorization so I is a monomorphism P is a cover, then the morphism is in the kernel. The morphism F is in the kernel. If and only if I is in the kernel and the diagonal of F is in the is in the current. So this is a more or less elementary argument using left exactness. I'll give some, I'll give it sort of very briefly. So in one direction suppose F is in the kernel. Well, that means that L of F is an isomorphism. L is the sheification, the left actor. L of a monomorphism is automatically a monomorphism, because it's a left exact localization. The hypothesis is that L of F is an isomorphism. And now all of a sudden when I apply L I've got a monomorphism, which has a section, the inverse of L of F composed with the with L of P. Well, it is true even if any categories that monomorphism with sections are isomorphisms. You know, this tells you that I isn't T. L of I is an isomorphism. And the other property is straightforward because L is left exact. The diagonal of L is L sorry L of the diagonal is the diagonal of L. So you can use that to show the delta F is T. Okay. So first of all, if I want to show that if these two things are true, then F is in the kernel. So first of all, if I form, if I think about the diagonal of F, which is a map from a times over VA. I can factor that through the diagonal of P where P is part of this mono cover factorization. The second map in that factorization is actually a pullback of I, in some sense, it is itself a monomorphism. And because it's formed by pullbacks from I, and localizations left exact. And since if I assume that I isn't even so is J. And since I'm going to assume here that the diagonal of F is in T, I actually have the diagonals P is in T. And I get to showing that F is in T. Well, I also have this diagram on the bottom. Here's the diagonal of P. It maps to this pullback, which is the square on the right. Well, if the diagonal of P is in T, since it's factors with Q, it factors the identity two out of three implies that Q is in T. Oh, but now I also have this P here, which is a cover. And I just prove this property that if this class this property of being in T is local. So because P isn't is a cover and Q isn't T I get that P isn't T. And since I've already assumed that I is in T. That's where part of my hypothesis the composite is T. Just I'd go through that argument. I don't think I didn't use, don't think I really used anything here that wouldn't work in say a one purpose. As a consequence. If I have one of these left exact localizations of an infinity topos with Colonel T. And for any in any finite in the, well, I haven't defined that for infinity but the intersection of T with it with the class of in truncated maps is determined entirely by the intersection with the monomorphisms, or to make it more explicit. If I have an in truncated map, which means that it's in plus two iterated diagonals nice and morphism. And the kernel, if and only if the sequence of maps I zero I one up to and I n minus one is in T, where those are constructed inductively, I guess like this. So these are the coming from let's just call this okay. These are parts of the cover mono factorizations of the various iterated diagrams. That's the, that's what we've learned here. Let's make a definition. This definition is due to Larry, a topological localization of an infinity category is a left exact. I should have also said accessible here. Actually, I don't have to say accessible. Every left exact localization of a left exact localization. Sorry, a topological left exact localization. A topological localization of an infinity categories is a left exact localization whose kernel is generated by some set of monomorphisms. That's a definition. And the theorem you can prove is that every left exact localization of oops, I should say something's a little bit wrong here. And that's because of an end of an end topos every left, left exact localization of an end topos with and less than infinity is topological. That's a combination of the fact, the fact that I proved here that the class, the kernel, the intricate maps and the kernel determined by the ones that are monomorphisms and the fact that all maps. In the category like an end topos are truncated at some level. They are all in fact in truncated in minus one truncated doesn't matter. They're all finally truncated for some given value. Okay. What we're actually interested in are the topological localization of the pre-sheaf category, meaning pre-sheaves of values and infinity group loads. So if I have a topological localization by definition that means that the kernel is generated it's the strong saturation of a set of monomorphisms. Now monomorphisms form a local class. In the pre-sheaf category we have a collection of objects which are the representable pre-sheaves and they're generators for that category into the usual sense, the infinity category. Everything's a co-limit of some small diagram of representables. So if I have a topological localization, I can argue so that I can replace my set S with a particular kind of set, a set of monomorphisms whose targets are the representables. I actually have to use the fact that the kernel is a local class in order to do this. So what I learned is that every topological localization of pre-sheaves determines a Groten-Dick topology on the infinity category C. So what's a Groten-Dick topology on C? Well, you can define it as a collection of sieves, that is to say a collection of sub-objects of representable pre-sheaves, which satisfy a list of properties which I won't give, but it's a standard list of properties that is familiar as one of the standard ways of describing a Groten-Dick topology on a one category. Very often people choose to describe the topology using a different language, using covering families, but any family of maps to an object can see determines a sieve. I'll note here that this is an infinity analog of the notion of a Groten-Dick topology, but it doesn't actually require any deep ideas. These are actually in bijective correspondence to Groten-Dick topologies on each one of C. So remember every infinity category has an associated initial one category that it maps to. It's called the homotopy category, the infinity category. And so in fact, Groten-Dick topologies on C correspond to Groten-Dick topologies on the homotopy category. So they actually correspond to conventional Groten-Dick topologies there in the usual sense. That's because you can read off sub-objects of representables from the homotopy category. And the conclusion is that every topological localization of pre-sheaves is of the form sheaves C S for some infinity site. So associated to an infinity site, which is an infinity category with a Groten-Dick topology, you can describe a full subcategory of sheaves. This will be exactly the collection of pre-sheaves. This is an isomorphism for every morphism in the set S of things in the Groten-Dick topology. So that characterizes the topological localizations of pre-sheaves. Now I can do something similar when n is less than or equal to infinity that as I look at localizations, left exact localizations of pre-sheaves of n minus one group whites. In this case, every left exact localization is topological. I think I said something like that here. So here, every left exact localization is a category of sheaves for some insight. So here C is an n category and S is a Groten-Dick topology on the n category. If n is less than infinity, everything is a category sheaves on a site. In the case of n equals one, this is the classical statement about Groten-Dick topology, which is usually actually taken as the definition. Now, of course, I've emphasized that in the infinity case, this is only the case for the topological localizations. So in notion of cotopological localization, what's a cotopological localization? A cotopological localization of a presentable infinity category, that's an accessible left exact localization with the property that its kernel is such that the only monomorphisms in the kernel are the isomorphisms. So it's sort of as far as you can be from being topological. And then you can show that an accessible left exact localization of an infinity topos is cotopological, if and only if its kernel is contained in the class of infinity connected maps. The proof I won't write it out, but it's by the same ideas that I've already shown that show that the topological localizations are determined by monomorphisms. Actually, there's a sketch of a proof here, but I'm not really going to give the full details. In one direction, it's obvious. If the kernel is consistent, actually in one direction, it's obvious. If it's cotopological, then well, you already know the infinity connected maps are, which are monomorphisms are the isomorphisms. In the other direction, if you have a, if the kernel contains any connected maps, you have to show it's containing the n connected maps for all in what I'm going the wrong way around. I think this page is again messed up somehow, proving this direction. You're approved this direction. Very good. So if you have a cotopological localization, you first show that all elements of the kernel are covers. And to do that, you just use the epi-mono factorization or the cover mono factorization of f. If f is in the kernel, well, that means that i is a monomorphism. We've seen this before, but l of i is a monomorphism. Therefore, an l of f is an isomorphism. But you can use that therefore to show that l of all these things are isomorphisms because you have a monomorphism with a section. So, or sorry, you can use that to show, oh, it's cotopological. This shows therefore that i is in t. But therefore, because it's cotopological, i is an isomorphism. Therefore, f is a cover. And then there's a similar argument that works inductively on n to show that t is containing the n connected maps for all n. You have to use a fact I haven't proved. It's the fact that if you have a cover, then f is in connective if and only if it's diagonal is n minus one connected. This is sort of an inductive argument that sort of comes ultimately from the inductive definition of truncatedness. Let me give an example of such a cotopological localization. If we use an infinity topos, we say that an object is hyper complete. If its projection is orthogonal to all the infinity connected maps. This determines a full subcategory of hyper complete objects. And that full subcategory is an example of a cotopological localization. The kernel of this localization is exactly the collection of all infinity connected maps. So this is in some sense, the maximal cotopological localization kill all the infinity connected maps. So it's a formal consequence of these definitions that if you have an in truncated object, it's, which is, it's automatically hyper complete. So the in truncated hyper complete objects are all the hyper, are all the in truncated objects if n is finite. So this hyper completion is is a phenomenon does it does not is not seen from finitely truncated objects. So for instance if you had a, if you looked at sort of less than equal to zero, you know, all the objects are hyper complete, hyper complete truncated objects are always hyper complete. So for example, this, this example that I gave not in the previous lecture but earlier. Well, previous hour I guess, of a particular topological space with an object that's not infinity connect that's infinity connected. That's not trivial. That's an example of an infinity topos that is not hyper complete. It has a non trivial hyper completion, because it has non trivial infinity connected maps. Now there's a theorem, which does describe all the accessible left exact localizations of an infinity topos they all factor essentially uniquely as a composite of two localizations. A topological localization. And then there's a co topological localization of that. So as a consequence, every infinity topos is a co topological localization of sheaves on some infinity site. So the idea is that he will be a co topological localization. The way it works is that ill inverts, at least some infinity connected maps. So, anyway, the consequence is that not every infinity topos issues on a site, but there is this co topological. Everything is a co topological localization. I should mention here is a historical thing so there's actually a class of infinity topo that were constructed. Before the infinity categorical language by entre all, then by Richard Dean. These are model categories of some official pre sheets on a one site one site. These model categories give you infinity categories and these actually give you hyper completions. They construct the hyper completion of what I'm calling sheaves on the site, sheaves on the one site viewed as an infinity site of course. The one site is actually a kind of infinity site. So, um, that's because if you look at their construction they actually define their weak equivalences in terms of a notion of homotopy group. Their notion of homotopy group cannot see infinity connected objects. So by the nature of homotopy groups. This ultimately leads to a question is every infinity topos equivalent to some infinity category of sheaves on an infinity site. It may seem like I've told you the answer is no. But what I told you, um, is that there are examples of infinity topo way, which I've constructed, you know, which admit non trivial co topological localizations. In some sense the proofs I've given you tell you that the canonical site. The canonical site of an infinity topos might not give it to give give you might not have E as a topological localization. The obvious thing topological localization. The obvious thing to do is to take a large, you know, a small but a full subcategory review that's closed under finance limits show that it's a left exact localization appreciates on that that's a canonical site. Those are often not topological localizations for a random infinity topos. So it's an open question, I believe as to whether this is true. I thought for a long time this, of course, the answer is no, because look at these examples I have but they weren't actually examples of this. It seems very hard to address that question, because I don't know anything that distinguishes the infinity topoi which come from infinity sites that are she's an entity site I don't know of any property those have that something else might not have. So I don't even know where to start to try to prove this. So it's in principle it's possible that the answer is yes I think the answer will likely be no but who knows. Okay. I see here I'm going still going a little bit over on time. I do want to talk about geometric morphisms. And this is very straightforward because you just do the things that work for one topoi. So a geometric morphism is between infinity topoi is a functor. It's an adjoint pair of functors, whereas the left adjoint is in fact left exact there's finite limits. This gives you an infinity category of geometric morphisms the notations people use here are fun lower star from E to F, but you can also have fun upper star from F to E. And that depends whether you prefer to think about the left adjoint or the right adjoint. This also gives you an infinity category of infinity topoi. So this has the property that the space of maps of infinity topoi from E to F is the maximal subgroupoid of this functor category. Of course, these are potentially large. And this is a large infinity category. In general. I actually showed you the recipe for computing geometric morphisms last time. To compute, for instance, a geometric morphism to pre chiefs on something. So let's compute the fun upper star so the left adjoints. So those are going to be the color preserving functors which are also left exact. So that's a full subcategory of the category of color preserving functors on pre chiefs are the coolant completion of C. So color preserving functors from pre chiefs are the same as just functors from C. The inverse is the left con extension. And you can describe what this full subcategory is these are the functors, color preserving functors such that well, they satisfy these conditions that I gave it takes the terminal object to the terminal object, and if preserves the pullbacks of co spans of representables. Rose. So this remarkably nice set of conditions. Consequences, well an immediate consequence, there exists a unique geometric morphism to infinity group oids. So that thing there is the terminal infinity topos infinity group oids. If you have an infinity topos a point is just a geometric morphism from us. So as you can do in one topoi, you can say that infinity topos has enough points. If the left adjoints, the stock functors for all points are jointly conservative. But there's a warrant warning that comes in here and it has to do with the fact that infinity group oids are hyper complete. So the infinity connected maps and infinity group oids are just the isomorphisms. So if I define having enough points in this sense, then he can have enough points, only if it's hyper complete. The stocks cannot see infinity connected objects, because if any group oids doesn't have any. In fact, in practice. At least if you look at some of the things that Jake has taken where he has done you take as a definition it has enough points. If it's hyper completion has enough points in the sense I've described. So that's an additional issue that shows up in the setting. Of course, even for a one topos, you could just fail straight up fail to have enough points. Oh, here's if you want to compute maps into a slice there's a recipe. Maps of infinity to those F into the slice of you over X, which is also in the topos that corresponds to GMM metric morphisms to E together with a section of the pullback of F in F. As a consequence. This actually gives you a fully faithful embedding of your infinity topos into the slice in infinity category of infinity top away over E. So if you just send an object to its slice together with its forgetful function to E. Oh, here's one more example. Torsers, this is sort of the classic example of a geometric morphism. Let's take a small infinity group or it might be easier to think of a group. And let's compute the geometric morphisms from E to pre she's actually compute the left adjoints. Okay, so these are going to be the, the, the funkers from G up to E, which satisfy my conditions takes the terminal object to the terminal object and preserves pullbacks of cost bands of representatives. They're automatically satisfied, because group oids have pullbacks. They just do, because everything's an isomorphism. So I only actually need the first condition so this will correspond to the funkers. P, such the co limit as a functor on G, I guess on G op is the terminal object. And actually the correct definition of a G torsor in the context of infinity topoi. This may not look like what you would call a G torsor. But here's why. This condition is equivalent to the following will certainly implies the following. First of all, if I have such a functor, and I have this property so that it's co limit is the terminal object. I obtain a cover as a morphism of E I take the co product of all the values of my functor at all objects of G. And the map from the co product is a cover by the general property told you before. Second of all, I have another thing that comes from descent. So let's suppose I have one of these P with this property. Let's consider for any pair of objects in my infinity group or a G. I can associate to that a map which you can think of as the action. So I have px, p y, and then I have the infinity group of maps from y to x in G, I guess in G. I can pull that back pie here is the projection is the unique geometric morphism training group or it's. So if you like this is a map P times G to P, if G is a group pullback G. It's the action map. This fits together. If I let why vary. So this actually is a functor gives me a functor from G up to my infinity topos the value at y is whatever and above. The X is fixed. So this is for each x p of x times p star the pullback of the representable pre chief on G to E maps to P. Now I can form the co limit with respect to G up. By definition because it's a G torsor the Coleman of P is the terminal object. You can compute the Coleman on the left hand side using university universality of co limits to get rid of this factor that's just constant. That's really p of x times the Coleman of the representable. So lots of representable funkers are terminal and pie star preserves co limits. So in fact this is just p of x. That's the co limit of that current transformation. Now the thing about this transformation it's actually a Cartesian natural transformation of funkers from G up to every morphism and G. You get up the you plug that into the transformation you get a pullback square and that should be just because it's an infinity. It's a group boy. You get a square where both arrows are isomorphisms. Well now I have descent. Descent tells me if I form this co limit and then I pull back again to each object. I get a pullback square. So I have a pullback square for every X and Y. If I write this out just for a group. So this P times the pullback of G to P to P to one is a pullback. In other words, this thing is equivalent to the product of P with P by sort of a tautological pair of maps, which are projection. And if you like the action that recovers the classical definition of a torsor in the one category case but it's a it's in some sense a different definition in the infinity category case. The torsors that oddly have a much easier deaf cleaner definition infinity. Alright, I'm very low on time. Let me run through a couple of one remaining topic. There's something called n localic reflection. So I have an infinity category of infinity to boy. And there's a functor, which is given by take your infinity topos and restrict to the full subcategory of n minus one truncated objects. That admits an adjoint which hopefully this is correct is a right adjoint. I'll call this R and F, let's call the n localic reflection, you can always promote for any in an end topos to infinity topos in a canonical way. And here's the formula if I start with an end topos. We said that those were always categories of sheets. So we have a category of n minus one group boards on n minus one site. I can actually pick this so that the underlying category see has finite limits. So the recipe to compute the n localic reflection is you pick such a site with finite limits that presents your end topos, and then the end localic reflection is sheaves of infinity group boards on the same site, which is also an infinity site. I'm consistent. So that's the formula for n localic reflection. Everybody forgets the fat well I always forget the fact that it has to be you have to use a category of finite limits otherwise you get the wrong answer. Okay. So I'm at the end of my time. I'll notice one more property here. So you can define co homology of infinity topos. So I have this unique morphism geometric morphism to infinity group words. So I have a pair of adjoint functors constant and the constant pre chief and the global sections. So if I take an island where McLean space in spaces. I can pull that back to eat and then push it forward again taking global sections, and then that defines the co homology of my infinity topos with constant coefficients which are in a dealing group will take the appropriate homotopy group of this thing I can use various in here, which are bigger than J. Notice I remember McLean spaces are in truncated. So this invariant it's certainly not a complete invariant, but it certainly does not distinguish on say E from its own hyper completion, because the pullback of the only more plane object is already going to be in truncated. The same answer for both in a type of completion. So this idea does lead to a very nice formulation of the idea of a shape of an infinity topos. Using this projection q, you get this composite functor. q, upper star falling q lower star. And this isn't an example of a functor from s to s. And it's in a full subcategory which is called pro s, which we can define to be the collection of functors from s to itself which are left exact, or more precisely defined to be the opposite and also it turns out a category of inverse limits of representatives of filtered interest limits so it's reasonable to call it pro s. It has a topological embedding by s itself by the native functor. Every infinity topos has a shape, which is a pro space in the sense, and you can prove for instance that if x is a nice enough space if it's para compact. Then shape of she's on x. Notice here I'm using space in the sense of actual topological space. If it's para compact and its shape is in fact determined by its underlying homotopy type. The next here is the infinity group void, which is the homotopy the usual homotopy type of the topological space x. Every topological space has a homotopy type that you think of as an infinity group. So para compact topological spaces have sort of classical shapes, but there's a general theory of shapes. Since I'm out of time I'll have to stop I wrote down some pages that talked about applications. One is you can talk about she's of infinity categories on infinity topos. By the way, do you want me to go on or do you want me to wrap things up. I can take five minutes and just fill in these pages. Yes, maybe just five minutes I think it's quick I can just fill in the definitions. I have an infinite category that's complete we can just define a she's on E with values and a to be a functor from E op to a, which preserves all limits, all small limits. If he is she's on a site, you can reinterpret this as in sort of a more conventional way she's on a site. And for instance, I can define things like she's of infinity categories, maybe large ones, even on E. So the example I wanted to mention is like the type of example that I mentioned in the very beginning, just to be specific if I have a scheme. I have for every open set some derived category of quasi coherent she's that's the one category. This doesn't form a sheaf. But derived categories come from infinity categories that derived category is the homotopy category of an infinity category. And the relation that sends you to its derived infinity category is in fact a sheaf in the sense this provides a language for talking about she's infinity categories. In some sense this is interesting even if he is just a one topos. Of course, if you have a one topos you can promote it to an infinity topos, and then talk about sheaves of infinity categories and infinity topos, and this is already useful in classical settings. You know, talk about these kinds of derived categories and people are doing that. The topos were really introduced by these authors for talking about derived geometry. So if you have a some category of ring like objects, which could be commutative DGA's or infinity rings vector if you like homotopy theory, then you obtain a notion of a ringed infinity topos and infinity topos together with a sheaf with values and a one of these categories of rings, and this leads to a notion of derived geometry. And to be honest, most of the interest is in this this category of generalized rings, but it's in some sense you need this notion of infinity topos in order to make sort of decent definitions. Differential comology this turns out to be a sort of an area where these have turned out to be useful. So in general comology, these are invariance of smooth manifolds that combine things like singular comology, say with the inter coefficients with Durham comology with the comology represented by differential forms. So what you can do is you can form an infinity topos. You can form a gross infinity topos a growth infinity topos of sheaves of infinity group oids on the large site of see infinity manifolds, which is actually essentially small so it's fine. This contains objects like a little bit of a plane objects which represents actually the singular comology of manifolds, but it also represents contains objects like omega n, which represents things like differential forms. And you can combine these to give things that represent differential comology. So this is a nice context where they're useful. Here's a page where I couldn't think of anything useful to say on one page so I won't. There's an interesting recent work by barric glassman and hey, and a paper called ex exodromy. I don't know how to pronounce the word exodromy. Which uses stratified infinity topoi and theories of constructible she was an infinity topoi to do some kind of generalizations of a sort of classical gal off theory in the in the context of algebraic geometry. Finally, I should mention the logical aspects of infinity topo and I didn't want to take much time in this because I don't understand this very much. So in one topoi. One topoi have an internal language. There's something for instance called the Mitchell Benabu language and that has an interpretation in a one topos. There's a notion of type theory due to Martin look called dependent type theory, which has an interesting aspect it introduces types for identity, instead of just having a sort of a relation. We have an identity type between any two terms of some other type. It also in many formulations introduces type families. So there's some sort of universal type. And it was noticed that these identity types behave like spaces of paths in a space, which suggests a homotopy theoretic interpretation, which was developed by everybody and Warren and boy volume of Ravadsky. And which led to the notion of univalent type theory. Let me not try to describe what univalent type theory is because honestly that's a whole lecturing itself and when I'm not competent to give. But we've asked he showed that infinity group points that is some official sets for my model for his univalent type theory. The univalent type theory has this universal type and that core corresponds to what I called the object classifier that the base of the universal morphism classifier. More generally, in fact it was early recognized that this was true and it's been proved by shulman that in some sense every infinity topos is a model for univalent type theory. So you can say that univalent type theory is the internal language of an entity topos in some sense. But you do have to be careful with that statement there is an issue here. It's not really an internal language in the way that I've been talking about infinity topoi. Type theory, including this univalent type theory that has functions, which can be composed, and that composition is actually associative on the nose. It's built into the way functions are described because they look like kind of look vaguely like functions of sets. So any model of such a type theory must be a one category. Must be a one category. That's the sort of model that these people have constructed. They've constructed models that are, for instance, a Quillen model category, which is only a one category. But the Quillen model category structure is used in describing how the model works. Fibrations play an important role. And so this theorem really says that it's a Quillen model category that can be chosen to be Quillen model categories whose corresponding infinity category, the infinity category you can extract from that one category is an infinity topos. So that's the sense in which every infinity topos is a model for a univalent type for univalent type theory. It's via one of these Quillen model categories and the interpretation is actually in the one category. I don't think anybody knows how to make this kind of linguistic or language or internal language that describes doing infinity category theory. This isn't really something I think anybody's really figured out to do in sort of a way that seems practical to me. So I think that's something that's an interesting subject if you like those things. All right, that's all I had to say thank you. Thank you so much for this very nice and very rich course.