C Mulvey / conversations: S Awodey (& others)

0:00 But the last two steps in the interim of the two sets is just p-sets, p-objects. So, whether this is what Carmichael is all about or not, certainly if he would admit some group structure, there's no point. In variant subgroups, meaning invariant under all other ones, then you seek that argument. Oh, and of course, the G-sets, unlike other topologies, have the nice virtue that the exponentials are preserved exactly near those between them. Because, as I said, they're a kind of agreement in terms of cardinality. Well, there we go. I mean, that's, that's certainly true, that expression is used every time I go into a lecture, as it really admitted to the structure of the lecture, you know, when you go to check one of the more recent lectures. So this, if this could be explained, it's a very good illustration now, I could just put it in a little bit of posterity, and that's something that's still about to be revealed. I've never been to India, but I've been to many places, maybe a year or so ago this was described, even in the journal, news journals, it was a big deal.

2:30 Maja, and so on and so forth. Yeah, I mean, if this all works out, we'll first of all mention that, essentially, you know, classical ring theory, not the apparatus under-associated. Now, they use this non-style polynomial ring. The symmetry, the kind of symmetry.

20:00 That's good behavior by now. It is. Okay.

22:30 You do, do you? I need to pay. Oh, okay. Well, we'll have to go. It's via San Egidio. Ask them to show you on the map where it is. No, no, no. No, it's close. It's near to the Duomo. It's a different location from where we've been before. How did you get on to objective number theory? I don't know quite. I think it was Bill going up to Milan, and this was some of the things he was going to be doing, so I don't know if you're following it. Ah, gosh, everyone's tired. It was noticeable how breakfast was quite empty this morning. There has been an awful lot of hard thinking this week, but also I think one does have the sense, after you've given your talk, everybody has a sense of anticlimax, and Dan Brace is less relaxed, as we all know, but what, if you don't mind asking, what was this result? The final question is whether a number is prime can be computed in polynomial time. Oh, that's interesting. That's the kind of thing they would give very big bucks to, just because of the obvious implications. We don't have the lunch actually included today. I mean, what is the window for the last lecture? The last lecture, well, there's only one lecture this morning, otherwise Bill's lecture, then Alberto Peruzzi.

25:00 Then that's the official end of the lectures, and then the afternoon is just the panel discussion, which I think is quite... It's on the program, I have to... Yeah, it's on the program. The actual panel discussion starts, I think, at 2... 2.10? 2.30. 2.30, okay. Yes, the Penrose stuff starts at 2.30, but check that. Well, can you think about it here, Ray? Yeah, certainly in part, yeah. It's always been, you know, number one. Certainly for leather goods in Italy. And, you know, don't go sort of buying expensive, you know, Gucci, Pucci stuff. I mean, some of this is very cheap, but I'm probably... Actually, you have to be careful with the really cheap stuff, because it tends to... the dye tends to... I suppose a lot of it is automatically brought in to sell it here because of the reputation of it. Yes, that's of course. Yes, you'll probably find that half of this stuff is probably cheap ingots, but it's just made. The definition of a topological rupoid. Topological rupoid, yes, sir. Great, help me. It's not that hard, but what you can do is you take the usual definition of a rupoid... Well, remind... yeah. There's a category in which every arrow has an inverse. So every arrow in the category is invertible. There's an actual operation on the arrows, just like on a group, which takes every arrow to another arrow, which is its inverse. So when you propose the two either way, you get the identity back. So that's the category in which every arrow is an inverse. That's a groupoid, because a group is one of those things. With only one object. With only one object, yes. So now we just generalize. Right, right, right. That's a groupoid in sets. And now a topological groupoid is simply one of those things in the category of topological spaces. That means that the set of arrows, the set of objects, first of all, is a topological space that has a topology on the object. And then not on each individual object, rather on the set of objects. Right, yeah. The same for the arrows. Right. And then operations, domain and co-domain and so forth, are all continuous. Right. Continuous maps in the category space. Right.

27:30 Okay, that's helpful. So kind of what's different than a topological group? A topological group is something like... I'm sure it is, but unfortunately we just don't have time. I'm sorry. I'm sure that's been a good cause. The topological group, like, say, the group of matrices, convertible matrices with real value. That will be pathologized, just like the reals, using the entries of the matrix, and then the group multiplication is going to be continuous, kind of algebraic operations in those coefficients, in those entries. Well, that's a duomo, aren't we? Right. See that, and then it should be about two more blocks. Okay, great. Yeah, these guys seem to know exactly where they're going. Yeah, okay, I see that. What's the connection? You need to have lots of different objects, and then the operations still have to be continuous. So actually in the canonical example, where you have a space, think of it as all spread out with its points. Those are the objects of the point, and now the arrows... These are going to be paths from one point to another, continual paths, short space, and then you have to have some equivalence relation on those because, well, the composition of two paths, if you just put them end-to-end, that's what you want the composition to be, but then you have to re-parameterize in order to get the points in. You can think of that as the arrow. There still has to be some equivalent relation here, though, because if you have the points that go out, then the inverse would be the same one going back. And then the composite of those two has got to be the identity at that point, which is just the trivial part.

30:00 Yeah, which is completely trivial. The whole point about this is it allows you to generalize that, to deal with things which are... Well, this is the connection with this idea of points having internal symmetries. Okay, so I begin to see where the connection is with non-convective geometry, which is the context in which Cartier was talking about it yesterday. I don't know, I was having a problem with Atiyah's stuff, but... Well, that's a relief to me, isn't it? No, he's talking privately with André Joyal about this recent work of Krynish and Murdai. Well, essentially they're trying to define a notion of thermology for topological rules, so apparently there's always been a big problem with that, with getting the right notion. It's very crucial in the context of non-commentative geometry. Well, I know you keep working on mathematics and geometry. I mean, I'd say it's very crucial in getting... Yeah, well, that's pretty crazy. You've got some kind of seminar this year. Well, that's right. I saw that. That's right. I remember seeing that on the web. It's in Holland, isn't it? Yeah, that's right. No, it's in Utrecht, in fact. So did I hear correctly that you've moved to France? Yeah, you did. You've moved? Yeah. Yeah. And what's the reason for that? Is it just more comfortable? Financial and I've got more room in France. I'm living in a house where for the first time I've actually got room for all my books and for plenty of friends to come and stay. Oh, good. And that's another, you know, urgent invitation, please. It's in a beautiful little town called, I say little town, it's not that small, it's about 30,000 people, well maybe a little bit smaller than that, actually I think Fougere is probably about 20,000, called Fougere in Brittany, just the other side of the Brittany-Normandy border, it's very close to Normandy, it's not deep Brittany, it's not out on the coast, it's as inland as you can get in Brittany. It's about half an hour, maybe a little bit less, from Mont-Saint-Michel. Right, it's about 25 minutes from Mont-Saint-Michel, about 25 minutes from Rennes.

32:30 Yeah, about 25 minutes from Rennes, in between Rennes and Laval, so going towards Paris. Yeah, it's convenient then for the tunnel to go to London as well, isn't it? Well, not really, no, because you've got to get all the way to Paris to get to the tunnel, which is... Oh, you can't go up to the end of it there? No, the end of the tunnel is Calais, which is almost the other end of France, from where I am. Well, not the other end of France, but... No, no, you have to go to Paris. ...to get the train to go through the town. Going all the way to Dover, that's a five and a half hour drive. But the TGV is so fast, you can get from Rennes to Paris in under two hours, which is fantastic because it's well over 300 miles. And then from there you can get the train. But actually it's much cheaper to fly because right now there are these very cheap flights from Dinard, which is the regional airport, which again is only about half an hour from where I am. Now here we are. This is? Yeah. Okay. Oh, this is magnificent. Oh good, we found the toilet. This is magnificent. Oh yes, this is splendid. This is splendid in a jalapeno. Oh yes, this is very splendid. Maybe Marta is here. It's okay, will everybody except Marta go on? Oh, in that case, that's everybody. You knew John had come on early with Peter. No, but that's okay. That's fine. Well, everybody's accounted for now, that's great. This is lovely, isn't it? This was the only place...

35:00 Which is also known as the coherent project of B, though it will always be the classifying work of the T expression. Geometric work is the classifying matter as far as the T. So the arc of the T is supposed to be stronger than B. And so the positive fragment that is the part that doesn't solve the universal quantifier equation of the left-hand side of the equation.

37:30 The notion, however, that we respect this constraint are regulated by the basic adjoins which connects one notion to the various arguments of mathematics according to the group of categories.

1:20:00 Feature can be this way. The basic ingredients of this are that the foundation is provided with a list of units expressed in terms of joint functions. We consider rational processes of the course of objects rather than just point-like entries of static points. We express principles which govern this kind, not the other way around. We have a more intense distribution of both symbols and semantics. We have a correlation of logic of each kind. And we have a structural meaning. Finally, we have also a historically important and philosophically important master, because we recover the foundational importance of algebraic geometry, which in the et cetera foundations is just one of the various islands living in the universe. Not the computational importance which it assumes within the spectrum, in particular the synthetic differential geometry. So these are of primary importance for the foundation of Einstein.

1:22:30 The point I have just suggested is structural communication. Such an idea can be made precise. To what the notion of structure means, the theory is, but actually it is the most generous theory we have of structure and structural concepts. The structure of objects is investigated by Max from the characterization of objects is up to as you want. Objects are equally primal and very definitional when you define the state equations between functions, so in the sense category there is a special theory of function composition, math more generally, but it's not that the objects are primal because when you define g , you require that composition, but the meaning of that is actualism. There are many different types of mathematical structures, and I would like to talk about one of them, which is some that dispose structurally as a philosophy of mathematics, according to which that structure is just the role it plays in a large structure, so it cannot be expressed.

1:25:00 So, I mean, the structures, they are very useful. All of them share this view, the view that the structure within which the author... This hierarchy, growing hierarchy, requires for something which is not identifiable from within, but from outside, from explore and instruction. Now, in linguistics, this view which goes back to the first decades of 20th century, ...was a terrible deal in the sense that it opened this phenomenon, but it turned out to be not correspondent to the facts as the 20 years of research. When you say, I understand the phenomenon, you say that you really understand this phenomenon. You have to consider the fact that we are all partners. All the faculties in which you can embed these categories and so on and so on. Because any mathematical logic is just in the role it plays, the role it has when it is combined with recursive methods. This does not mean that, of course, structure is not important. Structure that is based in mathematics and in developmental sciences and there it is, is actually a clear form as is structure and quantum transformation and quantum invariance.

1:27:30 But there is a substance. I mean, there is not just forms flowing in the air, where anything we talk about is implicitly defined. They were implicitly defined. There has been some building blocks. We stayed and we practiced as they are. Of course, there is a dialectics here, because you start from grasping the notion of... And then we can use various metaphysics characteristics to characterize this whole system, not only in one and the same category, but also through variations of the characteristics of the structure by a real number of study students. So, not just a few or four themes of structures, but there is the possibility of eliminating This view does not require you to live within one and the same. Even if you consider the so-called three topos, the base of the topos generated by just the terminal, the angular number of the object, and the two value objects. And there is a topos which is a very important problem, it is an issue within the category of topos since there is exactly one logical mode of distance from any other topos.

1:30:00 You can say, of course, that this is, in a sense, an ideal, this is reconciled to the formalist, the blackness and... But this is not a mathematical economy, I don't know what you think about. The idea of topos theory, the idea is not to live in one topos. The idea is to have a range of topos to which we consider one notion, its relation, its control relation from one to the other, and the advanced principles to relate this different picture of what, say, one topos. So, it's a democratic, more democratic view of mathematical computations, which is also seen by the physics of diagrams, okay? Sometimes I've heard comments saying that they're finding equations about symmetry, because in a quantum we have one dimension. In diagrams, we have many more equations. You may reconstruct that, but that's exactly what this means. That's what you need to do in a quantum. The other point about symmetries, and your point about this, is that we have a fantastic importance of the toposkeletic approach to logic, because then what I'm saying is that we don't have a giant, which is a form of symmetry, and we do understand the relation.

1:32:30 I would like to refer again to Poincare, because again Hilbert, who was told everything at the time, Poincare yelled, this is an architecture, this is an architecture. Even the proof, I will say that I found it again in the first part of it. Why are you putting that truth there? And he said, I see it. It was really paramount to me. It's a robust notion of universal history and makes a fundamental distinction between universal and abstract. Training was certainly for abstract. Nowadays, we make an important analogy between modern and abstract. We don't believe in universalism. It's like the declaration of universal human rights, which are very different, right? So you chase that diagram chain, but it's the fact that in this you can take it categorically without you being a poet or a scientist.

1:35:00 So because there are universal abstractions behind uniformity is more important than the uniqueness of this. The notion of sublimity is more basically than just the existential, the existence of such and such that we know what people have derived from. Do you agree that we need 10 categories for the 21st century? Do you mean it's correct? That we need 10 categories for the 21st century? Possible. Zero. I guess you're going to approach all that knowledge as yesterday, but particularly I think it's really now that we're avoiding this psychological protection from the ocean and make a tirage of focuses to define. So in turn, we're choosing what's important in this chronological project.

1:37:30 ...infinity category, extremely appealing from a point of view. Anyway, he said it is, for example, that it could be... And the crucial thing is that if you have two levels which are comparable but really distinct, they're the same, this whole thing would collapse into one. If you have two levels which are distinct, then that is a touquet. And saying that there are only two levels is equivalent to the answer. So in other words... In some sense, the notion of two categories here is kind of wrong, but in this dialectical way, it accounts for all the phenomena. So this is a problem to try to accentuate the notion of weak infinity categories. We can interpret this basic idea, dialectically, we need two levels. And so, in a sense, it's the idea that if you could compare two levels of precision, Maxwell is screwing up and down the levels of precision involved. The basic philosophical principle is that any theory is partial, but then, having recognized that, one must talk about theories and deeper theories. So this, and that idea can be used as a, as a, as a, here's an absolute, very last, common term.

1:40:00 There are two kinds of, one which says, any thing has itself, a thing is good at making itself and a person have relationships that lead to some kind of superstructure. The opposite view, which you have attributed to Structuralism, is probably right, and probably other philosophies have, too, in Durham, namely that really a thing is not just only all its relations, but there's a very simple principle which refutes both of those, and a thing is itself and the thing, all of them. Actually, if you compare that to that, what we need is an abundance, or an abundance. I just wanted to make a maybe trivial point about terminology. It seems to me that when you were talking about different views of foundations, it seems to me that there's a fundamental distinction to be made between architectomics and foundations. And the analogy is with the difference between an architect and people that work directly for him and the sort of structural engineers that make sure the thing doesn't topple over. It sort of confuses things to call all of them tonics. Clearly, two different aspects of swinging. I mean, saturated might arguably provide foundations, but it's got no architectonics built into it at all.

1:42:30 In that sense, it's completely inadequate as an architectonic. It's exactly that. The idea of having a gray tree grounded on one root, so we have, say, a forest. Explanations one group. This seems to be an important guiding principle for research, but in the end prevents to have an adequate, different branches of points, fields, or coverings. The idea that there is only one universe, the argument I put in this, was not intended to say that set theory is the universe. Are you talking about foundations or architectonics?

1:45:00 Yes. Sorry, the first speaker there is from this channel. He's from London. Yes, he's from London. He's a philosopher of the economy. Yes, yes, he's a philosopher. He's a... well, he's... well, no, no. He's not the same kind of doctor. He actually has, he has written quite a number of papers on combinatorial logic and he does, he does mainly talk about science. He's at the Economatic Library.

1:47:30 That was very interesting. Well, he's OK. He's a common journalist. But even within that, he could have said a lot more. But even within that, he could have said a lot more. Here's perhaps one of two interesting cases. He's published a couple of papers on the continuum hypothesis, actually. And he's very interested in Weyl, Herman Weyl. Yeah, he's written some good...