A description of free iterative theories (2nd part) / Fer-Jan de Vries: A lambda calculus for D (& others)

0:00 But perhaps the most intuitive one, which is equivalent for the category offsets to that piece, is that it is a quotient of a polynomial function. So, the function is finitary if you find a signature and you find a natural transformation which has epimodal components. Being epimorphic, they are completely described by their kernel, and the kernel of a component, well, it takes some couples indexed by operations, pairs, An operation symbol followed by n variables in this n-ary symbol and another operation symbol followed by k variables in this t-ary. I mean all such that epsilon brings these two elements into one element in H. This is a third way how to say a function is high-magnetic, namely, you say that you are given some operation symbols and some equations, but these equations have to have a special type that is only an operation applied to them. This is the case of a polynomial function which corresponds to one binary operation and you assign the recorded pair to the corresponding non-recorded pair. In other words, these special equations take here the form of the commutativity of the operation which is expressed by taking this formula.

2:30 There are no zero terms, there are no constants. Exactly, there are no constants, so this is not a very inspiring example as far as algebra is concerned. And so maybe I'll start immediately with a very interesting factor, the finite power set factor. This has a property that is a good interpretation of what co-archive are. They are monolingual transition systems of finite degree, so it's something that people work with. And here again I can express this as a quotient of a polynomial function where in this case sigma has a unit and an operation. The same, you take an interval and you interpret it as a set of all of them. So here, the operations, the equations I would consider, there are infinitely many such, but they all have all the following form. And, okay, let me clear this up. It's absolutely clear what I mean is, what is initial algebra. Algebra of a variety, namely of a variety of sigma algebras given by the equations that characterize my epitransformation.

5:00 In other words, I can describe it as the algebra of all More than all, the congruence that you get by saying, I apply star-finite union types. So, S is congruent to given if I'm going to give you S can be obtained from T by finite applications. Okay, and what is my goal? My goal is to assure you that an analogy, well, initial algebra, term of algebra, goes over. So, the expected answer is true, namely, you take the algebra of all finite and infinite trees and you divide it by the confines obtained from the above confines by dropping the finite one. So, the starting confines means s can be obtained from t by finite or infinite applications. Now I will stop. Except... Couldn't you describe I as the sets of finite rank, just starting with an empty set and taking finite sets and finite sets and finite sets? Absolutely, yes. That's another description, a more pretty one, and hopefully not useful for Michael, but that's the true rank. I wanted to illustrate through this example the general procedure, but you're completely right, yes, there is simply a better description than just saying this quote. Yeah, anyway, I can't stop here, I can't say that's all, because I need to define what does it mean to apply some, you know, some equations infinitely many times. And here comes my definition.

7:30 The definition concerns the fact that you can cut a tree at a certain level, except for that you really need a new, very constant symbol in the way that you cut the tree and you obtain a new leaf or tree. So, the first thing in the book is I introduce an enlarged signature, which is the original signature with a new symbol, which is newer. I now have the cutting at the level k function, which takes every finite tree, but in this I have takes every infinite tree, cuts it at level k, and puts label button where needed, new leaves. And so now I am able to tell you quite precisely what. What it means to apply equations infinitely many times? You cut your tree and apply it infinitely many times. But what are the depths you choose? So for every case, what one must hold is that the cutting of S is congruent to the cutting. This particular example is simply to remind you about. A very nice figure in TCS in 1994, trying to avoid non-recommended theory. Successfully solved in my opinion. But anyway, as a byproduct, he described the terminal co-algebra of finite power set functions. And this byproduct has never been followed afterwards until today. And that's my message. What I showed you on this particular example was absolutely general. So, I'll see you on a piece for every finite parameter. Every permanent co-algebra, permanent H-co-algebra can be described with co-algebra of all finite infinite C-matrix, factored by a congruence where, again, the congruence is S-EVAP, for every k, the cutting of S is congruent to the cutting of E,

10:00 Well, now these two trees live in the initial algebra of sigma volume, and this is the smallest congruence on the algebra, on the sigma algebra initial algebra, this enlarged signature, and you consider it as the sigma algebra, so we simply forget that there is an additional neural recuperation generated by So, Barre's example is simply a rule, not an exception, and a good example, well, just to illustrate what I mean by, so, by these infinite applications, they, by their operation, and they, their operation, let's say, sigma, and they, the equations to be commutativity, and it's not made clear on the first side, Whether the following tree is congruent to the tree which you get by completely reverting, and the answer is yes, and to provide quite exactly, I simply cut every level and observe that level k, I only need to use correlativity k times, so altogether I only use ruby.

12:30 For those of you who remember Elgott's iterative theories, let me recall the basic theorem of a paper written by Elgott, Bloom and Tingle in 1978, which said a free iterative theory is the theory of something in between finite and infinite, namely These are close infinite trees which, nonetheless, have only finitely many subtrees. A good example is this tree here. It's infinite, but altogether it has only two subtrees. This is one, and he himself is in his seventh one, because this one is already concomitant. Now, there is a deep connection between iterative theories and all algebras, and this is... And a recent discovery by Larry Moss and independently by a group of Peter Exel, Stefan Milius, Yuzi Velovel and myself, and it says approximately this, if you start with an arbitrary function h and have a good luck, which you always have if h is finite, that a terminal co-algebra will get an object x.

15:00 Then, this t of x is an object part. Now, there is a difference between iterative monad or iterative algebraic theories. Out of those, they are given an arbitrary system of recursive equations, finite system. You will always find the unique solution. And, for a completely iterative, you drop the finiteness. Any system, finite or infinite of recursive equations, has a free field. Start varying your answer age. For example, start varying it from a polynomial counter to a quotient. Then you can prove that also you are varying monad t. So without this, you will never obtain other completely iterative monads. Then, the three ones on signatures. And these are precisely the monads. If you start with chi-sigma, then your monad is the monad of finite and infinite sigma trees over x, in the sense that now the variables may appear on the leaves. And when you now make a quotient, then you will again get a quotient here. So there are no other completely iterative monads than monads of infinite trees for your some kind of congruence.

17:30 But if H is finite, and if you switch from completely iterative monads to iterative monads, then the main result is iterative monads have the form, you take rational trees and you portion them by this congruence of applying some basic equations, finitely, for the finite monads. Yeah, the surprising answer is no. It's such a big book. Yeah, it's a big book, and it has lots of examples of iteration theories. Yeah. But it is just one example of iterated theories, and that's our question on this. I was going to bring up non-law-bound executive because that's the terminal algebra for piece of it, yes. Yeah. So there is another connection to domain theory here. I haven't thought about using the congruences, but if you just take the absolutely free algebras without any equations, then you can set up some fairly simple power set domains such that the initial algebra is the least fixed point and the co-algebra is the maximum fixed point. The confining connection is the main theory and using fixed points of set theoretic, momental and continuous functions and get the co-element of the data. So I have to think about changing the equations because since everything is finite very quickly, it should lead to continuous functions. That sounds very interesting. So the equations you have don't vary that much. Not only that, they are even more primitive than that. Not only don't they vary, but that they also, for example, are not allowed to have any constantness. So they are all distributions as trees have not formed.

20:00 Here is an operation symbol, here is an operation symbol, and here are variables, but you cannot repeat variables, right? Absolutely, yes, I cannot repeat variables, but I cannot make associatives. Is this all you can add, or do you think that there's more which can be added? Good question. No, I really don't know. I was quite happy to get this answer for all the answers, and I did not fall for a lot of questions that can actually say something about other types of equations, they will not. Because his characterizing is all finite area. Right, yes. And last quick question. Okay, let's find the speaker again. Sorry. Thank you. Oh man, was that you? Okay, our next speaker is Ernst Heinrich Weiss, and he will announce his title as well. The title is simple. Long ago, I worked in infinity. There's a young worker, his father's married, who works in politics in the city of Torino,

22:30 and I'm currently working in NASA, which is dedicated to him. The aspect of this talk is that the infinity walls of the stop I'm 70 years old. First, the invitation was meant for lambda calculus, which is the data on the left-hand side. Just speak in the cloud, please. Speak louder. Speak louder, sorry. Our question is, what is the lambda calculus of these models? I will explain what I mean. The answer we will give is an extension of the finite lambda calculus. We will call this the finite lambda calculus. Our extension will be an infinitary lambda calculus with all kinds of things, which in a sense can be called a fully abstract language for its infinity model. And the unique normal forms in this infinitary language, two terms have the same normal form in this extension if and only if they are equal in the model. Beautiful, isn't it? This is an idea of trying to prove confidence. If you try to prove confidence, you start modest. You start with one step versus another step, a horizontal step and a vertical step. And then you have to, if you're a physicist, you have to do the horizontal step after doing the horizontal step, and that gives you a reduction in bandwidth. But you might have to do more than one step, you might have to do five hundred million steps. So this is a typical situation. I will give it our first time. In astrogramma, in inductive approach, you think you can do it until n, and then you use this extra diagram and you plug it there, and you can have the whole conference group by induction, but the fact that those vertical and horizontal arrows no longer are single sets, but they are families of three edges which you contract.

25:00 They are residuals of the one on the top and the one on the right. And somehow you have to take the descendants of the vertical family of redaxes over to this horizontal family. And given such a family, if I put the H or the V on top of the arrow, I mean the complete development of this set of redaxes. You can make this construction in nice rewrite systems if provided by your orthogonal. Lambda calculus is an example of that. The point is that in order for this to work, in the case of lambda calculus, these sets V and H have to be finely nested. So if you draw a picture of the... If you draw a picture of the tree, and here you see various of these ridges, then a family of ridges is finally nested on any path, only a finite number of ridges. And in case of the Landekopolis, that is the case. So, that makes it possible for us to do the finite proof of reduction of finite length. You start with this term and you're looking at a regression which continues and which goes on and which happens to be nice in the sense that you can think of the term as being a quotient sequence and then there might be a limit and suppose you have a situation where that happens and then you have two of those sequences, a horizontal one and a vertical one, then you might want to do the same construction again. But we were talking about, in these diagrams, about residuals. We needed the residuals of the vertical steps after the horizontal steps. So we need descendants. So if you only talk about quasi-converting sequences, something goes wrong. Here's an example. ix-identity term, you can make an infinite term.

27:30 So i apply to i, apply to i infinitely often. The first I of the term. Clearly all these terms, if you can do that, then you get a procedure sequence. There's a limit, the first thing. But you cannot say from the first I of this last term, this limit term, it descends from the I number N of the first term. The connection has gone lost. And that is because this is just a procedure-converting term. We need something stronger. But what we need in order to be able to talk about residuals in those limit points is the notion of a strongly converging reduction. And a strongly converging reduction is a reduction which is quasi-converging plus an extra property. Plus the property that the depths of the limit, when you look at the depths of the redaxes, which are contracted in the sequence. The successive dash of those limits has to go to infinity along the sequence. That is a quantum-computing sequence. And you can think of the elliptic sequence as approximating and making progress. So after a while you can start printing prefixes which the process will not be the same anymore. Procurement calculating the limit. In general, if you have a strongly converging reduction sequence, the limit does not have to be a normal form. It can contain the reduction, and you can continue rewriting. So then not only do you have a reduction sequence of length omega, but you can think of transparent rewriting sequences. So this kind of confidence proof, if you want to do what we needed, in principle has to be for arbitrary limits, arbitrary ordinal limits. If you start with a finite term, then the limit can be an infinite term.

30:00 A simple example is if you look at the fixed-point combinator applied to i. You get in one step i by i, the number of steps, and it repeats and in the end you get this limit i omega. So limit terms can be infinite. So if you are working in this context, it is natural to consider infinite terms of residuals of one step. The final term can be infinite. So these sets V and H are no longer finite, but can be infinite. And in a lot of calculus, these sets, no matter this complex, this finite domestic property goes on, even if you're talking about infinite terms. So how does the confidence proof work? You start doing the basic counts in a different setting, and then assume, as an induction hypothesis, that you can make such a whole diagram, but for the point, but for the final point, you can make all these, that means that you have constructed all these subparts, you get a reduction here. That will be, of course, equal to the reduction. The question is, will this be, and this will be a quasi-converting reduction. The question is, will these reductions have the same limit? That is not necessarily the case, but if the diagram, if those sequences, I don't say in the horizontal direction, are all strongly converging in a uniform way, then the bottom column is strongly converging and has a limit. And the limit of the right hand side will be the same as the limit of the bottom row. That is a nice property. We only need uniformly converted in one direction. And get them uniformly converted in the other direction as well. That is the basic tool of the proof.

32:30 The argument like this is there. What is curious is, I am talking about depth, but I have not told you what I mean by depth. Think of trees, terms as trees. There is an A-way of picturing trees and you have an E-way of picturing trees, but there are different ways of defining the depth of a tree. You can think of the exit of lambda, you can think of the left exit of an application node, and the right exit of an application node. So again, if you draw the thing a lot as a tree, you can say, well, we draw the exits of a Londonoid horizontal, and also the left exit of an applicationoid horizontal, and only the right exits that are down. So if you start with a head-normal form, drawing the tree in a fashion means that you get the left spine perfectly horizontal, and only the symptoms end, they dangle one level lower. But this is the home of geometry and its own interpretation of depth. All the things on this left panel in this picture are on depth 0. All the things on the next one, I will call depth 1. If you have a notion of depth, then you can define a notion of distance. If you have a notion of distance, then you can take the final terms, which is the notion of distance, you can have the metric of space, you can look at the metric of completion. It is natural to think of metric completions, so it is natural to think of infinite-thumb problems. Yes, so, why in the beginning, if not in the beginning, in the second reason? People had the intuition of infinite-thumb problems, but never made it really explicit. Maybe I have a misunderstanding here, but that wasn't the question I wanted to take.

35:00 This notion of geometry is directly related to the way an environment would... For example, if you want to denote a boundary, you would write the prefix bar lambda x y in the node as a label and then the boundaries of this m as defendants. This notion of depth relates to the notion of boundary. The only notion of tree in the game is the notion of lazy logo tree. And there is an easy way of drawing the tree is that the lambda goes down. The left exit and the right exit of the application go down. That's why it's called the lazy tree. On this track there is a whole class of things. When you start to find the lambda properties, you can make an extension with infinite terms. And I forgot to mention about the bottom. In all these formulas, n has a normal form, or n has a top-normal form, and you have also terms which don't have such a nice property. Those terms are in the boundary identified as bottom. For example, they have a number tree, they call it a bottom, or in the derivative tree, they call it a bottom. So we will extend lambda-couples. You can see that lambda-couple is a beta and lambda-couple is an echo. We will think of lambda-couples in an infinite extension with a bottom rule. And then there are these three variations. Turbines which don't have a head-normal form are mapped into bottom. Turbines which don't have a big head-normal form are mapped into bottom. Turbines which don't have a top-normal form are mapped into bottom.

37:30 With these three different calculators, we can prove that they are confident for infinite sequences, and the nice thing is that each term has a normal form, that is, the Herodotus tree in case of this calculus, the Phenomenon tree in case of this calculus, and the Bern tree in case of the identified terms without having a normal form. Is different from the Biotics tree of is bottom, and makes a difference from the Biotics tree of lambda x omega over lambda x bottom. And the Biotics tree of omega omega over bottom bottom. So it is, whereas in the Buntree, lambda x omega over lambda x bottom and omega omega over lambda x bottom. So that makes a difference. But, okay, so far so good. Now we are looking at simple-space airtime. So we want to add the airtime. Consider airtime as an intermediary complex. What immediately goes wrong is that you cannot add the airtime to the top or the middle of phone versions. Then you get silly critical maths and no confidence. The only sensible way of adding the airtime is adding the airtime to the calculus, and that's the idea of the boundary. In this calculus, you get the equivalent of normal boundaries. You can think of adding that addable, and then think of the boundary as the equivalent of the normal form, the addable form of the boundary. Then you get so-called add-up boundaries. That this calculus is confident is the content of our ITA series. What I want to do now... Consider the infinite add-adventures. If you look in the book of Land Department of the Environment, then you find a notion of infinite add-adventure, which is a way of an explicit element in the equivalence class of add-equivalent boundaries by the minimal person, but the definition is not here. It runs over several pages and you need several pages to explain things.

40:00 The notion of strong compartments allows us to simplify the definition and we will prove that this populace is confident and it has perhaps infinite number of votes and they will correspond to its infinite add-on electorate. Well, there are models which, in a sense, two terms of the same tree if and only if they are equal in those models. This is the pre-infinity model of Corcoran, Simon, and Sachse, and I'm guilty for that error. Not the Park model, which is a different guy. I wrote the Park model in the APA paper, too. You know this, and the guilty model might go also. The funny bit is that, if you have a cheat-freeze, you don't have a monotonicity. You don't have a model. So the argument of continuity, which plays a role in the construction of our rule of random powers, cannot be used for the periodic trees. What can be used is the fact that there are unique normal forms in this extended code graph. So, our construction helps us to give us a Böhm tree-like model, a Böhm kind of model for this data-layer instance of integer numbers over three. You cannot give a model, a domain model. So, let's now go to this corpus, and that's this funny error thing, f of three. So, these infinite error models If you take the term i or you compare it with the term j, the term j is an infinite abstract function of the i. You can calculate that that has the same notation in the infinity model as the i, but it's still not the same.

42:30 That is something you don't get from just looking at the arrow. So you need, in order to have a calculus in which those terms can be computed as having the same form, you need something stronger than just the eta. That is why we need the eta string at the end. So we look at a strengthening of the eta rule. We say lambda x m n goes to n. Provided that x does not occur to the pre-variable m. It can be expanded, perhaps in an infinite variety of things. It should be expanded in a strong and convergent way, so we call this n. So this n is an expansion of x. x as a variable. x is a variable. Oh, okay, but you can apply Ada Rule to a variable. Yes. That expansion can be applied to a variable. So if x can be extended to n, then we say lambda x mn can be reduced in one big step to m. That is a powerful construction. And then we can prove confidence to find out the infinite beta bottom in this eta regression. Every term is in order form. This is infant boundary, I don't know why I repeated it. The normal forms form an extensional variant of the bundle, with the help of this effect you can show that you can define applications in the model used to convert infant at the boundary, and you can prove... Syntactically, the two terms have the same normal form in this calculus, if and only if they are observationally equivalent with respect to their normal forms. We don't need the infinity model. In one direction, that is the blood shear, and in the other direction, we use the uniqueness of the boundary.

45:00 If and only if they have the same, so many tips in here, I'll stop them all. What are the ingredients of our proof? You need things like, if you look at a stony conversion, eta, range conversion versus beta that have commutes. Or that, if you look at an eta expansion, stony conversion eta expansion versus a stony conversion beta that have commutes. Here is the last step, the stony conversion eta expansion. In one direction, if you look at it from the other direction, it can be mimicked by its only converging alpha-1 regression. So this commutation diagram gives us a postponement diagram. If you do the deltas after an alpha-1, then you can also do the deltas before the alpha-1. To change the order a little bit. And the proofs of those level facts are similar for copper and alpha. The proofs of these facts are timing arguments. So, for instance, how do we prove postponement? You can postpone the eta after the delta voltage. So you start with an eta, a mixed, mixed thing, a mixed reduction with eta, deltas, and volatiles. You can recognize that there is a number of eta banks followed by some delta volatiles, etc., and it alternates. So instead of writing eta, bank, rewrite eta as function in the other direction, you have this kind of picture. We know that this is strongly converging. These little tiles, we can make all these small tiles, we get a kind of, no longer a square, but we get a triangle. We know that the diagonal is strongly converging, and they have a similar argument, therefore this regression is strongly converging and has the same limit as the limit of Witten's hand. And that shows us that a cross-bonding is an infinite setting, and that goes through. The closing in two steps, we want to prove, so suppose you have here a beta-bottom, eta-bottom, which is in this direction and in that direction, then we look at the bottom, beta-bottom normal forms, they exist because our theory is based on the beta-bottom, and in fact we have this diagram, eta-bottom postponed over the beta-bottom, that is this diagram.

47:30 And you have to go on the right hand side, so here you have the cure at the same relation, and here you have the cure at the same relation, so you have that response to get to the point of using the topology. Can you go back to your slide that shows the, the printed slide there that showed the context? Now, for the context saying head normal form, what kind of reduction is that? This is finite data. Just ordinary lambda calculus. Okay, so then you have a result... Ordinary with ordinary beta a. Okay, so when you put those together then you don't have to mention the infinitary lambda calculus. You have a result there about the infinity just with ordinary reduction. Yes. Yes. But your method of proof brings in the infinitary lambda components. Yes. So the method of immediate proof, the method of saying you get first is the statement for the infinitary terms. Also this infinitory context. And then you have to use some extra work to get the final form from this. Yeah. Right. But then to have a nice result there that doesn't mention the infinitary ones at all.

50:00 So, I think it's good to emphasize that, because that can be explained without bringing in the new land accountants. Yes, but can you explain the land accountants to the new land accountants? Oh, sure. Sure, I understand. In the 70s, you used the infinity model to turn the land accountants into mathematics. Does this humanized result also extend to the older case of the Amadana or the relation of the model with its observational equivalence? And the second question is, I mean, How do your burnt trees correspond to what some people call macadamia trees. I'm not absolutely certain what it is, and then the question is whether this observational order can be recaptured in terms of an inclusion of the macadamia or burnt trees. The first question I cannot answer. I just don't know. I have to look at it. The second question, that is something we are looking at, but macadamia trees are very strange. In our context, because we consider strongly convergent expansions, if you start with a branch tree, then it is finally branching. If you apply strongly convergent expansions, by the nature of strongly convergent, you can only apply a finite number of expansions at a given depth. So it keeps being finally branching. The Nakajima trees are a kind of saturation of expansion and they are infinitely branching. So we don't get them as limited in the natural metric. I have no idea that this future, maybe it's just impossible, to put them in a nice metric context. If you get the measured space and you rewrite that space, you might not ask the Nakajima trees are the... So in this context, you can define a large application of logarithms. But if you try to apply two macadamia trees to each other, the substitution becomes very strange. In a macadamia tree, you also have infinite prefixes of logarithms. And I have no idea how, if you try to do that, it becomes strange. I didn't understand what you said about the topological model of the world, but I don't know why, but do you think there is a proof there?

52:30 So do you know a proof that there is no one? Yes, I'm inclined to believe so because... Bottom is smaller than the lower x-axis, but if you put those terms in a context where you apply the right-hand side identity, then you notice that this inequality is not preserved on its own. It seems to suggest to me that you cannot have a model for it if you don't. Is it just a moment? Okay, let's thank the speaker. I don't think you need to worry about modeling. You can find really simple.

57:30 You can work with more theoretical data. But maybe you've also got to have a little bit of courage to do that. I don't need a lecture about mathematics. I'm a human being. I have an idea that you need. What is a calculator? A land calculator. A sort of land calculator. You might well have some philomena in there, but it's like waiting for a train to catch you. So at some point they decide that they want my package, which means that I'm going to pay the fees, which means I'm going to have to cheat forever. So it's like a test of your breakfast kit. Yes, yes. It's a test of your sense of confidence. I guess it's inevitable that a bunch of people who've been working on their own language, their family, decide you want to pull the horses. The commitment seems to be for everyone to want their sorts of features in that sense when it comes to the union of mathematics and physics. So now the thing is, how many universities are there?

1:00:00 More generally, you see that A over T is an actual fraction, where A is any object, but T is the one that's tiny object, so it simply refers to A times 1 over T, which means A times 1 over T, which means raise any power, raise any power, A over T. So your picture is that the possibility of these powers is just given by left-handers. Well, not end of story. See, that's sort of the broad framework. Now I'm going to the bottom. I'm building up again. Of course, any object. But then a tiny object taken backwards also. Any composite of left-advanced and right-advanced. The composite will again have its right-advanced. There's also another besides composing. So in some sense, multiplying is not commutative. Special case of objects, it is communicated, but for the more general, I call them actually distributional objects, because they have the effect of assigning to every point a distribution, in a sense, sort of like a gravity map. The next thing is you can add the objects, obviously, but the sum of two exponents in general should be what? As a functor, it's just a Cartesian product of two functors. So there is this commutative addition operation on the distribution of objects, which is just of such. So you have the problem of adding fractions. If the fractions have a common denominator, you can add two fractions. If you don't have, it becomes an even more subtle question. You don't have the quality of it. A times 1 over A. The reduction is just an adjunction, so there's a whole fraction on that, adding fractions of different denominators to the word.

1:02:30 So these are more general than simple fractions. So maybe the presumptive of this is exhaustive knowledge. Most likely, there are more. Some of it's destructible. Products of practical class. Sums or fractions of science. All these things have a meaning. It's true that our order of differential equations denominated the entire economy. You know this result? It's in my forthcoming paper, but already understood by Gonzalo and a couple of his sons. Before I'm given a map, well, I can have the notion that the solongation operates on an object x, which means I look at x to the power of this map, and I give myself this section, so this is the kind of structure of the object, so there's a category. All of these prolongation operators, all with respect to the given, and the example that I started with, you see, is the inclusion of first-order infinitesimal into second-order infinitesimal. This prolongation operator is precisely a second-order differential equation. It tells how to extend, you see, a tiny path to a slightly longer path. The second order eucalyptus consists of some sort of position velocity and acceleration, so it's an extension, so everything is smooth if you do give it an eucalyptus. Well, what about this category, alpha prolongation operators, or differential equations of order alpha?

1:05:00 If the domain of the given map is a tiny object, then this will again be a topos. Yes, this is fantastic. The category of all second-order differential equations is topology. The exponential agency on the truth-value object can have differential equations. It's all, I mean, it needs to be decomposed into components and products and actions. These are arbitrary equations, including infinite-dimensional ones. Wave equations, Hamilton's equations. The only hypothesis is that the domain of the map along which you prolong should be itself amazing. So in fact what happens then is that the category of these prolongation operators is co-monadic, lex-co-monadic over the original topology. But to define the co-monad... We basically start with the phasing right adjoint and then sort of iterate it. It's the key ingredient that is there for a long time. It's actually an essential, and it's not just a homonoid, but since it has an adjoint, you get an essential geometric morphism to this new morphosis of actions. It's sort of like, you see, it's sort of like the actions of a monoid, except the monoid is a Galilean homo. It isn't really a monoid. There's a mysterious link between the first and second order. Oh yeah, in other words, okay, if the given map happens to start from one, you have a pointed object. A prolongation is the same thing as an action. And of course the actions form a topos, because they're the actions of the free monoid generated by this pointed object. And it's also co-monadic, co-free. So the thing is that there's one, you have the inter-decimal object. These actions, these prolongations are sort of like actions. They're essentially homonautic.

1:07:30 It seems to be an amazing possibility. See, this is not science fiction. These are real-life speculative equations. Although, of course, on much more general spaces, as well as the ordinary garden way, it really is essentially all equations of physics. I'm sure. No, I mean, I'm kind of, kind of, I'm genuine. I'm deeply intrigued because, I mean, there's a possibility of that giving a very clean picture of the physical world, in some sense, what geometry and physics is about in terms of that view of the world, and the situations, the whole thing is... These prolongations in the case of the first and second order laws of motion, so these are the objects, the morphisms in the category of motion, at least in the special case with the domain of sort of times, because time itself goes on steadily, and so a morphism Thank you for your attention. I hope you get the energy to write this, if that is the sound of it. Yeah, I hope you do. The paper is coming out. The volume of the J.P. A. A. Manor is meant to kill you. Oh! Brilliant. I thought it was a good match of stuff in the old days of physics, but you see, there's a phenomenon which I call the algebra of time, which I spoke about a long time ago, and also at the Hamilton assessment in San Antonio.

1:10:00 In this category of laws of motion, there should be, corresponding to an open interval of time, The problem is, you see, that what we mean by time is usually in terms of... Let's say from T1 to T2, X could be called the configuration space, the state space, and X is the power space. So you move into another category... So you have an ordinary first-order equation, as it turns out, or a correct analog of that, going on down there. Now for those, you know, we know what does represent solutions, the actual interval of time and the infinitesimal shift. We've got the first order, so this is clear. But now it's also clear that to go back, we have to take the left agile. So the object in the category of laws itself is that the left adjoint of this associated state space applied to for a behavior in the first order of time and of course it's going to be, I don't know exactly what it is, but it's obviously something much more complex because as was pointed out I think by Gonzalo, I mean there is an obvious example of the law of motion.

1:12:30 Basically, this acceleration of zero, but then the lawful maps, the lawful motions of that sort of domain, in any case, are really only special ones that sort of follow, sort of pretend that the given law is actually an affine connection, some kind of maybe curve, but motion is uniform. There's only part of the state space of the configuration that's hit. So, no, it's something that's generated, in some sense, by the integral, by your integral, as it's non-zero, to sort of take account of... In other words, our abstract idea of time, we abstract it from all kinds of dynamic systems that are happening in the world anytime, anywhere. So, in that sense, it concentrates all these possibilities. And that's why... That's why the colonization, you know, fattened up the interval to register. I've lost my briefcase which I'm just going to search. It's still on the stairs, isn't it? It's still on the stairs, yes. Ah, okay. Underscored already? I'm pretty sure it is. Well, he wasn't just going to stay just for the morning, was he? Was he planning to go back there? I don't know. I thought he just came for his doctorate. Ah, well, I'd hoped to have a chance to chat with him. He said he's on vacation. Right. Is that Anas? I wondered if he'd left already. He's a great camper. He's going on his crazy bike trips. Yeah, he's biking every day. I remember when he came to Cambridge for our meeting there, he insisted on my providing a bike for him on condition before he agreed to attend.

1:15:00 With a list of all the cycle paths, you know, the routes he could take while he was there. Yes, now it's real dedication, because Cambridge isn't a very attractive place to cycle for a club. I think it was more cycling out of Cambridge and, you know, across the Fens that he was interested in. So I suppose you can get to, if you like, Lapp. Yes, yes. Actually, it's in many, many ways a far better place to cycle than many parts of England. I remember he took quite considerable pleasure in his cycling trips on that occasion. It hasn't changed since. No, either. But this idea of the generalization of time, staying in the differential setting is absolutely fascinating. Geometry is also a large part of geometry and also included in the picture because an athletic connection is really just a second part of the equation, especially when you're dealing with each other. They are a subject I'd come to if I was supposed to. I take a certain amount of, I mean, I should really like to do this, because I do a certain amount, I mean, I haven't been teaching category theory in Cambridge, but when I do, I do a good deal of, as it were, roughly speaking. Taking things of yours just to challenge the young. Here's a way that we're thinking about things such that. And there's some kind of war that goes on about whether I'm really interrupting the youth or something like this. A lot of these people go off and become geometers and none of them has ever come and said to me, you know, I really regret having done category. Yes, and tiny objects. I mean, it's just a wonderful kind of exercise in sort of basic techniques. It's always on the examples.

1:17:30 A lot of those cold talks about stuff I've never heard you talk about, about topos with zero and things like this. I think this is sort of, you must have given a series of talks once in which questions about the relationship between the two adjoins. Related to, I mean, you know, if you've got a category with a zero object, then you're clear. Speak of the devil. Yeah, right, okay. Those kinds of things. Can I ask a question? Here we go. I don't know if we can talk about this, but I kind of, I acquired a list of, as it were, facts of this kind. Those are very good. Just example, I get students just to do some first calculations. Just to compare left and right adjuvants, if we happen to have them. Now you know there's, I've heard me talk about this, but there's a sort of dual notion, a term notion, which is a pair of inclusions, a concomitant, and a fraction of all things that match up to each other. There is a dual notion where you have one inclusion and a left and right adjunct in front. For such a thing, you always have a crossover map that goes to the right as you went to the left. So you can take the image of that map and write something new. Now, what is that something? Well, I prefer to call it a model. I'm thinking of following an example of chain complexes. Sequences of Eulerian groups construed as chain complexes of zero. Now, the cycle has a sort of, I forget what it's called, but there's two adjuncts. And then the image, the homology. There's the usual homology. But you see, there are all sorts of non-linear examples of this phenomenon as well. There's a whole standard new table of instructions you can make as you try to take the average between the left and the right adjectives.

1:20:00 That's what he's doing. It's funny because many books on numerology tell you this is actually the best way to think about it, namely it's... Not cycles, not boundaries, but cycles, mod, whatever they are, co-cycles, whatever names they attach to it. Usually they could be dropped back to the boundaries as an image. I know a couple of books on logical algebra where something like that happens in the first couple of chapters. They say that Hilbert's algebra is the most symmetric in its own dual way, and should use this systematically. But actually, it definitely should be the same. So, that's a general category for this one. And you want to go and meditate for a minute? Yes, I need to. Good luck! Not something special to linear algebra. That's an interesting... I need to think about that a bit more because I've always been, I mean I've been worried about, I mean about, I think I can talk more. When I was first, you know, doing things at Hall, I think I probably had a conversation, it probably came from idea of Gavin Rees, but you should have things like the homology of theories and things like that in which you think, well, this thing could be presented in this free way and so, I mean, plenty, I mean. And on the other hand, it doesn't seem completely natural to take that thing and, so to speak, make it linear algebra. Exactly, yes, yes, yes, yes. And so, I don't, I don't, that's such a crude answer, it's not really interesting to do that. But what you're saying is just maybe there's a way to go back and rethink the whole of that. But the idea of homology is kind of measuring some essence, throwing away a lot that we... Well, here's an example. Linear algebra, but not usual homology. Take a subjective homomorphism of rings. So in other words, you have some ring, and you want to impose some additional equations. So that obviously gives you a subtext. Those modules are satisfied with further equations. It's always reflective and co-reflective of how many tests we do.

1:22:30 And then you have your classes, which will be taken in. So that when there's a sub-module where the equation is true and the quotient module where you force it to be true, there's a much smaller intermediate to sort of measure how serious this difference was. But do you know what I'm talking about? No, I don't believe it. And number theory in general. I don't think we have a lot to do with it exactly, but I still think that these are formulations that are coming out. Do you have another example of this? One reason why this is important is because there's some feeling that it would be good to do ordinary homological algebra, but not for a real group, but for, you know, instead of rings, wigs, in other words, so you don't have subtraction. So instead of a boundary operator, you have sort of a front and back operator. Right. But you could imagine, you know, sort of a boundary that's based on that without actually doing the subtraction. This comes up there. For example, I learned there is a theorem that bordism, which is a generalized theory, cannot possibly be given by a chain complex. There is no functor from stasis to chain complexes, so it's more or less bordism. But, in fact, there's a very precise positive thing about that. Any generalized homology theory, which does factor through chain calculations, In the sense that you have this coefficient, the sequence of groups which is the value at the point, you just take ordinary homology and make a convolution with that, that's the most general thing that can be derived from a chain complex of abelian groups. Which does give rise to more difficult questions. So, I thought, well now this is some complicated construction. At least one ingredient would be to see if these could have changed on classes.

1:25:00 The idea is, now here you have the subcategory, you have all these pairs of maps going on as you start talking. Special cases where they're equal. So you want to force them to be equal or take the part where they're equal. And again, I took the image. So I'm pretty sure that this known subtractive homology is exactly my image of these two graduates in that case. So the next step is the earlier one, after you get those spaces to those spots, you can understand the paper. But at least it's a very striking idea that you really get the geometric information. All sorts of issues that you thought were already settled when you were a student until you realized, oh no, it could really be quite different. Possibly it might even be better. It's the one that our current culture misses all the time. We go to these talks and somebody says, this is what, I mean, we're algebraic, this is what we do. The answer that was handed down from exactly in this room to all of us might be better. It's just coming up to half past, yeah, just coming up to half past one. Not quite, but twenty-eight past. When do we start? Two o'clock I think. It's clocking. That's why I can't meet him. He asked me a question. Gordon always asks me these questions in the form. Something like, you know, do you know if X is the case? And then X is something tremendously complicated.

1:27:30 He won't wait for me to think, you know, what does X actually mean in high terms. He says, you know, do you know or not? So it's best if I just say I don't. But then actually it turned out that he asked me something which, as it were, I trivially know. In fact, he can wait to answer us. Or perhaps I can catch him before he doesn't say anything unfortunate about me. But your construction of what you term this unity and identity of agent opposite levels, of course, shows up in, in fact, in more practical cases than this very interesting way, obviously, of looking at homology things. It also shows up in category of directed graphs, doesn't it? It's, you know, skeleta and co-skeleta. Well, that's the dual one. That's the dual, yeah. Two inclusions in one. But also with graphs, well, you can obviously... Because you've got so many adjoints in half of these places, you don't have too many of these things going on. But you see, you can have an aspect of graphs. This is an entire whole loop. Again, you can force that, or you can take the loop, and you can take the image, and you get a new one-shift. There are several loops that measure how far they've been different since the beginning of the movement. I mean, I'm a student who's writing about cyclical immunity that operates, and that's something which, well this is connected with this thing about what arities really are, I mean, exactly. From this very abstract perspective, arities are objects in some multiplicity, and the maps of the multiplicity are a way of putting things of such and such an arity together to make things of such and such an arity, and that works very neatly for a whole lot of things that are in operand theory, but the thing that you would sort of like to do is to have a completely clean way of

1:30:00 Explaining, you know, put this graph together with these cycles, so to speak, without just keeping them the same. Yeah, I make that equation. Genetically. That's interesting. Any given object can be made into a larger topos. This is clear from the point of view of the site. You take a site of definition which contains this object, and then if you appreciate topos, of course it's okay. The smallest topos still works. So this seems to be, at least in principle, advantageous because... Another sort of example of a prolongation operator is simply the whole idea of solving any kind of boundary value problem. Then, you know, the idea of, say, a harmonic function. There's a prolongation operator. I look forward to thinking about this paper.

1:32:30 I mean, all that question about prolongation and how many prolongations there are plays an absolutely fundamental role in string theorizing and what to do with the field theory and how to perform it and what sort of is to do with it when there's only going to be one sort of argument. The whole question of actually... And, I mean, in a lot of these cases, people talk about these things rather wonderfully. I went to a, there was a sort of meeting, meeting in Oxford for Graham Seagate's birthday, and Tom Savage came and he's a perfectly nice person, as it were, and there was a slight kind of flu in this grand person from Paris to give a talk about this thing, and he obviously is able to take. I mean to take abstract mathematics on a reasonably serious way. The only time I've talked to him he asked me whether I'd ever known your son, who I'd never heard of myself, and somebody in South Africa who'd been some fact about the book in completion on some category of mathematics. Something quite technical. And he gave a talk about, you know, what he believed sort of an abstract general one of those few things should be, and he was pretty abstract by those details, though, so to speak, the objects of the category were still sort of something like germs of manifolds and such on such a bound or something like this, but, I mean... It's good to talk with clear intuitions. And then, towards the end of the interview, you say, well, the problem is that it seems a wonderful structure, but it really isn't what it's like. On the other hand, maybe, as it were, in this sort of world, there are a wealth of much more abstract examples. I mean, maybe that if you're forced to make the damn things completely familiar manifolds or something like this, then you've got a difficult solution. Oh, that's definitely an extra complication. Maybe there's more to this than it can be.

1:35:00 Yes, I can't help but think of those are the subjects and those suggestions about enlarging the algebra of time. We really ought to have been talking about in the workshop at Imperial the other day, rather than this, well, we've already had this conversation with all this so-called non-community of topology. Well, the trouble with that meeting is that, I mean, I tried to say this, so to speak, to people there, but just from my experience. It's not enough to, as it were, simply to go in there and pick pieces of category theory off your shelf. You have really to learn the subject and see which bits of ideas and how you want to develop them for your own purposes. Which seems to me to be very much what's going on in some of the... although, yeah, this is quite the guy then, yes, very much. See, you don't kind of grab that, just take a little bit of... take one of the gadgets here and another from there and you just sort of cock up... And then see if you can do the whole thing over this graded algebra, because the physicists have got this idea that they don't want to have manifolds because of the diffeomorphism group buggering up the dynamics. But on the other hand, they haven't really asked themselves anything like what a manifold is in fact and really what differential geometry itself is about in terms of things. Yes, rather than say, let's just go through differential geometry, or let's try to generalize it in that completely abstract setting, simply in order to provide gadgets for solving these specific... Problems posed by the theories that we're looking for. It seems to me that there are just too many turnings in the way there where they haven't stopped in question deeply enough, which is what the motivation is for taking that particular direction rather than going back and thinking more deeply about the structure in the category of smooth spaces.

1:37:30 Thank you very much for your attention. Yes, I have to say you would have found rather a lot of science fiction going on in that workshop. Yes, I would have been very interested indeed if you had been there. Objects that we need to investigate. Little Darwin's way to investigate things that are going on in electrical engineering, but these are there as a full sub-category. It's not something weird. No, no, no, no. I would have been very interested if you'd been there to get your reaction to remember that moment when Chris Ayersham rather lost his... Thank you for your patience with Yanni. I mean, he's a research student. Stop using the word dynamics. This has nothing to do with dynamics. This is completely... All this elaborate borders and co-borders and theories to do with the way that you're supposed to be able to generalize differential geometry in such a way that it's no longer dependent on the notion of a manifold, all of this has gone very, very, very close to a machine. In fact, as I say, Chris Isle just really lost his patience. He said, listen, will you stop using the words dynamics? This has got nothing to do with time or motion. I would have been quite entrusted to... Turn it into a moment on this category which takes A to B to A to T events. Still, we'll have a chance. It's algebra, so it's the object of an algebra.

1:40:00 It's a set-in-depth family of algebras. So that's a kind of, that's a kind of, you know, a kind of, you know, a kind of, you know, a kind of, you know, a kind of, you know, a kind of, you know, a kind of, you know, a kind of, you know, a kind of, Fet together is seen to be categorised as the Asian variation in the world of differential geometry before they start to produce. Abstract substitutes for the manifold are the stuff we keep hearing about pre-geometry and free space. These seem to be almost entirely empty formal algebraic generalizations. I think they do believe that there is actually a structure in place in which they should be doing physics in place of the manifold, because for some reason, because of all this stuff that was supposed to happen at the Planck scale, the manifold structure will break down completely, so they're going to need some kind of incidence algebra instead on which they will try to define. The equivalent of the sheaves. But it's very, very, as you say, the problem is it's not subjective idealism, but it is science fiction. It's science fiction. Sure, I need to go anyway, actually. So do you have any more questions? There must be something. What is it to, if you project? There must be something. There must be something. You've got to, you've got to. You've got to join in. I'm just going to speak up for myself. It's an equation that's properly used at the estate.

1:42:30 So the algebra... So, as it were, you can obtain, you can obtain two, it must be a... No, it's okay, it's just I know I've got to get to the station to change my ticket for, I think it's before five o'clock. So presumably he'll finish about three o'clock. Oh yes, there should be a... Thank you for watching this video, I hope you enjoyed it, and I will see you in the next one. That's not a big deal, I'm just thinking that I didn't have one so I guess I'll have to go. I'm just finished now.