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.  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.


everyone thank you for coming I want to talk today about change and I want to talk about it in the framework of Virginia satir's process for change we'll talk about other things as well because we're going to weave some things into that Virginia satir was a family psychologist therapist she pretty much kind of invented something that people also refer to as systems psychology in which you're really looking at people as a product of their interaction with other people whether it's a family or whether it's an organization an individual team that sort of thing but change has patterns she noticed and that's we'll take a look at that so why is change so difficult sometimes and part of the part of the problem is sometimes we don't recognize it for what it is so I learned about satir years ago from Jerry Weinberg who picked up her ideas and applied them to the software world and from there I guess there's a lot of material out there now but Jerry has a book called the secrets of consulting which is a good read whether you're an internal consultant or an external consultant a lot of good ideas it may look like a crisis but it's only the end of an illusion and so let's take a look at where that illusion comes into play so when we look at the process of change now maybe some of you were in the last session next door with Utah X Dean Eckstein I'm probably mangling her name but she was talking about retrospectives and she introduced what I would call the classic model that you see of Virginia satir's change model which is you've got status quo you've got new status quo old and new and then you have this wiggly line that says there's a lot of friction lot of static a lot of annoyance that happens in between here and there so I've transformed it a little bit with the help of Sharon lotion Sharon lotion I meant I spent some time on the board of satir global organization so and Sharon is currently the president of satir global you can find them at secure global org and there's some interesting references there but in talking with Sharon what I discovered what she discovered or what she uses a lot is the fact that in all of this chaos in the center where everything is seemingly out of control there actually are choices implied you always have a choice so sometimes what happens is when that foreign element comes in something unexpected maybe but sometimes expected comes in you have a choice to react to it old status quo is very comfortable it's the familiar it's what you know it's the things that make you feel warm and cozy and it could be related to your level of confidence what your expertise is in your job or your family life or wherever that is and that foreign element is just going to push you right out of that you can try to resist but when you resist the change what happens you go back you say no I'm going to stay where I'm comfortable and you go back here but you're going to be bounced right back there again so this is a loop that just keeps pushing you back into chaos chaos being that area of discomfort things are out of control it's not safe and you're looking for a way to feel comfortable where however you're going to do that I wanted to tell you a story about some consulting I was doing years ago this was a big aha moment for me I was doing some consulting with and actually I was doing a workshop with a class set of engineers guys working on some automotive controls and I had a class of 24 people and the subject matter at the time was we were there we were helping them build executable models at the time and those executable models could be used to run simulations and so forth and so on so in this formal modeling technique you know the different iconography so that they would build something out and then would turn the crank and run the simulation well early in the class I wanted people to build something very simple simple model Tom was in the back of the class and Tom presented his model to me he took one of the icons and it really doesn't matter which one but when he presented it to me it had arrows going through it and knives being thrust into it and blood dripping out of it and it was I was just oh my is that his impression of what's going on so I had been on the road a while so I was tired and it was like oh my my last week before I get to go home and relax and I have somebody melting down at any rate later that day I went back to Tom before he left and asked to talk with him it turns out that Tom went to a university in the town that I lived in at the time so we talked about a few things his favorite hot dog stand and stuff like that and I talked to Tom about the knives and the arrows and all of that it turns out that Tom was one of the more senior developers and this whole new idea of an executable model that could run simulations he kept seeing as something that threatened his level of expertise and his seniority in his organization and so that's what all the knives and the arrows were about once I knew that and I knew that Tom was here and trying to go back here then I could help him I did not know this model at the time when I learned this model I realized oh my as a consultant I am the foreign element that comes in when their company makes a decision to make a change and I'm the one that helps move them or seemingly is perceived as moving them into chaos so but what I did at the time with Tom was I sat down with him and I explained how his expertise was extremely valuable to the work we were doing because all of the technology overlay was was an important piece so I gave Tom a few more choices as it turned out which was good sometimes something that you some of you might be able to relate to too sometimes the foreign element is is a good thing it's not perceived necessarily as bad like Tom being asked to do something a whole new skill totally different than what he was comfortable with you have your spouse and you decide to have a baby you don't have a family yet so this is your first child it's a good thing right however if you don't have any children all of a sudden you go from being a couple to being a family and maybe that baby cries a lot at night and there's a whole adjustment there's that whole period of adjustment saying oh my gosh how do we do the right thing I don't want to do the wrong thing and so a little bit of a little bit of noise while you kind of settle down into a new routine so what you're looking for when you're in that chaotic period is what are the possibilities that might take me to some other comfortable place some place where maybe it's just as comfortable as the old place one of the things about the classic satir model is it always has old status quo low new status quo high but that's not always the case you could be you could be in a neighborhood in the country that is being torn apart by war old status quo was much higher if you think of that in a scale new status quo would be much lower but you have to learn to do something to cope with it or survive it so I just made them equal because in this particular view I wanted to say I've got something here that's comfortable and then I want to get over to here because I've got to learn how to deal with that new thing however I do that so when you look at new possibility so as I started thinking about what Sharon was doing with choice I and and the idea of new possibilities I was thinking we need to look at opportunities so when we look at the opportunity of what what could be different so oh the thing that occurred to me when I was thinking about this this morning oops the lemon tart dropped so did if you were in Josh's presentation yesterday the chef the sous chef who dropped the lemon tart and expected the world to fall apart in fact the chef owner of the restaurant came by and said oh look oops the lemon truck lemon tart dropped and he came up with a whole different way of presenting that to so that was a transforming idea that took them to a new status quo the transforming idea one of the problems and I've seen this in organizations and with teams of people that sometimes people state what they think that new possibility is going to be like or how they're going to get here and what happens is management in the organization or colleagues have expectations and sometimes those expectations just can't be delivered for one reason or another safety reasons organizational structure I thought there were a lot of good ideas that I've heard this week on this week yesterday on ways that you can come up with well new possibilities and maybe a way to better manage the expectation I really liked the sociocracy ideas with you know taking that organization and seeing if you can get people at all levels that are participating and involved and building that up in stages whatever makes it safe in the organization I thought that helped you get here down this path rather than getting bounced back problem with a lot of organizations is the fear aspect just as Tom was afraid that he would lose his seniority and his skills would no longer be needed there's a lot of problems I think in with fear people fearing that if they don't master the new skills sufficiently they're going to be punished they won't be able to progress in their career they're going to have to find another job so that's something that I think you really need to think about when you've you've got everyone in that center chaos so I wanted to talk a little bit then about what we've got here so this is just the piece where we kind of want to generate we want to generate something that's going to take us up the green path and not put us back there so that choice power is really something you want to think about so one option is a trap two options are a dilemma which way do I go three options are a real choice so now I have a real choice and four and five and the more the better so part of the trick I think when you're here is thinking about what your choices are and so I've heard people talking about generating ideas trying to get maybe to the outliers or the different way of thinking about how we could do what we do the more you can put out there and say these are all viable I think and what is the one that we think we can support to get to that new status quo and past past the gate right here that is least likely to be something where we are going to get bounced back here so one of the things when I was looking at this that seemed interesting to me there's this is a little off the chart here but there's a woman named there was a woman named Danella meadows and she had an article she was somebody who did systems analysis in social systems and looking at how to make things better environmental issues and she has a little article that you can find places to intervene in a system then Danella meadows don a meadows org in her archives and leverage points places to intervene in a system so that's you can poke around in her archives and find it there hers was 14 different points very detailed very specific many times to physical systems so if I'm looking at environmental issues and I want to make a change I might be looking at physical power plants or lots of things like that so what I did was I wanted to take it and look at it through the lens of software and agile development and what she was noticing and I and I chunked these into categories these are not her categories there they're kind of mine but she she saw and observed that some of those measures are the least effective places to intervene they're easy to collect so I think we've heard a couple of times recently that velocity isn't always a good thing you have organizations that want to say well why can't your team be as good as that team why can't you guys deliver the same velocity and why are your story points so different and so on and so forth so sometimes we take things that I liked Josh's word I think he called it busyness accounting you know people looking at these things and it's kind of an accounting exercise but I did love the love metric that he talked about the idea of what is the is the customer using the product what is that feedback on the customer using the product so going up the scale to the more effective places to intervene feedback and I think this is where agile does a fantastic job of working towards getting things moving and things changing so there's demos there's retrospectives there's things that are probably a lot more transparent the fact that you're also engaging business people engaging people who care about the end product looking for the value one of the traps I often see is people don't think they have enough time for the retrospective yeah yeah yeah yeah yeah okay let's get on to the next project because everybody wants all their resources allocated so Mary talked about that just a couple hours ago that resource utilization not always the most efficient but a lot of opportunity here how can you make a change and suggest a change to actually make these happen so I think everybody goes through the demos but certainly when you think about assistance here you can think about things like getting the tooling and the infrastructure together to do continuous development you know that gives you that power more power on that feedback making things transparent I think one of the biggest challenges I've seen is not being in the same room for many companies that is a luxury you can't do that you have people all over the world so you have to be creative in terms of how you make the work transparent and there are a lot of tools out there some better some not so good but in the end when you take something and say well this could really help us in a worldwide implementation then you need to think about it from your perspective because these tool vendors make things and and those tools are built for anybody who wants to buy it so the anybody may or may not be the way you want to work so very important that you think about how you want to use those to be effective and and keeping it simple let's keep it simple and build up retrospectives build the time in if you can get somebody to change the way time is allocated build the time in getting to the more powerful levers rules of the system safety is big I had a colleague I worked with who this works better in a room and in a room one of the things he used to do is get the group together and give everybody a slip of paper the slips of paper were identical and everybody had a pen or a pencil he had a big hat and he would say on a scale of one to five with one being the least safe in your mind five being the most safe write down a number fold it up put it in the hat and what we're going to do is kind of take a temperature reading of how safe people really feel everybody was anonymous but when you if you saw one or two people feeling unsafe then there was an indication that you had to pay attention to that trust issue so now how do we solve that problem how do we solve the problem in the utas retrospectives she talked about the paradox I forget the yes paradox intervention I like that how could we make it worse everybody can think about how it would be worse so you can also use those creative juices also to then turn it and reposition it and say well if it looked exactly like we wanted it to look what would that look like and imagine create that imagined world forget if we you think we can't get there today doesn't matter because we're just pretending so what if you can use a what if to start thinking about that and then generating those choices generating the options communication is an important piece and I think there's been a number of talks about communication which I think is is an important piece you want to make sure that you've got an incentive program I once worked with an organization where I was I was so impressed with this clever little way this was a a an organization that developed software for for commercial sale and they were having a problem with quality and they were having a problem motivating people to change the way they worked it was pre-agile but they're you know there's always been quality problems that people are trying to address making it different making it better making it faster and then somebody in the organization decided let's make it different let's say we've got different teams working on different product lines let's tell the team that comes up with the software that does the best that they're actually going to get 2% back on the profit from that the sales and boom things turned around it was a little chunk of change for the organization itself but it was a huge incentive for the team because maybe if the team was like five or seven people they each actually got a nice bonus in there very simple very small but just changing the incentive system and it what it meant was you now have teams competing so everybody wanted to do better everyone but he wanted their product to be the winning product most difficult changing the system and so you know anyone that's tried to change a system and this is where I thought the sociocracy ideas were very powerful because you're looking at changing the organizational structure just trying to start to inject some of the teams back into the upper levels of a hierarchy it's a particular this is a particularly difficult problem in big hierarchies I was also very impressed with Mr. Chan's opening statements about how he purposefully took the team and moved them to a separate location and in doing that you know then protected them from the outside so the other place I had heard that kind of approach as well was in talking with a colonel in the US Air Force who had some software guys under him and they were trying to do something very very different in the way they developed software to support fighter planes things like that you know the heads-up display and he had done the same thing as well because he said you know everything in the military everything is very regimented lots of rules which is what I was hearing echoed with Mr. Chan in the government systems so if you can find a champion one champion that can help you make that kind of move that can be very very helpful so that that those are a lot of the places that you can start looking for your team to come up with ideas looking for choices obviously when you look at those missed expectations they could be around the same thing like the problem with measuring not really understanding what makes something work I had a colleague and friend somebody I've known a long time and he worked in an organization where they made radar systems and the particular product that his company had on the market was kind of stretched thin and they wanted to just slap more code in it and get this thing out the door and he said no no you've got to trust me I want to redesign the architecture and once we redesign the architecture then we're going to be able to make changes more easily and so if any of you have been in some of Mary's talks and she talks about that distributed federated kind of architecture I mean that's what my friend was talking about at the time I worked in organizations where sometimes they call these Lego diagrams and they can pick up the blocks and move them around and so what he did was he convinced his management to take four of their best people and put them in a special place so they found this little room in a building that was unlike the main headquarters and they used to sit around they got easy chairs and they brought in these interesting lamps from home with you know colorful shades and a little bit softer lighting than the fluorescence that are everywhere and so they had their cozy chairs and they had their lamps and they worked on their new architecture and the vice president of sales would come around and the vice president of engineering would come around and they would look in the room what are these guys doing they're sitting they're sitting in their chairs eating dried apricots and oh look at those lamps on the table and in the end what happened was the architecture was a success and what happened was of course then their cycles for change improved quite a bit the interesting thing that I thought was very funny was those vice presidents walking around trying to understand what made the team so successful what they did was they then passed a catalog around to other people in the organization asking them to pick out a lamp of their choice so I just thought that was such a funny story he was thinking that somehow the magic was in the lamp and not that unifying goal and mission to create an architecture that was flexible and changeable so people that don't get it sometimes luckily that was not a missed expectation that kind of worked out well feedback I had a class on design that I was teaching and it was a new introducing new things into an organization and the man who was the chief architect was very excited about the rollout of this class and how this was going to be accepted and it was a normal class there were a couple of people in the front that were asking a whole lot of questions and at one point I saw open door in the back nobody else could see him I saw the chief architect standing there listening in the back I didn't say anything and you know the class proceeded people were asking questions well that won't work because of this or this won't work because of that and you know we got the class to a close at the end of the class Bob the chief architect walked up to me and he said I was really concerned with all those questions being asked so you know all I could think of was he's going to go out and start punishing the architects who are trying to learn these new design approaches I said no no no I said you don't understand that's very important feedback that feedback tells me what are all the issues that people have what are the things that stop them from understanding or they believe would stop them from doing you know what we would all like them to do so he accepted that that was going to be okay so we were able to to twist that a little bit so that people wouldn't get changed rules of the system can be very tough some of you may already experience the problem of you know the time that people want people want you to meet the deadline and that becomes the most important thing as opposed to the quality and the value that you provide and that's a toughie but I think one of the one one person told me this is a colleague something that he tried was something he called a throw down and he said okay you think these agile methods aren't going to be as effective as what you've been doing let's get some data on the projects from what you've been doing and we'll start gathering some data on our turnaround time and defects and they they did a number of measures and in the beginning people were like oh this is not going to make any difference but what happened was he said at the end you have to had to take it out about six to 12 months because you needed some trending to see it start to change but he said the throw down was his idea for looking at how the trend improves I thought that was very clever because that was helping to change the rules of the system so but these are the areas where you're going to see the pushback and so I my thing is cut off there were five things but I think collaboration is important because you can generate opportunities and however you choose to use that collaboration and I think one of the ideas for instance that you had in her session yesterday a session I attended yesterday was sometimes the the group may not have to you know move along slowly as everybody makes the decision all at once sometimes everybody says well you know you're the expert in this why don't you help us solve this particular piece of the problem and so sometimes collaboration does not mean everybody moves lock step altogether sometimes it means you have a trust and agreement with your other folks management support is important like just like what I was saying with Mr. Chen the US Air Force Colonel the people getting and my friend who got the trust of one manager in his organization to be able to get the architects to re-architect the entire system so you want to look for those champions that are also going to help you leverage things continuous integration important to learn what works what doesn't if you could read number four I believe it says transparency figure out a way to share information and make it visible and I don't know if I even remember what number five was but I'll have to get these out to you but the ideas those are just five ideas there's a lot of ways that you can start to look at this those are some powerful things that I have seen and in the end you just want to keep remembering that you need to use the power of choice it may feel like a crisis but it's only the end of an illusion that you're going to remain in the same place so things are always going to be moving we use words like disruption innovation it's inevitable so recognizing that sometimes you're going to be pushed out of your comfort zone and generate ideas to change that choosing again so is the title and so choosing again is if something doesn't work go back go back go back generate more choices and I believe I am at the end yeah oh look there's a page that pops up there's my transparency but and trust and permission to fail I don't know how I got those two slides at the end that's and that is kind of like my friend in also in the working on the radar systems he he got a permission to you know we may not do it right somebody was their champion and bought into that but permission to fail sometimes you need to come up with process pieces that allow you to say stop let's find out what's going a little wrong with what we're doing that's the power of the retrospective also to talk about that talk about what did not go right as well as what did go right you never always want to preserve the things that that helped but maybe work in a little process into how you how you get that permission to fail okay so I don't believe if I have any more here I'd be very surprised I don't know how that must have that must have been a earlier version where I did but that's that's it so we're at the point where if you have questions I'd be most happy to answer answer them yes to ask about you talked about the safety I guess yeah whoa people didn't feel safe yes in terms of you know as you're helping them with change as you're yeah looking at that I'm kind of curious I mean what do you look for though to know that lack of safety I mean what are the size of the indicator especially if you have very quiet yes anything yeah so the people have talked about a whole lot of ways so if people don't feel safe how do you particularly if everyone's quiet anyway how do you assess that I think in some cases you're looking at how far they're willing to go outside their comfort zone so you kind of have to know something about the people that you're working with and what you think their tolerance may be um I would look at some of the presentations on drawing people out there were a lot of ideas about getting things down and writing Mary's idea of you know buying into the idea of doing sketches design sketches before talking about it so all those people that are quieter good facilitation techniques where you ask everyone to submit some ideas so everybody has a voice to some extent that's one of the ways to do it but certainly being quiet is can be an indication of lack of safety how big a team are you typically talking about usually the team sizes might be five to ten and multiple teams yeah so I think you want to get the people that really are familiar with the teams to kind of get a sense of those people maybe talking to people individually I've done that outside of meetings where just to get a sense of you know what somebody's take is on what's going on it's not everything but those are some ideas other questions how to increase the trust level inside the team so they uh can give the feedback faster yeah no matter negative feedback or positive yeah team trust is a is is an interesting cycle and it has its own cycle so there was another psychologist Bruce Tuckman who had done some of you may be familiar with this or have run into it where he said that when a team forms there are stages that it goes through and the first it's forming storming norming and performing and so when a team forms it's not atypical it frequently you will see some friction as people jockey for position in the team who's who's going to be the best in the team or who's going to have control over different issues and so sometimes it's just time and having somebody who is a good leader to help kind of facilitate the team into you know getting to a norming stage norming stage is really where you start to see comfort and trust and performing is then they're taking everything and going on so forming storming those are the two stages where you're probably just going to have to figure out how to get through that to get to the norming so that's something worthwhile reading too other questions so if there's no more questions thank you everyone for coming thank you


Thank you colleagues. Okay, so our project title, as I've already heard, is Technology Enhance Assessment Methods Team in Science and Health Practical Settings. As you'll have seen from the introductions, we are made up of four institutes of technology and four schools of, I suppose, science and health. How we started working on this project together, I suppose, is through the interactions that our four schools have through a forum known as the Head of Schools of Science. We're very conscious that this project is a discipline-led project, and for this reason we have four schools all from the same discipline, and particularly important, it's being led by the four heads of schools or the four senior managers in those four institutes of technology. So that, I suppose, was the initial driver for the project. In addition, there's been a very good track record of our institutes working together in teaching and learning projects. As far back as 2010, Carlo IT and Dundalk IT were involved in a project around learner assessment, and currently Dundalk IT and Atlone IT with two university partners are involved in two projects funded in last year's teaching and learning enhancement call. This is the project team. The majority of the team are here today, as you've seen. Our colleagues who are not able to be here today are being well represented by their colleagues from the different institutes. Our team is made up of the heads of school in the four institutes, also represented us from the teaching and learning units, and also a number of lecturers who are involved in the delivery of science and health education in the four colleges. So we feel that we have a number of very key partners involved in the project team already at this stage. So what's the project actually about? Well, in science and health disciplines, key to student employability is their practical skills. In nearly all science and health disciplines, practical skills are a key outcome for our graduates. And in fact, the Institute of Technology in particular plays a major factor on practical skills for their graduates, and it's a fact that's often comment on by industry when they employ our graduates. So I suppose what we want to try and do is enhance what we think we're already doing very well in this sector by using the collaboration that we've established through this project. So we want to really look at the practical element and how practicals are delivered and assessed in the science and health disciplines. So we can bring it down to two key questions. First of all, how can we use practical assessment to facilitate student learning and skill development? And then secondly, how can we enhance the design of practical assessment in order to shift the emphasis to an assessment for learning approach? So essentially we want to develop a framework to apply the principles of good assessment and feedback to practical learning and do this by a dialogue with all the relevant stakeholders. Why are we doing this? Well, the literature has suggested that there are some concerns with over-assessment in the science and health disciplines, that there is a reliance on using conventional kind of recipe protocol following techniques when you're delivering in the science and health disciplines. Although we recognize that it's very much an active learning environment when you're in a practical session, so the question is, is assessment actively assessed? And is there an over-assessment? Are we looking enough at skills development and synthesis? For example, my colleague Ronan Brie from Dendork IT in his publication 2014 looked at this whole area of over-assessment and looked at putting in place a very simple incremental marking system where he showed that he evaluated this study and showed that it really improved the student experience within the laboratory situation. Assessment is key to the student's learning process. It's very important in terms of providing the student with the relevant skills that are relevant for industry and also for enhancing their actual learning experience in the science and health disciplines. So if you want to change the way you do assessment, I suppose there are a number of frameworks that you can look at. And I suppose at a high level the framework we're going to follow in this project is that laid out in the JASIC framework. Basically we'll identify, first of all, what good practice is. Then we will engage with all the relevant stakeholders to see what do they actually need. Then we will have an appreciative inquiry phase where we will look at change in a very positive way. We look at the baseline situation, we put in changes, we evaluate it and we'll continue trying to refine it and evaluate it as we go through the project. And what's key to this is the change management strategy. How do we actually embed this change into our four colleges? So when you are involved in I suppose science and health practicals, there are kind of three components to it. The first part is the design of the practical. The second part is how you actually carry it out, how you actually have the practical session. And the final part is an assessment. And it's those three practices that we're going to look at in this project. And we intend as one of the components of this project to use digital technologies that we're already using in the four colleges to enhance what we're doing in terms of how we design practicals, in terms of how we actually carry them out and how we actually assess them. So for example, in terms of the practical sessions, using interactive lab manuals, electronic lab notebooks in designing and looking at problem-based learning and the assessment using things like online quizzes, self-assessment, peer-assisted learning. So these are a number of the practices that we will use in order to, I suppose, enhance the experience for our students within the four colleges. So there are a number of key stakeholders in this project. And I think the key word here is the student-academic partnership. And I suppose we want to have an ongoing feedback dialogue between all of these partners. Today I suppose we have representatives from the management side, the head of schools, who will be the big drivers of the change around the four schools. We also have representatives from teaching and learning units who play key role in staff development that might be required as the project develops. And we will have support from, obviously, from our lecturers who are going to be delivering these different types of assessment practices who are going to be piloting them and evaluating them. But key to this is going to be the student voice. The student is going to be involved from the very beginning of the project and as it goes through because we think that the student is key here in terms of their learning experience. Industry will also play a key part because we want to be sure that our students have the skills that they need in order to gain employment in the relevant sector. So the project group is very multidisciplinary and we feel that that really will enhance the overall project. And it's important that every single stakeholder has ownership. So how are we actually going to do it? Well, we're going to divide this project, I suppose, into three phases. The first phase is a baseline review of the practices amongst the four partner institutions. Obviously we'll establish the project, we'll set up the project governance and it'll have an advisory board which will consist of students from the four colleges and industry representatives identified with the four colleges. We'll carry out a literature review to look at international best practice in the field of assessment and science and health. We will also do an analysis of what is actually going on in the four colleges. We have a sense of what's going on already but we want to just kind of pin that down so we can potentially, I suppose, partner up similar like-minded academics within the four colleges. Essentially, we want to be sure that our enhancement that we're going to put into practice is very much grounded in evidence and this is really the first phase of the project that will hopefully ensure its success. The second phase, then, is after we've identified what we think are the practices we want to pilot and evaluate in our four colleges, we will then carry out a pilot and evaluation study in the four colleges. So what we could be talking about there is we might look at a particular assessment practice in microbiology that's happening in Dundalk, for example. We might then partner up that lecture with the lecturer at Lone and Sligo who might be doing something similar and together they'll work on a collaborative project to pilot this new assessment project and then evaluate it. And this is the way in which we think we'll be able to assess the impact on the ground with the frontline lecturers delivering the project and all the time looking for student feedback as the project goes on. So all the time as we're rolling out these pilots and evaluating the student who will be consulted and the lecturers who are actually carrying out the pilots will be consulted. In phase three, then, we will then look at disseminating and sharing the resources that we have obtained from these pilot studies by setting up a project website where we will share the assessment resources by dissemination through maybe organizing master classes in the four different partner institutes and inviting in the lecturers from across those colleges and then we will also organize a final workshop. And I think one of the main outcomes of this project we feel is that we will be establishing or enhancing an already established community of practice between lecturers who are working in the science and health disciplines between the four colleges. And we think that there is a huge potential to this to expand into our other partner institute technologies and indeed university partners within our different regional clusters. So what will we actually deliver? Well, at the end of year one, we hope that we will have our project governance established. We have our peer network of academics formally established. We will have a baseline report detailing practical assessment approaches across the four partners, a detailed literature review of best practices and pilot studies identified scope data, ethical approval and staff trained where required. At the end of the second year, then, we will have a number of these practical assessment approaches piloted and evaluated. We hope to generate some peer-reviewed publications between the partners. We have launched our websites. We have organized master classes and have a project workshop and look at ways of sustaining the peer support network. In terms of resourcing this project, it's a two-year project. We will put in some admin support to it. One of the key resources in this project is the academic lecturer buyout. Our academic lecturers have a very high teaching load and for them to be actively involved in additional research or projects like this for northern to give them, give it as much time as needed. It's important that we allow them to buy out their time. That's a major important factor of the costing of this project. We will need to purchase some digital technologies and we will bring a learning technology stand near the latter part of the project which will allow us, I suppose, to develop resources to put up on our website and to be able to have ways of which we can disseminate the information further. There will be obviously some travel costs and also staff development is very important there. We're very privileged in our institutions that a major priority is put on the staff development around teaching and learning. So we have a lot of experience in working together in developing CPD programs. So this will be an important part for getting buy-in from all of the academic staff in the four partners. In terms of, I suppose, our national impact, we feel that we're four institutes of technology that are quite geographically located amongst the country. We're four out of the 13 institutes of technology. We think that if this project, assuming this project moves forward, that it has the potential to allow interaction between all of the institute technologies and the relevant university partners in this discipline. Because it's really a way of taking assessment practices that we know are working very well, enhancing them, and then sharing the techniques with colleagues and allowing colleagues a network by which they can work together. Because we're going to involve industry at all stages of this, hopefully we'll produce graduates that will have skills that are really relevant to industry that will improve the employability of our graduates to Indigenous and local companies in the regions. I think it's important, I suppose, to state maybe as a concluding point that we think this project will succeed for a number of reasons. First of all, there is a very strong commitment and leadership from the senior management in the four institutes of technology who are going to be the drivers of change in these assessment practices. And that's, I suppose, evidence by the involvement in this project proposal. There's going to be a very strong ownership of staff and students in this, and they're going to be consulted at all stages of the project, as will industry partners. We think that what we're looking at here is an evidence-based enhancement of practices that we are doing already and that we want to put them into a mode that they can be shared with all colleagues and will overall improve the student-learner experience. And finally then, we think through our ongoing evaluation we have a chance to refine and improve what we're doing and we feel that it has a fantastic potential for capacity building in the health and science disciplines in Ireland and further afield. So, thank you very much.