Discussions, incl. FW Lawvere C McLarty, A MacIntyre, M Wright (contd.)

0:00 You see in Penrose, how you find that dimensional. Find that dimensional things obviously have lots of linear stuff. Imagine you have this cubic curve. I want to define a section. I can't do it because there's an ambiguity. There are two points.

2:30 I'll just take the average of the two. For any function, I will just take the average of the points. That will be a stochastic section, even though it seems like it would work. I would say any time you apply Newton-Berger's normalization and arrange a thing in that way over a field, notice that like most of the topologies, it's the same thing, namely the bar topology, everything is a cover. Finite fields, as is well known, has a retraction linear over the base. The linear one is a vector space over the smaller one. And that's how it should be so in order to turn the locally separable category of pre-sheets on the fields to the Galois, the Galois telephone is exactly what emerges from the barge apology applied to the site of finite field extensions. But that's going to be the natural restriction.

7:30 I could even repeat that right now. I don't know if I'd be able to repeat it in an hour, and I don't get it exactly, but you know, but yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah,

10:00 And it works as a general I mean there are other there are other contexts in which this notion of stochastic section is going to be well one would think so yeah but it's only at the moment it's only an algebra yeah I can prove you can show it really works you know but the next step of course is I guess I'd like to apply it to this See, once you get it, this is the kind of solid proof that something is going to be an epimorphism. You actually have a section. Yes, you've got a section. It's probability math that proves it. So of course it's going to be stable under all kinds of things. Yeah, yeah, yeah. But only in the algebraic, the polynomial content. I guess I'd rather say you've got this that's not a section but proves it's epimorphic. Well, they don't preserve products, you see. Yeah, yeah, that's what I mean. So it's a stancastic map, you see. There are a lot of categories in which you have maps, but also stancastic maps. Yeah, yeah, but that's something people aren't very used to, including me. Well, for example, I'm telling you, distributions are very important, even in algebraic geometry. Yeah, yeah, yeah. I don't even see infinitives. It's just a matter of distributions. And how is our understanding of the fibers of the map affected by this stochastic generalization?

12:30 Well, that means that we're sort of averaging over the fibers. Yeah, exactly, kind of averaging over the fibers. Because there is no actual section of the map which induces the polynomial, the whole of them work with it. We can't do it where there is no actual section, because polynomials have only a finite number of solutions. Or more generally, in higher dimensions, there is always this nurture-normalization thing whereby the... ...given variety is fibered over another one by finite fibers, where of course finite means both infinitesimal and field extension, so the infinitesimal has already come into Nurture's normalization here, but, you know, so, by the same argument one would imagine that finite dimensional have to do with finite dimensional algebras over a field extension. What you do is you see you have, well you have seven solutions, it's a seventh degree equation, so you have potentially seven solutions. You can't choose them, but you can choose to take the average of all of them. That is to say, a function, its expectation is a certain linear combination of its values at those points. So in that way, this is now a function on the total space. It's pushed down to a function on the base, which is what averaging means. Pushing down is just a sort of section, retraction condition. In the category, averaging is not a ring homomorphism. That's why it's not in the category, it's only stochastically in it. But in principle, you can average over the fibers of any nurtured normalization, I think. So, and you can weight these averages, including, you could say, we'll just take the value on this brain an actual second. Yeah, that's right, that's right. Right, right. Yeah, you could take various weights, or maybe you would park the function on the plane by taking its values at one point.

15:00 But now, what am I going to do over here where there is no one? In the complex domain, it should sort of exist almost all the time. It may degenerate, there's no problem. So something, something... Well, sometimes you get a total catastrophe there, which again, you could average over. That's for sure. And you say it is not... Which degenerates in a certain region to give you...

17:30 I think in terms of either in terms of real numbers or the rig where one plus one is one. Yeah. I mean, I might have to cover it less to get a stochastic section than I would in order to get an actual. You mix up the two, whatever topology it is you have. And then what is it we'll be able to use? The complex case is just a visualization. It's just a visualization. It's harder in the real case than it was in the complex case. In a topos, you can always define the space concept as invariant under pullback.

20:00 It's a statement that you can factor through sections. The question about a map that reads topos and as such, there's a sense of C-infinity. But they don't have this trick that C-infinity modules are. Did you ever think about that? In general, abstract modules have anything to do with the actual C-infinity structure of, I mean... The fact that vector bundles are retracts of trivial bundles means that, in some sense, that these are finite-dimensional vector bundles.

22:30 If you're dealing with faces and things where you don't have retracts for you, there you have the retracts for your chart. That's a marvelous, a marvelous, fortuitous tool that you have in the C infinity case. And there are retracts all over the place, none of which exist in algebraic geometry, but I don't think they would suffice to give a sufficiently geometrical meaning to nearly abstract modules of scalar that happen to be in the arrangement. I believe that was my, that's why I said all these things. That's my intuition that there's something defective there. You should be able to explain the fact that you can take a linear endomorphism and take e to that power. It's a very concrete thing that engineers use all the time. Simple C-infinity functions, like the exponential function, not only apply them to linear transformations, so unless the space, unless the category, there's some kind of compatibility with this. And the mere fact that the C-infinity functions on some manifold might be operating, so to speak, fiber-wise, would not give you any reason to think twice. In other words, there's a smoothness of the... All of these people are good to see in fencing, but I think as I say, even for finite dimensional ones, it will retract to other trivial ones. ...and giving a meaning to.

25:00 You want to take, you want to apply the exponential functions to differential operators. If you have a differential, you start with a one-dimensional space, but then you're in a space with smooth functions on it or distributions on it. And within that you have differential operators that you want to apply some kind of mysterious lifting. Smooth functions on the original one-dimensional space you want to apply to these operators. And I never thought of that. And convergent sequences. Sequences of analytic functions. This is the only way I've thought of applied to operators. Right, and insofar as the existing foundation is concerned, which of course we'd like to overthrow or supplant by something better, the way that one deals with that is precisely that. Why does e to the d exist where d is some differential operator? It's all because we can look at the formal power series that's supposed to represent it, and we can show that that converges because of this, doesn't it? So you use the convergence, right? That's the way the usual justification, according to the usual mathematics, would go. Yeah, and I guess I learned that from Stephen Smale's book on dynamic systems. Right, you're right. So in other words, to actually carry out what I just said requires dragging in Bonac's based sequences of both synthetic methods. Yeah. I'm sure that Deligno was able to read my book up to 1958 without supposing that it was based on the aritism category theory, but it is. I mean, basically my line on Groton, because he has this concept of simplification, and I mean, I'm not...

27:30 I'm trying to sneak something in about you under his name, but I really, the way I understand that is what I've learned from you. In what way is, because I'm always trying to get in touch with, say from, what does he think when he sees Brogdie claim that toposies simplify algebraic geometry? Well, he thinks geniuses are mad. I mean, it's just, you know. Are geniuses that super-powered? Yeah. You know, which I often invoke myself. Yeah. As a reason why... I told Cartier and other people, well, yes, we've introduced this logic, but Rotendieck didn't really need it, because he saw the structures in terms of direct limits and finite inverse limits directly. But he even calls it logic sometimes. So he didn't need... There's a lot he doesn't know. I think the word push-out doesn't occur in logic, but we can see directly that if it's a push-out, then it's something that's representable by a push-out. So I think he could do that, and therefore, in a certain sense, he could go very far with classifying phylococcus without introducing anything in the way of a formal treatment of it, because you see it involves... And it involves a sort of ritual deduction, namely the ritual deduction of every kind of structure into sub-objects, maybe of Cartesian products and all that, which logic in the narrow sense we're talking about, which is really just the algebra of sub-objects and how they transform. Well, why should we make that reduction and then do some complicated calculations and then try to come back to the real world? I mean, this is, from that point of view, circular.

30:00 But I think this is what he and Dudenay are talking about already in the first version of the first five of ETA. Of course, that's presupposing that the, you know, this classifying topos is the only thing, the only relation between topos here. Of course, it's not an entire order of business. Yeah. Excuse me, I just finished writing. Yeah, yeah, yeah. But they say that the problem the reader will probably have with this, and that we have ourselves, is learning how to lift constructions that are familiar in the case of sets into these. Other categories, and sometimes I want to pin down some grad student at some place like Harvard and say, you talk about a sheaf of modules, why do you call it a sheaf of modules? Because of course they don't make it explicit. They take this complicated machine and they're very happy and they then succeed in doing it. But why do they call this a module? They did this much in those definitions. Thank you for watching.

32:30 All of these are propositions of quantum mechanics, and I'm not going to go into too much detail about them, because it's just a good way to get to the bottom of the topic, and I'm not going to go into too much detail about them, because it's just a good way Push-ups, like the category we just watched, and the way that we can do these two topics together at the same time.

35:00 There's another thing that we could have to have coming back, which is to talk about the conditions of mathematics and physics. And these two categories are going to happen to be very popular in the category of mathematics and physics. Thank you for watching. In the period of the early 20th century, there was a lot of interest in the subject matter, so we started to develop the library of mathematics, and we started to develop the journal of mathematics, and we started to develop the journal of physics, and we started to develop the journal of physics, and we started to develop the journal of physics, and we started to develop the journal of physics. When we do a conclusion, not what we need, we do what we need and we do it well, and we do it well, and we do it well, and we do it well, and we do it well,

37:30 As soon as you're already in that world, you start to discover precisely the exactness of the people that you have, prevent their being, prevent their being in the modern world, and somehow make them, that's what we have on the planet. I was going to ask you at some point, the general issue of kernels of math that have an active effect, particularly of the effect of quills. I'm in connection with the Natal 2. I'd like to correct you. I'm not a scientist. I'm a professor in Scotland, and I'm not very interested in mathematics, so I'm an expert in mathematics, but I'm interested in the kind of... I mean, the reason that co-equivalence is not so important. Thank you. Well, you had a student who read a thesis on it. Well, I could be where I could hold my ideas from. Well, I wanted to get clear on it myself.

40:00 And particularly, they remarked that they were... Well, you said at one point that... I probably did, I'm sorry. Well, I thought that... Well, I might have misunderstood, too, but I thought that... Yeah, and I thought that, you know, this was an audible action, this kind of... I think that they had this sort of put together in the U.S. once, quite certainly, in the case of the islanders, especially if they had two topologies in the U.S. Yeah. Yeah, two topologies in the U.S., which is the reason that they're in, in a sense, topology. It has nothing to do with, in a sense, topology, particularly, because they're the top of the set. I mean, I'm sorry. I don't want to try to say it wrong, but I can see that the idea is that we're all supposed to have one or the other, and the other one is a bad one. It's very important. You can't do that. I'm going to tell you a few more things. Well, it's very much related to mathematics, but I mean, if you have some sense of mathematics, then you can ask me about it at a certain time, too. But if you have some sense of mathematics, then you can ask me about it.

42:30 There is a story of how to get up there. You know how to get up there? Well, there are all these things in nature. I'm sure we've done that before. That's right. You're quick-witted. You're not quick-witted. You're quick-witted. You're quick-witted. You're quick-witted. You're quick-witted. You're quick-witted. That's right. Thank you for watching. There's a very little, not a rich theory, but there's a lot of things that fit into all of them. Except, you know, just short of the metallics, there's not a clue to that, of course. There's hardly any generalization that comes to that. It doesn't all have to be the same. It's just that both have to be common. So that's what we have to do. There are many different types of mathematics in the world of physics, and I'm sure you've all heard of it, but there are many different types of mathematics in the world of physics. I'm sure if you found me in Chile, you'd be glad to let me pick you up there. Yeah, I was going to say, you've got your final couple. Absolutely. Well, it's probably your final couple. All right.

45:00 I think there's too many of us here. There's a lot. I don't recognize all of you. I'm afraid that you can't find me. I can tell how many of these students, whatever their names are, they wouldn't see what they know. They don't seem to have any memory of what they've learned. Probably they've been to different classes. So, I don't know. It's all over. They don't have to tell you what they've learned. Thank you for watching. I see. Do the Japanese have the same problem? Yeah, very, very high. Yeah. I guess it depends exactly on what you do. It's like a difficult thing to do. Which components are you throwing at it? Thank you for watching. Thank you for watching. This was the passage that I was caught with the question. It was such a bad record question.

47:30 Or even exists. I wouldn't try something like that. No, I'd never get on that one. I'll try to look a little more accurate at it. Hang on a second. Yeah, this was the... Here I have it. The last part of the variable consciousness. It's a rather small, reduced combination. Yeah, yeah, sure. Let's go and... Do you want to leave this until we have... Well, I was just, I was just looking for a kind of clarification in my, I'm not saying it's not entirely clear, it's just my own sort of weakness. Yeah, okay, there you go. There it is. Okay. Yeah, you know, you say... Definition of an atom is why it's an atom due to the existence of a man-man and a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to the existence of a man-man due to However, the property of being a topological space is additive. Thank you for your attention.

50:00 You might as well offer the commonplace of this text, you know, but you couldn't buy it before. Right. But there are ways. Yeah. Right. But the fact is, it's precisely the same. Yeah. Yeah. That is the thing which picked out the next thing. The fact is, it's the same. Right. The fact is, the terms are sort of trivial. Yeah. So it's all information. Yeah. Thank you. Thank you. Thank you. That's all I'm saying. Well, there's further... I didn't have the... There's more. There's more. The first example of an... I appreciate that. The first example of an economy seems to have been the space of moduli by Albrecht Kurz, which is prevented from being globally spaced due to the action of Galois groups within each coin. I'll get caught. And then something vaguely reminiscent of Parkinson's and the most naked person. There's a single point to be any of those two things. So nonetheless be, yeah. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. There's a single point to be any of those two things. Or in fact, they don't have any questions. Nonetheless, they have a sense. Nonetheless, they can have your phone. They have a sense. They've got to have a phone. So they're not taking notice. They've got to have a phone. Yeah. Yeah. And the most naked form is that for any group G in the category, a general explanation of why Aton will arrive in this policy is that the inclusion front does not in general preserve co-equivalence. In particular, if those groups of the axons are based on a bit of a co-equalizer diagram for the motion of organs, then if the action is good, if it's unrecognized, then the axon of the right mechanism will also be the two co-equalizers.

52:30 Well, if the action is bad, the latter co-equalizer tends to be an axon of the hypnotic group. And then there's the very interesting thing about Markov. And then there's the point. I think you're right. You know, if you look at the scale, you've got the mission of the second half of the first half of the second half of the first half of the second half of the second half of the first half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half

55:00 If I take the rest of the problem, you can't help it, but you discover one of the three is called the metric field, yes, and the other one you'd like to give to the study is the metric field, which is the metric field, exactly the metric field, and the metric field itself, exactly the metric field. Actually, this is the amount that I was physically drunk. I was literally drunk. That's right. I was drunk. But we have seen that many domains of variation are ubiquitous yet, though it would be even worse if they were flat-strung classes. For example, since part of their essence is trigonometry, and since part of their essence is a very strong 2-problematic algorithm, the 2-problematic algorithm of the right, I don't, so I think I'm talking about the variation of the 2-problematic algorithm of the left. Thank you very much. They typically involve mathematics, which is good for the humanities, but things are a new thing, so you need to be careful. I see. Okay. I'm sorry I've eaten that thing. Okay. Got it. Got it now. Okay. That's what I have got. So there's two parts to all of the math. Well, no, it's just my, I'm just very, very slow at things, you know, came to this start, you know. Oh, okay, okay, no, no, no, no, you don't have to justify or defend it. I just needed to be sure that I could be totally candid. Again, I, you know, I had thought that, you know, you meant that they involved some relationship between, you know, coming straight to the point.

57:30 Thank you for your attention. I was still trying to force men on the particular field to try to identify what was the bar that they had to shave on. Yeah. Okay. Yeah. Okay. That's it. I couldn't see the distinction with the general alphabet. What the philosophers have said is that the string theory and the relativity will preserve the theory of mathematics and physics. Too large to be exact, but small enough to be exact. So draw from the beginning. Draw from the beginning.

1:00:00 If you need any knowledge, you can contact me. I'll let you know if you need it. A couple of problems. The first one is that you need to have a lot of knowledge. You need to have a hands-on. You need to be able to make it so that it's very good. That's what you need to do. You need to have a lot of knowledge. You've got to have a lot of knowledge. You've got to have a lot of knowledge. You've got to have a lot of knowledge. You've got to have a lot of knowledge. Thank you for watching. We go up on the lawn later on and have a nice back out on a bottle of wine, do you mind if I? Okay, well thanks for explaining that Bill, because it had been confusing me for a long time. Okay, shall we have a short discussion?

1:02:30 Well, you'd actually say we're going to try to avoid intensive bilaterals, but I'll point them to you. Yeah, I'll say we've demolished them, haven't we? Yeah. We're completely a dead horse. I suspect we'll probably take us around the same course again, Mark. No, actually, I was going to ask you, Robert, about the 81-kilometer line. Yeah, it's a great line. The brain is the core of the brain. The brain is the subject matter, the subject matter of the brain. The development of the brain, the development of the brain, the development of the brain, is more basic. The brain is the core of the brain. The brain is the subject matter, the subject matter of the brain. The brain is the experience of the brain.

1:05:00 Are there any tools to be better on it?

1:15:00 Peter Sarnak and my... But when you read the books, the books all like... They don't like integer.

1:17:30 They like sparse matrices. Yeah, random integers. Yeah, I mean, they're bad. Gallinger says, he's interested in Gallinger, this program's been around a while, he's got a paper, a plain book, this is a cohomological, it's a bit related to this period thing that people like to have in Japanese, they're trying to do algebraic geometry, one element here, but if you could somehow, what it really means is to give it. Give sense to the notion of a variety of scheme being defined. There's no one on the field of that. Some of it, the scheme is defined over. And there are interesting, rather random ideas. Certainly there is such mystic talk. What definition of scheme are they using? Replacing by a perfectly explicit totalist system. I think so, yeah. I haven't thought about it. I mean, Suley's papers are quite, in a really odd way, quite nice, yeah, yeah, yeah, I mean, I don't know who first came up with this, but there are a number of papers, I've heard about Manning and all that, I mean, Manning's... Yeah, I knew Manning had published it, I would work on it. It's clearly a very interesting thing to look at, because they can track back some of the ideas to work with Jacques Keats on giving meaning to things like, yeah, the idea of a sort of family, some of the defending of that. The data is a parameter that's like the symmetrical test and or maybe the GLN and so on. They try to give a meaning from an n to a zero. And sometimes you give a very indirect meaning by these counts, computing the order of groups, different n's.

1:20:00 And then when the n goes to zero, you see it's the order of something else. It's a kind of numerology about it. But the game about the Riemann Accord, about the curves over the, not curves, varieties over the field of one element is that the game was somewhat, if they had a picture, the response to the picture that they had, the curves over a finite field, and to prove the analysis, then, is this with them quite close to prove the actual Riemann Accord, but it's a very sporadic, and many people think it's normal, including from what you were saying, Atiyah. Atiyah, this would not be his game at all. But, I mean, Denninger's approach is slightly different. It's more involved, more of those data functions of bits of a very complicated cohomology. It's very similar to Kierkegaard's cohomologies. Yeah. Anyway, Denninger has a long and, I think, not terribly obscure paper for Lenin. You know, one historian sagely remarked that the 98th ICM didn't do anything with schemes. Well, okay, right. Nobody presented EDA 7 as a plenary session. But, I mean, you mentioned it in your paper, and there were some other things. Sure, basic scheme theory is not cutting-edge research. It is bread and butter. Yeah, I will. Why don't you speak basic scheme theory? Sorry. You can get to zero and one by taking the infinity. The thing that's missing in all that kind of discussion is the three of them. It's like taking the number of elements of the S-group and moving them down. Precisely, the numbers are too abstract. No, no.

1:22:30 It's the objective numbers theory. One point about it is that it gives you... It ties down things that are infinite, objects that are infinite. ...that you knew only about, say, abstract Cantorian sets, or even only about recursive functions, because both of those categories have an incredible, you know, flexibility, so to take practically any, any simple reason and express two polynomials, they're isomorphic in terms of the calculation of the collapses, so you think that, you think that precise calculations are limited to finite sets, in this case, and the analog of them... But if you don't make the final abstraction, if you stick to the things as actual objects, then you can have categories where isomorphisms don't happen unless they're provable in Greek theory, for example. So even with analytic number theory, data functions, for example, I proved... The Euler's infinite product point has an isomorphism of monoid objects. These objects are infinite, so you'd think, well, there's no content to this. As abstract numbers, that would be true. They're all countable in carnality, and you'd finish, right? But, because you just maintain just a little bit. With the structure you treat them as objects, you can have precise, stable, proven, precise. The product for the zeta function is an isomorphism. All the objects involved are infinite. You can't apply the zeta function, you know, that's why I call this the Dirichlet topology. It's an incredibly simple thing. It's almost the simplest topology you can possibly think of. It's obsessed.

1:25:00 A single idempotent two-element monad acting on sets. The objective number theory of that topos, well, I mean, if you do make the abstraction, the rig that you get is the Dirichlet series. The other thing here is series, i.e., like an infinite version of the monoid ring of the multiplicative monoid of the number series that's contained in some sense. So then, compare this with castles. A typical object in that is topos. The union, it's a disjoint sum of connected objects. The connected object has a unique point. But surrounding that point, surrounding that point, there's a whole bunch of elements which, when you apply the operation, go zap and stop there. So he applied this to medieval Europe, the dukes who had their castles and all the presences. When hostilities break out and all the peasants have to retreat to the castle, it's kind of an image of what's going on. A very appropriate one, in a second sense, is the way we have it now, but it's extremely abstract at this point. Any number of dukes, and each duke can have any number of dukes, so then there's a unique connective... So you have this expansion. Any object is uniquely the sum of certain abstract set coefficients times these connected objects, and these connected objects have a special simple form, which obviously multiply exactly like the natural numbers, take two pointed sets, because they're now equivalent to a Cartesian product, two pointed sets.

1:27:30 Another point is that the number of elements now is a product, but that fact is rigidly maintained just by the fiscal structure that things have. So you take the ordinary Cartesian product of two objects and a total, in terms of this expansion, it's exactly the multiplication of the Dirichlet series. So the Dirichlet, the zeta function is which Dirichlet series? All of these coefficients are one, so therefore it's the object that has exactly one connected component for every possible cardinality, finite cardinality, because there's eight of them. But it is also an infinite product. That has to be equated with a little more. It's not a categorical product, it's rather a direct limit of finite categorical products. So it's not an inverse limit, it's not... And the individual terms are just pre-monoid. So because you know the typical, the expression 1 over 1 minus x, what's the algebraic equation that characterizes that? Exactly the characterization is pre-monoid. Any element of a pre-monoid is either a unit or else. There's a unique decomposition, there's a letter followed by a unit, you just write down, and that is a polynomial equation, it's the same one as the one in character, just 1 over 1 minus f, where I keep the alpha there, so instead of 1 over 1 minus p to the minus s, I identify these primes, just with the object that happens to be prime, and I take the free monoid. So therefore, you know, Euler's product form in some way is revealed, who have as its, sort of its basic guts, you have several free monoids, you take the Cartesian monoid, that's the question, it's actually the co-product again, sort of drastically mixed up the commutative and the non-commutative, by taking the product of three things, and now you...

1:30:00 You take bigger and bigger products, you take any given set of primes that you like, and there is some kind of expression, but then you enlarge these sets of primes until you've got them all, but you have to take the direct limit instead of an inverse limit, but you have those, not for general objects, but for monoid objects, because of the unit element. We can use that as a trivial injection, so if a smaller product maps into a bigger product, boolean, of course, then we're just projecting because we have a unit, so take the direct limit and you'll find out, well, actually that's isomorphic to the object, that's exactly one connected component for a charmality, so it's a completely different concept in a way, but is it really different? That's the question. It's all about infinite objects, but taking account... ...of their structure, both as an object in this topos, but also a certain level of the monoid structure. Now, of course, you're using the fact that the natural numbers themselves, by fundamental general arithmetic, is the free commutative monoid. So that's why, you know, the left-hand side of the Euler's formula is a sum. It's a sum over... The free commutative monoid generated by the generators on the other side. So in fact it's divorced really, you can apply it, as I said, divorced from any notion of prime. Take any set of objects. You can take the product of the free monoids on each. You can also form, and that will then have not merely a length function but a multi-length function. So far, this is just a curiosity, you know, it's mathematically precise in a way that the field of one element is not, but I feel it's really the same flavor of things that they're trying to get at.

1:32:30 For another example, I was telling you before about this. The rigs, the rigs that have a top element, then there's actually a field where a ring at the top has all the stuff on the former ring automatically. The unnatural members are sort of left out below. You get a projection that you have a section that preserves addition and multiplication, but not zero and one. Well, what I neglected to mention is that this is not really an abstract construction. The rig arises as the first rig of the... I mean, that's an important way. Definite categories constructed by Robbie Gates, an Australian.