Ordered face structures in N-category theory / exchanges G Reyes, A Kock & R Kostecki

0:00 These two terms have a very nice collaboration, so definitely Alex plays a role in the research experience, but then also he, on time, organized several things, one of which I can't remember even now, and was explaining to me why if you have a topos, you have a geometric morphing of the topos, you cannot say that it's the point of the topos. For each of the definitions, some of the journals are mainly for Svert and Wuhoff, and some of them being a nice feature on other playbooks and other things here. And several things like that I learned in Harvard at that time. Certainly, I knew this was a very nice start for me if I was working at that time. Okay, now back to the talk. I will talk about all the six structures and I will try to place it in the context, but already I know that the context is not entirely apt to see, but I am yet to say that there are important ingredients to this. All the six structures are sort of describing shapes of cells and then to the spacing diagram. The first one I would say was $5 of a topic set and then...

2:30 The Hermitian-Martin powers, they had these mathematics sets. Mathematics sets were based on operations. Mathematics sets were based on two-level, amalgamated, multi-categories. And then, technically, computers were before things, but they were maybe not so clear that they had anything to do with it. So, I put the names Street and Bassani because... And then there are three algebraic approaches, and there are two other combinatorial approaches, so now there is a third one, and then the fourth algebraic approach. So Thorsten Paul had his dendrotopic test, and then I will talk today about all the space factors, what they are. I'm not quite sure what are the connections. I know that, you know, I was able to connect order space structure to many-to-one puncture sets and say that they do this. I will explain exactly what is the connection, but three sheets of certain order space structure are exactly many-to-one puncture sets, okay? Then Mr. Tarnik and Mikhail wanted a fixed set and many-to-one puncture sets were the same. Well, I don't understand what this does and found in connecting multi-topic settings with them. I thought some understanding of physics is sort of equivalent to this data, because you said that that is a kind of good construction one way, but I don't know what to go back on with the question mark here. My student, Krzysztof Kapulski, here, is working on connecting these two approaches. This is a placement of my order of extraction. There are two other approaches which are not mentioned here.

5:00 These three approaches are like algebraic approaches which are software approaches. And there are also these two of Tostan and mine are combinatorial, so they refer from the very bottom, so they have a hardware approach, and from this hardware, low-level language is developed, and from there you can eventually convert to the software, but from the bottom. So this is how you relate my work. What are the talks? These things are not going to appear in the talk, but just for the lecture. To understand what such a thing is, is that you basically have a set of certain dimensions, so one dimension, so two dimensions, so you can keep on drawing as long as you want. Except that not all such diagrams are reasonable in the sense that you can produce one composition. For example, if this diagram is a good diagram, you can compose no matter what and how, you will end up always with the same answer. But for example, such a diagram like this, it's not good, but you completely cannot compose yourself, and that's the same for the composition also. For example, loop would be also not a good thing, because you don't know how many times you should compose the loop to get the composition. So there are certain... There are some restrictions which are pretty clear here, and then there are some combinatorial structures that describe what exactly this should be, and then these are like the simple omega graphs, and exactly described by Tarkinian, Planariou, Friedrichs, Orgel, with Hitcher on the technical dimension, and we can use these two categories of features, like the dual category of this and the category of this. So, now we want to, a little bit more, we want to make

7:30 As rich sheets as possible, but to get certain control over them. So the first idea is that, first idea is take everything, but it doesn't work. So let me say that, okay, in the domain of one cell, in the domain of the cell, we can allow more. But what does it mean more? Really more or really also less? If it can be less, it's many to one. If it must be more, it's positive to many, in effect. This codomain might be a loop, so... Change is very small, but it generates lots of things that I will address, but... The first thing I want to describe is exactly what combinatorial structure underlines this. This is one of the possible... In my talk, I will basically describe this structure, and you will say how it relates to... Don't use that, and then I will... This theory is parallel to that. I think it will be better to see exactly what you have here, and then we go as much as... A little bit maybe hairy to start with the definitions. So the structure which I described that, I named them positive space structure and order space structure.

10:00 I didn't mention the domain and order. It is because you need some kind of order as an additional structure to explain what came up. First I will describe positive and then order. So we start with positive at once. I will... What are the data to describe such things? You have to say what are the states on the finite domain of the first one, and then we have two functions, and one is called codomain, just for a cell, this function attaches a codomain, and the domain is treated also as a function that has a relation, that you relate an N cell with... All the cells in its domain. So, for example, if you have the cell, three cells A, then the domain of the cell, all is delta, right, because nothing is actually in its domain, right, if you think about that. And gamma is like C, at least, of course, let alone three parts of it. So the delta of A is just a set, this and that, and gamma is just one thing. We have two derived notions, and these are what they are. They are defined out of domain relations, and how they come about.

12:30 So, A, B, and A goes before B, jumps from the domain to the co-domain, from the domain to the co-domain, and of this one to the co-domain, and jumping to the upper, higher dimension, and then jump from A to B. The definition is precisely that whatever is in the domain is more in the upper order than what is in the subdomain. This is a one-step relation. And upper order is just the disclosure which generates the relation in a positive interaction model. And there is another way that you can compare these arrows. For example, from B you can go to C, right? But this is going through the lower dimension. Go from B, not up, but down to the co-domain. And this co-domain sits in the domain of another cell. And the co-domain of the cell sits in the domain of C. This kind of comparison is the lower order. The definition is here. The comparison is the co-domain of A with the domain of B. And the lower order, not the order of the axiom, but the lower order is, for example, These are just three technical notions and there are two derived notions and this I will tell you what is that. Could you explain what upper level is? Yes, because you can go from A to B, sort of upper.

15:00 I can explain that you are going from A to B by looking at the cells which are of higher dimension. So one dimension up. B is smaller than C, goes up. Then there are definitive directions, this and that. This is not a picture for one-dimensional, it has hundreds of hundreds of... I mean, at the end of the day, it's a partial strict order, right? And then you have to say that at the zero level, if all definitions on that slide are fundamentally complicated, but still, I mean, first of all, gamma, gamma of A, what is this? This is just the codomain here, and then the codomain is just another thing you can do, and then gamma.

17:30 And then you take just domains. There is a third particular subtraction, then like words are said. You see that if you subtract, you will get exactly the lower. So, and this is kind of fundamental relation which describes more what is important. And other reactions sort of just to make obvious things. The second one actually there is a, has a clear meaning which appears in many orders to be seen. And what it essentially means is that there are no loops and no even circles, which is sort of known as loop greenness in some approaches, and you cannot come back to the place after one journey. So this is fixed. The next axiom is saying that these two orders which we have, the relation is the relation of comparability. In this relation, you can compare it one way or another, but you cannot compare it in both ways, right?

20:00 Rather, it's just trying to put things that were coming and coming, several combinations, converging to something, and then I decided this is the nicest example you can put to you. What does this say? It says that if you have a set of... they might be comparable, or comparable, and... In fact, you have to have a linear order on this, given by this upper relation. And the same thing if you take six elements in the domain of all of them, they also should be comparable in this left relation. But these are all equal, so there's just one element in this. So these are the axioms, and that's all. From that on, I will just describe everything. For example, I mean, the good thing about that is that there is no retraction. So, these are comments, but I mostly made them for Hilde. And then what is next? I mean, I have to say what are the morphemes of the space structures, and they are very easy ones. They just have to preserve domains and colonies, and in this case it is rather clear what it means. And morphemes are rather poor, Hilde.

22:30 There are some technical consequences to show you how these axioms can be used, in particular I will introduce a little notion here that once you have a cell, the alpha, the kind of important cells go way down, it's called internal phases, so those phases which are... All these terms are derived from the domain, but you don't see them in co-domain. The definition is here, that you take the domain, well, go to the domain with alpha, and then intersect whatever you can get as a co-domain of such a cell with whatever you can get as a domain of such a cell. And the cell itself is like a kind of important object. They are in the domain of the co-domain, or the co-domain of the co-domain is the one cell, or they are in turn of nature. There are several such systems of going, because this one is kind of interesting. Delta is something that tends to be big, and gamma is just the one cell, and then this is much bigger than that one. If you want to see the co-dimension in any space... Alpha is controlled through gamma in one cell, and at the end, the last two times apply delta, and then you will get everything which gets applying delta.

25:00 This is a kind of method we could use for the second time. Uh, you've had 25, uh, you've got another question. Oh, okay. Maybe I will give you a little argument to show you how to use this reaction. Suppose that we have one cell in a dimension k greater than zero, and then we have two cells in the domain of it. And then the claim is that they cannot be comparable in the sense of upper order. It's a kind of reasonable thing that you think about it. What I draw is certainly comparable, but then I will put a preview from those axioms and show you how the axioms look. They are comparable. If they are comparable, there is sort of a chain of them. One dimension higher than those of A1 and A2, so that we go from A1 to A2 in the sense that A1 is in the domain of B1, the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and the co-domain of B1 is in the domain of B1, and We have D1 and alpha, witnessing that D1 is smaller in the lower order than alpha, so we have that D1 is smaller than alpha in the lower order, but A1 is in D1 here, but it's also in D2, in D1, but it's also in alpha, right, someone's saying that this and that, and in fact the linearity says that they should be comparable in the plus order, comparable in plus order.

27:30 And there are comparatively as well in the lower of that, by disjointness we get contradictions. I won't go through this, but there are, for example, we can, the main of this, and this is big people from the polarity condition and then, and much, much, much we have to do with that, so I'm going to try to take structures like, just as a shape of a cell in a... But if you have a shape of such a cell, there is a meaningful thing between what is the shape of the domain, what is the operation of such and such thing, which gives you that, and they are pretty easy, the domain of such a thing will be this, co-domain of such a thing will be that, that's the level one, and that's the level two, will be like this. In our homology, we have actually precise definition of what it is. I mean, not only just drawing it, but this is the definition. We count all cells at the, up to the level calculated domain, and on the level K we take almost everything except for those cells which are exactly there. Calculate co-domain is slightly more complicated because you have to subtract everything into the domain of some figure, but then...

30:00 On the co-dimension tool, subtract also some faces, because co-dimensional cells should be thrown away, and this is the way to say we throw away all internal cells of K plus 2, and that's how you have domain-intra-domain operation, and then you should be able to explain why you have a certain type and a certain other type, and they are supposed to be composed. What is the definition, what is the shape of the composite? I thought of a special push-out here. The situation is such that at the level of k, the shape of the domain degree, or the co-domain degree, the level of k is the domain. And so I can embed things of shape at the domain and at the co-domain correct. And then in such a situation, we have a push-out. But there are very few of them. In particular case, an example such as we showed is a cell here. We get something called monoidal globular category in the sense of a tiny. And since objects are rigid, it's in fact omega category object in what's true about them.

32:30 For any such a positive space structure, I can't produce omega categories, The sub-positive states form an omega category, and the N-cells of such an omega category are those positive state structures that have a smaller N-identity. The operation I described will tell you what are the composition, dominance, condominium, and all the activities of this site, which says that these state structures can be embedded into computers. And then we can keep on going and get into omega categories and get something which is called omega types. We'll be back into omega types, specifically on objects, but definitely not going back to space. The thing that I have here is that the concrete that can be expressed as from all the positive state structures it has, that preserve expression for better than that, produce a notion of a sign. This is a sequence of natural numbers, and the precise definition is like this, you take the cardinality of the set of those spaces which are not in the domain of any other space at a zero level, because at a zero level it would be always one.

35:00 All of these which are not in the co-domain are in the domain of time. So at the level 1 you have 3, at the level 2 you have 2, because this one is dead. So at the level 3 you have 1, and then it's 0. So actually it's a very good way of measuring things. In particular what is called polytops. I call them critical, and they are such that each level is... If we take a category of critical, we shift on that, it would be exactly the same. I probably won't have much time to say about many of the one case, so I will tell you in this case exactly what it is, how this story continues. There is certain monadic info, yes, indeed. Positive to one concept, monadic over, well, many are categorical monadic over positive to one concept, as well as positive. And here this picture can be sort of made a little bit more precise because we have this embedding of politicized factors and then there is an adjunction between politicized factors

37:30 But then, on the other side, we have that basically all these are the same, but all these are omega functions, but not necessarily preserving indeterminate. We consider functions from this category, which is much more, more than this exact omega category, which is only loaded. This embedding becomes just left-hand extension on the right-hand geometry, and the same on the left-hand geometry. This is not a trivial method, but... This requires very separate arguments, which takes a lot of time. Okay, so that's the story of politics and structures, and I think I will tell you what I want to tell you. What should I do to this entity-domain statement? I have to say that, you know, and then codomain functions, as it was, the domain function marks that.

40:00 A lower order is something that there is no way to make it a strict order. Now, I declare that I have another lower order, really, and it would be emergent, the other one, and sort of sub-actualist. Globularity condition becomes a little bit more hairy because it's essentially the same thing except for the three little modifications. One thing is that for some reason from one data here you have to... Then there are three other actions. Local discreteness is something that doesn't appear in the other situations because I was able to scroll through the things I wanted and actually this is the only thing that I proved to you that we have local discreteness in the domain that they are not comparable in them. All of these terms may be used to describe the

42:30 There is a sort of incident that they must be comparable one way or another and here it is not clear. Well, we cannot say that the plus comparables are one and a half percent or lower, but they must be one way comparable. The last condition is something that is kind of obvious thing, but you are not allowed to have such a loop. You can have a loop, but there should be something in it. The first one is the co-domain of something which is not a loop in the statement and you could think that you could say here that it's an anti-domain phase but in this loop it's not an anti-domain phase but in that one in the domain there is this and eventually you will get another moment and basically with this notion you can do whatever you could do with the other one except that... Things are a little bit more settled and there are two kinds of mortgages which are important for this recent lockdown and one of them we will define them now. But basically, whatever is true for positive space fractures, computer is positive too, and computer is true for other space fractures, which we expect from many point of view.

45:00 In the set of co-domains of something which is not a loop, if we have a loop, then we know that there is something inside, it might be a space like this, and in that space there might be a space like that, for this loop. People are going like this, sometimes, because the orders are strict, and the factors are finite, and then you think of a loop. Well, I mean, you have that, you have that book style, right? No, no, no, no. Nothing from nothing. You can't have a four-cell, nothing from nothing. There is nothing on this. And I mean that's all. This is the...

47:30 Well, there is a, yes, there are some relations. I mean, it's a kind of disjunction comparing both ways. Lower order must be sort of a, some kind of part of what is the lower and our final support. And then there is one condition here which is telling you that These two terms are at par. Additional data is exactly what comes first in YAML. The slower order is mainly to tell you which of the groups comes first. Which one will come first. You want to keep track of that. For example, if I say that I know that this comes first. For a part, it's consistent. In fact, if I would suggest... This order is created to look, in one point, at how we recover the right to make an arrangement.

50:00 Unfortunately, my good colleague, the historian of the Atiyah Institute, Ioannis Napout, my international student, didn't have a chance to hear about the category theory trade, so he made an arrangement for lunch. The point is to have some important assignments for lunch. If you write it in this way, if you go from here to here, Then I have trouble, I cannot do that anymore, because you see, I cannot write this composition in terms of the f of i, so these are different, right, these are very specific. If I take a projection here, so I will have f of i, and so I want to write this in terms of the f of i and the alpha j's, so how can I do that?

52:30 This. So it's F-Circle-Alpha. Yeah, F-Circle-Alpha in terms of different F of I. Yeah. You see, the most obvious idea would be something that would be something like F of Alpha, Alpha-1, and then the other would be F of 2, Alpha-2, and so on. TM here. So this is a question of unifactorization. So in other words, this is the question. So this is the question. In order to follow this and to transform this equation into a system of linear equations... By the way, I would start trying with W, the ring of dual numbers, I mean, the simplest value of algebra, and see what is this conditioning in these terms, so you really have a, well, just a deformation, a first order deformation of alpha and beta, the data for F. I think that for this it works, if I'm not mistaken. Oh, if it does, then I can... it will take me less than ten hours to work on it. Well, maybe even less than two hours to work it out. In general, if you can show me how it works for D. Yeah, but the trouble is... yeah. Well, let me take it back, because... I mean, certainly that should... It should be... Is it clear what it translates? If you, for instance, take the W just to the B, the problem must be a very clear problem for your calculus students, for the second year at least, I mean, the system of differential equations, as you suggested.

55:00 Have you made that explicit, what differential equation that amounts to? No, no. I really don't know whether you can reduce even a differential equation, right? If I could reduce it, then that's... I remember I once wrote a little essay which was never published. I think that infinitesimal deformations of complete vector fields are complete, and somehow by an infinitesimal deformation I mean one with parameter space d, not a general valence. And a complete vector field is one where you have a global solution at every initial value. And then I proved that for any such infinitesimal deformation... i.e. you change the vector field just as you do here by an infinitesimal object, then all these are also complete. That was not completely trivial and hinged on the fact that for linear clusters, frontier equations, you have global solutions by an explicit formula. Of course, here we have to translate everything into external terms, which is something you can ask the calculus students. So it's actually something like, you mean that the solution would be based on, like its lemma is encoded. I mean it's just assumed and and say that model satisfies it and you mean that this this call this this what is your average paper is something based on a similar way of of approaches say you say that we just assume what we the result we assume that on the classical level the result holds yes but that's not an assumption this we we know i went when we Consider models that are based on classical data. So whatever holds classically, at least constructively, when we reduce, I mean, what Gonzalo is doing now is to use the semantics to get reduced to an external classical problem.

57:30 Which is known. Okay, you see what happens is that if you take here alpha, it would be three, right? And then we can safely make m into one point. Right. So it's really just m point. Right. At least to start with. Right. And then this is f of alpha. First of all, I'm going to start right in general. So that we can understand what is going on, right? Now, this is pretty clear how to do it, okay? So this is the composition. Now, what I am proposing is to describe this composition in terms of this. Right, so these are the different, right. And so R to the K or whatever. And then, so then how are we going to describe, so this amounts to this question. Of course now there is no map from here. This diagram is not the diagram obtained by transpiring down from this one. So this, well maybe this is called it that way. We have this. But then, what is the meaning of this in terms of this? Well, let's see. Let me just scratch it carefully. M is a parameter space, and if we do it sufficiently constructively, things will automatically be smooth in parameters, so it's enough to do it for M a point, at least to zero.

1:00:00 So, what we really have... Now, alpha, beta, and F will take generalized elements, but now defined at stage LW, and if we simplify as much as possible, this will be stage D, L of general numbers. So, these three are just three global functions together with, for each of them, a deformation. Which is, again, by this r double, it's a couple of functions. So the data you start with is just the six tuple of functions, alpha, beta, f, and alpha 1, beta 2, alpha 1, alpha 2, alpha 1, alpha 2, alpha 2, alpha 1, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha 2, alpha So what is the meaning of the composition, right, as an also, we have to get it as a... Thank you for your attention and see you in the next lecture. New alphabets are coupled with a couple. Alpha, alpha, one, yes. And if... I think that I want to get rid of diagrams as fast as possible and put it in terms of the calculus, I mean not the generalized elements, but just global functions, so I mean the initial data here is that you want...

1:02:30 Oh, alpha and beta are the supposed solutions, right, but you see, I was just, first of all, to solve this question, how to express alpha, yes, so f of alpha, yes, yes, okay, yes, I'm just doing this one step. Okay, so if you want to, we are considering the differential equations, the differential equation f. Sorry, alpha dot equals f of alpha. Yeah, sorry, f of alpha. That's a different thing. Now it takes place with a couple of functions, because we are considering not global functions, but functions, not global elements, but deep parametric elements. That's right, yes. Alpha is, maybe I should write, alpha 0 plus epsilon alpha 1. Similarly, F, where alpha 0 and alpha 1 are just global, ordinary global. F is F0 plus epsilon F1. So what is... Thank you for your attention and see you in the next lecture. To write without thinking is sometimes the best thing to do. F plus epsilon, F plus, F0 plus F, F0 plus 1. Apply, so, F0 plus epsilon alpha 1 of T. Yes. Now everything, the only real... Let us now change this f0 of alpha, alpha 0 t plus epsilon alpha 1 t plus epsilon f1 alpha 0 t plus epsilon alpha 1 t, and this is f0 of alpha 0 t, that's what it says, plus f prime.

1:05:00 These are the number 05 and the number 06 of the column. Thank you for your attention and see you in the next lecture. Now I can write it with T. Plus epsilon times epsilon prime, composed with alpha zero, plus... Subtitles by the Amara.org community So, that seems to be the answer that we took. If zero comes close to alpha zero plus epsilon into epsilon prime alpha zero times alpha one. That looks like the standard chain rule. Well, plus F1 over alpha zero. Well, that's not quite... Stand up because you are making two variations, both on head and on both. But this seems to, you know, This seems to be the answer, in case the W is the simplest one. Right, so this is the first and this is the second function, right? Yeah, right. Okay, so in this case, then, it is, you see, alpha prime, it would be just alpha, it would be just a couple, right? Alpha zero, prime alpha, one prime. So then the equation, the first equation would be here, alpha 0, alpha 0 prime plus epsilon alpha 1 prime equals this.

1:07:30 Turn the page. And so then we have these two equations, right? Alpha 0 prime is equal to f of 0 times alpha 0 alpha 1. And alpha 1, the other one. It's not there. No, it's here. I'm sorry. I'm sorry. I'm sorry. I'm sorry. Alpha 0 prime is just one very simple thing. It's F0 composed of alpha 0, right? And then the other is alpha 1 prime. Yeah. It's equal to F0 prime. Yeah. And then the other is alpha 1 prime. Yeah. It's equal to F0 prime. Yeah. And then the other is alpha 1 prime. Yeah. It's equal to F0 prime. Yeah. And then the other is alpha 1 prime. Yeah. It's equal to F0 prime. Yeah. It's equal to F0 prime. Yeah. It's equal to F0 prime. Yeah. Okay, so this should be a system, differential equation, in two solutions, I mean in two variables. Yes, right, okay. System. Yes. Because this should be written in the form alpha 0 is equal to f of alpha 0 and alpha 1. Of course, then this is... Yeah. And alpha 1 prime is equal to f, and this is again in alpha 0 and alpha 1. Is that right? It's another geology. And so now we have ordinary differential equations and therefore we have uniqueness in this case.

1:10:00 That gives me faith in the project. So then this, if we put more things, you see, hopefully we will be clear. You mean reducing the array always to the case of... So that then this will reduce to a system, you see, of ordinary differential equations and they are... They are unique because everything now takes place into the real world, you know, in the so-called real world. So this gives... And, in fact, does reduce it to plain calculus. To plain calculus, that's right. So let me see what was G again. G was this, that thing, that's right. But if you would consider, for example, D2, yes, as a, I mean, the next step, this G would grow up here. This, I mean... Yes, let's see. No, that was going to be yours. We don't get into second order, you see, for some things. That's my trouble. No, because it doesn't matter whether we get second order. See, here's the G, and F appears with one prime. It doesn't matter. It doesn't matter because it's alpha that appears only one. Only one. Okay. However, if I have a G, it doesn't matter because several derivatives of F occur there, the data. In any case, this is the function. Yeah. Yeah. No, but the problem is the unknown function, so to speak. The unknown, tubular functions are the alphas. Yes, but you see the alphas in more general cases.

1:12:30 What do we have? Epsilon? Oh, different, different, different. It's not alpha prime. There will be alpha i's. Several alphas. Yes, but only one prime. Very good. So this would seem to show that the system will be bigger, but in any way it will be first order, but that would be very nice because if one realises these things as it seems, then that would be true in the career talk, as I have the conjecture. That would be... Well, for the other things I don't know. The complication arises that the uniqueness... Classically, it's not really a sort of global thing. Yes, but here it doesn't matter because it's enough for my purposes. It's not locally. Yeah, it's enough to have this local thing. I mean, it may reduce, but still, on the general level, it will be right. I think the level of generalization is what we need, yes, because we need just to have something that is local, yes, not infinitesimal. But anyway, the point is really not here an existence, but a uniqueness. So to the extent that alpha and beta both solve the equation. And so on, on some connected set, they agree there. That's right. So it's really not a question of existence. Right. So we don't... The geom problem doesn't really come up. It's wherever they are defined, on a connected set, and solve the differential equation. That's what we want. That's right. Because existence, for instance, can be the proof of the book. Thank you for your attention.

1:15:00 Deductions, where you don't really have to go too deep into models, but make internal deductions, is McLarty. Colin McLarty. For instance, he has some remarks about how to deduce... Thank you for your attention. I lost a lot of the work because of this parallel transport that I made from my office to and from. I will have to call Hannah and Andrea and see what times they are. Whether they want to join us. And probably Marie will be also with them. Oh, I hope they can come. Actually, just on a sort of administrative thing, I need to talk to you both about Colin, because I was speaking to him just before coming here. And he has been trying to organize, he's been trying to get the funding for this history of chief theory and category theory institute in Oberwolfach. For which now the best, it seems the best chance will be in March or February of 2009 and he wanted to know very much whether both of you would be... I'm not quite sure what format they're going to have because the problem is they've originally they were hoping to have it this summer in August and then they they got turned down for that. And they wanted to do something for the anniversary of the growth of Egon Esmeralda and they're doing a whole program for that with many different workshops but he wanted them to agree to have one.

1:17:30 It was very easy to go to Oberwolfach when I used to go to Morus and then we would go to Oberwolfach or Morus. Yeah, or vice versa. That was part of the basis of a lot of the summer activity in Oberwolfach. Every year there was a category meeting in Oberwolfach in the summer and I saw satellite events. Many category theorists went to Aarhus afterwards for a week or two because it's a nice place in the summer, unless it's raining. It's terrible. I miss terribly Aarhus. So does Marie. Anyway, just to plant it, just stick it in your diary. I can't give you the exact date, but February or March 2009. He would particularly like to get you there. Yes, but I would be very happy to see category theory reinstalled in all. We were kicked out, so to speak. I remember that because I said, look, we have something very important in topology. Would you be kind enough, you know, to let us this time, you know, to take over topology rather than that and then we'll return to the usual next time, to the usual schedule. Next time, they say, oh, but I think... Since you were not interested in having at that moment, therefore, you know, they have been eliminated because it's clear that, you know, you just gave this... No, it's a deeper reason. Really? Yeah, I mean, category theory was put down, but partly, I mean, it was also the quality of the meetings in the World War getting... There was really not science enough to have one week every year, or at least then they didn't get the right people.

1:20:00 That's what Panaghi said, which is just coming up now. I must get your email and all your coordinates and address and everything because there's loads of stuff I want to send you. I'll give you a little memo before I go of the stuff that you particularly like and I'll try to explain it as soon as possible. There's still lots of things I wanted to ask you about. And, um, torsion. Actually, you know, what's interesting... Is it true that Cartan had this... played with this idea of a unified field theory? Yes, yes, but... Well, I don't know anything about this, about the incorporation of quantum field theory, but I was speaking about Tartar. Actually, you know, there are several nice properties of this Einstein-Cartan theory which are very attractive. Actually, loop quantum gravity is really more close to Einstein-Cartan theory than to original Einsteinian theory. Right. Well, you've taken the question right out of my mouth, which is, is there any connection to the kind of motivation of loop quantum gravity? So, loop quantum gravity is very... Tell me. Actually, you know, in Einstein-Tutton theory, you do not assume that you have zero torsion, and this leads to a situation where you don't have a big bang, because the extra spin...

1:22:30 ...forbids you for collapsing to singularity. That's right, yes. And this one appears in... Oh, alright, so what is the result of this? So, when you introduce spin structures... You use the double covering group of Lorentz group in order to introduce speed. And the same story appears in quantum gravity, however, because quantum gravity tries to start from Hamiltonian approach, so it considers actually splitting, it considers two space, but at the end of the day it doesn't consider SO3, yes? But SV2, which is also a double comedy. So, and, well, the idea is, well, you know, this SV2, you can... We were performing calculations coming from all this spin stuff, and at the end of the day, you obtain the result that there is no Big Bang. So it was actually interesting, and it's no Big Bang that comes from quite, you know, hard and mysterious calculations. Personally, I do not believe in this model. This has been the covering of the original Rollins group. Actually, it's a special part of it. It is impossible to formulate in a singular way, so it seems more geometrical to some extent. On the other hand, it is also a little bit more general, because you cannot consider that something is equal to zero. It is always more general, so if you can assume something more general and obtain quite considerable results, this is always more valid. Loop quantum gravity is still a very, very specific model which has many... Sorry, did you say a very specific model? Yes, it's a very specific model and it has several assumptions which are very questionable from the physical perspective. And it's of course...

1:25:00 And these being? I mean... Oh, many. I mean, you can talk later about this. Okay, sure. Okay, good. Just give me a general one. But there are several things. However, of course, there is a big difference between loops and string theory, because string theory is just an indefinite way of approaching things, but this is not a coherent theory. ...based on questionable assumptions. It's better to have a model which you can question and you can deal with assumptions than something that is actually not a model. It's just a mental framework for making sentences. Well, yes, just basically a collection of recipes for... ...for likes, yes. ...however, you know... ...which they claim make contact with. Well, let's not go on about string theory. Let's not go on about string theory in any way. I have to say, I've just again suspected some of the most recent work on M-theory that you have with these kind of... I think that nobody understands it really. I mean, people ask us many times about people speaking about it, but at the end of the day, all of them just know me somehow, which is stressful for me, but thanks, but, yeah, it's nice to be here today, and it's nice to talk to you, and it's nice to be here today, and it's nice to be here today, and it's nice to be here today. There are two papers. One paper is called Deconstructing String Theory and it's written by Bert Schroer. Bert Schroer is one of the persons who, beginning from the 60s, is involved in algebraical quantum theory. And he really understands what is the big difference between... Something what is nowadays called, you know, quantum field theory, almost everything is quantum field theory, and just, you know, inverse functors to vector spaces. Really, you know, there is still not a physical theory. It's just a name, so... But actually, he...

1:27:30 Yeah, but in any case, you know, he knows what are the real contents of real quantum theory, yes? And from this perspective, he shows what are, you know... Those papers are one thing that's not... This is Shura. That's Shura. That's Shura. That's Shura. I heard the name, but I'm not quite sure what it's called. I don't know what it's called. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. Oh, Schroer, sorry, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer, Schroer There are very good, because, I mean, he, it appears in quite the same time as Smollett's book, but probably before it, or somewhere in between, and, but you know, it's something, perspective of person who spent, you know... The kids are really thinking about the... Yes, really working in the field, which is, yes, yes, and trying to make things precise, yes. So this is quite good, because Smolin rather, you know, Smolin is a kind of person who... Smolin can be very hasty, to say. Smalling skates on thin ice a lot of the time, automatically, and, you know, I'm not saying he's not a great guy, he is, and he's obviously very good, but the field, just in terms of having raised his profile, and just having lots of right-wing young people interested, and he's a very expositor, but he has become a bit of a... I say this in earlier, in a kind of light-hearted way, he has become a bit of a mediatard, and he is... And he is inclined, as I say, to point out in his papers, I know, in the last few years, that he does seem sometimes to rather skate on thin ice than do mathematics. And considering, you know, how much of his own rhetoric is directed against the Hathaway string theorists, this is part of his schedule.

1:30:00 And one of the reasons, of course, that's hard is the fact that... I'd like to say that gravity is not worth starting with, which is really stupid, but some of the string theory is very, very, very powerful. I'd like to understand more about... I mean, I read the original papers by Hashtag, I can't potentially follow them as closely as I would like to. I think it would like to get something what is called a technical but more concrete, still accessible, and on the other hand... There are two papers. One paper is by Hermann Nicolai and some of his probably students and collaborators. It's called Global Quantum Gravity and Outside View. And actually this is a critic of global quantum gravity, but critic of mathematical, critic of physics, very good. And later, the year later appeared... The answer of Tiemann, maybe I will write that down as well. Who's the guy who wrote the critical survey paper? Nikolai? Thank you very much for your time, and I hope to see you again soon. No? We're not going yet? Okay, it just seems there seems to be a general sort of movement of everybody in that direction. Of course, we have to get going, but Anders and Porzem are going to bed with them. Oh, I see. Ah, that's where they're going. Okay, right. And others? Others are coming. But if we just want to walk to lunch, we should just stick around. Well, actually, I thought also we could go to the hotel first, do you think? Oh, no, I was just going to go to the hotel first. In any case, those two papers should be, you know...

1:32:30 Right, okay, that's very interesting. I read a couple of quite good survey papers on... There are a number of different approaches to QG, but they were written, well, Jeremy Butterfield actually wrote a paper, but it's very, obviously, pure expository. Tell me a little bit more about the connection with the covering space of the Lorentz group and the connection with the Cartan, Einstein Cartan. It obviously connects with the way that the singularities get treated, singularity of homologous treated with ordinaryity. When we walk to lunch, are we walking from here or are we going from the first floor? From here, but I hope mechanics don't stick around. I'm not sure that I'm going to eat now because I'm overeating. There's Gonzalo and Anderson. They've got a car. You can obviously share it with them. I'll just pass. So in that case, you're going to stay here? I'm going to stay here, I think, and then go. So obviously continue the conversation this afternoon. Great, once again, thanks. Thanks for the teaching.