Pre-session exchanges FW Lawvere / morning session

5:00 I wasn't concerned about Olivia because I wasn't shocked. Probably, she must have been a student, yes. Yes, she must have been one of the economists. I think everybody's here. Oh, you talked to Richard. I talked to him before going away. He said that after we got back we could do the class thing. Well, yes, that was what I was doing. No, no, no, I understand exactly why you were thinking about the reading. I started down in 3 and 2. When you were at Yarns and stuff. I think that's right. I know Anders is in number 73, he's 72, but I don't know which number to leave you with. Because I'm going to leave him with the number that's wrong. The thing is, I'm not sure if I want to go back. Oh, no. I have to go back. Oh, awesome, I believe. Was this done from Cambridge to London or from London to Cambridge to London? From Cambridge to London. No, I was, you see, I was first, then I did the meeting with, the afternoon I was born, and I was at the end of 111, so it was 1115. It isn't really associated with meetings, but it kind of implies it's the month of November.

7:30 And then all the destruction. In the course of the course of the course of the course of the course of There is the mother thing in the theory, and the human beings inside themselves, which are very soft. I'm afraid people never think about that. They think they're so wrapped up in their own mystery. It does tend to be a bit of both. Oh, you've got to crawl down on the old... Oh, you did? Wow. If I'd been there to take an exam you would have done it anyway. That's not quite so easy. Yeah, I'm really sorry because I had gone out to the courthouse about making, you know, transferring these recordings and I hadn't realised... So you think we've got everything straight now about the things that we were discussing with you on the first day in the context of...

10:00 If it does appear to be... I remember you telling me that... I mean, there's so much coming up in that discussion. I hate very much that you run it on for most of us. He seemed to need some convincing that the conferment had got one point. Didn't he think that it was going to require some... Oh, yeah, there is a real problem with it. Mainly there's the mere fact that the G itself is a light-missile object, not only with self, but with self-heating power. Yes, yes. So in other words, let's just say the terminal in the homogeneous pattern, the object of math for many things in the homogeneous order is one, which in itself is at the homogeneous order of one. Yes, that's it. But it's clear enough that maybe it's not the object, it's more of an experimental one. The fact that it's an experimental operation takes care of it. And so on and so forth.

12:30 The moderator is definitely not in the interval, but that doesn't matter. You can use it for different wordings. That's related. Is that related to the ation monide as any direct product? No, no. That's the most special one. This is just a cartesian-type category, and the rest of the subcategories are reflected and preserved by new products. The definitions connected with evolution are by using the evolution that can also be contractible. So it's next year you can have a moonlight outfit with space moreover equipped with a mass from the square into itself. It doesn't really have to be associative, it just has to have a unit, and also be pointed towards the one side of zero, so then you can do things like that, but not only this, but any stage in which you act in such a way that zero is connected, hence the fact that then you have... If you act on A, you also act on A to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to the B to But then there's another realm of distorted theory that says that the point is to say that it suffices to have x to the x, that is, the self x.

15:00 The idea is universal quantification, but it turns out that if only the self x... x to the x is a concrete example of a lone wagon when we just argue that it flies. Now that's very, you should write that down, because it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, it's a very, So just saying that D is connected does not seem to imply that. No, not at all. So that's the whole struggle is to figure out the simplest way to intend it, even D. Yes, that's as understood as what your discussion was about that session the other day. Whereupon R will also be connected, it's a retract. Yeah, it's a retract because it's a new thing. I mean, it has really to do with... And then the equalizer is coming from forcing it to be common. Thank you for your attention.

17:30 In fact, there are many, many, many, many, many, many, many, many, many, many, many, Yeah, yeah, that's what I'm saying. So, you're saying that you didn't bother to observe it a little bit, but it turned out to be so interesting. Yeah, it does seem to be interesting. Yeah, it does, it does. It's absolutely fascinating. But I'm happy to say that you've got so many of those questions. But anyway, there was somehow a belief, you see, that these groups are actually the right ones. Yeah, yeah, yeah. You know, they're the ones who exist. Yes, exactly. In fact, originally, algebra is to be grouped in one way, for a simple case, in one way to understand it. It's the construction of the addition. But this spreads out very strongly in multiple cases, in one case it's an addition. Indeed, as you pointed out, in several occasions. In several, it's a fairly different, truly different concept. But in this case, it's a relationship of additions that come out naturally.

20:00 There are still two birds with one stone, and one way or the other they are, and the fact that there's a retraction given a straight line, puts it over to the identity of the militant, since the leader of the militant is known to have these derivatives, and for that retraction, there's a monarchy, and all these things are subtle, because the general differentiation satisfies the same rule, which is not a straightforward multiplication. Because it's so simple here, it's literally a whole more thing. But those two properties came out as one statement for the map that's there anyway as an inverse. Yes, yes, and turns out to be, let's say, very strongly unified. Yeah, so those two things. I mean, who cares? I mean, the problem is that you're in some people's reading field and you didn't get to something so far. That's the way to do a lot of things. Enough and unifying. Can't really make it. Some of the principles are more powerful than others. As we've just made this major discovery. Well, it is a discovery. Well, it is wrong. One thing which the philosophers, I think,

22:30 I don't know if it's exactly anything worth it, is the element of R. So that's the meaning, that's the real definite ratio. So that's the ratio of lambda, and it's one over one minus lambda times... You can work it out behind the level. Times the... we'll tell you the time the chord gets written. They're changing it. Yeah, there's a good way to take credit for it. Yeah, yeah. Sometimes there isn't even an addition there, right? There's a symbol one over one minus lambda. There's another ratio. Yes, the eraser, the eraser. So it is, again, on the line of the point of some sort, because it's directly related to something else. Indeed, that's the very point which was overlooked in the original conundrum, which was based on the idea that if you wanted to summarise conservation as a slogan, where it was that Zeno made the mistake, it was complicated.

25:00 Well, this is something which, this is where, as I said, this is where the very important technical on the program that connects with, you know, quite distinct from the nice combination. I do hope that you feel that I have been making a very

27:30 Curve my garrulousness, of which I am only too painfully aware to make a fault, and to allow you the space to develop the mathematical tasks that you carry with you. I do understand the necessity, but I really do apologize privately, so when I have to prepare something, I'm really very slow thinker in a way. Thank you for your attention. I was hoping we were going to hear a lot more about that. Well, I can think of a discussion about it even if it's wrong. By the way, I'm a completely packet of things. Do you understand that I don't know, I don't understand, how you get, how you don't like this?

30:00 I don't have an answer, but I can't, I don't know what to get my in there. It was online, yes, it was online. I'm so familiar with PCs, the crossbar on a PC, but it's quite different than that. I've clicked all the obvious things, but not... Would I just tap my own email address on there? No, not there. Ah, okay, we're going in the direction. This is the shortest way. You go along there, not the very next building, but straight down the street, Meridian Close, that takes you into Queen's Street, Queen's Avenue, and then if we go that way, we come along Queen's Road, which is slightly along there.

40:00 Oh no, in fact, okay, Queen's Avenue is what I meant. If you go that way, you go the way you feel. Many of the victims appear to have been students because the main hall of residence of...

42:30 The university seems to have collapsed. A lot of people have been buried under the rubble. Ah, you're going to go down that way? Okay, well the shop I need to go to is actually up there, so why don't I just join you at the department? Okay, I'll catch up with you. Okay, cheers. No, that's fine. You're quite right, this is the quickest way. I'll catch up with you, don't worry. I hear Bill now, so we can discuss this now.

45:00 Oh, maybe I misheard. Walking here together, but I had to take a detour to go the long way around because I needed to go to the shops to get some more batteries and things. So, in that case, please carry on. I didn't mean to interrupt you, it's just that I thought they were just outside. You were saying about the things that the the the the the classical picture the dependence on yeah yeah oh here he is yes perfect timing hello olivia hi giving your phone smooth man suicide on the tracks awful and also he was saying he came as the day before We had a speculation when I was talking to Azendi and Vickers about this during Aysham mess and I thought very much the need of your presence. We speculated that in fact what you were doing was what you'd indicated me at lunchtime you might be doing which was meeting this guy to sit down over Wikipedia and to go over all the things that needed correcting. We had visions that when we logged on in the morning, all of the topos theory articles in Wikipedia would have been straightened out completely.

47:30 I was immediately in an area where there were all these street names, none of which were on the map. I made a couple of turns and it took at least an hour or maybe two hours. I wound up at the Cavendish Laboratory. Which is a very long way further west. You must have walked at least two miles to get there. And the observatory, you said, also. Yeah, sure, but right out of Maddingley. As Mr. Rumsfeld says, stuff happens. The discussions afterwards, as I say, would have benefited a great deal from your input. I kept a good set of notes, I guess we could. Well, I think Steve Vickers is trying his best not to do that. He's trying his best, but... Alex was just saying, obviously, since he's leaving today, I mean, he can't come in tomorrow, apparently. I have to do that on the walk, so I think I'd better come at all tomorrow. So this is the last day we'll have Anders. So the question was what did you and Anders regard as the priority, given that he's going to be here one more day?

50:00 Well, I regard it as the priority that he should talk as much as possible. I agree. I haven't seen him yet. I'm skimming through it. This is a first-hand report from Aquila itself. It's five pages of reports. The entire village almost leveled, but which village? The village of Ona, five miles outside Aquila. Ona, O-N-N-A, whose church was destroyed. The village was almost completely leveled by the earthquake, with at least 25 people killed from a population of 300. That seems, that's the only village of which I've so far seen the name, but there are five or six pages of coverage. No, I imagine they probably have been completely preoccupied with this.

52:30 According to this coverage, which obviously is very hastily put together, there's a controversy going on between the seismologists and at least some professors of geophysics who claim that it is now possible within a certain There's a range of statistical probability to predict earthquakes like this. They're saying that in the previous three years, sorry, the previous three days before the earthquake, they had seen a very large increase of radon in the rocks, and that this is now recognized as a classic indicator of the buildup of the stresses that generated earthquakes. And that they issued a warning, but because they're not seismologists, it wasn't. Apparently the seismologists don't take this particular predictive as seriously as they're saying they should and this is the, at least that's the gist of the report that I'm looking at. Could we possibly use it to ring Richard? He might have, yes that's very kind of you, I'm sorry to, yes, because he might, because he's the one person whose number Davide had and would have been his point of contact, so I don't think.

55:00 Oh, you have? Oh, even better. Okay. Oh, right. You have it too. Oh, great. Is that? Oh, no, but that's Richard's, but in fact we can ring Davide directly. If you let Bill have the phone, he can call Davide directly. Yeah, yeah, that's it. That's even better. It's compatible, and there really is some unique state that's being described. I have to try to make a plausible story out of that some day. There is that sort of thing and possibly a computational interest. And also just by analogy with Isbell's stereotype, he used urinary operations. If I had put N here instead of 2, or instead of taking the monoid of N-domax of N and taking the algebraic theory of... I don't think that would give a different result. I doubt it that would give a different result. You think that it would give the same result? The same result. Which is an algebraic theory, not just a monoid, and in this case you will take the full algebraic theory of the two-valve mass set. Now this is really an example of the Reiss paradigm, as some people call it. Double globalization, but with a restriction on the external double globalization. So for instance, with vector spaces, for any space, the linear mass form. For N-space N, the linear maps from R-M to R, in a suitable smooth context, are the distributions of compact support.

57:30 The actual points are the direct distributions. Distributions of compact support allow us to close points. Now, this makes sense in the character of communion vector spaces, and is, in that case, actually the distribution of compact support, One thing I would like to know is to what extent, suppose Envy is a Ferdinand space or something like that, to what extent is, I believe there are some remarks on that in Miklos' book, but to what extent is the Walbrook theorem true, in other words, beyond the inclusion by the monetary association, the remark distribution, to what extent is this a free vector space? If this is the key vector space, then, because the theory of vector space is a commutative theory, all operations are commutative, this means that these two maps I was describing, the other, finding monoid like three, In vector space, you have two canonical maps, psi and psi tilde. But if this is really free vector space, then it's a commutative algebraic theory, hence the monad is commutative, which means that these two are equal, which is something which I believe they are very interested in.

1:00:00 So this is really, in their context, this is a ban. I don't know whether you know it. I mean, as a question on community vector space. Don't waste your time trying to... No, no, I... I don't get it. What I would like to know is E. Surely it's a strong monad. It's a strong monad, and it's not the free vector space, certainly. That's the big problem. I don't know. Well, at least I think that's what we... The theorem says that many objects believe that it is, or perceive this to be the free vector space, but I don't accept it. Well, it's the free, complete vector space. That's an additional. Well, in the category of F, isn't any vector space complete? There are many problems around what we urge. It's this. Well, there's one thing about that. Say these two things are opposite. Are they united, the entity of opposites? Which one's the map coming back, these two maps? Oh, these two. No, no, no. They're parallel to two things. CMC and TILDA are opposite.

1:02:30 United opposites. Is it formally a UIU in the sense that there's a map coming back that is splitting so? Well, I mean, in a special case... If these are the modes, if you put n and n equal 1, then this is a monoid, and these are just the multiplication and the dual multiplication, and there are no special maps between a monoid and toposix. First of all, about the ghost maps, well, the general idea about distributions is too far. There's no hope that a general distribution... The fundamental prejudice of statistics, which is largely true, is that if you have multiplicative ones. Multiplicative is the same thing as having standard deviation zero, because to say that there's products will say that if you apply it to the case where two are equal and subtract, you get the variance. Basic intuition, which is very often true, but not always, is that if you have a distribution, but moreover it has standard deviation zero, then it is concentrated at a point. Is that a prejudice, you say? I mean, that's a fact, isn't it? It's a fact about a lot of situations. Yeah, I mean, if you take... Algebras, then this is at least in good models, then this is just a very multiplicative. Right, right, right. But this is Gell-Planck's, so-called theory.

1:05:00 Yeah, I mean, it's... Then you have the algebra ones, and as I say, the basic idea is that every space is a spectrum of algebra. But this is already false in other great geometry. Predictor space is not a perspective. So what I'm saying is that it's one of those things that's often true, but you have to check in a particular case whether it is. And you've already set up to smooth the tree so that it largely is. But somehow just the fact though that the descent from here to here somehow tied up nearly in one thing. Well, you can come up with different linear combinations of that. Just one thing, namely the product itself, you don't have to drop down to one thing, namely this r to the m, has no one to consider. In the positive situation, that's the same as probability. That's total expectation of the constant should be the same constant. But that's the probability thing. There is a commons section that just with merely this little change preserves the constants.

1:07:30 And this in fact has nothing to do with linear economics. Is it clear that, well, I'm not asking whether E1 is a monadic one. Because I suppose you're just talking about the restriction of the already existing psi to the subspace E1. Is that so? Yeah. Okay, but is it clear that when you restrict psi to E1 times E1 that you end up in... Well, this cross polar two distributions is what psi usually is. Well, what I'm saying, maybe not even what I'm talking about really, is just any kind of average and geometric basis of our arithmetic. Sorry, it's not this, but any kind of average, i.e. any kind of section, just preserving bearing constants. In other words, a way of reducing variable to constant, some procedure group. Whether it's linear, or problem multiplicative, or whatever the minimum requirement is, if it's constant, you get a special variable for the same, and as soon as you do that, you do get three areas. As you say, there's a strong monad, and there's two ways to perform that, let's just consider time, one of them, it doesn't matter which way you have it.

1:10:00 The product distribution, there's the product distribution, so the probability is in one direction also, the other, the Fobini problem, which is to combine distributions on the margins to the one in the rectangle, and what you're saying... There are sort of two equally good, but in general definitely different ways to do that. Yeah, sure, I mean, that is, I equally couldn't have seen, discovered that from different kinds of terms, but even then, there's also the question of clarity. I mean, there may be some M such that these two equal, or M, for instance, if M is a, there's a choice. Then it is so small, it's a kind of smallness condition, where to say that these two are equal is in some sense to say that we could be filled in quotes by functions, entity functions, just the two different ways of integrating, two orders in which you can integrate two variables.

1:12:30 And it's done without the third thing, which is the product measure, and these are two ways of computing it. But somehow in this setting you don't have an outright third thing that both are computing. All you have is something funny about that too. This is concretely in the context of real valued stuff. The reals are commutative. I guess it was Ahrens who first discovered this type of thing. To do something that turns out to be non-commutative doesn't seem like it should have been non-commutative. Although I think I asked some analyst friends that they don't usually deal with cases which Guglielmi's theorem fails. Guglielmi's theorem is true for L1, for functions that are not only measurable but actually at absolute value is integrable. So if absolute value is integrable on both sides, then it will set it. There's not some extra condition on the relationship, although in some sense the requirement that the functions have their absolute value, integrable, is a kind of strange condition because you're not sort of dealing with that.

1:15:00 M and M are affine schemes, so M is A bar, or spec A, or commuter scale of ring A, and similarly M is B bar. I want to calculate E. I think that for affine schemes M and M, these two will be equal, because affine schemes are so small that to be the theorem holds. You can calculate the E of the spectrum of A quite explicitly because this is the symmetric algebra of K of the underlying K1U. Yeah, this U is for underlying, the underlying K1U. And I think, well, yeah, that's the basis for asserting. I mean, it's just... So to be twisting the universal property of the symmetric algebra, this is a commutative gradient between the two spectrums. Maybe this is not finite, we can send it anymore. Right, it's not finite, yeah. Only say it's infinitesimal.

1:17:30 Consider it's gradient is the same as the spectrum having an action by r. An action, a linear action on the covariance side, r is kx, so that's a co. The question is the commutativity of them. You have a strong monad by double utilization.

1:20:00 Yeah, but that was... You wanted to know whether you got a commuter extension or not. Ulbricht's paper is a starting point for the C infinity case. 1965, first volume of the journal called Functional Analysis. The two papers, one about bornology in general and the second about applying bornology to the C infinity case, showing the distribution of compact support on the manifolds. What is the free, complete chronological vector space? Complete meaning that every complete chronological vector space perceives it as free. Right. Oh, okay. Well... That it is a complete vector space. There's a category of free, complete. The category of complete chronological vector space is... You paired it with classical manifolds, tuned functors. No, that's more...

1:22:30 Extended to the other side and it seems always to be complete so there there's a oh hi the gentleman from back that's okay because everybody is just discussing monads and whether they're commutative or not you may have heard of them good good so i was just saying that i sort of The natural prejudice that comes out of staring at bornological vector spaces in contrast with smooth manifolds, C-infinity manifolds, is that any dual space ought to be complete. If you have a bornological vector space V and you look at Ham-V-R, or the ground ring, that ought to be complete. This is one of those naive conjectures that there's a long tradition of proofs. But it's false in the language of functional analysis as it's usually practiced, because the sort of thing that can happen, if completion, if complete means every Cauchy filter is convergent, then that may fail for a dual space, but only because the Cauchy filter is not automatically bounded. I mean, we're accustomed to the idea that the Cauchy sequences are bounded, and so... But in the more logical context, that should be a further condition, and the complete, the Cauchy filters that are bounded do converge. And so somehow the, what it means is that somehow the idea of being a dual space, going back the other way, it's already a notion of completeness.

1:25:00 It's already a notion of completeness, but it's hard to convince functional analysts. I was speaking recently a couple of times with Christian Huzel in Paris, who thought about these things the most, and he agrees with me. He said, yes, if we set up things properly, then dual spaces should always be complete. And in fact, we can sort of duel with dual spaces with completeness, as though being a dual space is what it's all about. It's still not solved that way, but it means you have to realize that it's the complete ones that are, in any case, the free. Well, I asked you this the other day, and you said there is indeed a general theorem to the effect that any monad is contained in a double dualization monad. Could you reconstruct that? You can always make the construction, but then they proved, I think they proved, concerning a double dualization, and one that's in a closed category, again this of having one single object into which you dualize, I don't think that many are very serious.

1:27:30 That way, but there's a generalization of that idea, namely, well, you double pluralize into n, but then you may take the n over all n, so that's sort of an inverse metric of double pluralization of mathematics, and I think here really many algebraic theories sit inside mathematics of this kind. To have one single D, it has to be, in some sense, to be generic or something. I mean, to give a monad map from a monad C into such a common realisation monad is... You do have to live in the end and see how to construct it. So, that is an illusion. All n-ary operations are faithfully represented as operations on n, on the object n. I shouldn't have called it n. D, not the D of SDG, but the D in the object in which you don't equalize, one single D will do, then that D is, if T is embedded in something, if theory of T is embedded in something, not a realization, 1F, this is equivalent to saying that D is in general, is first of all just that we have a 1F mathematical, by the way, a theory mathematical, from that aspect of the topic.

1:30:00 But it is better to think in terms of monad. The monad T is an algebraic theory. To give a monad on T, as a monad, is the same structure as putting a T-algebra structure on D. And to say that it is monic is evidently a way of saying that D is a generic T-algebra, in the sense that you can test all T by... Thinking of them as concrete operations in D. The theory you should be taking is primitive. That doesn't mean every production in Monad is useless. No, no, no. It's just, you are so polemic in terms of it. Of course you have, Boris. I'm just asking a clarification, but I think the concept is useful. Is it not primitive to D, or are you trying to say, boom, D? No, I'm saying that, from what I've learned from Is the same as giving B a T out of 5? Sure, sure. In the sense of 1S or T is my wife. So what was the question? I'm just hanging on to a little bit of what I'm looking for here, right? Because we're trying to get a D and then say for all T there's a 1S from T into... I don't believe either is true, but certainly the direction that I understood your statement was that for every T that is a D-subject, T, in other words, for every algebraic theory, there is a generic algebra.

1:32:30 Yeah, that was my understanding. I don't know whether that is... I guess I may be misremembering, and there may be side-positions as well. That might be one. T of one might be one. The free group in one generator is commutative, so it has too many equations. Or even T of one can be one. Yeah, yeah. But that wasn't the classic example. Like my averaging operator. He was talking about... Free at-line space in one generator. Yeah. Well, no. Yeah, that wasn't the classic example. Pre-algebra, uncountably many generators, or generally, it's basically a question of small support. You have a small support, and by taking those objects where the support lives, you can somehow either construct or dualize it up. May I inject something to the agenda? It's the last day of the VEDA, and I would really like to hear... There's more about continuum mechanics which was sort of a headline for... Right. Me too. Okay, I think, well, it's just about 12, right? And I want to abstain that.