 So welcome everybody. So it is my pleasure to introduce Professor Charles Resk from University of Illinois at Urbana-Champaign. We will talk about higher topoi, which of course is a way to make topos theory even more rich than what we have seen so far. So you can imagine. Okay, so it's up to you. Okay, thank you. So I assume delighted to try to give these talks. So I'm going to give an introduction to this notion of higher topoi, which lives in this world of infinity categories. I certainly don't want to assume that everyone is an expert in infinity categories. So in this first hour, I'm going to give an introduction and give you an idea of not so much, teach you how to do it, but to give you an idea of how it looks like. All right, so first some notes on infinity topoi. So as I said, they generalize topoi to infinity categories. Those are one word that people use to these, they're higher stacks. So this theory, I didn't put it in the list of references here, but that has roots in the work of Grotendijk and his famous paper on pursuing stacks or his famous non-paper on pursuing stacks. One question I should answer is, why am I giving these talks? I don't believe I've ever published anything ever published anything about infinity topoi per se. I did give one of the earliest, I'm one of the earliest people to give a formulation of higher topos and I recognize some of its properties which may have influenced some other people. Carlos Simpson gave a similar formulation around the same time. The first place this theory was actually presented was in by Antonin Vizzosi. And then most famously, it's the subject of a very long book by Jacob Lurie. So in my discussions, I'm going to basically follow Lurie's definitions and terminology because that's become the standard. All right, so what is this higher stack just to give a brief idea? So very often you want to play the following game given a topological space, you'd like to look at some notion of sheaves of let's say categories on that space, maybe sheaves of group oids. So just to give some example of possible candidates of what that should mean, for instance, if I have a space and then if I have a group G then for every open subset I have the collection of say principal G bundles it doesn't really matter. They're principal G bundles. The point is that that's a groupoid. The groupoid that you associate to each open set in the space, it behaves like a pre-sheaf of group oids and you'd like it to be some sort of sheaf. Here's a fancier in some respects much more difficult example. Let's say I have something just to be specific, a scheme and then for every Zariski open set there's something I can associate to it which is called the derived category of quasi-coherent sheaves. Could be other examples but it's some category you associate to an open set and you get functors when you include one open set into another restriction functors. So these are candidates possibly for sheaves on some space that take values and categories. So in the first example, when you think about the sheaf condition it's not literally a sheaf of group oids or it need not be. What you really want is what's called a weak pullback of group oids. So this is a groupoid. This is a groupoid. This is a groupoid. An object in the weak pullback of this part of the diagram is a pair of objects and f of u1 and f of u2 which maybe do not go to the same object on the intersection but for which you've chosen made a choice of isomorphism between their images inside the intersection. Those are the objects of what's called the weak pullback. And the sense in which you want this to be a sheaf is that you'd like the value on the unit of open sets to be equivalent to the weak pullback. This is a property that this example of principle G bundles actually has. You can glue together G bundles if you have isomorphisms on restrictions. An example too, you can't do it in the same way but you kind of wish that you could. That's not even going to be a sheaf of categories in this weak sense. Now this is all very fine and this leads to the notion of a stack but you'd really like to generalize this even further to what are called higher group voids. I won't try to discuss all the motivations of that. One comes from a relationship between this stuff and co-homology. When you wanna classify things like elements in this stack of bundles and wanna classify global sections you're often led to some co-homology often co-homology in degree one. And then if you'd like to have higher co-homology classify something you probably want to have higher group voids. A yoga based on that idea. Let's just accept that that's what we want to do. We'd like to put higher group voids into this game. This is going to be harder because higher group voids are far more complex both in how you define them and then also the sort of mathematics for manipulating them. So we'll do this using the connection between the higher group voids and the homotopy theory which is I guess nowadays well known. So if I have a one group void that is a normal group void, a category of objects and morphisms that is all the options are isomorphisms. There's an associated topological space called it's classifying space and the fundamental group void of that space is the group void and then the higher homotopy groups are trivial. K greater than one, so this is the fundamental group void. This actually characterizes the classifying space up to homotopy equivalents. So it's not really a space that I'm interesting it's the homotopy type of that space. Now there's an observation which I've attributed to Hopf because the basic trick of the proof goes back to him. If you have two group voids you can look at the category of functors between them, functors from D to H, that is also a group void. So we can take its classifying space and then what we learn is that that is homotopy equivalent also a weekly equivalent because that's the particular kind of equivalents that homotopy theorists use. Mapx induces isomorphisms on homotopy. So you can think of homotopy equivalents. That's homotopy equivalent to the mapping space, the space of continuous maps between the classifying spaces. So in some sense you can recover all the information about one group voids including the functor categories between them from topology, really from a homotopy theory. So this leads to the following hypothesis on what n group void should be and group void should correspond exactly to in truncated spaces. So those are spaces whose higher homotopy groups vanish above dimension here. And then of course you could let n go to infinity because why not? And then you're not putting a condition on your space anymore. You just have a space. But I should of course emphasize that we don't, it's not true topology in the sense that we care about exotic spaces. It's topology in the sense of homotopy theory. We care about the homotopy types of spaces. So that's one proposal for what higher group voids are and that infinity group voids as well. Now there are a number of different ways to approach homotopy theory. In addition to using topological spaces, there's a way called simplicial sets developed in the four years and fifties, which gives another more combinatorial way to talk about the homotopy theory of spaces. In particular, pick their simplicial sets known as con complexes. So what we do is we generalize this theory, which is a theory of infinity group voids, according to what I told you, to a theory of infinity categories. So that's the plan. I don't know, I guess I'm not taking questions during the talk, but if there are any, I guess I can answer them afterwards. So what I'm going to talk about here are quasi categories. A quasi category is an explicit model for what's called an infinity category. And I should emphasize here in this context, infinity means infinity comma one. So when you have an infinity category of morphisms going all the way up, morphisms with higher heights, the one means that they're all isomorphisms above degree one. The two, three, four, et cetera morphisms are invertible in some sense. So this is the sort of more generic term for this, but this is the term that people tend to use for infinity one categories who do infinity one categories. This is terminology due to Lurie. And of course this includes the infinity group voids, which are really the infinity zero categories. Confusingly, the word infinity category is also used for quasi category, which is a particular model for the theory of infinity categories. There are other models, which go by names like Siegel categories, complete Siegel spaces, categories enriched over topological spaces. There are all sorts of different models for infinity category theory. And so you might use infinity categories as the generic term for all of those, but it's also used for this particular model of quasi categories, which is probably the best one. So this notion, well it was introduced by Bourbon and Vogue, but they didn't really use it for higher categories. The key thing was the work of Joyol who showed that this would be a viable model for infinity categories. And then Jacob Lurie came and wrote a book. It's a book on higher topo theory and it's 900 pages, but 600 of them are simply establishing the foundations of higher category theory using quasi categories. Now, as you might expect, it's a 600 page, 600 very dense pages on infinity categories in this book and it's not the, it's only the beginning of the story. It can be a difficult area to learn because there's a lot of stuff in it. I'm gonna try to give you a guide to sort of how you think about what's going on. I've listed on the right here some textbooks. I really like Sosinski's book as an introduction. There's a book by Marcus Land, which I haven't really looked at, but seems good. Jacob Lurie himself is essentially rewriting his book to make it better and it's online. It's an online rewrite of his book, kind of in the style of the stacks part, but it's just Jacob writing it. Real and verity are a number of things that I'll put this into some context that's useful. All right, so those are quasi categories. All right, so let's try to give a definition. So we have this notion of a simplicial set. It's a functor from this category delta op to sets. So that's the full subcategory of categories which are non-empty linear partial orders. Really a skeleton of that. And so a simplicial set is a pre-sheaf on delta. It's a functor from delta op to sets. If I have a simplicial set, I'll write C sub N for the value at what I'm calling brackets N. I'll think of this as the set of N cells of my simplicial set. Sometimes people call them in simplities. So for instance, an example of a simplicial set is the nerve. If I have a category C, I can form its nerve, which is a simplicial set. Its N cells are the set of functors from brackets N to C. That's the nerve. This is actually a fully faithful functor. So categories are embedded in some special sets. So in fact, I will think of a category as a kind of simplicial set. So now I can define a quasi category, a quasi category or infinity category because a simplicial set with what are called inner horn extensions. So delta N, I'll use this symbol for the representable pre-sheaf represented by the object brackets N. It's called the N-simplex. And then that has a number of sub-objects. There's one called lambda N sub I for every I from zero to N. It's called the Ith horn. So it's the largest sub-object not containing a particular N minus one cell. You picture a simplex as a geometric simplex and then what you're doing is removing one face. And a simplicial set is a simplicial set such that for every map of an inner horn into the simplicial set C, there exists an extension to delta N. Inner horn extension condition. Inner means this condition. I exclude two of the horns. This is the picture when N equals two. There's a two simplex, which I represent as a two simplex, but think of it as basically a category with three objects, two composable arrows and a composite and the identities. Inside there, I have a simple set, which is not a category, just has those two arrows zero to one and zero to two. And the first horn condition, the only one in dimension two says that I can always extend the picture on top and C to one on the bottom. So I can think of the arrow on top as a choice of two different arrows inside the category, if it's a category, from the category. And I'm saying there needs to exist some extension. The image of that third edge is what's called a composite of those two morphisms or edges, one cell is called the edge. So in a quasi category, one condition is for any two one cells or morphisms, you have a notion of composite, but it's not unique. It nearly exists. So there's non-unique composition. The entire map of the two simplex, and to see the entire two cell is a witness of the composition. There can be many different witnesses. So when you compose one cells, you compose arrows, there's many choices, and then there's some information that recognizes that that was a choice. Then the three cells in some sense, witness the associativity of composites. So the composition is not actually defined as a function, it should be associativity. And associativity, therefore, is not strictly speaking an identity, but it's a kind of relation. So there's some picture, special set, the zero cells, if it's a quasi category, the zero cells are called objects. The one cells are called morphisms. The higher cells don't have special names, but they witness composition, associativity, some kind of higher associativity, and so forth. I'll notice here, if I have a zero cell, that always determines a particular one cell, which is the identity of that cell. That's part of the structure of a simplex set. And that does behave like an identity in the sense that there always exists a canonical witness for the composite of F with the identity to give the identity to both sides. So you always have that. Identities serve working nicely. We have this more complicated compositional structure. Anyway, so a quasi category C is actually a one category. It's actually a nerve. If only its inner horns extensions are all unique. That characterizes the one categories among infinity categories. Now starting from this, we can start to develop some theory and we can make some definitions that are like the ones we have for one categories. Some of them are very easy. For instance, a functor of quasi categories is just a map of some official sets, which are quasi categories. If there are actually nerves of one categories, that actually is the same thing as a functor. Here's an example of a quasi category. So it's actually kind of hard to give explicit examples of quasi categories that are of interest. It's just one of the difficulties in the subject, but this is what I can give. This is the quasi category of one categories. So it's a simplex set. It's zero cells or just categories. It's one cells or just the functors between arbitrary categories. The two cells are diagrams like so. I have three categories, functors going around the edges of the triangle, but it doesn't compute. Instead, I have the additional, the data of a natural isomorphism from GF to H. So I put a natural isomorphism in every two cell. And then a three cell is the tetrahedron where each of the four faces has one of these two cell pictures. And furthermore, the diagram has to commute in the sense that if you patch those natural isomorphisms together, you get a commuting diagram of natural isomorphisms from that. The higher cells are similar, but there's nothing more to say. You'll have higher dimensional pictures whose faces will look like this. And there's no additional condition for a higher cell. You just have all those pictures on the faces. So I see here, people are talking about large and small. So that's an issue, but here I'm just giving you a definition that makes sense to me. So when we say simplex set, sometimes it's not a set. Sometimes we allow a simplex class or for people who like to do things in growth and deque universes, you can talk about a simplex set in a higher universe. In fact, that's the formalism all implicitly used. Okay, so for instance, by the way, this is a version of something called the dusk and nerve. This is a known construction which is itself a special case of the street nerve. So for instance, we can already say something. If I have an honest one category, viewed as a simplex set, if I have a functor to the infinity category of one categories, that's the same thing as what's called a pseudo functor. Sorry, a pseudo functor from the one category to the one category of one categories, which is not the same as the infinity category of one categories. So this is already a nice formalism that gives you an easy way to think about pseudo functors. All right, there's sort of a standard way to construct infinity categories which unfortunately is complex. Any situation where you can do homotopy theory, you can lift up to quasi categories. The typical situation is when you have a one category and then a class of morphisms called weak equivalences in your one category. For instance, topological spaces and weak equivalences. Often this comes as part of a fancier structure called the Quillen model category on C, structure on C. But even just with these pairs, you can build a one category generally large, usually do this for large things, which is the category of fractions where you formally invert the class W of weak equivalences. That's often called the homotopy category. But it turns out you can lift this construction to give you an infinity category, a quasi category. And that sort of encodes higher data that is lost if you just think about the category of fractions. So the ones that we care about in homotopy theory arise this way, but this is a fairly complicated and sort of difficult to understand construction. So I'll say more about it. Okay, so those are quasi categories. So given this, we can start to try to do category three. So for instance, we can make some definitions. We have a quasi category C. We have morphisms, which are one cells. We can talk about isomorphisms. An isomorphism in a quasi category is a one cell, such there exists, well, another one cell and composites. So G compose F is equivalent to the identity on X and the other way equivalent to Y. The only wrinkle here is that composition is not a function, it's a relation. So there has to exist some two cell witnessing these compositions. But if that data exists, then you say that F is an isomorphism. It's in that sense, just like the usual definition. Then an infinity group is an infinity category where all the morphisms are isomorphisms. An important fact, which was proved by Joyol and got this thing started, is that those are the same as the con complexes, which are the things that have horn extensions for all horns, not just the inner horns. So I can be anywhere from zero to N, including those endpoints. And as I said, that's a model for homotopy theory, the clutch of con complexes. So this is basically verifying the homotopy hypothesis for this model of infinity categories. So the infinity group boards are really the same as homotopy types. All right, let's continue. We can define a functor quasi-category. If I have two quasi-categories, C and D, there's an internal function complex in some partial sets because it's pre-sheaves, that's a small category. So the n cells of the internal function complex are the maps from C times delta N to D. Remember that's the representable functor. Delta zero is the terminal object. Zero cells, therefore, are just the functors, the maps. One cells are natural transformations. And it turns out that if my simplicial sets are quasi-categories, then this is also a quasi-category. Now, here I have to warn you about something which is just all over the theory. This was, so far, we've just had some fairly easy-looking definitions. I define an internal function complex. Everybody knows how to do that. But there's a fact I need. I need to know that if I plug in quasi-categories, I get a quasi-category. That's already kind of non-trivial. I mean, it's not super hard, but unlike, say, constructing the usual functor category between two categories, it's not an exercise left for the reader. It's a somewhat involved combinatorial argument that takes several pages and you won't come up with just by thinking about it. This is very common. There's some construction, often it's sort of an easy definition to make, but then you need to show that it actually behaves properly. So one of the reasons that it takes hundreds of pages to sort of establish the theory is because you have to verify long lists of statements like these, showing something like this. But once you're done, so the interesting thing, of course, about it is, once you do that, turns out you never actually care about the proof. Certainly in this case, I've never had to think about why this was true after I saw the proof. And this is also a phenomenon that often is the case. But once you have that, now you have a functor quasi-category. Now that means you can define natural isomorphisms of functors. They're just isomorphisms in the functor quasi-category. Once you can define natural isomorphisms of functors, you can define equivalences of quasi-categories, just like you would for categories. It's exactly the same. And I hope it's clear, a one category is a kind of quasi-category and these definitions do specialize to the usual ones. I'm gonna go a little bit further into the theory, just to give you an idea of how some of these constructions, how some of these definitions are made. So there's a notion of the join of two one categories. So if I have one category C and D, the join, we can see start D, is a one category with C and D as full subcategories. The objects of the join are the disjoint union of the objects. The morphisms of the join, we have the morphisms of C and D but you also have a unique morphism put in for every object of C to every object of D. That's called the joining of categories. A very important example of joins of categories are what are called the left and right cones where you join with delta zero. Delta zero is the terminal object. It's the terminal category actually. Here's a simple set. So the left cone is where you formally put in an initial object. The right cone is where you formally adjoin a terminal object. Now this definition is very common and you can extend it to some partial sets. In fact, the representable some partial sets, the delta P and delta Q are already categories. So the join gives you a formula for that. And so you can extend the whole definition of a joined as some sets by a very easy formula. And then you can verify that the join of quasi categories is a quasi category. That one actually turns out to be easy to show. So I can use this to define the notion of a slice quasi category. So let's say I have a quasi category, even a simple set and I have a zero cell X, then I can build something called the slice the slice simple set. It's characterized by this property to give a function from T into the slice, slice over X is to give a function from T joined delta zero, the right cone into C so that it extends the map from delta zero that classifies that zero cell. So if I draw this using category pictures, the objects of C over X, that's a quasi category are the morphisms in C from a random object to X. The one cells are a little commutative triangles where the terminal thing is X and so forth. And then you can check that if you plug in a quasi category that can be used for every simple set. And the slice is also a quasi category. And if you plug in one categories, you get actually the usual notion. So for instance, I'll use this to define the notion of a terminal object in a quasi category. So we'll say that a zero cell is a terminal object if the forgetful functor from the slice back to C is an equivalence. If you look at this definition, you can figure out exactly how to define the forgetful functor, it's obvious. And so we'll say that since I've defined equivalence, I can use that to define the notion of a terminal object. That's not the only definition of a terminal object. Let me show you one other one because it's perhaps a little closer to the sort of more elementary definition we would consider. So I have this functor category and I'll take functors from delta one. Delta one is the walking morphism here. I have evaluation functors where I evaluate my arrow at source or target. And then I can pick out, let's say a pair of objects, let's say Y and X. And then I can form the pullback and supplicial sets. And this pullback is called map C, Y over X. And if C is a quasi category, then this is an infinity group point or con complex. And it's called the mapping space. Space because it's an infinity group point and we call them spaces. Well, I do because I'm a typologist. So whenever you have a quasi category for any pair of objects, there's an infinity group void worth of morphisms between those two objects. And then it turns out that you have a terminal object if and only if that mapping space is equivalent to the terminal quasi category for all the objects for all the zero cells. These are equivalent formulations of terminal object. It's not obvious to show that those are equivalent. You have to go a couple of hundred pages into your textbook on quasi categories in order to do that. So an equivalence of quasi categories to address the question is the definition that I defined back here. I just defined it for you wherever it was. I gave you enough information that if you write down the usual definition of a equivalence as a pair of functors in opposite directions and natural isomorphism between the composites, that's an equivalent. That's literally what it is. You can think of it as a homotopy equivalence, but it looks like the usual notion of equivalence. All right, I wanna give an example. Let's think about sheaves. So let's take a topological space X. I can talk about a pre-sheaf. So that's gonna be a functor from the opposite of the partial order set. So category of open sets to A. Well, I can allow A to be an infinity category, not just a one category, because I've defined functors. So now I wanna say what it means to be a sheaf. So let's suppose I have an open cover consisting of UIs. So let's see, any open cover gives me a functor from what I'll call P, excuse me, P-fin of I. I is the indexing set. So this is the set of finite subsets of I. Let's call this alpha U. So I have a functor that takes a finite subset of I to the intersection of the elements of the open, of the collection of open sets, indexed by that finite subset. So this category, it's a one category. It's actually an example of a left cone. So it's the join of delta zero with some subcategory, which is actually the non-empty subsets of the indexing set. Oops, I actually, oops, I'm sorry, I skipped a slide here. Okay, I'm gonna go back to this. So F is gonna be a sheaf exactly if this is a limit. So let me write this down here and then I'll go back and pick that up. F is a sheaf if the composite F with this functor of one category is alpha U is a limit cone for all collections of open sets. It's the usual definition except now we have a notion of a limit cone and now we have to be careful because the targets an infinity category. So I have defined the notion of the slice over an object but there's a more general notion of a slice over a functor. So if I have a functor from J to a quasi category C, J is also a quasi category, then I can form the slice of that functor. It's characterized that by this mapping property, T into the slice is the same as maps of T join J into the slice where the restriction to J is just the given functor F. So if J equals delta zero, I get the definition from the previous page. And then you can say what a limit cone is. A limit cone is a given a functor from J to C, a limit will be a extension of that functor called F hat to this left cone with a property. So I can say, just say that the forgetful functor from the slice over F hat to the slice over F is an equivalence of quasi categories. That is a definition of limit. If you check that for a one category, it's equivalent to the usual definition. Another way you can say this by the way is that F hat, this actually corresponds to an object of the slice over F. So that's a zero cell. And that, so being a limit is the same as being terminal in this slice. That's another way you can say, okay. So this gives you a definition of limit and using that I can make a definition of sheets. It's an easy definition. Proving things about it may not be easy. A good exercise by the way here is to plug in your favorite infinity category, which isn't a one category. And the only one we have so far as this one, the infinity category of one categories. So unwinding this and understanding what a sheet is is a very interesting exercise. All right, I think I need to speed up a little bit because I'm going a little bit behind schedule. So that's just gives an idea of sort of how you develop this theory. You start making definitions, which often look like the ones that you would make for one categories. However, there's often a far more work that goes into making sure those definitions are viable. Of course, I'm not showing you any of that work. Now there's one more infinity category that you need. It's the infinity category of infinity categories. So so far I've introduced some plitial sets in this subcategory of quasi-categories, which it contains categories faithfully. But that's just a one category. What we really want is something called cat infinity, which should be the quasi-category of infinity categories. To receive a functor from the one category of quasi-categories, but it should have this additional structure. This is a somewhat tricky object to construct. There's a construction due to Lurie. I don't want to say it's in elegance, actually a very elegant construction, but it's a little bit ad hoc. It's not obvious what this thing needs to look like. In fact, there are potentially many different constructions which will not be isomorphic to each other. They'll only be equivalent to each other. And in case people are wondering about size, this meal will be a large quasi-category, so it lives in a higher universe. There are some other work on constructing the infinity category of infinity categories, but even giving a construction doesn't really tell you how to work with it. In order to work with the infinity category of infinity categories, you need something called straightening and unstraightening. Let me just give you an idea of what's going on here. So, classically, that is in the world of one categories, there's a notion of a functor called a Groten-Dick-Op vibration. Let me not give you the definition of that. You can look it up if you're not familiar with it. There's Groten-Dick vibration and Groten-Dick-Op vibration. You can show that the Groten-Dick-Op vibrations over C correspond exactly to pseudo-functors from a given one category into the category of categories. So there's a correspondence. So you can think of one category system kind of classifying the Groten-Dick-Op vibrations. And I mentioned earlier that pseudo-fibrations from C into cat are the same as just plain old functors into cat one, where this guy's only a quasi-category. So that's how we get at cat infinity. We generalize this correspondence. So there's a notion of what's called a co-cartesian vibration in this context. That's a functor between quasi-categories whose definition I won't give, but it's entirely analogous to the definition of Groten-Dick-Op vibration. And it turns out that there is a correspondence between the co-cartesian vibrations over C and the functors from your quasi-category C to this cat infinity. And I'm being vague about what I mean by correspondence here. In fact, there are various, stronger or weaker versions, you can say it. A weak thing you can say is that it's a correspondence between equivalence classes. On the left, I can just look at co-cartesian vibrations that are equivalent if there's a commutative diagram in suplicial sets where the top map's an equivalence. On the right, I can just say it's just functors up to natural isomorphism. But there are more refined notions of that correspondence. So this in some sense characterizes cat infinity by sort of a universal property. It tells you what it represents. There's actually one more thing you need to know here to actually get sort of viable information. This cat infinity, which is large, is equivalent to a union of infinity categories, which I'll call cat infinity kappa, which are all essentially small, equivalent to small infinity categories. You can take kappa to be a size down. Kappa is a regular card. This is the sort of game you often have to play in this context. So there's the infinity categories, infinity categories. And then you get the infinity category of infinity group boards. That's a full sub-infinity category of cat infinity, which classifies the co-cartesian vibrations whose fibers are group boards. I should have said here, by the way, in this correspondence, how does it, when you have a functor, the associated co-cartesian vibration, the value of the functor at some object is the fiber of the co-cartesian vibration at the object or at least as equivalent to it. So this is like a bundle of categories, of infinity categories overseas whose fibers are the value of the functor. So we have an infinity category of infinity group boards. Here's a fact about that. Every functor between infinity group boards is equivalent to one of these co-cartesian vibrations, which has a funny consequence. It means that if I take the infinity category of infinity categories and take the slice over a particular infinity groupoid, that is equivalent as a quasi category to the functor category from C into my infinity group board, into infinity group boards. Possibly it's nicer to say C op, but it's an infinity group board, so it doesn't really matter. This is analogous to, for instance, and you know, if you have a set and you take the slice over the set, you think about it as functors from that set into sets where a set is just a very discreet category. Let me make a warning about what's going on here. When you go and look at this theory, you discover you have two things. You have a one category of quasi categories. You also have an quasi category of infinity categories and you're constantly going back and forth between these two things. You often have to make two constructions whenever you want to do something. First, you construct something in quasi categories, but then you discover it wasn't really in infinity categories and so you have to do it again. Here's an example that I actually was wondering about the other day. So I told you that there's an internal function object in quasi categories. It's just the internal function object in simple sets. That's a one category that's pretty easy to talk about. But of course, you don't just want that. You want an internal function object a functor, functorial internal function object on the infinity category of infinity categories. And I have not told you how to build that and I didn't even know how to build it. I asked on the algebraic topology discord and somebody pointed me to this reference which gives you the machinery that lets you define things like that. This, by the way, is a nice expository paper about some of these ideas involved in these vibrations, co-critical vibrations and others. All right, so slowly I'm building up some of the things you need to do interesting things. You of course need a Yoneta functor. So if I have a quasi category, you can construct a functor, I'll call it HD. It goes from C op times C to infinity group points. And it has this property. If you evaluated it to objects of C, you get a infinity group point which should be equivalent to that mapping space that I constructed earlier. The mapping space construction I gave is not functorial but there's this variant of it that you can define which is functorial. And one way you can define it is to simply say which co-critical vibration it classifies. I'll tell you the answer in this case because it's kind of nice. It classifies the co-critical vibration associated to the twisted arrow category. So this is the quasi category. Well, it's a simple set. You can show it's a quasi category with some work whose end cells are those. And this is actually generalized to something you may have seen in one categories. An object of the twisted arrow category is an arrow and then a morphism is a diagram like this. You think it was a morphism from F to G. The arrows go this funny. It's not a commuter square. So that turns out to be a co-critical vibration. It classifies therefore some functor and that's the function this, if you like the ham functor. Okay, that's literally a functor from C up times C to S. So I can adjoint it. That gives me a unada functor. You can show it's fully faithful. That's the unada embedding. You can also use this to define adjoint functors of infinity categories. If I have two functors, F and G, an adjoint functor involves a natural isomorphism between these two things. That definition looks like the one for one categories, but it's built from this more complicated construction. I like Sosinski's book, by the way, for describing how this works out. It's a very nice story. All right, so this is a sort of machinery you build. The moral here is that it looks like one category. Everything I did was just a generalization, so you can do with a one category. To me, this doesn't feel like higher category theory. It's really sort of category theory, plus homotopy theory. Of course, I'm a homotopy theorist, so that's what I would think, but I'm just sort of using the same language I use in one categories, but there's this sort of extra homotopy theoretic component floating around in everything. For that reason, when I describe things in the later lectures, it's gonna look like I'm talking about one category. I'm talking about categories, functors, objects, limits and co-limits. The usual cast of actors will appear. In fact, it may be unclear where the infinity is sometimes. Of course, you may ask, can you do infinity and categories? And the answer is yes. There are many models of these. There's a bunch that are known to be equivalent to each other. I don't understand on what the models are. People are still trying to learn what the best model to work in is. I won't say anything about it. Of course, we've seen some infinity two categorical phenomena sneak in already, like the functor category between two infinity categories. But to talk about that requires some extra theory. And I won't complicate things by describing that extra theory. Let's see. I would like to conclude this discussion by talking about some potential issues and pitfalls that happen when you go from thinking about one category to thinking about these infinity categories. Now notice I'm calling them infinity categories, not quasi categories anymore. Cause I don't really want to concentrate on the model that I introduced, but rather on the idea of a infinity category. Here's one of the main tricky ones. I may often say that I have some commutative square in a quasi category. And that just saying that it's fraught with danger. I've written down the square. I've written down four morphisms. Maybe I've told you what those morphisms are, but I have not told you what the square is. I've not given you complete information because although I told you the edges of the square, besides I've told you the one morphisms, I have not told you the additional data that has to be associated whenever I have a commutative diagram. So if I draw this as some official set, it actually has two triangles. And for each of those triangles, there's some additional witness of composition. But I haven't told you that data, so you don't know which square this is. And that is a potential source of misunderstanding. So there's this famous example that is of course well known to homotopy theorists. So I'll say in infinity group boards because that's homotopy theory. If you have an X map from the terminal object, I can consider it, let's call this map, I don't know, F, then there's a commutative square involving something called omega X, and these will be called P. So it's a square both of who got ways of going around and all the same two maps. There's a sense in which the square trivially commutes. So there's a trivially commutative square because it's the same maps on both sides. But there's another square, which is a pullback square. And that does not trivially commute usually. I wrote omega X because it's loops. In topology, in the homotopy theory of spaces, this limit is called the loop space on X. And it's a famous sort of sort of error to think, well, if you just write this diagram, it commutes as an epitopological spaces. Well, it does, but that's not the one we mean, when we wanna say that loops are the homotopy pullback of the rest of this diagram. It's epitopological, there's actually a homotopy between these two maps, both of which are the constant map, but it's a non-trivial homotopy theory, homotopy that encodes the fact this is some sort of pullback. But when you talk about in Finney-Karris, I might just draw a diagram like to say it's a pullback, it's not the trivially commuting diagram. This is just, it's just a feature of the thing. You can never say what all the data is in a finite amount of time usually, not understandable. So you run into this potential pitfall. I'm often gonna talk about finite things. So there's a notion of a finite infinity category, like there's for a finite category. In particular, you can talk about finite limits and co-limits. I won't define that, it's not hard, but I'll tell you what operation is the thing you need to know. An infinity category is finite complete if and only if it has a terminal object and it has all pullbacks. Every functor from the co-span, I guess it's called into C admits a pullback in the sense that I described earlier. Okay, so finite complete looks like the sort of the notion we know in one categories. And a functor preserves finite limits and it's between finite complete categories if and only if it preserves these. There's a funny warning here. Thing to be aware of, a finite one category is not necessarily finite as an infinity category. The standard example is a group, a finite group. That's a finite one category, but if it's not the trivial group, it's not finite as an infinity category. As a topologist, I know this because the classifying space is infinite dimensional. That's what that means, that's why that's true. Or if you like the nerve of a finite group has non-trivial cells in arbitrary large dimensions. On the other hand, infinite groups are actually finite. So here's another thing. So we can define the notion of a monomorphism in an infinity category. We say it's a monomorphism, F is, if this diagram is a pullback. Now, I've just told you the tricky issue. In this case, it is the trivially commutative square. I'm asserting that's a pullback if that square is also a pullback. That's equivalent to the definition in one categories. Or if I'm finite complete, I'll just say that diagonal map is an isomorphism. So for instance, in infinity group points, which is the homotopy theory of spaces, a map is a monomorphism. If and only if it looks like this, it's a homotopy equivalence to a subspace of y consisting of a union of some subset of the path components. That's what a monomorphism is in homotopy theory. Since homotopy theory is also infinity group points, you would call this a fully faithful and functor, a fully faithful functor of infinity group points. That's also a monomorphism. That's the same thing as a monomorphism infinity group points. I'm using this script S for infinity group points. Of course, it's because it's spaces. Monomorphisms don't behave always as you expect. Here's a warning, it's a serious one. If I have a diagram like this, where the composite is a monomorphism, it does not follow that the first morphism is a monomorphism. So in particular, if GFC identity or something, the inclusion will not, you know, the first, the retraction, you know, retracts are not sub-objects. This is kind of non-intuitive. I only realized while preparing these, how non-intuitive this is. In particular, the diagonal map can fail to be monomorphism. So it might not be monomorph. Some things are true. For instance, if you have the composites and isomorphism and the second map is a monomorph. In fact, even if the composites are monomorphism and the second map is a monomorphism, then it is true that the first map is a monomorphism. That one works. But this first thing just doesn't work. So this leads to higher notions of a monomorphism, which is called truncation. So we say that a morphism is intrunkated. I'm gonna assume my category is finite and complete. It just makes me talk about, it's intrunkated if the iterated diagonal is an isomorphism. So you get this sequence of classes of intrunkated morphisms for various n's. So the monomorphisms are the minus one truncated with this number. So for instance, a morphism in infinity group voids is encrunkated, if you think of it as a space. So let's say the morphism to the terminal object is encrunkated if and only if the higher homotopy groups are trivial. They vanish above dimensions n. So we get a full subcategory of infinity group voids that are in truncated. And those are what we think of as the n group voids. That's the infinity category of n group voids. There's a dual notion of epimorphisms and n-code truncated morphisms, which is formally dual, just turn the arrows around, but often behaves very weirdly, stranger even than epimorphisms in familiar categories like set. So for instance, in infinity group voids, the map from x to the terminal object is an epimorphism if and only if x is what's called in a cyclic space, which means that it's homology is the same as the homology of a point. And non-trivial cyclic spaces exist and they're interesting in topology, actually. But they're fairly rare. For instance, they all have sort of only one path component. A consequence of this is that you do not have any kind of epim, well, you don't have epimono factorization in infinity group voids, unlike what you do in sets. We'll talk about what replaces that later. Here's just a heuristic. Very often when you're talking about one category, you often run into equalizers and co-equalizers, which are limits on this diagram. And that's a perfectly fine thing to do in infinity categories. But when you see this in a one category and you wanna generalize, you will often generalize not to an equalizer, but to a simplicial or co-simplicial object. So a functor on this indexing category. This is just a very common pattern and we'll see that later on. Okay, so those are just a few little pitfalls that you can worry about. So this concludes, I guess, the first hour. So in the second hour, I'll actually tell you what an infinity topos is and start telling you things about it. Thanks. Okay, thank you.