OK, so I'm very pleased to introduce Professor Resk 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 presheaf category, presheaves of infinity groupoids on an infinity groupoid. And then I spent some time describing an equivalent characterization. It's a presentable infinity category with universal co-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 are homeomorphic to F, or more precisely, is 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 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 homeomorphisms of the space F.

And then it's a group. You can form a principal bundle, the universal principal bundle of G, but we want the universal associated bundle with fiber F, which is constructed

as a Borel construction. 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 homotopy. 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 a homotopy theorist,

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 these fibers to be homeomorphic to F, what I'll arrange for is for the homotopy fibers to be weakly 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 G will be something which I'll call H-aught F. And this exists, and it has the property. If I look at arbitrary maps, E to B, with homotopy fiber weakly equivalent to F,

those correspond to maps from the base into BG. Here, again, it's up to equivalence over the base. Here, it would be weak equivalence. So the equivalence relation generated by that property.

And here, up to homotopy. So these exist. So this is not just an exercise in pure thought, by the way. Examples of this were important early on in areas like surgery theory. Turns out sphere bundles up to fiber homotopy equivalence, as it was called, was something you had to understand.

So this was actually something that started to be developed very early on in the subject. So what is this H-aught F? Well, I'll take the space of all maps from F 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 the 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 component 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 Eilenberg-McLean space.

So maybe G is an abelian group, and N 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 to 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 Eilenberg-McLean space itself. It's actually an, it's equivalent to a topological group

that's an extension of two groups, the discrete automorphism group of G, and the Eilenberg-McLean space itself, which can be up to homotopy, given the structure of an abelian group. And then you learn that if you wanna classify maps whose homotopy fiber is an Eilenberg-McLean space,

that is, whose homotopy fiber only has a homotopy group in a given dimension, 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 as classifying space. For instance, this classifying space

only has two non-trivial homotopy groups. In pi one, it's the automorphism group of G, and in pi n plus one, it's G. That's all if n is greater than equal to two. And you can do similar things when n 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 groupoid of one groupoids that look like G. You can describe its homotopy groups. Pi one is actually the outer automorphisms of G,

and pi 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 specifying the fiber, that seems like making things a little bit difficult.

Let's take not maps with a given homotopy fiber, which is arbitrary maps, up to the equivalence relation given by diagrams of this type. Can you do diagrams? Well, those are gonna correspond to maps up to homotopy, to the disjoint union of all these classifying spaces

of the homotopy automorphism monoids, where I take the co-product over the collection of isomorphism classes, or rather the collection of weak equivalence classes of spaces, 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. So you have to make sense of what that means.

This is kind of a characteristic property, I think of it. Characteristic of infinity groupoids. It's not general shared by infinity categories.

You're not gonna find in most infinity categories, a universal map of this type, but it does work in infinity groupoids. And well, of course, it's gonna work. You can do this in infinity topos. I should say here as an aside, so up to this point, I was working with sort of an explicit model

of infinity groupoids like topological 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 groupoid, there's another large infinity groupoid I can think about. It's this. So this is an infinity category of infinity groupoids.

This symbol means I'm taking the maximal infinity groupoid inside this infinity category of infinity groupoids. 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 the correspondence works for infinity groupoids,

but it need not work in arbitrary infinity category.

But you can build these kinds of things in infinity groupoids. 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 the Cartesian subcategory, 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 sub terminal object of that infinity category. And this is what I'll call a universal family.

So it's sub terminal or what's an equivalent terminology minus when truncated. So what this means is if I take morphisms, the mapping space in this category of some arbitrary object into U, this is going to be either equivalent

to the terminal infinity groupoid or the empty infinity groupoid. Oops. If it was actually terminal, it would always be equivalent to the terminal infinity if it was a terminal object. Okay. So the collection of universal families,

assuming they exist as a partial order inside cart because of this property, equivalent to a partial order. So of course I can, that definition makes us perfectly sensible in a one category.

And then one topos, a Groton-D 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 B up to isomorphism correspond to maps from the base to this classifier.

That's a sub object classifier. That is an example of a universal family. In set, it's the largest universal family. So every universal family in set is a monomorphism.

They're 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 if somebody can figure that out for me. All right. 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 could formulate that as saying equivalently as saying that the forgetful functor from the slice over you back to the Cartesian category is fully faithful. If that was an equivalence, that would actually say it was a terminal object. So this is saying it's sub terminal.

So I can talk about the essential image of this functor, which will be a full sub category of cart E, which I'll call that L sub U. And that's an example of what I'm gonna call a local class. So a local class is a full sub category of cart E arrow

with two properties. The first is that it's closed under base change, which since the morphisms in cart E arrow are all pullbacks anyway, I can just say that if I have a map F to 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, that's actually equivalent to 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 U. That's the collection of all morphisms in E that are pullbacks that can be obtained as a pullback of U. That's actually what'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 are 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 B, that pullback is gonna be the slice E over B,

it's equivalent to the slice. Except that I'm actually looking at Cartesian squares, which are lying over the identity map of B. And because they're Cartesian squares, they're pullbacks, the top map's also an equivalent. So this is actually the maximal subgroupoid of the slice.

So that's an infinity groupoid, 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 infinity groupoid of,

you know, whose spanned by objects in the slice, which are in L. So it's the morphisms over B, with target B in the local class.

And so to say that the local class is bounded is just to say that each of these things, these infinity groupoids of things that are in the local class is essentially small. So it's an equivalent to a small infinity groupoid.

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 groupoid is none other

than the space of maps of B into 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 of these are, you know, things like ER.

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, sorry. This is for an infinity topos. The universal families correspond exactly to the bounded local classes. The idea here is, well, to each B,

I can assign this essentially small infinity groupoid, which is determined by a local class. Oh, so I should say here, the direction from left to right, I've already described how a universal family gives you a bounded local class. So I need to tell you how to go the other way.

So if I start with my bounded local class L, then I get this functor, and it's a functor from EOP. Well, it's two infinity groupoids, and it's just small infinity groupoids, really essentially small. Let's just pretend those are the same as small.

Furthermore, because you have these properties, universal co-limits and descent in your infinity topos. This functor preserves limits. There's a functor from EOP to S,

so it looks like a representable functor. And then since E is actually presentable, some nonsense that you have for presentable infinity categories tells you it's representable. And you can use that to build the universal family. It's actually representable by an object U. That'll be the co-domain of the universal family.

So that's telling you that, well, tell me what it's telling you.

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 arrow category

is a union of bounded local classes, let's say L Kappa, where L Kappa is defined as the bounded local class of relatively K Kappa compact morphisms. Kappa is some regular cardinal. So you put some kind of size restriction, you get a bounded local class. And if the size has become large, you get everything.

So 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. It's kind of 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 an infinity topos. E is an infinity topos if and only if it's a presentable infinity category. It has universal co-limits, oops.

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. 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 a sub-terminal object.

There's a more intrinsic characterization of a universal family. And that's as something called a univalent map. So if I have a morphism

in an infinity topos, I can form something that I'll call isope going to B times B, infinity topos. 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 isope is. Let's say it's a map over this projection, it's called pi.

Let's say I know what F and G are, 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.

Those are the pullbacks of P along either F or G. And I can look at the collection of all maps that make a commutative triangle where this is an isomorphism. So it's an isomorphism classifier for this map if you like.

So this thing exists. In fact, you can make the same construction in the one topos. And we say that a map P prime to U is univalent if and only if...

So there's one more map that exists here. I should have mentioned, let's call it I. So there's a tautological isomorphism of two pullbacks along the identity map, which is the identity map. So this in some sense classifies the identity map as an isomorphism. So P is said to be univalent if the map from the base,

maybe I should use the same letters here just to be consistent, if and only if B going along I into ISO P is an isomorphism itself. And then the theorem that you can prove is that P 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 end.

I did wanna make that definition. I'm going a little bit behind, but I think that's okay. All right, let me pause for a second and get my wits back. Okay, so let me describe yet another

closely related characterization. So there's a notion of a van Kempen 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 functor.

Let's look at a co-limit cone inside my infinity category. So J is a infinity category, probably small, and I'll take the right cone. So formally join a terminal object, and then you can have a notion of a co-limit cone. It's a van Kempen co-limit if the induced functor

from the opposite, oops, 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 Kempen co-limit. So this makes sense even in the one category.

And so for instance, in a one topos co-products are van Kempen. If X is an object in E, that's a co-product of some things, then the slice over X is equivalent to the product of slices over the X I's. In general, push-outs or other co-limits

are not van Kempen in a one topos. Sometimes they can be, as we saw earlier, push-outs along monomorphisms are van Kempen, but in general, that's not the case. However, guess what?

In infinity topos, all push-outs are van Kempen. In fact, we have the following theorem. E is an infinity topos if and only if it's presentable, and all small co-limits are van Kempen. 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 infinity 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. Well, I got functors in both directions. So there's a functor from left to right that's built from the pullbacks of the sort of the tautological maps from X I to X. By the way, I'm getting to the points where, if you wanna 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 functorially related, the fact that this represents an object of the limit means that it's a Cartesian natural transformation. So you can form the co-limits. And the fact that these two things

give you the identity amount exactly to universality of co-limits 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 colons and clean co-products are universal,

but there's another property that coproducts are disjoint. And those two coproducts, universal coproducts, disjoint give you this property in a one topos. All right, so you can say a little bit more even.

So I have this functor from EOP 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 sub-category 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 groupoid. This functor, well, it preserves limits using descent

and well using universality of co-limits. And because E is a presentable infinity category,

you can show that it's representable by an internal infinity category object. It's called U dot L of the infinity topos.

So let me not tell you what an internal infinity category object is. There is such a definition. And this even leads to another characterization due to Russach that E is an infinity topos if and only if it's presentable and every morphism in a local class is represented by an internal infinity category object.

So that's a pretty story. So around this point, or maybe earlier, people start wondering, well, we're talking about infinity topoi, which are analogs of grotony 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. It often has a connotation

of sort of using finite constructions. People talk about first order of 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. And in infinity categories, it's very hard to get away with anything that sort of has a truly finite nature

in the sense of say, first order of logic. I don't know how you would do that. Nonetheless, people have proposed definitions of an elementary infinity topos. This is Rothfech's definition. So you take an infinity, 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 axiom, by the way, because it's not implied by the last axiom. 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 you need that extra property. Rothfech 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 topos. They're Cartesian closed and even locally Cartesian closed. Okay, so I'm gonna shift gears now.

I wanna 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 morphisms being orthogonal, as we can do in a one category. So we'll 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 the top and bottom can be anything and there's a unique lift making, in any such 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 homsets

or in an infinity groupoid 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 contractible choice. That is, there's an infinity groupoid worth of choices of lift. We want that to be contractible and 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 will even say surjection. However, I need to emphasize something which something that makes this kind of an awkward term is which is that covers are usually not epimorphisms.

In the first hour, I pointed out that epimorphisms and infinity groupoids 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 an infinity groupoid, which is sort of a 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 in spaces.

The class of monomorphisms is stable under pullback because it's defined 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 case. Now, the thing about a monomorphism is that we're actually asking about maps in the slices. So the maps in the slice from F to U,

from F to G is already either empty or contractible. I don't know why I wrote something wrong here. Let's put this in, that's 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 contractible.

So, sorry, I'm trying to interpret my notes here,

which I'm not understanding. Probably somebody can figure out what I mean to say. 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, F is a cover if and only if in the slice category, the space of maps is non-empty and therefore contractible, what's the saying non-empty for all monomorphisms into B. 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 surjection. No, that's wrong. I've hit 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 non-empty only if G is an isomorphism. They're 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 being served that property. So F is a cover in E, if not only if it's a cover in the slice because of this pullback property. I wanna 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 throughout monomorphism, then it's also a monomorphism, but covers do have the complimentary property.

Okay. There's a consequence here. If I have any co-limit in my infinity topos, actually this is in any infinity category, then the induced map from the co-product of evaluations

at objects in I is a cover. And I've described a proof here, but I'm gonna pass over that. So in a presentable infinity category, the classes cover in mono of covers and monomorphisms

form a factorization system. So they're mutually orthogonal in the sense that I described. Both classes are stable under 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 write like this, where I have a cover and a monomorphism. I'll call the intermediate object the image. And this factorization is essentially unique.

Meaning unique up to contractual choice. I might just say unique. Notion of uniqueness you have in any of the categories. So this is the replacement for epi-mono factorization, but these are not, remember covers are not epis.

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.

So it's a functor from the right cone of the delta op, the category that the index of simplicial sets 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. This is really a kind of right con extension along the inclusion of the sub category 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 op, the sub category, the simplicial indexing sub category. And that gives me a factorization.

The co-limit maps to the cone point, which is B. And it has a map from A, cause 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.

This is actually 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, you usually cut off the diagram right here, cause the part of the diagram to the left of that line is irrelevant when you're 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. So to show that this works, well, without loss of generality, I can assume that it's a map from A to the terminal object cause I can work on the slice.

So here's what the check nerve looks like. If I take the 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 simplicial object. So really I'm extending the functor along some inclusion to a larger category, which indexes that.

And it turns out that diagrams like that are given example of an absolute co-limit in infinity categories. Every composite that factors through this extension is a co-limit cone 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-limit in one category. Split co-equalizers are not an absolute co-limit

in infinity categories. You must instead use this simplicial object with a contracting homotopy if you'd like a similar result. Anyway, you can use this to derive the result. So E here is gonna be this co-limit.

If I take A times E, I can put it inside the co-limit because I have universality of co-limits. So I have A times CF, but that's a co-limit of iterated products. And anyway, I just showed you that that co-limit was gonna 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 determinant of the co-limit involves at least one copy of A, A times E is A. So in fact, 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 E is a co-limit, because of something I said earlier, the co-product of all the values of this diagram, mapping to E is a cover. But 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 in one category. I only used universality of co-limits.

I did not use descent in this argument. So 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. That's the first property. And second of all, for any pullback diagram of 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 this third condition says that if a map is locally in the local class, then it's in the local class, hence the name.

All right, I apparently have a few pages here where I described, 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 upper F will be the class of all maps, G E double prime to B double prime,

such that the following is true. There exists a diagram of the following form. There exists some pullback of G along a cover for which is also some kind of pullback of F. So G looks locally like F.

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. So there 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 wanna show that local classes have these three properties, and two of them are actually immediate. Local classes are closed under co-limits.

In particular, 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 co-limit of a check nerve,

of its own check nerve. So if I have a G, which is in my, and I wanna know, I wanna figure out if it's in a local class, but I know its pullback is in a local class, I'll just 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. F is in the local class, therefore, so are all its pullbacks,

cause that's part of the definition of a local class. So all these FK are in L, but then I get G by forming the co-limit along this Cartesian natural transformation of simplicial 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 local class at the closed under base change and closed under co-limits in the Cartesian arrow category. Base change is automatic.

So we already have this 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, it's just a morphism. All of these,

because of the descent property for every G, for every I, each of these diagrams is gonna 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 hypothesis,

the fi's are all, or the fi's are all in L and that class is closed 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 the class is closed under co-limits

in the Cartesian arrow category. Anyway, the point of giving these arguments is to show that all I'm ever really using is university of co-limits and descent. Those two properties really let 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 sketched the proofs here. 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've said, the notion of monomorphism is the first of a sequence of conditions called intruncated.

So remember in an infinity category, a map is intruncated if the iterated diagonal, so you iterate the diagonal construction on a map, if the iterated diagonal is an isomorphism, I have to do it in plus two times. And then an object is intruncated if the map to the terminology is intruncated. I'm gonna write C less than or equal to N

for the full subcategory of intruncated objects. Let me notice something here. If I have an intruncated object, then if I look at the mapping space into that intruncated object, that has to be an intruncated infinity group void that is an N group void.

So this full subcategory of intruncated objects is an example of what's called an N plus one category. Or a better term is probably N plus one comma N category. Larry's term is N.

category. That is, it's a category whose function mapping spaces are actually just in-group weights. 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 complementary classes of n connected maps. I don't seem to have the definition n connected here but I hope it's obvious. F is, oh it's right down here, a map is n connected if it's orthogonal to the n minus one truncated maps. There's other terminology that's

used here. Sometimes this is called n connected and sometimes this is called n minus one connected. So that's great. I'm going to follow Jacob Leary and call it n connected

so I don't have to deal with this confusion. An object is n connected if it's mapped to the terminal object is n connected. In a presentable infinity category it's a factorization system. In particular you get factorizations essentially canonical of any map into an n plus one

connected map followed by an n truncated map. I'll call it the n image. I'll call it the relative n truncation and then if I apply that to the map to the terminal object it's called the absolute n truncation. I'll call it tr of x. That'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 in infinity groupoids a space is truncated, n 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 n. And similarly a space is n connective if and only if its homotopy groups are trivial for every choice of base point and every k less than or equal to n. Here I have to be careful. I have to also make

sure that the space is not empty again assuming n is zero. This is what's classically known as an n minus one connected space. So there's an off by one in the terminology.

And then you discover that in infinity groupoids a map is either n truncated or n connective if and only if all its homotopy fibers over all points in y are n truncated or n connective as the case may be.

This condition in homotopy theory is classically called an n-connected map. 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 intronation of a space. You kill off the homotopy in high dimensions by attaching cells of large dimensions. In a presentable infinity category this construction exists formally. I'll note here that if you want to compute this in a pre-sheaf 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. And then 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 n truncation because they both preserve finite limits and particular pullbacks which is all I need to define n truncation.

So to n truncate an object in e you just n truncate as a pre-sheaf and then sheave of i. 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 to say you can see

how some of these fit together and they relate to other mathematics. So let's think about n-gerbs. I'm going to follow Larry here. I'm going to define an n-gerb n is the intersection of the classes of n truncated and n-connective maps. According to my sources apparently these

are just the Eimelberg-McLean n-gerbs and that n-gerb is actually something much more general which I was unaware of but I'll just call it this. This is sort of the first non-trivial example of an intersection. If I took the n truncated maps and the n plus one connective maps

the intersection is just the isomorphisms because they're orthogonal classes so this is sort of the first non-trivial example not just isomorphisms. So if I have an

e to the terminal object which are n-gerb n 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. So what is actually true is that I could if I look at the infinity category of

pointed objects in e sub-gerb n 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

this may look like a familiar story. The pointed n-gerbs for n-gerb and e to the two are equivalent to the one category of abelian group objects in the one category

e less than or equal to zero of zero truncated objects in your infinity groupoid 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 these zero truncated objects that is an example of a one topos

as we will see. So these are abelian groups objects in a one topos associated to our infinity topos. So the construction of I won't prove this proposition but I'll construct the functor from e sub-gerb n star from the pointed n-gerbs to the zero truncated objects.

So I won't give the abelian group structure. Well if I have this pointed object I can take the n-fold iterated diagonal of the point inclusion or the map 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. 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 n truncated and you can use that to

show that if I take the n-fold iterated 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 n-fold iterated loop space functor

of a base space. So that's the construction of that functor and then this has an inverse functor which takes an abelian group object to something that's usually called k a n. And that's the Eilenberg-McLean object associated today. So for instance,

suppose I pick my abelian group object then I get an Eilenberg-McLean object in E. It's actually an n-gerb. As we saw to associate to every map there's 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 gerbs because gerb is itself a local class so it's in that class but it's of gerbs that locally look like p. So for instance, just to orient

yourselves if you know something about gerbs, here's this universal family. We have our sort of typical example which is the pointed gerb associated to the abelian group A. This actually

factors through another gerb which is the inclusion of a base point into k n plus one. That's also an n-gerb. So we have pullback diagrams like this. There's a pullback diagram also like this. This is not the trivially commuting diagram. This is the pullback diagram

that relates k an with k n 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

gerbs banded by A. A notion I won't mention but this is just to orient you if you've heard about gerbs. A banding is a structure you put on this gerb which in some

sense is a local identification of its fibers with k an in a certain sense sort of a choice of localization of identification. Anyway so for instance we get a very cleanly a theory of these em gerbs from this from these notions in an infinity topos. Here's one more example

that I'd like to mention. It's 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 n image is the is equivalent to the codomain for all n. It's an intersection of local classes so it's also a local class. We'll say an object is infinity connected if the terminal object is.

In infinity topoi like infinity group boys and more generally sheaves of infinity topoi, infinity connected is the same as isomorphism. That's some version of the whitehead theorem in homotopy theory. Weak equivalences are by definition determined by homotopy groups. That's a consequence of this

fact. The interesting 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 connected object. First I need the infinity topos. I'll start with a topological space which has this lattice of open sets. I could also give a point set for the topological space but I don't need to in this to tell you what a sheaf is.

So I'm just telling you the locale. So I can define a pre-sheaf of infinity group on this so that its values at u0 plus minus u1 plus minus u2 plus minus are equivalent to the terminal object and whose values at the v's are in Eilenberg-McLean space. Let's say kzn

where z is the 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 the homotopy pullback of such a diagram or more classically it's the loop space on 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 its values are not contracted its values at the vn. So it's not

to compute the let's say the m truncation of f for any m 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 are these claim spaces but the m-truncation of an item of a 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 vi's. 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 topos. So that is a phenomenon that does happen.