 Okay, so I'm very pleased to introduce Professor Reske again. All right, thank you. So, I will continue today to talk about some properties of infinity topoi. So, let's see so last time. I defined the notion of an infinity topos. It's an infinity category. That's a left exact localization of a pre-sheave category, pre-sheaves of infinity group oids on an infinity groupoid. And then I spent some time describing an equivalent characterization. It's a presentable infinity category with universal code limits and descent. So it'll be useful. So, I want to return just briefly to topology again to motivate one more idea. So, in topology, you have the notion of a fiber bundle. So it's a map P whose fibers over any point or homomorphic to F, or more precisely, it's a locally equivalent to a product of F with the base. That's a fiber bundle. So, one of the great theorems of topology is that you have universal bundles. And we have even have a universal bundle with fiber F, usually under some hypotheses, but I will not even attempt to state those. So, one way to describe this is it's a bundle associated to a space called BG, which is the classifying space of a topological group. It's the topological group of homomorphisms of the space F. And then it's a group. You can form a principle bundle, the universal principle bundle of G, but we want the universal associated bundle with fiber F, which is constructed as a Burrell construction. All right, and then you get a correspondence bundles, fiber bundles with fiber F, correspond to maps from the base space into this classifying space BG. And this is, say, up to equivalence or up to isomorphism of spaces over the base. And on the other side, up to homotop. And of course, there are hypotheses B usually has to have some niceness property. It's not fully general, but it's great. It's a great theorem because it connects something that's not actually operate about homotopy theory. That's about topological spaces of particular form to isomorphism. And it says they're actually classified by homotopy. However, so if you're homotopy, if you're homotopy theory, you kind of want everything to be homotopy theory. So you can actually think about modifying the problem. Instead of looking at fiber bundles, I'll just look at arbitrary maps. Instead of trying to arrange for the fibers to be homomorphic to F, what I'll arrange for is for the homotopy fibers to be weekly equivalent to F. So I can form a homotopy fiber, which is the homotopy pullback of the map along any point in the base. And I can try to classify these. So I could ask for a universal bundle, which will have a similar form, but will G will be something which I'll call H or F. It exists and it has the property. If I look at arbitrary maps, E to be with homotopy fiber, weekly equivalent to F. Those correspond to maps from the base into BG. Here again, it's up to, you know, equivalence over the base here to be weak equivalence. Equivalence relation generated by that property here up to homotopy. So these exist. So it's this is not just an exercise and pure thought, by the way, examples of this were important early on in areas like surgery theory, sphere bundles up to fiber homotopy equivalence as it was called, was something you had to understand. It was actually something that was started to be developed very early on in the subject. So what is this H or F? Well, I'll take the space of all maps may have to F and then I'll notice that there's a subspace which consists of homotopy equivalences. It's actually a union of components of that space. This is not a topological group, but it's a topological monoid. Although it isn't a group, it's what's usually called group like as a monoid, because although it doesn't have inverses, it has inverses up to homotopy by definition. Any point in that monoid for any point in that monoid, there's another point such that the product is in the same past complaint as the idea. Okay, so this was, as I said, the work in the subject. By various people, there's a good formulation in 1975. Okay. Well, Oh, here's an example just to sort of orient you to how this looks in practice. So, for instance, let's take F to be an Islandberg McLean space. So maybe G is an abelian group and it is at least two. This is a space whose Nth homotopy group is isomorphic to the group G and whose other homotopy groups are trivial. Then you can actually compute what the monoid topological monoid of homotopy automorphism looks like up the homotopy. It's got a subgroup, if you like, that's actually the automorphisms of the abelian group that acts, but there's also a part that's the Islandberg McLean space is actually an extent it's equivalent to a topological group that's an extension of two groups, the discrete automorphism group of G, and the islandberg McLean space itself which can be up the homotopy given the structure of an abelian group. And then you learn that if you want to classify maps whose homotopy fibers and islandberg McLean space. That is, whose homotopy fiber only has homotopy group in a given dimension. And then that's equivalent to maps into this classifying space and that's useful because since we have a description of this classifying space we can describe. We have a description of this monoid we can describe this classifying space for instance this classifying space only has two non trivial homotopy groups. And pie one it's the automorphism group of G and then pie and plus one. It's G. And you can do similar things when in is one and G is just a group. In this case the answer is more complicated. So, this is basically what we might call a two group way of one group points that look like G, you can describe its homotopy groups. And pie one is actually the outer automorphisms of G, and pie two is the center of G, and the other homotopy groups are trivial. This gadget is related to the problem of classifying extensions of groups where the kernel is G. Okay, so that just gives you an example of how this might work in practice. So it's very tempting at this point since you can do this just to put everything together. By specify the fiber that seems like making things a little bit difficult let's take not maps with a given homotopy fiber just arbitrary maps up to the equivalence relation given by diagrams of this type. Well, those are going to correspond to maps up to homotopy to the disjoint union of all these classifying spaces of the homotopy automorphism monoliths where I take the co product over the collection of isomorphism classes, or rather the equivalence classes of spaces on call this thing I don't know omega. It's kind of a universal map. So over this guy there'll be some universal example of a map. So I'll call it the domain of that omega star. Of course, this is not really in the category anymore it's large to make sense of what that means. It's kind of a characteristic property. I think of it characteristic of infinity group it's it's not general shared by infinity categories you're not going to find in most infinity categories, a universal map of this type. But it does work in infinity group points. And, well, of course it's going to work. You can do this in infinity topos. There's an aside. So, up to up to this point I was working with sort of an explicit model of infinity group it's like top of spaces. Here I switched to think about an infinity category. So I'm using a different language. But if I take this Omega which is some kind of large infinity group where there's another large infinity group what I can think about. It's this. So this is an infinity category of infinity group points. This symbol means I'm taking the maximal infinity group void inside this infinity category of infinity group points. Well, that's actually what Omega is. I actually kind of said something like this in the first lecture if you go back I said, functors into s are equivalent to the slice over s which is a very strange thing to say, but it's true. And I said that I said the correspondence works for infinity group points. But it need not work in arbitrary infinity category. But you can build these kinds of things. Okay, so let me return to a more formal setting where I'm talking about infinity categories. So recall if I had an infinity category, E, I have the arrow category, you have an arrow. And inside there I had this subcategory not full of Cartesian the Cartesian subcategory whose, which has all the arrows as objects, but the morphisms are just the pullback squares. And what I'm asking for the dream is that you should have or would like to have a terminal object of this category that's what I'm asking for I'm asking for a universal morphism. I cannot generally have this. So I'll ask for something close. What I'll ask for is a subterminal object that infinity category. And this is what I'll call a universal family. So it's subterminal or what's an equivalent terminology minus one truncated. So what this means is if I take morphisms, the mapping space in this category of some arbitrary object arbitrary object and to you. This is going to be either equivalent to the terminal infinity group or the empty infinity group. If it was actually terminal it would always be equivalent to the terminal infinity was a terminal object. So the collection of universal families, I mean they exist as a partial order inside cart. Because of this property equivalent to a partial. So, of course I can that definition makes us perfectly sensible in a one category. So if you take a one topos a growth in the topos for instance, you have a sub object classifier which I'll write as a morphism from the terminal object to omega upper mono. And of course that has the property that monomorphisms e to be up to isomorphism correspond to maps from the base to this classifier. That's a sub object class right that is an example of a universal family. Set on it's the largest universal family. So every universal family in set is a monomorphism there in fact not very many that are all sub objects of the universal classifier. I'm sorry of the sub object classifier. When I was preparing this I was actually going to say that that's true in any one topos but I couldn't actually come up with a proof so maybe that's not true. I don't know. Somebody can figure that out for me. Okay, so I've defined this notion of a universal family it's a particular kind of morphism that's a sub terminal object of this Cartesian category. I can formulate that as saying equivalently as saying that the, the, the forgetful functor from the slice over you back to the Cartesian categories fully face. If that was an equivalent so to actually say was a terminal object. So this is saying it's sub terminal. So I can talk about the essential image of this functor which is will be a full subcategory of cart E, which I'll call that else of you. And that's an example of what I'm going to call a local class. The local class is a full subcategory of cart E arrow with two properties the first is that it's closed under base change, which since the morphisms and cart E arrow or all pullbacks anyway I can just say that if I if I map f f prime in there, and f prime is in my local class and so is f. And then the second property is that L has co limits, and the obvious functor from L back to the arrow category preserves co limits. That's the definition of a local class. If you remember from last time this notion of descent I talked about. The equivalent of saying that this thing itself cart E arrow is a local class. So if I have a universal family I get this corresponding local class L sub u. So that's the collection of morphisms that are pullbacks of you. That's the collection of all morphisms in E that are pullbacks that can be obtained as a pullback of you. So it's called a bounded local class. So what does that mean. So, let's suppose I have a local class L it's a full subcategory of cart E, and there's a functor from cart E arrow back to E which is the target functor so objects or morphisms. Just send it to its target. Let's pick an object of E, and I'll form pullbacks of infinity categories. These are pullbacks in infinity categories. So, here so if I look at the arrows whose target is be that pullback is going to be the slice you over be it's equivalent to the slice. Except that I'm actually looking at Cartesian squares, which are lying over the identity map of D. So the Cartesian squares are pullbacks the top maps also an equivalent so this is actually the maximal subgroupoid of the slice. So that's an infinity group or potentially a large one. But now I have this full subcategory. And so I can restrict to the subgroupoid. Which I'll denote with this lower L. So this is the group is infinity groupoid of, you know, whose object span by objects in the slice, which are in L. So it's the morphisms over B was target B in in the in the local class. And so to say that the local class is bounded is just to say that each of these things is infinity group of things that are in the local class is essentially small. So it's an equivalent to a small infinity group. That's what it means to be bounded. Now, in the case where I actually have a class that comes from a universal family. Well you can actually compute what this thing looks like. This infinity group is none other than the space of maps of B into the domain, the co domain of the universal family. And well it's a space of maps and those are always essentially small. If we have a locally small infinity category which all these are things like you are. So that's a balance it's kind of a size restriction on the local class. Okay, with this notion of a bounded local class. I can now assert a correspondence. The universal so of course this is for an infinity groupoid. This is for an infinity topos the universal families correspond exactly to the bounded local classes. The idea here is well to each be I can assign this essentially small insanity groupoid, which is determined by a local class also I should say here that the direction from left to right I've already described how a universal family gives you a local bounded local class. So I need to tell you how to go the other way. I'm going to start with my bounded local class L. Then I get this functor and it's a functor from E op. Well it's to infinity groupways and it's just small infinity group it's really essentially small. Just pretend there's the same as small. Or because you have these properties universal co limits and descent in your infinity topos this functor preserves limits as a functor from E op to s so it looks like a representable functor and then since he is actually presentable. Some nonsense that you have for presentable infinity categories tells you it's representable. You can use that to build the universal family it's actually representable by an object you that will be the codomain of the universal family. Okay, so that's telling you that. So finally, it turns out that in fact every morphism in infinity topos is contained in some bounded local class. So again this requires a proof which ultimately depends on the presentability hypothesis. So I can talk about sizes. So the idea here is that this category of this Cartesian. Cartesian arrow category can is a union of bounded local classes let's say ill Kappa, where ill Kappa is defined as the bound local class of relatively K Kappa compact morphisms Kappa some regular card. So you put some kind of size restriction, you get a bounded local class as the sizes become large, you get everything. And as a consequence you get an exhaustive collection of universal families, every morphism is in the bounded local class associated to some universal families, every morphism in E is a pullback of some universal class. And it's kind of, you know, nice to think about the union of those universal families as giving you a map which lives in a higher universe, which is this object or morphism classifier the universal map, let's say. This also gives you yet another characterization of infinity topos it's an ease and infinity topos if and only if it's a presentable infinity category. It has universal code limits. And it has enough universal families. That is every morphism is a pullback of some universal family. I feel like at this point there's a slide here I didn't make and maybe I feel like I should talk about it's sort of important. Let me just say something here briefly. So, this characterization of universal family. It's a universal property right it's something it's a subterminal object. There's a more intrinsic characterization of a universal family. And that's something called a univalent map. So, if I have a morphism in an infinity topos. I can form something that I'll call ISO P. Going to be times B. Or actually you can do this in a one topos as well. So what is this guy. So, to tell you what this is. I'll tell you what a map into ISO P is. Okay, it's a map over this projections pie. Let's say I know what f and gr that's two maps from T into B. Those correspond exactly to giving a diagram of the following type I can take. P and pull it back along either F or G so I'll call it F upper star V or G upper star V. And I can look at the collection of all maps that make a commuter triangle where this is an isomorphism. It's an isomorphism classifier for this map if you like. So this thing exists. In fact, you can make the same construction the one topos. And we say that a map P. Prime to you is univalent. If and only if. So there's one more map that exists here I should have mentioned. I. There's a tautological isomorphism of two pullbacks along the identity map, which is the identity maps this in some sense classifies the identity map as an isomorphism. So he is said to be univalent. If the map from the base maybe should use the same letters here consistent. If and only if B, going along I into ISO P is an isomorphism itself. And then the theorem that you can prove. He is a universal family, if and only if it's univalent. So this is the univalent that appears in the univalent type theory. This is how they recognize their version of universal objects. I'll say a little bit more about that at the, at the end. I did want to make that definition. I'm going a little bit behind, but I think it's okay. All right, let me pause for a second and get my wits back. Okay, so let me describe yet another closer related characterization. So, there's a notion of a van Kampen Co limit. So let's say I have an infinity category which has pullbacks. Then, as we've noted if I have a morphism in E, then I get an associated pullback funer. Let's look at a Co limit cone inside my infinity category. So J is an infinity category, probably small, and I'll take the right cone. So formally join a terminal object. And then you kind of have a notion of a Co limit cone. It's a van Kampen Co limit if the induced funer. From the opposite. To large infinity categories, the hat means it's large, which sends J an object to the slice is a limit cone in infinity categories. So slice takes co limits to limits of this type, or rather it takes this co limit to a limit. That's called a van Kampen Co limit. So this makes sense even in the one category. And so for instance, you know one topos co products are van Kampen. If X is an object in E that's a co product of some things than the slice over X is equivalent to the product of slices over the exercise. In general, pushouts or other co limits are not van Kampen. Sometimes they can be as we saw earlier, pushouts long monomorphisms are van Kampen. But in general, that's not the case. However, guess what an infinity topos, all pushouts are present our van Kampen in fact we have the following theorem ease and infinity topos if and only if it's presentable. And all small co limits. And I'll just briefly sketch the idea. So let's say I have an object in E that's a co limit of some functor from a small category I so I can form the slice over X. Or I could form the slices over the X I and then form this limit in infinity categories of those slices. And I got functors in both directions so that there's a functor from left to right that's built from the pullbacks of the sort of the tautological maps and X sign the X. By the way, did I'm getting the points where you know if you want to actually make this rigorous in infinity categories you have to do a lot of work. The way I'm talking about this is rather imprecise. So there exists such a pullback, but its definition requires some thought, but nonetheless you can define that. And there's a functor going the other way which is taking co limits. If you have a collection of maps, which are sort of in this functorily related. The fact that they this represents an object of the limit means that it's a Cartesian natural transformation so you can form the co limit. The fact that these two things give you the identity amount exactly to universality of columnets in this way. And in this way. That's descent. Modules are actually carrying out that proof that's the idea. I can answer that question now yeah so in one topos. In a one topos all columns including co products are universal, but there's another property that co products are disjoint and those two co products universal co products to join this joint give you this property in a one to us. So you can say a little bit more even. So, I have this functor from E op to large infinity categories that takes me to the slice. I could instead introduce a bounded local class and not use the whole slice but rather the full subcategory of the slice that is spanned by arrows over a which are in my bounded local class. So, of course the point of here is that this is actually a small infinity category. We saw already that it's maximal subgroupoid is a small infinity group. This functor. Well it preserves limits using descent and well using university out university of co limits. And because he is a presentable infinity category. You can show that it's representable by an internal. Infinity category object called you dot L of the infinity topos. So I let me not tell you what an internal infinity category object is, there is such a definition. And this even leads to another characterization to the rustic that ease and infinity topos if only if it's presentable and every morphism in the local classes represented by an internal infinity category object. That's a pretty story. So around this point, or maybe earlier, people start wondering, well, you're talking about infinity topoi which are analogs of growing the topoi what about elementary infinity topoi. And my answer has always been that I don't know what elementary means. And other people know what elementary means I don't know what elementary means. And that's the connotation of sort of using finite constructions people talk about first order logic or things like this. That's kind of hard to do an infinity topoi because if you notice here infinity is in the name. In infinity categories it's very hard to get away with anything that it sort of has a truly finite nature. In the sense of say first order logic, I don't know how you would do that. The last people have proposed definitions of an elementary infinity topos. This is rough definition. So you take an infinite you say it's an elementary infinity topos if it's finite complete and co complete. It has a sub object classifier, and then this property every morphism is contained in a local class which is represented by an internal infinity category object. That's one possible definition. You want this action by the way because it's not implied by the last action. Every monomorphism will be contained in some local class, but that local bounded, which is represented by internal infinity category object. But there may not be a single one that represents all the monomorphism so that extra property. This shows that this is good enough for instance to show that your infinity category is locally Cartesian closed for instance. So you do actually recover things that you would like to have. I haven't talked much about the Cartesian closure of infinity topoi their Cartesian closed and even locally Cartesian closed. Okay. So I'm going to shift gears now. I want to talk about some particular examples of local classes that are important. So let's start by reminding ourselves what monomorphisms are. So in an infinity category with pullbacks, F is said to be a monomorphism. If it's diagonal is an isomorphism. Now, in any infinity category we can talk about a pair of maps being morphisms being orthogonal, as we can do in a one category. So let's say that the two maps are orthogonal. If a unique lifts exists in any commutative square, where the left and right sides are F and G. So top bottom can be anything and there's a unique lift making in any situation there's a unique dotted arrow making both triangles commute. Another way to formulate this is that you can form a commutative square of home sets. In an infinity group void mapping spaces, where you have composition with G, or composition with F, and this should be a pullback that's an equivalent condition. Unique in an infinity category means unique up to contractable choice. That is, there's an infinity group void worth of choices of lift. We want that to be contractable. That is equivalent to the terminal infinity. That's equivalent to that diagram on the right being a pullback. So we can define a cover in an infinity category to be a morphism, which is orthogonal left orthogonal to any monomorphism. There are other terms used here. The most common one is actually effective epimorphism, instead of cover, and some people even say surjection. However, I need to emphasize something, which is something that makes this kind of an awkward term, which is that covers are usually not epimorphisms. In the first hour I pointed out that epimorphisms in infinity group voids are kind of rare and a little bit strange covers are not generally epimorphisms. So effective epimorphism for that reason is an inconvenient term. For example, if I'm in infinity group voids, which is sort of the homotopy theory of spaces, then a map turns out to be a cover if and only if the induced map on the sets of path components is surjective. So these are the sets of path components. Or one way to say that is you can lift a point in the space up to homotopy. That's what a cover is. The class of monomorphisms is stable under pullback, because it's a find using limits. So a map is a cover, if and only if a unique lifts in every diagram where I only need to use squares where the bottom is the identity of B here, because I can pull back everything to that set to that case. Now, the thing about a monomorphism is that the that we're asked actually asking about maps in the slices so the maps in the slice from F to you from F to G is already either empty or contract. I don't know why I write, I think I wrote something wrong here. This is what I meant to say. A lift exists. I'm sorry, I kind of, I guess I'm not thinking correctly. This was correct, but I haven't completed the thought. Because this is a monomorphism maps of anything into G in the slice is either empty or contract. So, sorry, I'm trying to interpret my notes or sort of not understanding. Probably so we can figure out what I mean to say. Um, I get, let me let me sort of just cut through this because of this, it turns out the uniqueness condition is not necessary. And ultimately we get a statement which has this form. If there's a cover, if and only if, in the slice category, the space of maps is non empty, and therefore contractable was the same non empty for all monomorphisms into be. In other words, if that's a cover. That's the case. That sounds right does that sound right. I think that sounds right. Yeah, that's what happens when you have a suggestion. No, that's wrong. That's I'm I've had exactly backwards. I'm sorry. I apologize. I think I'm confused about something here. What do I mean to say, I think what I mean to say, let's just say what I mean to say, this is, this is non empty. Only if G is an isomorphism. Assuming that G is a monomorphism. That's a cover. So the point is there's a very elementary description of what a cover is it's like games or that property. So F is a cover and if it's a cover in the slice because of this pullback property. I want to note one other condition that you have. If a composite is a cover. That implies that G is a cover. If I have a commutative diagram like this. And that's a cover. That automatically implies that that arrow is also a cover. This is kind of interesting. This is a kind of asymmetry with monomorphism. So I told you in the first hour that if you have that in the infinity world. Monomorphisms do not have the property that if something factors as the first map through a monomorphism then it's also a monomorph but covers do have the complimentary property. There's a consequence here if I have any co limit in my infinity topos. Actually this isn't any infinity category, then they do snap from the co product of evaluations that objects and I is a cover. And I've described a proof here. I'm going to pass over that. So in a presentable infinity category. The classes cover and mono of covers and monomorphisms form a factorization system. So they're mutually orthogonal in the sense that I described, both classes are stable in the retracts and every morphism can be factored as a cover followed by a monomorphism. So for any morphism from a to B, you get a factorization which I'll probably like right like this, where I have a cover, and a monomorph and I'll call the intermediate object the image. And this factorization is essentially unique, meaning unique up to contractable choice, I might just say unique, the only notion of uniqueness you have categories. This is the replacement for epi mono factorization, but these are not remember covers are not at ease. Now, in an infinity topos you can actually construct cover mono factorization directly using the check nerve. So I'll sketch this briefly. If I have a morphism F. I can form the check nerve this is what's called an augmented simplicial object. The object is a factor from the right cone of the delta ops a category that indexes that's for instance, and it looks like this. The augmentation is the map F, and then you complete the diagram by putting in iterated fiber products of F along itself. So I can kind of write current extension along along the inclusion of the subcategory that just has this map. So what I can do is, well, I can take the co limit of the check nerve restricted to delta off the subcategory, the simplicial index and subcategory. The co limit maps to the cone point which is B, and it has a map from a because that's one of the objects that I'm taking the diagram taking a co limit of. And this actually turns out to be the cover mono factorization so that's the image of my map. It's precisely analogous to something you can do in a one topos, except, in fact, the same thing is true in a one topos only in a one topos usually cut off the diagram right here, because the part of the diagram to the left of that line is irrelevant. When you do calculating this co limit in a one topos. I've sketched a proof out here, I'll just tell you some bits of the argument. I'm going to show that this works well without loss of generality I can assume that it's a map from a to the terminal object because I can work on the slice. So here's what the check nerve looks like. If I have some product with a everywhere, just the constant diagram a, then I have some extra structure here I have some additional maps that go in the reverse direction, which is classically called a contracting homotopy for the augmented object. So really I'm extending the functor along some inclusion to a larger category index is that it turns out that diagrams like that are given example of an absolute column in infinity categories, every composite that factors through this extension is a column in any infinity category where you can do this. No other conditions needed. This is precisely analogous. In fact, I probably implies a more classical statement in one categories about split co equalizers, which probably many of you are aware of split co equalizers are an absolute co in one split co equalizers are not an absolute co limit infinity categories, you must instead use this simple logic with a contracting on the top if you'd like a similar result. Anyway, you can use this to derive the results. So, he here is going to be this co limit. If I take a times a times e I can put it inside the co limit because I have universality of co limits. I can put a times CF but see that's a co limit of iterated products. And anyway, I just showed you that that column was going to be a. So the projection map from a times e to a is a. Then I can do the same thing by taking e times e, put it inside the co limit, each terminal of the co limit involves at least one copy of a times is a. I can use that to show I get e. So actually the projection map from e times e to e is an isomorphism, therefore the diagonal map is an isomorphism, so e maps monomorphically to one. And then because he is a co limit, because of something I said earlier on the co product of all the values of this diagram, mapping to e is a cover. And actually it's a little bit better. All those maps factor through this given map. They factor through a, and therefore we can conclude that P is a cover. That's the proof. That's a fairly straightforward proof it's really something that you could have done a one category. I've only used universality of co limits I did not use descent in this art. So a cover covers give you another characterization of local classes. So, if I have a class in the Cartesian arrow category. It's a local class if and only if it has these three properties. It has to be closed under co products. And second of all for any pullback diagram that shape. Well, if G is in the class and F is in the class. But also, if P is a cover if the bottom map is a cover if you pull back along a cover, then F is an L implies G is an L. So the third condition says that if a map is locally in the local class, then it's in the local class hence the name. I apparently have a few pages here where I describe Oh sorry here's a consequence first of all. Suppose I have an arbitrary morphism in my infinity topos. I can define a class of maps. So L or F will be the class of all maps G double prime to be double prime, such that the following is true there exists a diagram of the following form. So some pullback of G along a cover. Which for which is also some kind of pullback of F. So G looks locally. An immediate consequence of the characterization I just gave you is that LF is a local class in fact it's the smallest local class containing F. Every local class is contained in a bounded local class and this is the smallest therefore it's also bounded. It exists, which we might call a universal family, which classifies maps, which locally look like F. Okay so here I'm giving a characterization of local classes. Let me sort of run through this very quickly. So in one direction I want to show that local classes have these three properties and two of them are actually immediate local classes are closed under co limits. They have to be closed under co products and base change is part of the definition. The key part is to show that in the local class you have this pullback along cover property. So the idea is if you have a cover. Well, it's the code limit of a check nerve of its own check nerve. So if I have a G, which is in my, and I want to know, I want to figure out if it's in a local class but I know it's pulled back as in a local class. So let's pull back the whole thing along the check nerve. So I get some business like this, actually get another check nerve. The stuff running along the top is also a co limit that's universality of co limits on F is in the local class. Therefore, so are all its pullbacks because that's part of the definition of a local class so all these fk or an L. So by forming the co limit along this car keys and natural transformation of some special objects. So G is in the local class as well. That's how you prove the locality property. And here's the proof in the other direction. So the you're already so local classes to close under base change and closed under co limits along involve of the Cartesian arrow category base changes automatic. So we already have that's closed under co products. If I have a general Cartesian natural transformation from some category with the property that each morphism that natural transformation is in the local class. So really I have a functor into L, viewed as a subcategory full subcategory of a Cartesian arrow category. Well, I can form the co limit. So that's a morphism in the Cartesian arrow category system morphism. All of these, because of the descent property for every GI for every eye. Each of these diagrams is going to be a pullback. That's what descent gives me. I form the co limit of a Cartesian natural transformation. And then I pull back, I pull back again I get where I started. Therefore, if I take the co product of all these maps indexed by I, that's also a pullback and that's also using descent. Oh, but P is a cover. So. Oh, well by my hypothesis the f eyes are all or the fi eyes are all an L that's that class is closing under co products that's one. That's the first property. And then, well, property three gives me that G is in the class. So that shows that on the classes closed under co limits in the Cartesian arrow category. The point of giving these arguments is to show that all I'm ever really using is University of Co limits and descent those those two properties really lets you prove all these things. I'm coming close to an hour. So, even though I have more in this particular lecture I might pause very soon. These ideas also show that both monomorphism and covers are themselves local classes and I've sketch the proofs here. All right, let me see how much more I have here. Oops. Let me do. Let me take about three more minutes, and I'll talk about truncation. So as I said the notion of monomorphism is the first of a sequence of conditions called in truncated. So remember in an infinity category a map is in truncated. If the iterated diagonal. So you iterate the diagonal construction on a map. If the iterate diagonal is an isomorphism. I have to do it in plus two times. Then an object is in truncated if the map to the terminology is in truncated. I'm going to write C less than or equal to n for the full subcategory of in truncated objects. I'm going to use something here. If I have an in truncated object. Then, if I look at the mapping space into that in truncated object that has to be an in truncated infinity group void, that is an n group void. So, this full subcategory of in truncated objects is an example of what's called an n plus one category, or a better term is probably in plus one comma in category. This term is in category. That is, it's a category whose function mapping spaces are actually just in the voids. So I get from this, this chain of classes of morphisms in my infinity category. Isomorphisms the monomorphisms are the minus one truncated maps. And by orthogonality, I get a complimentary classes of in connected maps. I don't seem to have the definition and connected here but I hope it's obvious if is. Oh, it's right down here. A map is in connective, if it's orthogonal to the n minus one truncated maps. There's other terminology that's used here sometimes this is called in connected. And sometimes this is called in minus one connected. So that's great. I'm going to follow Jacob Luria and call it in connective. So I don't have to deal with this confusion. When objects in connective if it's mapped to the terminal object is in connective in a presentable infinity category. It's a factorization system. In particular, you get factorizations essentially canonical of any map into an in plus one connected map, followed by an in truncated map I'll call it the image. I'll call it the relative in truncation. And then if I apply that to the map to the terminal object, it's called the absolute in truncation. TRF of X. So it's really just a special case of the first one. As with cover and mono these are local classes. I'll note one more thing. They come in a tower, because it's a nested sequence of classes. So if I take an object, I have associated to it, a tower of truncations. So for example an infinity group oids, a space is truncated in truncated if and only if its homotopy groups are trivial for every choice of base point in the space and every K strictly bigger than in. And similarly, a space is in connective. If and only if it's homotopy groups are trivial for every choice of base point, and every K less than or equal to in here I have to be careful I have to also make sure that the space is not empty. Assuming N is at least zero. This is what's classically known as an N minus one connected space. There's an off by one in the terminology. And then you discover that an infinity group or it's a map is either in truncated or in connective, if and only if all it's homotopy fibers. Over all points and why are in truncated. Or in connective as the case may be this condition in homotopy theory is classically called an in connected. Now you see the source of the confusion involved in the terminology comes from homotopy theory. In spaces classically there's a construction of the introduction of a space you kill off the homotopy and high dimensions by attaching cells of large dimension. In a presentable infinity category this construction exists formally. I'll note here that if you want to compute this and appreciate in a pre chief category, you can just compute it point wise, because truncation is given by a limit condition, being truncated. If you just truncate point wise it's a functor. And it turns out to compute the truncation in pre sheaves. So if I have a left exact localization, I can use this adjunction to compute the truncation in E, in terms of the truncation in pre sheaves and that's because both L and I preserve the property of entruncation, because they both preserve finite limits, in particular pullbacks, which is all I need to define entruncation. If you just entruncate an object in E, you just entruncate as a pre sheaves and then she's alive. Okay, I will pause here, and then we can have our break. Let me finish my discussion of truncation and connectivity by briefly describing an example. Just so you can see how some of these fit together. So let's think about n gerbs. I'm going to follow Lurie here I'm going to find an engine in gerb in is the intersection of the classes of entruncated and in connective maps. According to my sources, apparently these are just the only room acclaimed in gerbs and that injured was actually something much more general, which I was unaware of. This is sort of the first non trivial example of an intersection if I took the entruncated maps and the n plus one connective maps. The intersection is just the isomorphisms, because they're orthogonal class orthogonal classes. So this is sort of the first non trivial example, not just isomorphisms. So let's write infinity topos e let's write e subs gerbin for the full subcategory spanned by maps from e to the terminal object which are in gerbin. So the interesting property of this full subcategory it's an infinity category, but the interesting property is that it's almost a one category in some sense. What's actually true is that I could if I look at the infinity category of pointed objects in e subs gerbin. So things equipped with a section. This is actually is a one category, really equivalent to a one category, but I will say is. In fact, you can describe it. It may look like a familiar story. The pointed n sharps for anger Nick but to our equivalent to the one category of a billion group objects in the one category. E less than or equal to zero of zero truncated objects in your infinity group or sorry in your infinity topos. If n equals one, you use group objects. And if n equals zero there's not much to say you just say you're using pointed objects, as we'll see E sub less than or equal to zero the zero truncated objects. That is an example of a one topos. As we will see. So the construction of I won't prove this proposition but I'll construct the functor from E sub gerbin star from the pointed n sharps to the zero truncated objects. So I won't give the billing group structure. If I have this pointed object, I can take the infold iterated diagonal of the point inclusion or be mapped from the point. Let me be careful and not call it an inclusion because it may not be a monomorphism. Nonetheless, I have this diagonal on I generally I'm going to write omega n of s for the target of the iterated diagonal. That turns out to be well, P is in truncated and you can use that to show that if I take the infold your diagonal of s s is actually n minus one truncated and therefore this thing. This object is actually going to be zero truncated. I call it omega n because in fact in spaces this is the infold iterated space functor of a base space. So that's the construction of that functor. And then this has an inverse functor which takes an appealing group object to something that's usually called can. And that's the I remember McLean object. So for instance, suppose I pick my appealing group object. Then I get an I remember McLean object. In E, it's actually an engine. As we saw to associate to every map. A universal family classifying the maps that locally look like your given map. So we get a universal family uan star to uan. It's a universal family of jerbs, because jerb is itself a little class so it's in that class, but it's of jerbs that locally look like P. So, for instance, just to orient yourselves if you know something about jerbs. So here's this universal family. We have our sort of typical example, which is the pointed Jerb associated to the dealing group a, this actually factors through another Jerb, which is the inclusion of a base point and the KN plus one. That's also an injured. So we have pullback diagrams like this. There's a pullback diagram also like this. What the trivial compute trivially commuting diagram. This is the pullback diagram that relates KN with KN plus one one is loops on the other. I mentioned this just because this thing in the middle. P is not a universal family. It's merely a pullback of a universal family, but it does classify something it classifies. It's bounded by a notion I won't mention. But this is just to orient you. If you've heard about shares of banding is a is a structure you put on this Jerb, which is sometimes is a local identification of its fibers with a certain sense or a choice of identification. Anyway, so for instance, we get a very cleanly a theory of these EM Jerbs from this from these notions in an infinity dopus. Here's one more example that I'd like to mention. This is the class of infinity connected maps. So the classes of in connective maps form a descending chain of classes. So we can take the inner intersection. So these are the things, if you like, they're the maps such that the image factorization, the image is equivalent to the code domain for all in. It's an intersection of local classes. So it's also a local class. So the matrix is infinity connected. If the terminal object is in infinity to boy like infinity group boys are more generally sheaves infinity to boy infinity connected is the same as isomorphism. That's some version of the Whitehead theorem in homotopy theory. We could call this is our by definition determined by homotopy groups. That's a consequence of this fact. The first thing is that in an infinity topos there can exist non trivial infinity connected objects and I'd like to give you an example. So here's my example of a non trivial infinity connected object first I need the infinity topos I'll start with a topological space, which has this lattice of open sets. I don't have a point set for the topological space but I don't need to tell you what a sheaf is. So I'm telling you the locale. So I can define a pre sheaf of infinity group or it's on this so that it's values at you zero plus minus you one plus minus you two plus minus are equivalent to the terminal object. The values at the V's are an Eilenberg McLean space let's say KZN or ZZ integers just to be specific. That's an Eilenberg McLean space. So, of course I'm going to do it so that it's actually a sheaf. So here's a picture of this functor. The squares that I'm going to draw here are going to be pullback squares because KZN is is the homotopy pullback of such a diagram or more classically it's the loop space and KZN plus one. So this is actually a sheaf on X, which as I've told you is an example of an infinity topos. Now, the claim here is that this F is actually in infinity connected. However, it's not the equivalent to the terminal object. You know that because it's a sheaf, but you know it's values are not contract values at the VN so it's not the terminal object to it. So if you prove something like this, well, if I want to compute the, let's say the M truncation of F for NEM in sheaves. Well, what I do is I compute it in pre sheaves on open subsets, and then I sheafify, and I compute the truncation on pre sheaves point wise. Well, the values are either already contractible, or these Euler-Berber claim spaces. But the M truncation of an Euler-Berber claim space for a fixed M, this is going to be trivial when N is large. I guess N is bigger than M. So if I fix an M and I trunk M truncate everything in this picture up here, almost everything becomes contractible. The first few values at V aren't, but eventually they'll all be contractible. And then it's fairly easy to show from that point that the sheafification will just be the contractible object, because you can actually recover the sheafification without the first few values at the VIs. Oops. By an argument that I won't show. You can explicitly say what L looks like in this case. So that's an example of a non-trivial infinity connected object in an infinity table. So that is a phenomenon that does happen.