Morning talk

0:00 What are your plans for this morning, Matthias? What topic are you actually going to talk about? Two or three technical questions. Ah, well, that's far more than just technical, I think. This has an additional dimension of interest over and above the already very interesting technical questions. Very interesting. Well, at least two of those topics I don't think of as being just, I mean, obviously they are technical, but I don't think of them as just technical at all. In fact, I'd say that they're clarifying the, really clarifying properly the distinction between Grosvenor and Petitopolis is one of the most, you know, the things most loaded with significance for philosophy of math. That's fine. That's exactly what I thought you might be in mind. That's great. Unfortunately I may have to miss at least part of it because I simply got to go and do this application and can only do it on Richard's computer. If he's not around I'm going to have to go to an internet cafe and do it because everything's going to be closed from tomorrow and I'm supposed to have it in by today.

2:30 Bloody nuisance. I'm hoping Richard will be there so at least there'll be somebody who can keep a really good set of notes. Okay, well I'd be very grateful to get a copy of those off you. Oh, by the way, did you leave, I think you left some notes in there the other day, didn't you, which I have, in the seminar room where we are, or maybe it was, maybe they were Davidi's, they were left behind at the end of the second session, not today, but the day before yesterday, maybe they were Davidi's, anyway, oh, sorry guys, we stopped off on the way for coffee, hi, sorry about that. Well, it wouldn't have done me any good, but I do have a mobile computer which will always ring me. Unfortunately, I've been using my mobile as an alarm clock. I forgot to stick it on my pocket. Oh, well. I'm Richard. Oh, hang on. Yeah, actually, I need to... Sorry, I didn't mean to... When... You've finished. I couldn't get your printer to work. Oh. But actually, if your thing is short, you can do yours first. No, mine is longer, so it's better that you do yours first. Right. I'll show you where it is. Can you show me? Sorry about that. I had to have a 45-minute conversation with Ben. It's just a passing moment. It means that the algebraic theory of rings is very different qualitatively from the non-communicative, from the non-distributive lattices, whereas distributive lattices is very similar.

5:00 It's also co-extensive. Extensive in quotes. Maybe that's better than co-extension. That simply means that the cohomology do the categories. So that's the general thing that opposites of algebraic categories in general are geometrical? No. They have to be extensive. Category proofs is not at all like that. Okay, so extensiveness is what makes that work. It's almost like we're providing some of those spaces. Okay, I'm getting the picture. The idea Bill was commenting can be simplified in this particular extensive categories which have a terminal object to test for extensivity. You don't need to try to pull back along any coproduct. It's enough to check whether that coproduct gets pulled back. Turning the whole thing around, you can see the The product of the initial object itself, with its projections, and then you multiply the photons here in this ring, and if you invert them, and by the result, it was floating, this is a product diagram.

7:30 If you invert an ion photon, it goes to 1, so it is mute. The same goes for here. And that's why the category of rings is coextensive. Strictly said means it's all coextensive. It is exactly that. Algebraic theories, the category of algebras, can take its form. For example, one can consider examples of extensive categories, one can consider an interesting plot for distributive analysis. Right, I mean, just to remark, the irrelevant idempotents come in pairs, so in rings we think, well, given one idempotent, we can take one minus that and we have a pair. In a rig, we don't have that operation, but anyway, the relevant thing is pairs of hippotons, or triples, or whatever, or partitions of unity, which I like to say, and the rigs have those just as much as the rings. You mean a rig is like a ring, huh? Well, no, a negative, yeah. Oh, yes. Yeah. It's only that you can't start with a single hippoton. You have to start with a pair.

10:00 But not with non-commutative geometry. No, it won't work. Non-commutative geometry doesn't work. But it's not good. Well, usually they have you. They claim that they have something to do with geometry, but the present establishment at the IHVS is based on that. Non-commutative geometry, does it exist? They're doing a lot of non-commutative geometry and saying, well, this is somehow, there's a lot of non-commutative algebra and saying this is somehow geometry, but there's no clear, no clear notion of the Cartesian product of these spaces. What they use always is the tensor product, which is commutative, not the free product. So, just to point that out, there doesn't seem to be any easy way to... Finally discover what non-commutative geometry is by finding these ideas. Although they, basically they're saying, well vaguely it's like this, you see, the classical algebraic geometry, but it's a little too vague. They don't seem to have a good notion of a map. It's not even a good notion of a map. It's just a general way of mapping between spaces. It costs a number of people with an extensive category, community groups, unit, and non-community notion to know. Non-commutative affine space, so to speak.

12:30 Well, I mean, Steve Shanierow had a very excellent construction, which had never been exploited by anybody that I know of. He published about it in the Journal of Pure and Applied Algebra, but it was not based on, well, it was based on, he has a topos. Topos was a rain object, which is clearly non-commutative, even though its global sections are commutative now. Consider, let's say, over the complex numbers or the real numbers, all algebras, non-communicative algebras, which are nonetheless finite dimensional as vector spaces, so in some sense non-communicative infinitesimals. And then there's one trace of cohesion that you put in, namely there's a notion of bounded set in these algebras. All the morphisms are going to be monological. You have a functor, the sort of underlying set functor for these algebras is considered to live in the category of monological sets, telcos of monological sets, which is built over monology by taking an internal. So now, this has the beautiful result that the natural structure, the endomorphisms of this object. The underlying object is sort of a line. The external endomorphisms are precisely arbitrary analytic functions. You get the idea of analyticity pops out without putting it in. And if you, more generally, if you considered matrices which have a certain, a given restriction on the spectrum, the spectrum in a certain open set of a complex plane. You look only at those that have that property. That's again, that's a sub, that's a sub-functor because any homomorphism will preserve spectrum, you know, as the, those numbers lambda, where lambda minus x is inverted over x, so you take all those x for which this lambda exists in a certain open region, so there are a whole lot of sub-functors of this one which correspond to.

15:00 These are open subsets of the complex plane, so that part of the geometry. And the idea extends without any real problem to several variables. So you can take that ring object to the power k and map some of that into r, and now suddenly you have a notion unknown, I think, even to IHS, a straightforward definition of analytic function of some non-communic variables. It's a funny sort of boundary, this non-commutative but finite dimension, because if you considered things that weren't finite dimensional, then you're basically just getting classifying topos for algebras as abstract algebras, and it's just polynomial functions. On the other hand, if you could consider arbitrary non-commutative algebras, well... Or bornology or not bornology. There are various choices you might think of in trying to extend the basic concept of spectral theory from the commutative case. Known aspects of you. But so this is the unique good one that doesn't give some kind of tautological answer. And it's highly non-tautological because who would expect that the precise notion of... But if you look at it in detail, again, you look at the actual calculation, it's not so surprising because the point is you're talking about a functional, a natural endomorphism is a sort of functional calculus. It means in every one of these finite dimensional algebras, you assign to every element some other element called f of x. That depends on the algebra A.

17:30 But now, it says that it would be natural with respect to arbitrary holomorphisms, A to B. Then you can apply it in particular to triangular matrices, the algebra of 3 by 3 matrices with a 0 in the front. And the sort of thing that comes out of the algebra is that F of something is going to have the values of the function in two points on the diagonal. And some upper diagonal element, which is really a difference quotient, is forced to be, a mere matrix calculus forces the concept of a difference quotient. So now if you put the two points equal, that difference quotient becomes the derivative. So just the, up in the corner, so the diagonalization, again that's the basic contradiction behind the derivative in the first place. You then consider the function of two variables where they both vary when you do this quotient business, and then you pass back to the case where they're equal. Sometimes one does that with limits, but you don't have to use limits. In this case, the bornology, the bornological conservation of near-bounded sex is sufficient to ensure that that equi-pointer exists. And therefore, these natural endomorphisms are automatically differentiable. Automatically see infinity, automatically...analytic change. General Pienaar applied algebra? Do you remember him? A long time ago, no. You have this? I do. I just started doing pencil work ever since you gave the same clear text position. That's fine. That's fine. You have the record? It's...I think it's about 1918? Might be. It's in the ages. I could check it, but I don't know what it is. That's right. If you limited yourself to a commutative case, if you limit yourself to

20:00 fine-interventional algebra, well, that's an extensive category. Good. But somehow it's just about the abstract formal power series. But it's the non-commutativity plus bornology instead of... For actual derivatives instead of formal derivatives. Mathematics, the matrix is kind of like that. No, that's just a further example. In other words, if you don't put any restrictions, then you've got the whole complex playing, there's no restriction on the spectra. You've got the entire function, the ring of entire functions. But it's not just that. As I was saying, there are many sub-objects, or related objects, in the same topos. So this central object is somehow... The complex plane, but viewed not to be a civility, but then all the sub-objects you can think of, that should be there, are there, as other objects in the same category. I've forgotten that in terms of this. Sorry, I didn't mean to. As I say, no one's ever really pursued this, and obviously is very pregnant with suggestions that, so, excuse me for, no, I don't think there's any relation. A lot of examples of spaces come up by turning around categories of algebras. Of course you can go this way and get a classifying topos, but that would give you an embedding of the opposite of the category of algebras into...

22:30 That's the first step towards at least really the intention of getting every category of spaces that you can think of inside a topos model. Well, you can do much better than that. The embedding forgets a lot about limits. So, there's another construction, which is subtopos that generate the reserves of products. You can not only embed your categories, and this is for any extensive category. Let's forget about algebras for a second. They are not the only source of examples. If you have an extensive category, it's always embedded into the preaching topos. You can put this joint-covered topology beneath that provides a shift subtopos, which is subcanonical, so that means that you never embed any factors to this position, and this preserves the coordinates. Put differently, you can always embed your category of spaces into the topos, preserving the coordinates, which applies in particular to commextensive. Sorry, two extensive fields. I wish I hadn't rubbed off time. I'm thinking about just the second, the second half of the semester.

25:00 There are these many, many preserved components. The components that you recognize as being the right ones are some of the preserved ones. It has a very nice outlook to the next year. It seems that this distinction between the field and the world was known from the very beginning. It's precise. We're gross topos over a base. Recortism, which is essential and local. Essential, in other words, as the connected components. And local, providing the body's creative function. Plus some extra-arithms of inner cohesion. But the point... Would you just remind or tell me what the original manifestation of creativity was?

27:30 The subcategory of topological space is really very different from sheaths over a single phase, while still being a topos was rather different from sheaths over a small subcategory of topological space. I'm not sure what idea they had in mind, but they could see that they had different versions. These are sites, every Mount Mono, and these are sites with every Mount Eppie. I don't know how much this picture they could see, but they did. It's clear from the literature of the agency that these things would be very different. The simplest example was the distinction between Reuzex and Graves, sorry, Precious of the Lays and Neri Zeffi, and of each rise to a very different total. So the first occurrence was in Giroux. There's a book by Giroux on non-Nobelian cohomology. I've done it before that.

30:00 The point is that one kept seeing that certain categories, which turned out to be topos, had the character of all spaces of a certain ilk. In fact, I think the first example of that, in 1960, the Cartan Cinema wrote and approached the idea of a general analytic space of all possible dimensions, including even whole-market maps between them. And so on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on and on And then linearizing that by taking modules over the sheaf of rings, and then vector bundles were in that. This was all just a technology which was spreading out, but was implicit in one space. And the set value level of it was again a topos. And so, and in algebraic geometry it's the same thing. Of the nature of all algebraic spaces, there's some precise ilk of algebraic spaces, but of all dimensions and of all kinds of maps. Whereas, of course, classically the sheaves, the ordinary sheaves on a space, they could be viewed as, spread out, as an espace de taille. The projection map is local homeomorphism, and all the maps between those are local homeomorphisms, rather than arbitrary smooth maps. The idea of Petit also contained this idea that the fibers were discrete.

32:30 What's in the case of the Etendus? Well, the actual definition of Etendus was these were toposes which were locally given by a single topological space. In other words, the operation of taking the sheaves on any given space, And so forth. So, in particular, the possible quotient spaces of a given topological space in that larger realm are more. There are more things that look like quotients. One of those is locally a topological space if it's a quotient in that larger world. It is a topological space, but it's locally a topological space. That was the definition. The actions of any, for example, any category of modern morphisms has that property that you can cover it, transition from natural numbers. This addition of natural numbers has cancellation, i.e., it's both epic and monic. So the site of dynamical systems is, in that discrete dynamic of systems, is locally a topology. Namely, what space should you think of? Well, you just make a... You had a monoid with only one point, and we added a monoid and natural number, and you spread this out to become an ordered set with as many points as there are elements of the monoid, and then you have an order relation between them if there exists. So this is just sort of the spreading out of an analytical system as considered as a family of sets parameterized by the natural numbers and equipped with...

35:00 In general, a sequence of spaces and, in general, the different maps have no particular relationship except, of course, that the longer ones are composites of the shorter ones, but the generality of the shorter ones is not compromised. The topos of actions on a single set of the natural numbers is gotten from that spread out one by squeezing it together again, collapsing it. There's a sort of intuitive relationship. So that's why the etendus and in many, many other ways, they sort of behave the way you would expect one of these petite things. The QD ones were, by contrast, what I would say is an example of both. You have cancellation on both sides, and they behave like a single space. By contrast, it seems to me gradually that one feature that everything which is like a category of spaces... My discovery was that this tiny little difference looking at reflexive graphs instead of graphs seems to make that... Irreflexive graphs, for example, on this topological space, even the ordinary sets of sheaves actually covers the graphs.

37:30 This one is locally like that. It's a very strong sense of problem. Reflexive graphs are just sets, just diagrams of this shape. It's a set of arrows and a set of vertices. The source and target theory of graphs is called the source. Any structure of that sort, you could always imagine picturing inside as a graph. Whereas the reflexive ones have a chosen loop at each point. The structure of that's the key. I'm just here showing you why this particular example is at HL2. Because this is locally that, this is really one of these five open sets that's at a covering point. This is the fact that the value on the whole space is something, some set, and the value on each... And it turns out that, well, of course, the value at the empty set is empty, and godless third point is there. So there's no further condition. Values on the five open sets with the covering condition is the same thing. These values are the two generating open sets in no condition. Now it's clear that you can get a diagram of this sort. The approach and operation is much more careful. It identifies these two dots. Without identifying these two restrictions, they're still free to be any arbitrary concept of source and time.

40:00 I'd like to detail this one. Vibration is in the case of the past distinctly. It's an aspect of... I'm sorry, I'm trying to understand the way that there's domains of variation. Again, it's a qualitative one. It has something to do with unique lifting paths. In situations where you have unique lifting paths, it's a very special category thing over the space. The space itself and the things above are required to be connected. And then you take the topos generated. Johnstone defined the QD property by... Coming at it from a logical angle, every object in the topo, not just itself, every object in it should be covered by something good, namely something that's decidable, something that's diagonal, as an honest, knowing, silent, compliment.

42:30 Actually, that's what I said. Chat Noir and I decided to call it separable. Yes, I was going to say. Because it's an algebraic geometry, it's exactly the same. The sort of thing traditionally called separable is the polynomial. The polynomial is separable. It doesn't have repeated roots. But if you take the polynomial that does have repeated roots, its graph is such that, you know, that the ambiguity at the actual zero is actually an infinitesimal which is implicit. A minus A, A minus A1, A minus A2 times something else as your polynomial. If you look at it as an algebraic space, it ought to have, it ought to have, not this, but you let, you let A y approach A 2, and you get a repeated root. So the idea is that that's a null-potent curve. You're getting, in effect, you're getting, you know, x minus a squared zero. Potents are implicit in the polynomial. Do that repeatedly. Separable polynomials are known without repeated roots, meaning... Given any two roots, the difference between them is an invertible quantity, so they're definitely separate. On the other hand, a root with power, you could have varied it slightly. You couldn't separate the two from each other. You could separate logically. Instead of a1 equals a2, I'll say the difference is square zero. Then I have an infinitesimal, right? So infinitesimal or not is sort of tied up.

45:00 Decidability. Decidability in the sense of the diagonal. Diagonal being where all the equal things exist. And the complement of that is well-defined or not. Not well-defined, you have no focus. There's a fuzz. There's a fuzz on the diagonal. So there have to be no neutral in order to have the... Yeah, yeah, and all of that is definability. The separability is just sort of a geometrical aspect. Connected with. And there was a further aspect which you drew attention to. In other words, you don't, that's right, you sort of don't introduce any extra parts like going up along that region. That aspect was the un-amplified. Well, un-amplified simply means separable over a base. Yeah. But that says it again, you see. So you can take the objects over a certain base, x, and it's exactly the same QD condition. Unramified is the same thing as the QD dimension, it doesn't apply to the slice categories, and so again, the name unramified says, well, there's no branch above, although both spaces might have no focus, the big one sort of doesn't have any more. In SGA IV, they had the philosophical interest in this. In fact, they never addressed the question, is there a qualitative difference between these, they're all topos. So these petitoposites, are they really any different kind of thing? As far as the criteria that they were applying, no, you couldn't detect this intuitively.

47:30 It should be very, very different. But this is where I come in. I say, well, actually, it should be different. What is this difference? They didn't solve it. They didn't even address it, almost. Directly, they were interested in the relationship, not just in the nature of the... Poemology was an obsession. Poemology of a space could be computed in terms of its petitopos or in terms of its grotopos. Originally, grotopos was grotopos of, grotopos of a space. However, this always turned out to be just the original ilk of spaces slash that space. A kind of rotopos suffices to know the rotopos. So that's why it became, it was originally of rotopos, of retitopos, of single space. And the retitopos roughly identified with the space itself, just a more expanded picture of what's happening inside it. The cohomology was the same, so even though one, even though the big one, really bigger. We had the same pool of hours you provided for technical efficiency. This was why Giraud was interested. And then, in the same, in SGA IV, Giraud-Tendez has this exercise that we fondly refer to as the chocolate exercise. A list of ten, I think ten exercises, of which he was so proud when he got done that he said,

50:00 And when I asked him about it many years later, he was still proud of it. This was a key step, although I don't know of anybody else who really pursued this except Anders told us the other day that he had a student who actually worked all these exercises. I hope he sends it to him, having the solutions actually worked out. A site, or a protopos, in terms of its relationship to certain related categories, which are the sites for the protopos, at least of each representable functor, linking all these things together. Now these are small categories. There's a small category and several other categories related to each object and certain axioms. And then there's a discussion for construction. We give him a topo liquid called rotopos, also for each object at least in the site. Petit topos for every object in the topos is probably an easy to obtain, I forget what it is. Well, yeah, he does, he mentions that, he does that too. So he gets a petit topos not just for each object in the site, but for every object in the site. Jack, the construction is very interesting because the exercise is distinguished between... Schemes of dimension zero and schemes not of dimension zero have quite different properties with regard to this.

52:30 Sometimes the connection between the two causes even space in it. This is then the topotopos of the space. Schemes, again, I crudely call it, the schemes of the topotopos, there's some kind of interconnecting these. Chasts, a couple of adjoins, and so on. So the precise conditions seem to be, in some of the exercises, for certain conditions on a scheme X, for example, you get a local morphism, of course, Duncan was emphasizing the same as the local morphism method. On the other hand, in some other circumstances, different exercises, you would get instead a sum of all those which is actually essential for the precise exactness conditions. On the triple of counters, it could be, as I was saying in my lecture before that was here, no, problem one in that open problem is about the general wax idempotence. There are several categories of topos defined by idempotence, not necessarily adjoint on the left nor adjoint on the right, All toposes are defined over the same base. In particular, you know, if you see an interesting subcategory of a topos that's closed under finite limits, it might in fact be a subtopos if it has a further colloidal problem, or it might in fact be a quotient topos in the sense of the inverse image inclusion of a geometric model of a circuit, an epic. That depends on a different colloidal problem.

55:00 In essence, the T topos, whatever it is, whichever it is, there isn't involved in it a subcategory. There is a subtopos where the local quotient of essential inclusion amounts to further exactness conditions and further closure conditions on this subcategory. Presumably, the direct definition of a subcategory is easier. Then you can sort out that we accept that it's a subquotient, in some cases quotient, which led to this axiomatic cohesion, sort of very blatant properties that almost no particular course has, namely that the truth value object is connected. That's almost never true for any particular. The other thing is that, as you were perhaps starting to say earlier, the etendue, which is the monic sites, basically, into the etheric sites. The common generalization is sites that have no idempotence, no idempotent endograms. So that's a well-defined kind of cohesion. There was no idempotence, having idempotence.

57:30 I mean, certainly, the key and rule are not supposed to be just merely logically the opposite of negations. Lots of topologies are sort of a mixture of the two. But it seems that somehow the idea of degeneracy is a... Part of being a group of topos is we have something coming from generators, a site, a figure, a figure in X, X might be a very complicated space, but A is a circle or a line or a model that's discussed in the slice scanning where you call an incidence, to what extent a certain line intersects a certain circle, blah, blah, blah. So that's the complete picture of the inside. You can't see the inside of objects in a category, besides we see it that way, it's the same as the structure of this category, provided you know how to picture the individual A's, you have to have that data, but then the picture of X is just as an amalgamated complex of those special pictures, and the incidence relations tell you exactly how to amalgamate them using the global...

1:00:00 Okay, so anyway, by a degenerate figure here, simply one that's not a monomorphic, my teacher Eilenberg in the mid-40s introduced a singular homology that's based on applying all logical algebra to the complex of all figures of triangular triangle shape in a space, which are all continuous maps, not just a monomorphic map. Previous attempts received to deal with sub-objects... All kinds of useless trouble because to map one space into another will probably collapse some of those parts, so you simply take the singular figures as the basic ingredient and trivially those are mapped into their totals in their next space. But among the singular figures, there are special ones that can be called degenerate. If there's an item called A, then you can define X prime. If you can do that for a non-trivial hidden component, then the axis can be called degenerate. So, for example, if you have a triangle as your basic kind of shape, the particular figure in that shape might be a segment where it's all collapsed onto one end point. There's particularly simple ways of making a thing. I'm getting a singular figure out of a non-singular one by collapsing it along a hidden point. If that can be done, then the original one is degenerate.

1:02:30 So degeneracy is a special kind of very controlled singularity, controlled by these hidden points, algebraically calculating hidden points. The rest, the part that's left over, they still must be a monomorphism. Our sites have no input on the entry of T. We get the idea that in a Grotto post, every site has at least one singular figure, a non-trivial degenerate figure. This has something to do with the idea of an ilk of spaces, you know, as opposed to a single space, that when in an ilk you can always find this kind of situation. And that's certainly true of all the concrete examples. Do you mean by ilk anything technical, or do you mean just a kind of state? I got it from Max Kelly. He used to use technical terms. Well, they used to say the McKenzie of the British state, the McKenzie of that ilk. Yeah. The thing is that grammatically, as I understand it, you know, to say it of that same kind means the same thing as of that ilk. You don't have to put in the word same. Somehow, the tribe, you see, should have smooth blood or athletic blood or combinatorial blood. This is a really different type of substance. No problem. It's a really different tribe.

1:05:00 Plants, yes. So sorry to take, I just interrupted here with sort of a brief bit of a history lesson. I'd like to interrupt with a pee break. Do you realize you can get coffee here? All right, let me show you. This is scouting with their vengeance. You can't get coffee without money. This is England, after all, leaving capitalist countries in the world. They need 60p. Yeah, but any... John, I've got notes, but I used all my small change to get a copy before coming here. Just give me a second. I can throw one off you and... Let's see. I can murder that little copy. I've got nothing smaller than a fiver. Well, that seems a good deal. A fiver for 60p. Oh, yeah, well, Bill's got 60p. He's got... I've got 60p. There's 50. There's a 50. There's a 50. You'll have to liberate Bill's money without consulting him. No, I mean, I don't think... Are you going to speak on that? Good. Obviously not. Thank you very much. I'm the captain of the CCP. I don't care. It's okay. Follow me. Okay. Conference, gentlemen. I thought I'd laid out some hints of history. This is a conference with the other... I completely underestimated his... He was teasing me as we came here this morning. He said, he said, I know that you and... So I sort of groaned and said, what do you want to do?

1:07:30 No, it's going to be strictly in the sense that I can ask him a question. That's a very important subject. That's a huge one. And then I'm going to ask him a couple of other things, but of course the point is to get him to... I will consider his answers mathematical, he will consider them philosophical, we'll go through everything. I see, you just press this. Oh, I see, you just press it. You see, it comes out.

1:10:00 I don't think it's disappointed you. I also can't work out how to make this down. I really put the money in. Oh, yes, I'm sorry. I now see exactly what the strategy was. It is working for a little bit. It seems to be working very, very well indeed, and I think, I conjecture that we're ready to go. So I thank you.

1:15:00 The trouble is, it costs you 15 quid to buy the little patches, so in a week, especially if I'm not here in Bristol, I can go through the patch treatment, I've done it many times, it's very successful, but globally, it doesn't really work.

1:17:30 It did. I remember the first time I met you, you were rubbing your arm all the time. I thought that was a weird habit. Then I realized you were rubbing that bloody patch. But you haven't smoked since, have you? No, I haven't. I quit before, once after two years. Well, I had a funny experience with those patches. I lost all sense of addiction. I had a funny experience with those patches. I went on a prolonged, well, one morning I had a number one. I was sweating. I was sitting in a chair. But I was so agitated, I got up and paced around the room. I couldn't figure out what was going on, and I realized I was getting an overdose of me. Ah, I see. They graduate the content of the patches as you graduate and come off and lessens the dependence. The irony is that the times of the number threes cost exactly the same as the great big number ones. And this sounds suspicious to me, but they say that paying out that money is part of the treatment.

1:20:00 How much does it cost to save your cigarettes? Depending on how many cigarettes you smoke. Well, I mean, I'm going to go cold turkey soon. Not today. Oh, St. Augustine, St. Augustine. I'm sorry. Why does that remind me of St. Augustine? I mean, I don't want to be... You're making me chaste law, but not good. When all these Russian guys started turning up in the maths departments, and one guy... Nicotine interference. Nicotine. This guy who had a departmental meeting, I guess everybody probably didn't know him. We had a department meeting and it was decided that there wouldn't be any smoking inside the building at any point of the day. Of course, that's long since anybody was smoking inside the building at all. But I remember Vitaly Litskin, this Russian friend of mine, in the middle of it, when the vote went against him. Oh my God, he said, I don't see how I can come work in my office if I can't smoke. Well, we had a complete dickhead as head of department who couldn't distinguish between a cri de coeur and a threat, so he read this as I'm putting you on notice that I'm not going to appear in my office under these circumstances. So this guy wrote him a letter saying I understand you threatened not to come into your office. If this is not on, your contract requires blah, blah, blah. It was a long, unpleasant letter, which was then PP'd to the vice chancellor, to the dean, to the university visitor, Uncle Tom Copley and all, and this guy receives it.

1:22:30 Well, okay. He is already paranoid anyway. He's convinced that they're out to do him in. I mean, this guy just, I mean, I got these threatening letters from this guy. I remember one... Was that the same chairman that gave you? No, he was the guy that appointed that job. When he was kicked upstairs, you know, he hadn't reached his full level of adultness yet. He was kicked upstairs to be dean of science. But he sent me a letter. I mean, you know, this guy didn't, it never occurred to me to just take a colleague aside and talk to him. He sent me this letter saying, it's come to my attention that you are not... ...conforming to the department in marking your examination scripts. So the department put in a rule that you have to mark in red ink. Well, I mean, it's deadly, right? Because, I mean, you can't change your mind. You can't cross... or the record of your incompetent first marking is, you know, there for the whole world to see. And, of course, you have these external exam... So I always do... I always mark my stuff in pencil. So he sent me this stiff letter. It's come to my attention. And it was such a shock to him that anybody would openly say that they were defying, that he just never replied. I guess I had. But I didn't want that. It's just so... It's just a little subtler than this. The famous story about Turing, when Turing was signed, during the war when they were in Bletchley. They were all required to join the launch of the Herd Guard, which was a military... No, they weren't required. They wanted to learn how to shoot. That's right. He wanted to learn how to shoot, in case you know. So he won't join this thing. But when they joined up, they had to sign a document which said, by signing this document, you do understand that you are placing yourself under military discipline and under the king's regulations, you know, by joining the king's...

1:25:00 Yes, I, blah, blah, understand that I am placing myself under the king's regulations, etc., and the military discipline and the various codes inside this. Or do you understand it? So he just, of course, said no. Thank you very much for your time. The literary discipline when you join this organization. And he said, no, I didn't. I said, yes, you did. Because we've got it here. And then, of course, they bothered to look at it. And so he filled in when he said, no, I do not. Quite useful. Because, of course, 99 times out of 100, these bureaucrats never read those things. So he saved himself from being shot. Probably not being shot, but he probably would have been. But in theory, they could have done. I think even the British establishment would have been so stupid as to shoot their top codebreaker for having... I mean, the idea used to be that you were nobody else's business, however. It's fair. Speaking of community... Let go, yes, absolutely. The use will break. There's two of them, actually. It has now taken over the whole...

1:27:30 The whole country. The whole country? Yes, exactly. That's absolutely true. No, the whole country is just run by demented prefects, demented, the sort of people who were born to become secretaries of suburban golf clubs or tennis clubs or house prefects in very minor public schools. The sort of people. Can you imagine if they'd run Los Alamos on that basis? The military then would have gone crazy. That kind of person was getting... Absolutely. This is justly what we were saying, especially in this country. Oh, gotcha. It's a general disease. No, it isn't. We had for, um, and then the administration said his term should be another guy. A guy that I knew all along thought he was a regular guy. But as soon as he gets in that position, he starts behaving in that manner. Becomes a demented little jobs worker. Yeah, in that manner. Well, in Britain, it's called the New Labour Project.

1:30:00 You know that, isn't it? I mean, it seems like it's sort of like... This guy said... Absolutely. Absolutely. I had a more serious one. So he wrote me a letter and said, if you don't have any respect for me, at least have respect for my own people. Junior something. He was the head of a department in a provincial university. He was the Archbishop of Canberra, you know? I mean, it's just... No, I'm sorry, I'm not certain. It's not, there's certainly not a beach, no. No, it's not a cherry. Cherry has much spiner blossom. It's pretty trivial. Yeah, pretty, yes. It's not red, but they have red buds. So at least one non-trivial degenerate figure. It's a pretty dull one, it's not dull. Oh, yeah. It's pretty easy to do, if you can get rid of it. Well, the one goes with the other. It's because it's a... No, let's go, Ted. There's a relationship between this proposed doctrine...

1:32:30 Then, at least for, there's a standard thing, which is described in my Bogota paper on the same issue, and categories of space may not be generalized spaces, and it seems like a very, very, very special example there, which it is, but the procedure actually applies to any... We have discovered this input on x just to be again appreciated. Namely, we look at c slash x, which is the same site of this thing, but then we apply certain universal construction to it, namely killing all the inputs. So the square bracket is the fact that the category of categories, Left adjoint finite products along the bracket. Interpotents have this nice property that both the equation and the localization. If you want to think in terms of inverting things, you can just say, let's invert all the interpotents. Of course, that'll force them to equal one because they've already got their inverse built in. Or you can say, let's just consider the quote of the category modulo the congruence relation where we identify.

1:35:00 Any pair of parallel arrows for which, which follows, which would follow from identifying an integral with the identity. In general, the definitions, just logically the definition of this is x squared equals one entails as it has endo-camps in the category. So there's a lot of simple definitions, not just equational ones, but simple definitions of subcategories of caps. ...which have this property that you get that preserves products, which is equivalent to saying, you see, that if you have category D with that property and raise it to any A for arbitrary power, again, as that property, D has no impotence and D' has no impotence either for any A. What I'm trying to say is it's not totally hopeless to compute this. It's very confusing. You can try to actually start with a very little category that you know about and see what happens. Even though this condition seems to be localized at a certain object, of course it has consequences propagating possibly all over the category. If you actually compute this left adjoining, finding a way to compute it would be a good thing and maybe only found out the message. But you see, we first have to, right, I mean, we're going to apply that operation to the slice, you notice that we're looking, we're looking at all the figures contemplated in the original Grotto films. This is considered as an object now, so the incidentulations have become just the morphins and the figures are just the objects.

1:37:30 If you have an interval here and you kill it... What I wanted to say was that in that original computation, I started with the category all of whose maps were interpotens. Or that is, you know, splittings of interpotens. You have split epics and split monics and composites of those and sort of everything that's really all terms of interpotens. But then when you pass through the slice categories and kill all interpotens there, there's still a lot left. It's probably directly relevant to the traditional thinking of simplicial sets, because with simplicial sets, again, there's a question, what is the petit-tropos of the simplicial set? So the traditional analysis talked about the minimal vibrations. You look at the category of minimal vibrations, perhaps. Well, part of the minimality was precisely... In terms of the equivalent to saying that the one that you're looking at has no further impotent reductions, no further degeneracies. You take those of least possible singularity, least possible again in this sort of constructive sense of taking place entirely in the little category entirely in terms of impotence. So some of the these figures might still be singular but you couldn't tell them by your So, minimal in that sense, plus the vibration condition, this was a traditional way of getting very precise calculations about things of interest. So, in some sense, I think all that's going on inside the Petit Topos. The Petit Topos is maybe a bit bigger than most that are considered, but I think it's a passage from the general category to the minimal complex to describe very succinctly.

1:40:00 But it goes back to the 50s. Minimal objects in the category of figures over x are not so hard to compute, because the C and C' in that case are just what delved into the tetrahedra of the various dimensions. If x were 1, then everything is a retractor, everything else sets. Given figures here, then the minimality has more to do with the nature of x. That's exactly what goes on in the case of the sets. In delta? Yeah, delta. So in general, that seems to be a kind of boundary condition, maybe. But if you look at a grow topos, e over another base, u, like sets, then the petite topos of the one-point space is just u itself. That's the kind of boundary condition not realized by that prosodole that I made the other day. Sorry, let's let Matthias talk. Why don't you shut me down? Okay, let's continue around the time.

1:42:30 Well, my original plan was to go this way, but since this is the distinction, more importantly, let's go that way. We are in the mood to discuss the distinction between Rome and Haiti. Let me carry on. That would be my question. To complete, as before, some axioms were isolated in the moment of anger. And also a philosophical guide.