Isar Stubbe — recent work on left & right adjoint module morphisms etc. (contd.)

0:00 In fact, we've been working on it for, let's say, two years. I think we started two years ago. I can relate to that. And so I think that, well, I can write anything. But also, not only writing, but also transforming it from the first proof to the best proof. Yes, yes. Which is also a big work, which is important. So we're finalizing, and in fact it's about what we think is the notion of a school local home, a non-communicable local home. So I might, you know, go on. Oh, I think so, I really do think so. Very, very good. And there's a couple of other things I'd like to ask you. So, as I say, this is one of the subjects which came up in discussion with Crane about what the natural non-local, sorry, non-local... So the idea is always that when I look at the locale and look at the trees on the locale, then basically I'm interested in the category of all the sets in the suburb, because that is an ordinary suburb, so somehow it seems more natural when you try to generalize over mountains or fields.

2:30 There you are, quanta or quantaloid. Sheaves over only possible and less than an extra ingredient. Mimicking symmetry and information. But ordered sheaves is sort of a primitive thing, generalized. Generalized. I said the definition that I take. Sheaves over a quantaloid was precisely Crane's candidate. I talk about ordered sheaves over a quantaloid. And there is an elementary definition. But let me give you the mode, because you know everything. So, what I take is I take specific categories. Well, I don't necessarily know them, you know. I think I know a lot less than you kind of generously attribute to me. I mean, you know what specific categories I have. So, when you take the pantaloid... Actually, just remind me again what the definition is. Well, here I just take... It has to do... Well, these are risk categories. Ah, yes. So, they are all here. So, I joined the idea. And, well, in fact, here you have the splitting, you have the... It's implying the locality. So if you take this as a definition, then what I have shown is that, well, you know, these are ordered sets. What do you know about ordered sets in a lot of stuff? But one important thing that you know about ordered sets is that some ordered sets are better than others, mainly when you have complete lattices. And you put supremo-preserving mortises between them because then you know that there is a pre-construction with an induced monad impact. And this guy is the algebra. So what is the monad, I mean the monad for the sets is that you take, well I could have written, the couple where you take down sets, down quotes, if you want to see it primitively as a subplasticity, the union has a subplasticity, right?

5:00 So, but, situation has a part of it, so what, it's mainly that both functions are, and obviously, injectable objects. Meaning, usually we regard soup as a part of the core, really expressed by this. But you can also regard form as a part of soup. You can say, okay, you know, you're talking to an alien who knows about lattices but not about sets or sets, and you want to explain to him what an orbit set is, then how should you say it? Well, you should say it's a replete image of a sphincter. But you can do it intrinsically. That's a theory. It's actually really rather natural to do it the other way around, because you're thinking in terms of... yeah, I think... So you might be interested in studying the image of a sphincter. And you can do this, and this is a theory for algebraic lapses. So, in fact, what you want to describe, instead of describing what you do here, I don't know, more algebra for this guy, is now describing quasi-algebra. That's what you want to do. You don't describe the quasi-algebra.

7:30 But I have every right to put these guys on the right. And these guys are still the algebras, and I remember more algebras. And the construction is essentially, so if I think in terms of me pushing it into categories, is essentially doing some pre-sheet construction. Essentially. Because, you know, of course when you take a category and you construct pre-sheets, you get a new category over the same day. So there's a trick that goes like this. You have these categories here. You have some pre-sheet constructions. You end up with co-complete categories, pre-co-complete ones, so that you write this for co-complete ones and co-continuous ones, like you would write soup for soup modules, but you're going over the wrong base somehow, right? You're going to modules over cubes. So what you can do is you can play tricks because you can show that this is in fact equivalent to the category of modules. Modules, I mean... So two frontiers from the opposite of the guy, the enriched module thing. So here is two frontiers from QOPS. And this you can show is equivalent to this guy here, basically because, you know, here you want to split idempotens, but all idempotens fit in the base category, so you can equate these things. And it's sort of, it's this composition, a very natural thing, that is the last one. So, you know, it's the power of circumfraction. On the categorical left is the down section of the question, but now it's the category, when you finish it, it's the down section. Don't wrap it up. He'll come back, he'll come back. He'll come back, okay. Because I want to make notes, too. Do put that back when you're done. I really wanted to make notes on this. I'm only going to retain this stuff if I can make notes.

10:00 You have this guy here, and so essentially this guy is the... Yeah, yeah. You know, here you want to compute the repeat image of this guy, and this is the theory of super-algebraic suplattices. Here you want to compute the repeat image of this guy here. What does that mean? What does it literally mean? It literally means that you want to... Can you possibly save it just for now? So, what you literally want to do, when you want to compute the repeat image of something that comes here, So basically what you want is you want to single out certain modules and certain module morphisms so that you obtain an equivalent category. So what's the answer? So what is the question mark? Well, so here's the definition. I think it's a new definition, but it should be old because it's intriguing to me. If I have a module, so a module like that, let's call it, let's say that an element principle, if something is true, so to express what is true, I must introduce a little bit of notation, so Bayonetta, you know that an element of a module is the same thing as a natural transformation into F. I've even got you on my t-shirt.

12:30 And it goes from a representative, right? Let's say that, in fact, I write A in F, then I should write A for some big A. That's what I'm going to do. Write this. Now, let's also write, again, Jan-Eda, suppose you have an idempotent in Q. It doesn't split in Q. Not necessarily. But it does split when you look at the representative. You can have an idempotent in here. The representation of a mathematical transformation, it must split. So let's try to split it. I just, I know it splits, I just tried to split it. And in fact, there are very many ways to write this guy down. One very easy way to define it is to say that on an object x, the more pieces in Q from x to k, the more pieces in Q from x to k, the more pieces in Q from x to k, Let's me write this as a sigma E and a pi D to this notation. Then when do I call the element locally principled? When this composite, so I call it locally principled, let's say, at E, to mean that I have an eigenpotency. Moreover, this guy is a left-hander. After sigma E is a left-hander. That's the difference. So essentially what I'm saying is that there is an open. What it does mean in this case is that you have a principal value. Yeah, so it's a definition. I mean this notion of locally-principal element is often a definition that involves actual contoriality.

15:00 So let me go further with this. Let me say that f is locally. What do I want to express? Well, I want to express that every element of f... It's somehow a supremum of, somehow, of these special ones, locally principled ones. You can write it in two ways. There is sort of the, well, they both have an advantage. I want to be able to write how x, how, where a is, yeah, yeah. So what do I want to, what am I writing? So I have a little x. I know it is the same thing. I have a little a. I'm writing square brackets universally for you. These are the square brackets, meaning it is the biggest natural transformation from one to the other, so that the composition is still smaller than this one. So it does make sense, if you compose for any a, so a is variable, but x is fixed, you compose this bracket a, you get something which indeed goes from q... The representative of X to F, the taking of the joint is something which is parallel to tau X, so it doesn't matter. Certainly it is small, but it might not be actually equal. So the equality is the condition that is required. But in general it's a parameter and a sign.

17:30 And the thing is, you can prove that it's equivalent, that it's actually equivalent, to the following statement. The identity on F is superior for all these components. I can write it as follows. I can write it, well, compose it with your algebra. So I write it like this, with all the thetas composed with, where I take, where I take in fact E when I compose, in Q, and I take thetas, the left I don't do that. It's precisely the same thing as this is the identity of every x. The fact that you require it for every x. And so, what is more, what is in the question? Oh, no, don't grab anything off here. I won't do it. You've only got 25 minutes to stop me. The question was, left-adjoint, between modules, which happens to be... Like the principle generator. It's left-adjoint, and you see, the left-adjoint thing is recurring. Because you see, he wants a sort of generation by left-adjoint from objects of which I have reported to you. And what do you understand?

20:00 It's a hugely algebraic characterisation thing. I mean, it doesn't depend on the... Exactly, so it is now. Precisely, as you can only... Okay, actually, this is extremely interesting because I'm trying to recall that the... Crane gave the definition of a quantum process, and it was, and then he had some way of doing it long-term. Certainly, it certainly turned on this business of the, of this case. Well, oh, certainly, I don't think so. As I say, you can listen to it online, you know. Oh, this is extreme. Yeah, well, don't drop anything off until you do it. Well, I don't think there's anything wrong with it, but I don't know which evidence to believe, because I watched it, and it's very important, and it's not very... I'm not an analyst, unfortunately.

22:30 I mean, it'd be worth making this a journal, I'm telling you. I don't know them very well. I do. I'd certainly be happy to send this to you. Yeah, I'm just, what's the, what's the, what's this? Oh, that's quite good, of course, yeah. It's done on Mars Tierney, on Galois theory. Well, as you say, we've heard a great deal about Galois theory. Yeah, well, the thing is that this theory, which rules for any quantum, is the best example. The general hypothesis, if you want to use a model, is also a key to analysis. But is there a thing here of some kind of prior order that makes key differences between them?

25:00 Well, you know, when people learn to do that, and it connects with these things in their past, and they're bearing over them, rather than in terms of the beta values of variables, I think it's kind of like that, isn't it? Then we've kind of, you know, formed out what we have in the general.

27:30 Well, there is this amazing memoir coming out. Well, it's not so much a memoir, it's a... The idea is that there should be two classifying rings for every theory. One classifying Thomas for each classifying ring. And that therefore all the structures that we have, every kind of structure we can have, have a course in what we ask.

30:00 Can we write some? I can't read the second letter. Also, at the end, the first with Justin and the last with me.

32:30 I never think this is LPG or some other kind of thing. Well, it will be attended, so they're looking for that. Yeah, it's all a question if Freddie can do the magic trick of getting some software out of the academy or anywhere else.

35:00 I mean, that's a deal.