One out of many / Andrei Rodin; Isomorphism, identity, category theory (contd.)

0:00 What is quantum state as input, quantum state as output of a piece of classical data? And then I assume this rule... Anybody do that? There are some physicists here... Failed! This is what is called von Neumann's Projection Poster. If you do a measurement twice, then you will get the same outcomes. So it means here I do two times the measurement, here I do once the measurement, and I copy the outcome. It's von Neumann's Projection Poster. If you repeat the measurement the second time, you will get the same thing as you got the first time, so you can as well just copy the outcome. Bob, there is a subtlety, because of course, you have to do the copying simultaneously to do the theoretical consistency. Do the copying simultaneously? I can tell you, I have never seen any physics book where they actually use the word copying. No, no, no. What you call copying, the execution of the second measurement? No, no, no, that's not what I call copying. In this case, the execution of the second measurement in order to be theoretically consistent should be executed immediately after the first. OK, you don't want, yeah, but that's implicitly the ruling, there is no dynamics here. There's no dynamics, no time, it's in the language. OK, but that should be explicit, yeah. I have an abstract way like complete this time, and if I want something then I would have put a box in it. This is like a very nice abstract way of writing the expressive one's projectible. And then it turns out, this is a theoretical result, that if I, in addition, adjoin this rule, which I'm not going to explain, and this rule, which I'm not going to explain, but they have very clear physical interpretation in terms of minimal requirements you would like measurement to satisfy, Then you can prove that basically, this is again from the Caddivay theoreticians, that you exactly get quantum mechanical measurements in terms of self-joint operators in the other space. This is the presentation here. You exactly get spectra projectors in this way. So, just putting down these three pictures, if you articulate this, let me help you with space and linear mats, exactly gives you quantum measurements, which in the textbook they call self-joint operators.

2:30 the same thing. So this is the presentation here. And so these two things are like the projection possibilities. It's like the best we can hope for in quantum mechanics, you know? There's some connection you want between the outcome of the measurement and the state which comes out. It's like, if you wouldn't have that, then basically there would be no meaning to the measurement whatsoever. You know the measurement changes the state, so at least what you want is some correlation to whatever comes out in your measurement. And this, similarly, this has a This is like the nicest one, and I'm just going to leave you in complete confusion to say that this means no faster than light communication. So I want to thank Baal for the original and didactic presentation. and I'm sure there are questions so one, two Bob, this is really lovely stuff as you already know but just one quick comment and then a very general question which I can address first of all you did say at the end that all other approaches to the presentation of quantum logic really had been based on starting from measurement I don't think that's entirely fair I think there's one very nice approach which John Bell presented back in the 80s was really based on the vector space representation of the predicate algebra and connecting it with the behavior of covering spaces. And he did very directly connect it with this business of failure of transitivity of entailment and the idea that one had arguments based on decaying premises connected it with linear logic. But that's a side. That's a historical side remark. I mean, that's never been a big development. No, I agree. It's not being completely worked out. They're probably because, like you mentioned, linear logic, so they're definitely closer to this stuff. I absolutely agree. I think what you're doing here is indeed working out those ideas and connecting them with linear logic and other approaches in a very nice way. But that was just a kind of brief historical side remark. What I wanted to ask was, it's a big question I know, this graphical calculus was obviously developed to present symmetric, monoidal and biminoidal tens of categories. And it clearly works very nicely with the notion of the Frobenius construction.

5:00 and it seems to relate very well to categories which provide a natural environment for the representation of relations but have you considered how to extend it, I mean have you ever contemplated extending it to a general approach to the representation of category theory in general because it doesn't seem to relate quite so well to the categories which are the natural environment for the representation of functions so there is a yes, so on your historical remark linear, like linear logic give too much credit for this stuff to Jean-Yves Girard, because he doesn't deserve it. Here, here. Because it can be really a pain in the effort. Here, here, but that's again a side remark. A lot of his ideas, they actually started in computational linguistics, in 1956 with Jim Longmaker. So, his linear structures, and the way of, like, the way, the compositional way of thinking here about pump mechanics is very, very close to the compositional way of thinking about sentence structure types types essentially sentence structure and i actually recently you know the paper which is specifically about signing meaning to words which is actually issues the same language and this this is taped together with professional computational linguistics with just parsers and stuff like that like the analysis of documents so this logic is not it's just such a general sort of thing which which seems to pop up everywhere like no theory no theory is based on this sort of structure and a lot of a lot of other stuff Yeah, so secondly, where does the category of functions live? This was already in Carboni-Walter's sphere. This is actually an out-result due to Tom Fox. So if you look at the category of chromonoids, of symmetric chromonoid category, then this is always five or more Cartesian categories. So this is out-result. So your functional world emerges. Right, so this brings out, this is exactly what I want to get. You're really thinking of categories which are the natural environment for representing relations as the fundamental ones, and the ones which are the natural environment for functions as derivative from those. So this is an old result which should apply to general symmetrical neural categories. In this sort of language which I gave, you can extract all these species. So basically, you can extract all the relations, because I started from quantum world, so not from relational world. Within that, you can define what the relations, what is the stochastic map, the total map is what you were after.

7:30 The spartial map is, again, a completely different category. Yeah. Permutation, which is groupoid. So if you extract this, so I really think that this language, and another important difference between this sort of language and the category of relations is that here you laugh for multiple days. So then you get nice structures reading there, like this sort of stuff. These are two mutually unbiased observables, which are nicely written down in terms of biological laws. So here I work. It's actually this slide, I don't know. Biological. right, but the point I want to get across is that you're thinking of, for instance, Cartesian closed categories as filling in a corner of this spectra, rather than as being yeah, okay, right yeah, okay thanks I'll go back to the beginning when you were talking about composition of one in capital states and you said it's a process something happens here something happens there but at the end you have no dynamics so the question I do want to ask I mean, I have a degree in physics, but most of the physics in the history of the world has been done since I got my degree, so you can't stay on that. But the question I want to ask is really about how it's involved in the Lake of Geneva, because the Lake of Geneva is big compared to the laboratory, but it's rather small compared to the universe. And so you talk about a phenomenological presentation. I mean, we know that happens. It happens around the Lake of Geneva, whether it happens anywhere else. Can you say something about the relative size of the Lake of Geneva in connection with this process that you've built that? It sounds as if it's temporal. So, yes, this is discrete structure. This discrete structure, like that continuous process completely abstracted away. Time is discrete, it's discrete. So it's computational, it's very computational, like sort of computational logic. You would like to express all of these continuous dynamics. in this direction. There is a recent paper by a smart guy from Imperial College, sort of described the quantum harmonic oscillator. And this is very close actually to, his name is Jamie Vickery, and he's very close to generating dynamics. Because the thing is this copy, so you know your basic analysis? So, Taylor series, Taylor series, what is Taylor series? I'm just Let's give them one example here.

10:00 This is zero copies, one copies, two copies, three copies. Actually, it turns out that the monoid structure allows you to copy structurally can generate exponential behavior in a sort of high-level logical... So this is something which people really in linear urology came up with. So they are like developing within the scheme of linear logic differential calculus. This is definitely not done yet, But that's why you expect to say things about dynamics. So I must say, so what we are trying to do with this is basically doing a lot of mileage within the field of quantum information, quantum computing. So we basically can prove new results as compared to what people in that area prove using these methods and in a much simpler way. So that's where we are doing mileage, both for the sake of my students that they all can get nice jobs after they finish their PhD. Because I was in the opposite. dreamer when I was young and I had a horrible career. So I want to make them, give them some robust, also to get them managed to pick them, by the way. So it's extremely computational theory, it's extremely well suited for computational applications. Now this is very closely related to what people like John Byers and his action are also thinking about. And they of course have cosmology and quantum gravity and studied mind, which is big. I don't dare to speculate Or from there, it's just like such a despegulation. So, I think, Professor, you... Yeah, sort of a follow-up question. You mentioned the Feynman diagrams, eh? So, I understand in your diagrammatical language, you can sort of prove a lot of general theorems. Feynman diagrams are, of course, used to calculate numbers. So, if I give you, say, the Hamiltonian of the helium atom, can you calculate its spectrum with these techniques or not? So that's why I say that there is one paper now of a guy with a quantum harmonic oscillator. So this still needs to be analyzed in operation because the way I present things is very operational. I didn't want to put in mathematical assumptions too much like there. Yes, he makes all the two mathematical assumptions I'm not happy with. The steps are taken in that direction, let's put it like that. Now, structurally, he's not far away from the final diagram, by the way, although the heuristics is very different. Structurally, he's not far away. Also, people who are working in topological quantum field theories, they're technically at the same edge, like John Morris.

12:30 The first part, which I call quantum structure, let's use the same sort of structures. Just goes one level, higher dimensional leap category, theoretically. So, it's very related. It's not completely understood yet. Crane, incidentally, in his paper, Categorical Phrymonology has got a construction very likely. Absolutely. And this also traces back to Penrose's pictures in the early There he didn't put it at a formal scheme because the category theory was not rich enough. The available category theory was not right to give formal semantics to Penrose's pictures. But it's the same thing, basically. Sure. What is the relation between your graphical calculus and the languages, the more, let's say, the less transparent or more complex languages, like yoga, space formalism, blah, blah, blah, that can be expressed by it? You mean like sort of moral theoretical presentation terms? So, a fragment, the only result I know in this context, because they are very hard is that a fragment of this language is called traced monoidal category. So this thing comes with a trace structure, which is very important in urban spaces, too. Whatever you can prove in vector spaces, which is expressible in the language of traced monoidal categories, it's provable in the graphical language of traced monoidal category. Now this is a slightly more sophisticated language, but that result seems to say that this will be quite similar. So we have a paper in which we proved computational universality, We have to add one little bit more structural ingredient, but we can simulate so that in common computing there are proofs that with a limited number of gauges can simulate any unitary evolution. So we have such a proof within this graphical language that we can simulate as pictures any unitary. We don't have algebraic computers, meaning if there is some result you can use proof in matrix. It's not a quality you can prove using matrix. We are not sure whether we can get it. This is something we are working on. So, I would like to thank the speaker again. So, as I said, there's a change in the program, so our next speaker is André Rodin.

15:00 And I think we'll continue in a technical way. And the title of this talk is Isomorphism Doesn't Replace Identity in Categories. Okay, there's more. The philosophical talk, there is two parts. We can first talk on category theory. The whole talk is related to category theory. And first, I relate to the general issue of structuralism. And second, on identity. Okay, so I just start with a little quotation from description given by Veeam and Karine from description of the workshop. And I think it represents rather general views, or I'm not trying to catch that, which I'm going to criticize at a certain point. So they say the debate on whole part relationships, stability and change of time and the like, all start implicitly or explicitly from the idea that objects precede properties of the life. So it's kind of substantial, which is here. Of course, two recent developments in mathematics and category theory, that's why I'm sure stressing this quotation, open up roads to a formal treatment of a more structural viewpoint. Okay? Now, two remarks, first, just informal remark not that recent that at least if if we think about structuralism i just this is kind of official reference i took this helen probably not yet published i took this from internet uh his paper in the encyclopedia what is mathematical structural which it it traces it back to hilbert and i think it's quite correct yeah so it's first my remark that it's not that That's probably recent, and of course, in terms of mathematical practice, say, it is in heydays, I think in the 60s, in Burbank, you go. So but more serious claim that I want to make is that as far as recent mathematical practice is concerned, and by recent, I mean categorical mathematics, not even necessarily category theory per se, but just mathematics using language of categories as kind of instead along with language of set, then this structural viewpoint of mathematical structuring is kind of outdated, so it can be probably, yeah, but I tried to say that the category theory suggests a new view on mathematics, which reduce neither to mathematical structuring nor to a form of sub-tionalism, which we've seen in quotation of Karinian Vim, like object-received properties and so forth,

17:30 but just suggest something something new in my view which i'll try just to trace a little bit probably uh explore weight and that seems to be rather general meaning like structural like structural we have in social sciences in a sense probably physics now people talk a lot about structural physics and i think that this categorical view also has at least this potential to be standard to mathematical science and probably even there is something that this piece is now interest on physicists and category theory it's not kind of all the philosophical but but it turns to be really effective as an instrument in research in quantum mechanics and so forth okay now I'll try to defend that I start with very obvious think about structuralism just to how say to make my prejudices explicit about that so what is structure and I think it's very general definition as far as I know it agrees with all varieties of different definitions can you make it a little bit bigger this this and now of course I cannot oh it's not well said yeah okay I'll talk about structure is something you think determines optopisomorphism if somebody done object that there are notions of structure that doesn't doesn't apply this definition. I would be really interested to know about that. As far as I can see, that's kind of basic. That structure is something up to isomorphism. And then why it's problematized identity, because all this kind of expression borrowed from mathematical problems developed in this book by Kimi Mathematics of the 60s, unique up to isomorphism, equal up to and so forth, suggest actually either kind of relative identity, which Jean-Yves Miseu yesterday mentioned, and earlier was suggested by Peter Gitch, or introduction of, in more, how say, old-fashioned films, after Freddy's unusual object, probably like structure or something, like over and above this isomorphic quantities of models. And here is probably smaller than example,

20:00 but if you just call it example, structural setting suggested in early Gilbert, actually Gilbert changed his view rather dramatically later. And then it was made extremely influential popular, particularly in United States by Vatlin and other postulate theorists. And the idea that we have this formal theory and a bunch of isomorphic models, right? And Hilbert's point was exactly that, in a sense, this model is somehow more fundamental than models, which could be kind of subjective matter, right, a way to think about objects. and and the problem which he didn't see from to start with that's interesting historical that it doesn't work indeed i mean with if we put reasonable requirement on what is our logic and i mean if it would be say just on the logic we have this categoristic problem Categoristic in the sense that it's not to do with category theory, right? Categoristic in classical web and sense as all models are isomorphic. So we should explain our model as something like intendant model. And then still if you think that it works, the setting, then we have this kind of controversy between, say, assumption is view that still we have we should count identity of model something basic and this think about on formal level in terms of isomorphism between models and we derive this notion isomorphic from uh some kind of basic notion of identity applied to models right or what was the whole thing of course somehow reverse that relation and think in a way make isomorphism somehow more fundamental than identity right or replace in a way replace identity isomorphism or do something about that so and that constitute i think more than that problem about identity in structuralism, but probably this workshop is very much motivated, but instead of trying to resolve that problem, which probably might not be resolvable, right, I just say that

22:30 actually the structural improvement doesn't work, doesn't meet needs or mathematical case, kind of a little bit pragmatic element. And the reason for that is that not only isomorphism but all morphisms matter okay so the structural view somehow distinguishes isomorphism from other morphisms and constitute all that all that notion of structure and so on and categorical approach as I'll show exactly amounts to saying that we just should take notion morphism general notion which is as primitive and and go on then we should distinguish of course isomorphism particular kind of barfism and the thing is that also it works better it works better in a sense now i just remind what is general barfism and here is standard explanation which already made in the structuralist group like you say all right and uh okay i don't like to say this kind of condition that i i just took example of group We have a kind of underlying set, you know, the jargon goes, and then structure upon something like operation on it. And then we have something like the star condition, which, which would mean actually kind of coherency condition, right? We have this star operation in one's end, this plus operation in the dollar's end, and we have this whole quarters and we have some map which map one set to another set and we need that group structure as this structural setting applies as you say to be preserved. I would like to say that terminology is very misleading because what do we need to be preserved here, right? You see, structure, group, structure is preserved, so it would be group homomorphism, not naturally, isomorphism, right? But actually just think about kind of forgetful morphism, meaning just homomorphism from whatever group to just trivial group, when they have just one unit, right? So it forgets the structure, right? It destroys the structure if you want. So, the claim is that the standard terminology reflects the fact that in the structural setting, a general morphism is conceived of after the special case of isomorphism, and from at this category point of view, more general point of view, it's not justified at all, it's just wrong, and kind of general philosophical argument for that, then I go to more precise mathematical argument, it just says that this kind of start to healing condition is nothing to do with preservation of science.

25:00 invariant or something. Actually, notions of symmetry and invariant are much related, but I just can't go to detail now. It's just not a right way to think about that. Cohedence is just much more general. From that, these things, they imply cohedence, but not at all that will be rough. And more specific mathematical means that actually set group to go through space and many other bits of so-called structure, Yeah, because we still know about that normally, S structure from universities, whatever. They can be fully captured in terms of their general morphism, meaning S category, but they cannot be captured in that way if we just restrict ourselves to isomorphism. It's absolutely, I would say, unrealistic, almost ridiculous. And actually, so the carolery, the concept of groups of logical space, et cetera, as structures, one first takes for granted a model, intendant, so-called that one, of set theory. And this set theoretic matter, you know, there is this kind of folk structuralist metaphysics that sets and met in structures kind of forms. I still didn't like metaphysics, which is folk metaphysics. It's not seriously found at all, but quite popular. So we still need this kind of matter because of categoristic problem, right? We just should pick up that problem which was stressed by Scholem in the very beginning of history of SET-3, but somehow people got around them, Fruits and Melech, Sematic, and so on. And so, at least in that sense, in Scholem's sense, say, the structural view just doesn't go through, even with SETs. And category theory really allows for getting rid of this remainder, And so it makes the structures dream true, and probably that's the reason why Karine, I don't know, other people just put category theory as kind of defense of structure a bit. But I think it's just one, because, however, category theory treats isomorphism with putting with Marxism differences, it wouldn't work with isomorphism only, right? And this is why categorical reconstruction of mathematical structures, by which I mean,

27:30 say category of group with with a proper axiomatic it just describes that category they are not structural themselves in that sense there is nothing they are not defined up to isomorphism for instructional reconstruction one consists of reconstructed concept up to isomorphism but categorical it would be rather up to general of my and this makes this difference later i'll show you it's not a good way to put it up to morphine, not at all, but I just use it to make this link. Just a small, can you give a little bit more explanation why you need the set theoretic matter in a way to... This model theoretic stuff, because just when you say you have this set and then you put the structure upon it, right? And if you think seriously about set, you would need to choose your model or set, right? And you know, well, it's not categorical, So we just pick up, and that's kind of a reminder, you know, which in that setting, for clear mathematical reason, can it be somehow dispensed with? So, what does it mean to think after General Martin? Okay, first answer, just think about our concept as a category, right? So without saying set is such and such piece in this keyword setting, we just say if it is a category, then we should describe it, of course. And that works for sets for many things. And what I find that these things are not structures. It's quite common to think about category structure. a little bit of all sorts of interpretation when I say categories are not structured. It's not mathematically precise claim, of course. It's a matter of all sorts of interpretation. But still, I think it's just rather, how to say, not a good way to think about categories as structures. But for a strong reason to think about categories something more general, in a sense, than structures, in any event, something different. Because when we just choose these categories, are more, most vocable one, right? But which are set theoretically speaking suspicious because particular of this set, the groups, the category of set, the group, and so on. So we are supposed all sets to be there, or large category, big category, groups, and so on.

30:00 They cannot have anything like isomorphic corpus because it's just absurd. All sets will be there, right? And anicopia will be just another category of set, but it's already there. It's not a kind of thing, it's not a structure in the sense of this very general definition which I've just given above, something up to isomorphism. They are not, okay? And so actually here, I can go into detail, we have kind of an alternative setting, alternative to this Hilbert-Tarty setting. We have a category of models, we have a theory as kind of generic model, it's Biloviro's metaphor. And we have a category of model, but what's important, this old requirement of categorism, it just doesn't make sense, and it cannot work. And we don't need it, on the other hand, right? We just need this category of model to be kind of a good one, and we may ask it Cartesian or whatever, what about limits, and so on. But it's just absurd to require that it has basically one object after Eisenberg. It just never happens except through trivial, non-interesting cases. So it's no longer an issue, you know? And that's something rather, rather different from structural view, right? That we need this categoricity for, because otherwise it just doesn't describe our mathematical reality. So setting changes. But then, of course, the question is still, what's going on with identity here, right? And so, okay, yeah, I have some what I'll try to explain. Second part now about identity in categories. And of course, it's related to the first part. Okay, now we're in general remark about mathematics in general, I say, ambiguity is about identity and mathematical objects like numbers, genetic or fingers, algebra, group, et cetera, traditional systematic. There are exactly five platonic solids, say the cube is one of them, but still in different sense of z. There are many cubes, three group of permutation of three elements as always isomorphic corpus actually much later to talk about classical object right that we have this coping possibility of us and that's how classical say mathematical objects behave right and what happens in categories i think just category theory makes the problem more acute you know because normally we just don't have problem okay

32:30 there is the cube, right, five platonic solid in one context, and in different context, we say, well, of course, there are many cubes, and so it's just people like Freddy who had considered that as something is a big problem, right, and non-mathematicians just say, okay, it's of course a defense of context, Plato gave a rather sophisticated theory allowing for copying mathematical objects, but they don't go for that, and in category theory, as far as we have something like all sets, you know, in one big class category whatever we just cannot just be so careless about that right we should say something about that so that that way somehow makes this problem more cute but i don't only suggest it makes people probably consider more seriously that kind of structural solution but as i already told i don't think that the category really supports this kind of solution that's just a quotation how people go around that problem it's it's from how it's called handbook on logic topos logic from Fulman strictly speaking the canonical isomorphism are necessary I don't know how much context exactly and having realized it is best in the interest of clarity to forget them so just people need to they can't just ignore the problem but they have life saying stuff like this just somehow say okay there is a problem yeah so so that that's why i think why again this some people think that category theory somehow support this idea that isomorphism is more basic than i into whatever but i don't think it's thought that that's actually correct yeah now what we what we do have in category actually we have kind of two different identity concepts, okay. One is kind of usual as everywhere in mathematics we use this equality sign and it's nothing specific for that equality, right. And where it is important first to define the composition of morphisms, right. We write composition with equality and actually a little bit more than that it's very important to to determine which morphisms are composable which are not, right, because we need not all, the operation of composition is only partial, saying it's traditional algebraic terms, right, not all morphisms are composable, the target of first should be the same thing as the source of the second, right.

35:00 Yeah, that's first thing which is usual, right? The category is just brought to category theory from mathematics, say. And the second thing, a little bit more specific, it's called identity morphism. It's just one action of category theory that we have this identity morphism, which behaves in that specific way that for every incoming morphism is just limited to the theories, and for every outgoing morphism is again limited to the theories. What's interesting is kind of contextual, right, so definition of which morphism is identity, which is not, just depends on, like, kind of neighboring morphism. So we can think about topological reasonings here. Also important to know that there might be different morphisms from given object A to A, other than identity, of course. They could be also idomorphism. Yeah, also important to remind that actually it can be, it immediately follows from the definition that it is unique identity and existence required by the axiom and actually that allows completely get rid of objects, right, or what people are doing with that. Normally they are doing, so just identity is objects, so yeah. But of course in that standard setting we just cannot put it as fundamental because It is somehow defined through this first identity, right? It wanted to fail, right? So officially speaking, it's not identity, right? It's not logical identity, it's kind of specific morphism and as real identity, we just have equality signs as everywhere. So, yeah, I just put because I probably need, notions, equality identity and identity morphism, we can define what is isomorphism in that context. And that just means that we have morphism which is reversible, which has a reverse, which gives identity to the left in one composition and identity to the right to the other composition. Again in that morphism it has no sense to say somehow isomorphism is fundamental because we means identity both actually our logical identity and that it might seem to define what is as

37:30 what we do yeah yeah okay but what we can try to do is what seems to be really interesting probably and something corresponding to what is going on in category theory it somehow replace this get treat with this identity, God-given identity, and replace for this identity morphism in a way. And kind of straightforward way to make it, just instead of writing F, J, equal H, just replace by new morphism, right? Just replace our God-given identity by identity morphism and but here we would have like morphism between morphism that that's as people call shape is using the operate based higher categories theory but simple thing is standard more standard for just instead of set of morphism between two given object think about category of morphism between two given Okay, probably I'll skip it, just standard way to introduce by the further categories from enriching. But what I would like to say in that context, just rather make sense instead of, say, having associativity to have associativity morphism, right? And of course, it somehow we can correspond to say you're replacing, I didn't buy, isomorphism. Except, here we don't necessarily have isomorphism, right? All natural examples, something like homotopy theory, suggest us over this structuralist reading. It's very simple example. We have topological space, we have points as objects, we have paths between points as morphisms, we have composition of paths, and we have continuous information between paths as our second order of our piece. And the thing is that they are not composed, associativity is not strict here, it's a usual thing, yeah? We always need this rape revitalization, which makes all things up to homotopy. That's all, up to homotopy. And that example suggests this kind of, again what Jean-Yves mentioned yesterday, kind of weak structure, right?

40:00 We have here weak notion of identity, but weak still reversible, right? Because he, and that kind of example structuralist view but in a little bit more general setting we have what's usually called lapsed categories right and that would mean that we have the same thing but morphism are no longer reversible and so we we don't at all have this structural vision on the whole thing and last thing i just give you kind of down to an example of this high category i don't know it's really important mathematical or not but just take a symmetry group except n12 which is trivial and six which is kind of uh weird exception i'm just saying permutation group of three arms right and just try to think that we don't have a correction like one permutation from what each other equals again third permutation right but just try to think we have a kind of okay set of permutations and try to define the whole group from the group of automorphism of this group which would be this course trivial case of this like two categories right yeah and finally do anybody know the answer what this group of automorphism through mathematicians here of symmetric group do you know the answer and it's written here but no and the funny answer is that And except these two cases, one trivial case we don't have non-trivial automorphism, in case two we don't have non-trivial automorphism, in six things like explodes, but in all other cases it's the same thing, it's the same thing. So we just have permutation of three groups, and then we have permutation of permutation, and that's the same group. It's a funny thing, right? And so, just imagine we have this kind of, how to say, pillar, I don't know, group, all group automorphism, et cetera, et cetera. It comes to the same thing. And so still identity and that's a unit of the group, yeah, would be a distinguished element of all this infinite restructure. And actually all this infinite restructure would come from the same group, so it's kind of redundant to think about infinite. It's in fact what people in high category call stabilization, a very trivial example, and at least it makes it reasonable to think, okay, we have this unit or group kind of fixed point on the whole thing, right?

42:30 So we probably can't really get rid of this identity and say, okay, this identity somehow emerges as a fixed point in that context, okay? I just leave that example at this and come to the conclusion. First, like I say, structuralism inherits kind of platonic biased over mathematics of eternal forms, where category theory rather supports a white canon view according to which special temporal intuitions are from the mantle of mathematics. Categorias are generally not structured, but something more general than structure, and the categorical view doesn't reduce to the structural one, and Eisenhower identity. A second thing is just a quotation from who says, identity is a relation given to us in such a specific form that it is unconsimable that a very respondent should occur. And I think that that attitude must be definitely changed. Just, I was impressed by recent book, French and Crowder, that they give an example of bosons and fermions, which they interpreted as having different identity. I think it was very And I think that identity needs to be reintroduced in medieval theoretical concept, but not redo the piece in advance. Although category theory in its existing form doesn't allow actually for this. For this, category theoretical methods in high categories may be useful for future theory. Thank you. We have time maybe for one question? well it's already one o'clock so you should have been there at a quarter I can still discuss enough with him if somebody else you had a question yes it's more a comment than a question Andre, well first of all thank you for a fantastically rich and concentrated talk which obviously brings together so many different ideas at such a proud level that it's impossible to do justice to it in a brief discussion just want to bring out Your final point, perhaps more your penultimate point, because I'm afraid that the business about the fixed-point construction slightly passed me by, but this point about homotopy constructions as being... Hormotopy constructions, homotopy type theory, as providing potentially a completely general foundational framework, which of course connects with what I take to be the broad conceptual stance

45:00 that you take, which is that one should see logical constructions underlying geometrical background that provides their meaning. This is already being developed considerably at the level of the treatment of two cells in an equational semantics for two categories. You mentioned, however, that when you go to higher categories, to end categories, you may be in a situation where you lose the reversibility of the maps. with a point which I know Lord Veer has often made, I'm not certain if he's ever made it in print, but he's certainly made it in my hearing on more than one occasion which is that when we consider the requirements on a general category of spaces, categories of space in general, when we look at the required construction that's needed for the map spaces, the restriction to the reversible case, the case of isomorphisms is clearly far too restrictive because one needs notions like continuous defamation of path in order to provide any kind of general notion of morphism for the map space in geometry, and Klein can be seen from the perspective which we now occupy, to have really imposed a very, very restrictive and rather distorting viewpoint on the philosophy of geometry by trying to base the classification of geometry on symmetry groups. That was much more a comment than a question, I know, but you might want to bounce it back. The first thing, reversibility I don't know.