Evening discussions FW Lawvere

0:00 We mentioned that we're calculating some things about eggs, all these organized around one row of eggs.

2:30 The thing is, nobody really understands, in general in particular, what we're independent of. And what is not clear, really, is the precise relation between... You mentioned, in Cambridge, of having one of these things, and for each one of these things, associated together with a geometric morphism during the talk. The growth opus of shapes on manifolds and then considering a connectively group point and then what happens?

5:00 Nicholas defended himself as well as he could. In that case, as in a simplest case, much simpler than proposing, these are embedded in here, or any sequential space, a slice and then shifts over that sequential space in this vague sense. There is a well-defined equilibrium topos on the sub-objects. Some objects of X, one kind of example of this, and indeed the original example of Giroux, their applications received these etendus for a pen. In this case, G-sets for G a group ought to be not M-sets for M a monoid. An example that must go along with the Macaulay thing.

7:30 Universal covering spaces and sort of pops out the fundamental group, for the fundamental group is on the same footing, the topological space. The Petit Togo says you cannot be reduced to, should not be reduced to. On the other hand, this is a very instructive example that should be looked at. It's just not as typical. They do, yeah, they open, you know, topological space. Well, as I say, really that's just a special case. From any topos over sets, you can take the locale of sub-objects of one and get the topos, and for change to e over x, do the same thing, the variable x with xd, then you have a doctrine of the type being solved. Take the anchor of connective locale and locale factorization. That gives you, for general reasons, a locale topos out of here. But it's a naive thing, okay, you have factorization in the system. If you want to look at ectopos, you get one, but that doesn't mean that you can, that this is, that has all the information that it should have. So, it's too naive for... In fact, it may, probably does have all the information in the case of, so you're saying that's a subtopos. This is, it goes that way as a geometric... The inverse image along the, first of all, hyperconnected means what?

10:00 What is the truth value of an object, which is the thing in which direction it goes backwards? Again, in terms of a subcategory, it's just the subcategory generated by some objects of one, or some objects of x. Maybe one should mention the idea of germs. Traditionally, in chief theory, one always talks about germs, or germs of analytic functions, described by power series. The point is that you've got a space on this category of opens, e sub x, in one section. This is giving you an answer from x up. We use all the sub-objects of the facts. You can factor this, both the sub-categories of the pre-sheaths, the satisfied gluing conditions, this factor with its left agile.

12:30 Even going all the way to pre-sheaths on that post-heft, there is a left-exact left agile. You do not really want to, so you don't want to . And that comonad is giving forever e, nothing over e, to this new map of the germs. The fibers of this new map are the germs. And the adjunction, the adjunction, the evaluation, is more than a point, but at least you can evaluate it at a point. The value, practically the construction of the left adjoint, is precisely the filter co-limit over all the neighborhoods at the same point. Over all the neighborhoods is what's called germs.

15:00 For those of you who know general topology, the concept of the interior operator, it's exactly an interior operator, except it's at a categorical level instead of a post-step level. In other words, the interior of a set is contained in the set. Well, there's that math. On the other hand, if you do it twice, or in the post-step case, you get an impulse. But instead, in general, I would say a combonat is not impotent. You have relationships between the square as you pass through the associated co-set. And the fact that it's left exact corresponds to the fact that an interior operator preserves the finite intersections. A core operator that preserves finite intersections is impotent. That's what's going on with interior operators. The left exact cohomodad, more refined categorical version. What's the closure of it? There is not so much on the surface. I mean, it's an ordinary topology there. Yeah, but the whole sheaf theory, the idea of a classical sheaf is that it's a generalized open set. It's an open set that happens to have multiplicity at each point. We've got stars. And I'm just saying that in that same spirit, in fact, you could start with sets over, just abstract sets over the points of x, but you're assumed given a basis for topology, topology, but that's just free travel, but then the obvious adjoint pair exists, and you can take, you can take, in fact, there's a name for it, not necessarily continuous sections, and that's the new one word, I think.

17:30 You take the not necessarily continuous sections, the left adjoint of that, again, is a filtered co-element, hence its left exact, therefore the composite is left exact, hence the co-element, so the left exact co-element, i.e., in this sense, on just sets over x. But now you get the actual category of sheath by taking the image of that arrow. So this is a reformulation of the classical definition of sheath on a space, is to start really with... Just the set, you have sets over it. You need a basis in order to get this filtered. You have two neighbourhoods, two neighbourhoods that are the smaller one. That's why he left Adjelington. And then the ordinary category is C, factorization of two, something that has faithful inverse image, which is the same thing as coming from a profaner. And, you know, full and faithful on the other side. For a while, you know, for a while when people thought you can't actually generalize the classical topology until there's a vast area where you also have simple things on the left. So the germ, the germ point of view is to what extent can this germ interpretation be seen in a proposed doctrine of the T-group.

20:00 Please forgive me. This is the idea of the... Can I just... All this... It was just to say the following. At the time when this philosophical guide was formulated, it seemed to have already legitimized the distinction. It provided a way to organize the thing. And now we are at the point where we actually have an explicit definition of growth. At the time where right now all the intuitions that the community had have a precise place, and there should be a way. To understand all these possible associations of a petite thing to a gross thing, in the following way, you have ended in the axiomatic equation, and now we need out of that a theory of petite things, so you should come out of the gross setting, a cohesive universe over a base, a way of being, a general category of being over a certain base, and now all the categories of becoming, the naive things.

22:30 We already know that the work in general, the slides, be a general theory, taking care of all the different examples, requesting something here as additional structure, seems the wrong way of going, and in general we will not be local of examples as yet. The examples suggest that that's important. Localness will serve in some contexts, but not in others. I'm taking the hyperconnecting local effectorization into just one question. I'm sure Bill has refined the thoughts he had in his Colombian papers to put in place to come in mind concerning this problem.

25:00 It doesn't have to be right now, but that was the third question. Actually, the second question. The question is how, in that relatively specific AC paper, we define the doctrine of it. Do we have enough information? So do you have an idea of the key topos associated to a community ring? Not to a community ring in general, but let's go to the greater thing. A ring without... A ring without a classifier for rings. It's greater than some topos. It's not in general, but my impression is that this representation theory is saying that

27:30 Represents have a lot of associated shifts on its spectrum. That's the right petite topos associative because taking global sections give you the right thing every time. Is it just interflux or is it torsion in general that we're worried about? It's just interflux, right? In this case... Is torsion just completely... Much torsion? Multi-admitted. Unipotence. Oh, oh, unipotence. A torsion-free group is something that has nothing to do with anything else. Right. Unipotence. The unipotes are just, it has to do with describing, splitting things up. Well, there is something. There is an analogous torsion of x to the n-th. It doesn't seem to be right. Well, certainly not on the surface. I mean, there are such, those are mere examples at this point and not part of the general theory that I'm aware of. But it's popping out anyway. This is something that's associated to any extensive category. You have sheaves. Any extensive category may not have anything at all to do with rings or rigs or anything like that. It's wherever you got it from. There's a very superficial kind of topology, namely disjoint, finite disjoint covering topology, which is sort of...

30:00 It's the thing that makes the unative embedding of the original site preserve sums, as he said before. That's all. So in other words, the intuitive idea that the functions on a disjoint union are just arbitrary pairs of functions. That's the link. It doesn't matter what function means. It could be values in anything. That should not disturb the inclusion. And if you look, as you were doing, specifically at the rain case, Unravel what it means inside for it to be extensive is the story about orthogonal interposes, where it's nice, you know, it's nice to sum 1 equal to 0. Notice that that implies that each of them sums. You don't even have to say that. This is an internal description of a partition of unity. The extensive property is saying that we have that sort of thing externally that the objects of any extension have. But he's drawing, he started before we interrupted again. We decided to draw the correct conclusion that somehow the Petit version now should have something to do with that and nothing more. Because in the practice of algebraic geometry, they do more sophisticated things. They have stronger topologies which take into account things like locums, and they tell us about them. But to be sort of fair, you don't want to go into that. You've probably got to be somehow an idea of what Petit was just based on.

32:30 Is that right, what you were going to say next? I wasn't going to say it like that, but that's the impression of what he gets from this thing. Each category of the right indicates that it gives all the information that you need. Somehow the idea appears. The Sarinsky representation theorem says that if you are considering this category of being, the deep thing that you want to represent is the Sarinsky spectrum.

35:00 The theorem says that if you take global sections, it is always the case that the right comes above the right. Even if slightly modifying this, but by adding hypotheses, there should be a way of extracting the right that you've not shown. In typical cases, this manifests itself as a representation of the theorem. In a more general case, it manifests itself as a representation of the theorem. It's a kind of boundary condition, and if someone proposes a scheme as the answer to our problem, in general, we apply that to Z or G that get these two. Those are the two different things, particular things coming out correctly from one general concept. This is some support of the general concept. This part should work in some generality of ways. For any, if you take any extensive category, small extensive category, then that's a way of, as briefly mentioned in a paper like that, it has a kind of like a fundamental algebra. So that should serve as a spectrum. You can follow that line, grab that spectrum and check whether, in what generality, an extensive algebraic category gives the right name. It does in the case of rings, many more extensive algebraically.

37:30 Which should give, how do you say, some support for... Oh, we stay here. At some point I said I was going this way. We were talking about, it looks like you're going down to a bridge somehow. That's where I was going to say, I've got to go and fix my brother somewhere. I thought you were leaving early. Oh, no, I couldn't possibly say that. Well, you've got to write that. I don't get it now today. I don't get it now. Well, I didn't work very much then. I was here for one month. Did you succeed with what you wrote? Good. I was going to suggest that we eat at, maybe, the little place we ate at where I tried to duck out. Because I can get away with it this time. Yeah, that's reasonable. Oh, this evening, you mean? Yeah, yeah. When will we finish here? One quick question, John. I don't know where she wants to put her hand. I don't want to switch my briefs, but again, is this going to be again? A hundred and five times as I listen. Either we can't use this room if we use my office. I'm going to visit. I'm visiting. Well, okay. We can squeeze it if you're up.

40:00 Yeah, yeah. We should make it Santa Claus tomorrow. Yeah, but I'll see you at 7 o'clock. Oh, yeah, yeah, yeah. I think we should go... No, no, no, let's go. Yeah, thank you. These would be welcome, if somebody had a question. It's a close agreement. Actually, this is exactly the wrong thing to say. It's very fruitful, like proceeding. Yeah, at some point, I want to talk to you about my own stuff, because I've got questions about the connection between stuff I've been doing and synthetic differential geometry. Well, we've got... We'll get that in. Why don't you, we were actually saying last night that you have some ideas about how to find out how this works. Yeah, well, yeah. Well, I mean, what they're trying to teach me about this business is the flexibility of R in this synthetic, the symbol R. R just splits into lots of different things. Well, maybe we can talk about that tomorrow.

42:30 My problems are much more basic than that. That will never work in the finite context, precisely. I mean, you're talking about the finiteness? Yeah, yeah. Richard and I mull over this stuff. We get curious. Things were quite different as well. I mean, in the first place, we didn't start with the notion of square zero infinitesimal. But you get something very good. I'm sorry you won't be here. Thank you for your attention. The only thing I have to let you into the audience is that you're in there. You're going to leave them now on YouTube. Just to bring things out of the way. No search. No search. Okay, well, I don't know. Let's go do another piece. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you for your attention.

45:00 There must be, there must, there must be some change. And there's, there's, there's, there's, there's, there's, there's, there's, there's, there's, The generators themselves are already in existence, but the inclusion of generators is really an important point of reference. The way that these are mapped is very much the same. It's not the kind of circuit, it's not the one that sort of still tells the spell out. That's kind of the thing that has to be memorized throughout. I don't think it's a mere logical thing to talk about, because there's just some point at which, there's some observation at which this thing can actually be very strong, but just looking to have a two-pointed, very precise, very intuitive, I mean, sense. Totally, right. I mean, in this particular sense, the fact that it's a concept of being the unit for the universal population. Yeah, I mean, they're kind of uncomfortable because they're really all there within themselves tonight, and we're able to show which is connected to the other side of the balance, which is what prevents them from making sense of some of the other things that come up. Maybe, yeah, some of the other organizations today think that it should be like we did back in the old days. We understand that we try to give benefit to the aims and the aims to the aims. Well, I don't think that's the way round. Well, I suppose it's not the way round. Well, I don't think that's the way round.

47:30 Well, I suppose it's not the way round. Well, I suppose it's not the way round. Well, I suppose it's not the way round. Thank you very much for your time, and I look forward to seeing you again soon. Yes, that is so far the question, isn't it? Yeah, but that's, is that a question? Yeah, that is sort of, it's just simply, yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Okay, well. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Yeah, it's a good question. Thank you very much for your time, and I look forward to hearing from you again soon. Of course, every function is uniquely adjustable, at least it's not canonical, it's a canonical actualization in terms of its official status. I mean, that's a very strong notion, being a user, it's a very strong notion of the...

50:00 There's something definitive, I think, of speaking unit in terms of not being homomorphic. You can't imagine anything stronger than mathematics, which would allow mathematicians to somehow set all of those things down as content, but then something weaker than that would be not only that, but something that's unambiguous. I mean, it's very interesting. I mean, it's very interesting. I mean, it's very interesting. I mean, it's very interesting. I mean, it's very interesting. I mean, it's very interesting. Thank you very much. That's great. Thank you very much. Right, so I was just sort of generally interested in how you treat these things, foundations of history. I got interested in how you would treat Casco theory and science theory. But as far as I can see, and we will talk about this, it's not been so much explored. These natural number objects are weaker. They don't have full recursive properties. The definitions I've come across, yours and the ones that Fry's dealt with, they're all equivalent and you immediately get the tiny two natural number objects are isomorphic.

52:30 And that's somehow guaranteed by the strength of the axioms for the underlying topos and its constructions that are permitted by the fact that you... Well, no, I mean, it's just because they're, it's a, adjoints are unique up to isomorphism. Right. Recurgent property is stating that something is a left adjoint. Sure. Sort of independently of what the properties of the ambient category are. Universal within it, you'd be a universal endomorphism within it, determines a unique object. Right. But I think my speculations about finiteness haven't looked instead sort of at a level of the category of finite toposes, so that these toposes themselves don't contain natural number out there, but they're sort of, their corresponding bird-side rigs will be approximations toward non-standard toposes. I think he got a very, very good answer, namely that, first of all, it's obvious that it can't have, but the best approximation of it are, in fact, totals is, for example, a more general category that contain objects x for which x is isomorphic to 1 plus 2x. So if you abstract from that category of the corresponding rig, then this element, the element corresponding to that object, and then you go a step further and you force the rig to be a rig by joining negatives, then you can cancel in that equation. Subtract an x and you get x equals 1 plus, no, 0 equals x plus 1, so x is minus 1.

55:00 Yeah, yeah, same. Minus 1. You could put it that way, that in the virtual minus 1, you could always have, just by ringifying the rigs, this particular kind of category is not really virtual because it comes from an actual object, even though it doesn't have the full equation in the category, of course. Thank you for your attention. Yes, very good. So the only people I know in that particular area who can let mathematicians get away with it. The only people who have maths suffer from serious maths envy. No, I don't think anybody went into either. Topos theory or interphilosophy with the intention of making big bucks and thumbs on seats. I mean, you could have made some kind of lecture on that, occasionally having had ideas of making money. Yeah, but he doesn't need money. No, he doesn't. Well, of course, neither do most of the people who are obsessed with making it. It's supposed to be worth at least $100,000. I can't believe it. I mean, he was early on in Sun, right? And I remember he mentioned to me at one stage, you know, setting up a company or something. Who is this? Oh, Vaughan Pratt. I didn't know that he was Sun.

57:30 Well, he's done a lot of very good work. Well, some good work. Does he ever put any of his wealth back into research, into foundations, into, into research? Well, does he ever put any of his wealth back into research, into, into academe? He's effective in one or two of the positions at Stanford. He wouldn't do theory except for him. Oh, okay, so that's... Now, I don't know whether that directly is money, but... No, it's still... ...it's still in your capacity to be equal. That's interesting. There are such people even in philosophy, but the problem is they tend to be people with really rather wrong-headed agendas. The obvious example, of course, is Patrick Soufis, who was the bête noire of... Clifford Truesdell, he has a huge amount of money which he spreads around in Stanford, also in Stanford, but I don't think he'll ever spend a penny on category theory because I think he's still probably still morbidly resilient. I think you may be the leader there that we know. Yes, but in terms of proportion... I didn't know about Wesley. Wesley Floyd obviously had more money. Yes, yes, than I ever dreamed of having, or ever have had or would dream of having. But I didn't follow his career until I knew him in Sydney and briefly, in fact, meeting a few organizers in 89. Yeah, that's right. Again. This is great. Meeting at Cambridge, we just ended. He was a master's student at ANU.

1:00:00 That's right. He used to visit Sydney. Yeah. Then he went to do his PhD. And then Mark Storman, the guy who convinced him was the cash pyjamas and brought him to Edinburgh. In fact, he was absolutely super-fetishist. He was right on the border of being a charlatan. Are you shaking your head? No, I'm not shaking my head at all. I'm just a little... I mean, I knew it, and I saw it. No, I wouldn't presume to not qualify to join. They submitted one paper that went through, so there were five referees that got accepted somewhere, and you can't even talk about it now. A few years later, they wound up submitting the identical paper that went through all the referees, and it got accepted. At that point, a student of mine actually made that. ...the main theorem, but he couldn't figure out what the definitions were supposed to mean, and when he sorted out some reasonable notion of what the definitions were supposed to mean, not only could he still not prove the main theorem, but he couldn't see even a sketch of a possible proof that would make him sort of go about it in that way... And that was pretty symptomatic of what was going on, so he got out of bed and went back to Australia, and that was about the last thing I heard of him in March, until he went off to USA and got involved in high finance. It was in the US for his high finance, so this is what he was involved with. Other firms or financial stuff. Yeah, which is what he's been doing for the last few years. When in Edinburgh we found out that he was involved with derivatives, we thought it was hilarious, we thought it was just perfect, perfect wisdom, right? I mean, all the high flying, all the kind of loud sort of stuff, there's basically no content beneath it, right?

1:02:30 That's sort of the nature of derivatives and that's the nature of his research. I certainly agree with you about the mathematics of derivatives and but I didn't know anything about his research. But was he involved in the mathematics of derivatives? No, no, no. He was Martin's student. Martin just didn't realize at this particular time that he was giving Wesley all these ideas. They weren't coming from Wesley, they were coming from Martin, but he didn't realize it. And then of course Wesley would go around and tell other people these ideas, but Wesley thought they were coming from himself, and Martin thought they were coming from himself, and so people thought, oh, he's very bright. I had a student like that once, and he also went into high finance. Well, he's dead now, but more precisely, he would talk to Steve Shanwell, and then he would come to me with these brand new ideas. He would talk to me, and then he would go to Steve with these very aggressive ideas. In escalating like that, he was sort of a communicator, but while he was presenting this as some advantage of himself, he made it a previous. Well, that's an old trick. I think people can convince themselves. Well, you see, that's why I've raised my eyebrows a little bit, because I only remember talking to Martin when Wesley was a student in Cambridge, and he certainly had a quite high opinion of him. Simply because of that, I had thought that he was at Cambridge and doing a PhD after Martin, and I think even Peter said some positive things, but of course they may just not have realised that. Well, even so, he could still do good things in terms of supporting mathematics.

1:05:00 But I'm very touched by your kind words, because proportionately, I mean, I have spent a lot more, but only proportionately, unfortunately. In absolute terms, I wouldn't even be a drop in the ocean in terms of what Wesley Fowler is able to spend. There was this meeting with Ross in Macquarie a little while ago, and I know he was involved with some of the funding for that. There's a joint thing between Macquarie and ANU. I don't know how much funding was provided for this meeting with Peter Dawson and Martin, but it might not have been very expensive. Right. Most of the expenses were taken care of by the registration fees. That's what I imagine he was, yeah. Well, he certainly did make a contribution, a significant contribution. I think we should be generous in acknowledging that. Even if he may not be a pucker categorist himself, he has done what he can to support the field. It's interesting to see, because I've seen a few instances of things that are a little in that direction. We were in academia for a while, but then go into industry, but somehow treasure the link with academia, and so they'll find one way or another, and then find that link with academia, which is very important. Like David King, he was a professor at Bauer, but he now works full-time for Hewlett-Packard Laboratories in Bristol. He's managed to arrange things so that officially he works 100% for Hewlett-Packard, 5% for the University of Bath, he gets no funding from the University of Bath, but he still calls himself a professor.

1:07:30 That means he still visits the department basically once a week or that, and so clearly for him the attraction is... ...being able to identify with the university. Because he could go and talk to people anywhere. Yes, yes. Rude. Yes. He could still talk to people. He would search with people all the time without being a 5% professor. Yeah, yeah, yeah. But, you know, he went through all this trouble to make himself officially a 5% professor. Yeah. And so... Well, that sounds a much more questionable motive and that just sounds like he's obsessed with QDOS and status. Yeah, but I mean, I would, you know, I think there's probably an element of that in other people, such as Wesley, as well. You know, they like, they love being involved in universities, you know, like the... Well, that doesn't have to be motivated just by their design. It may be concerned with their biography, how their biographies are looked at. Yeah, so I'm sure that's why people like Gates have ploughed so much money into academic research at one point. No, that's true. There are also instances of people who have gone into the, particularly into finance, and then come back into academia. It's quite striking, as you pointed out yourself, in the case of mathematical physicists, but the guy who is now the head of the department at King's College London, Which has always had a very strong reputation in general relativity, in fact it's always been certainly one of the two or three leading departments of general relativity in the UK, certainly for the last 40 years. The guy who is the head of the department there now, George Sparling, was a doctoral student of Roger Penrose's in the 60s, early 70s, and stayed in Oxford I think till about 78, 79, and then went off and made a huge fortune in the city. He was a financial mathematician but he actually became the head of one of these and then about five years ago he, rather sensible in view of what's happened, he took his golden parachute and applied for a job rather to the amazement of people in the field and having published anything for nearly 20 years he was given a position at King's College and isn't it now the head of the department there and I'm told that his work in GR is extremely good.

1:10:00 His thesis was in twistor theory, which is, of course, as a program, not really gone anywhere, and it's not a field in which people tend to get appointments, but he succeeded in coming back into academia in his, I guess, well, he must be the same age, isn't he? He must be 60 now, he must be about 54, 56 when he came back in. This is very unusual indeed. He was a very good relativist. A major example in the States is Simon's string theory. He found the technical renaissance technologies, which was and is a derivatives firm, which promised to take 40% return. Rapidly growing. I don't know what Wesley was doing. But this was a huge, one of the leading firms, and so recently he had millions to Stony Brook to build a new building, a big building devoted to string theory. Yeah, the little point was that he hadn't published anything for years and years, so he and Dennis Sullivan put on the archive a little paper. Basically some very trivial category theory within a differential setting. But anyway, the point is that he's back, you see. On the basis of this little scrap, he can claim to be back, maintaining his link, certainly maintaining his link with the academic field. Not only if you're giving too much money to it, but actually being active in it.

1:12:30 I imagine that Sullivan probably did most of the work. Interesting. I was curious, George Sparling has a slightly similar story, but in his case I think he was helped by the fact that, frankly, twistor theory, the twistor program had not really made much significant progress since the time he left it 20 years before. It was more or less stuck where it was, and they were just going round and round in circles discussing the nonlinear graviton. I think I saw Max again, because this was a big trip with Max, and I said to him, oh, there was a former student of yours at this meeting, you know, a former undergraduate student, and he asked me to send him his regards, and Max was there, and I said, well, you know, I mean, I... I can't remember all of them. You know, undergraduate students, I can't remember them all. I mean, you know, I don't, I don't, I don't remember all of them. It's nice that he said his regards, but, you know, I don't really remember them. And then a little while later, Max found out, you know, the significance of Vaughan, right? And he suddenly was, my student, Vaughan! I may have been the one who told him. In any case, yeah, yeah. That's right. Obviously, it was Peter Fry who came up with this estimate of a hundred million by looking at some reports, you know. Well, he'd be pretty well qualified to judge, wouldn't he? To see himself as a man who... Peter Fryder's not unconnected with the world of mathematics. I don't know, I thought he was... No, he's not really connected. He just likes to argue. Well, he always would give me the impression that... I'd always have the impression that he was... he was a banker, you know? No, I don't think he was ever a mathematician. He was once an advertising man. Ah, he's grown slightly in the telly. I remember that night we went out on the boat. Do you remember in 1990 in Como, the conference? Yeah. They had that excursion on the lake, on the boat. Lago di Como. I remember you sitting and looking at the machinery inside the engine room of the Patasima

1:15:00 and saying lots of interesting things about materials and science. But along the way, I remember having a conversation with Peter in the prow of the boat, and he was telling me about his time as a banker, but maybe it was actually as an advertising executive or something. I'm still a publisher. Thank you for watching. But he still sends in things to the archive, to TAC, and he still takes a very active part in the categories list, and he still comes to category 3 meetings. Well, I mean, he wasn't at the last one in Calais, but he might not travel all that often now, but he certainly, in the 90s, he was still pretty well-ordered. Strange guy. I remember last time I saw him was actually at the Topos III summer school in Belgium, in the Ardennes, in Haute Boudure, that Francis... ...was sort of put together in 2005, yes, 2005, and yes, he was at that, because I remember he and Bill having a long, interesting conversation about the natural numbers, well, because they've been, the natural numbers object being their joint baby, and also, so he's still there. Or in the sports process, soccer is looking at stuff. Well, he's a very interesting guy. Yeah, he has been a great deal. In fact, he would have done a great deal more if he hadn't been completely out of it. But he wouldn't be here if he hadn't been out of it. Very, very interesting. It could either be that or what I was going to say. Colin, we saw quite a lot of him. He said he really shows up in his book on allegories. Yeah. He doesn't look at me. Yeah, we were saying the other day, about 15 years to finish it. That's because he never was able to, yeah, it was shed wrong. Clichés. Clichés. You bet. That's what it is. You have to know all these clichés.

1:17:30 All right. And the fright cover of the topos, he did all of that. It's kind of a way of nothing. Yeah, thank you. He did a huge amount. He has some great paper. I remember having a long chat with him. His wife is a very nice person. She's very fond of him. We can be a bit of a, you know, tends to, when you first meet him, he tends to come across, you know, very aggressively, but in fact when you get down there, he's fine. He has a reputation. In your face, yeah, he's going very gung-ho. His style is certainly very gung-ho. When he lectures, he's very gung-ho. ...appearance of the colourful by using some historical example. Or another one we were thinking of before was taking the wind out of your sails. It means you've been rested with popular weather. That has a reasonably precise meaning. It's not just plus or minus. That one, no. It would be depleted. Yeah. Which is the most visual representation. Yeah. That's fine. Cat's pyjamas, it has to be the cat's pyjamas. I wonder where that comes from. There's a British or maybe Cockney retirement on this. You have these always stock cliches and so in order to say The whole content, you say the, for example, I'll see you later out here, it's a way of farewell, or after a while, but you can just say alligator. Yeah, it gets even more subtle than that, because sometimes you don't even say alligator, you say a word which rhymes with alligator, and they're supposed to know, because it is rhyming slang that they're referring to.

1:20:00 That's right. And there are other sorts of results, that most of them are rather obscene and can't be a rhyming sign, that's the way it is. Radiator. The cat's pyjamas, of course, has become in, well I don't know if it is still teen speak today, but certainly ten years ago, young people used to talk about something as being the dog's bollocks if they wanted to say, try and pose it very highly. Yes, also inexplicable, that's a cliché. Well, that's the one. There must have been some allusion, there must have been some story, but eventually, you know, it explains where it came from. Yeah, it could just be, see you later, L.A. Yeah, it could be rhyming slang as well. No, I kind of resent this culture in a way. To be honest, to admit, I often think in these clichés, okay, when I'm thinking about something. Well, this actually prevents me from coming up with an honest, straightforward, creative description of the problem I'm working on. It's a lot of, it's sort of like the music, you know, the music that grabs you and goes round and round and round and round and round and round and round and takes up a lot of your mental energy when you quit the game. I guess the cliches are designed as though, at least the people think that those are designed to save you mental energy because you use them without thinking instead of course, exactly, you use them without thinking. Yeah, it's become a very stale way of expressing myself, but it happens to everybody, I think. One thing which did come out of the twistor programme that was quite, I thought was interesting, certainly from the point of view of the things that my archive or this archive contains,

1:22:30 is that in the 1970s, Penrose sat down in Oxford. ...to teach himself sheaf cohomology, which he decided he needed in order to push forward the twistor programme to make it rigorous. And I still have recordings of the twelve sessions that he and Atiyah had together, where Atiyah was teaching him sheaf cohomology. That was, just as a historical fact, quite fascinating. Very fascinating, because obviously... Well, everyone thinks Atiyah's pronouncements on beauty and things like that. He's a great, very great mathematician, and to see him teaching somebody who's also a great mathematician, but he just didn't know that subject, and it was quite a fascinating, well, a very, very interesting exchange. He wrote some things that were in some basic topos theory. I don't think there was any people who wrote them. It's just Wesley Cushman and E. K. R. Sainz wrote them. So the overarching picture would be fine, but then if you actually try to find, where does he actually find something, you'll have a lot of trouble finding a definition. You know, I'd like to try to find out precisely if I stayed in theatre, you know, I'd like to try to find out precisely if I stayed in proof, you know, on the level that I've been complete from. But, I mean, the stuff, I think, the best I recall, I mean, it's now been 15, 20 years ago, the best I recall, the stuff, you know, the things that we had about topos theory and stuff, and we've got the topos in that work.

1:25:00 You know, the reasonable things that would mean work. I think a lot of them, it's very much like writing down what Martin Scroggins, and it's fine, but I mean, you know, people find it very misleading to listen to Martin, because when Martin calls, it sounds as though he hasn't actually calculated details, but he actually does, I mean, I've got about 12 or 13 children's papers with him, and you know, occasionally, they'll be, yeah, I mean. When it comes to detail, almost all of the time, you know, if we disagree, basically I'm right, but there have been a few occasions when I haven't seen how to do something and he says, well, that's not what I mean, I mean, he's aliphating, and then he gives a bit more detail and I say, well, yes, he does. There are a number of different types of students who are interested in the field of mathematics and physics, and we have a number of students who are interested in the field of mathematics and physics, and we have a number of students who are interested in the field of mathematics and physics. And I think being a Cambridge student doesn't help. I mean, somehow or other, the anti-Peter Johnston students, I think they're... Peter Johnston is Martin's alter ego. So, I mean, the ones who think it's alter ego, they pick up on it a lot, and a lot of them actually don't do deep math. Yeah, because from the speeches you get it. General feeling of varied and intercised speculative field. Exactly, of all this imprecision around it. So Wesley was one of those ones who'd sort of pick up on all the debate and all that stuff and managed to somehow absorb how to cut out with the words and all that sort of thing. And then, not without actually ever calculating anything, there was the opposite of that with someone like Eugenia Chegg. I mean, I've examined most of these theses, by the way. I mean, not Wesley. I've examined about six of them since then.

1:27:30 But Eugenia was the opposite. I mean, in her thesis, it was very, very carefully done. And we've ever been sort of out exactly right. But again, when she speaks, it's completely... So I had no idea that she could do anything because I'm just from listening to her talk. Yeah, well her thesis was great, but then after that it sort of scrolled gradually kind of downhill. I see. And she's been making finer and finer cuts on what she tried to publish. And it's becoming more and more trying to explain the ideas to an audience who doesn't know her. But then she went to Chicago and had a great time there and Peter May was very happy with her. He's not the kind to be happy with someone who can't do the detail. So that went very, very well. And Peter Johnston was the internal examiner, I was the external examiner, and he thought physics was perfect. And so, whereas there are other ones, Peter and I have both kind of pulled our motors, struggled through them. I think the funniest one, and he was by no means the worst, he was arrogant, but the funniest one was Tom Lenscott. He was the first one I examined. External examiners keep wanting to prove their worth being a Cambridge student. The way they do it is by being very tough and doing these incredibly long things and asking lots of questions. But then the question I ask is, we'll find out, does the student understand the thesis and the topic of the thesis? No, they've already heard Tom give us some talks, so they knew perfectly well that they understood the thesis and the topic of the thesis.

1:30:00 I was sort of nervous that I'd be going on for three hours or so. And I said, none. And he looked sort of stunned. He said, none. I said, yeah, well, they ask this stuff, and I already know the answers, so I don't actually need to ask him if I know those answers, because I already know them. And he said, there's a convention at Cambridge that all the things that have to last a minimum of half an hour. Can you ask questions to ask a half an hour? I said, yeah, no problem. And there's five-dimensional categories, and they've gone on forever. So I said, no problem. As it turned out, he was right, that there was no such convention, but he just panicked because for years afterwards I was telling people, oh, Peter Johnston says there's this convention at Cambridge, and my mother, and Peter would say, ah, well, not exactly, well, you know, you're understating a bit here. He's just panicked and just needed something to say, there's no convention on it, but anyway, so we get to the old fence. Peter had already written the post-Weibach report with the paragraph saying that we will... Anyway, so we get there, Peter asks his two questions, five minutes has passed. So I've been asked, of course, to kill off another 25 minutes. So he had this joint visualisation of this thesis of, you know, categories in which the monoidal categories, you know, like maths, and categories in which the bicategories. Have you read Max Kelly's book? And he said, no, it's not available in Cambridge. When we were sitting in Peter Johnston's office and it was on the wall, I said, well, actually, have you read the literature about categories in which you write categories? No, it's not available in Cambridge. What I want to do is this. You've got this very general motion in which you're trying to enrich the body of theory surrounding enrichment in the nodal categories or the body of theory surrounding enrichment in the bi-categories, because they're two different bodies of theory. And he said, I don't understand the question. And so as I explained it in more detail, I said, well, if you're trying to do this, if you're trying to do that, if you're trying to do something else, I said, what theorems have you got? To which he said, what theorems do you want?

1:32:30 This day, Peter Johnston was in fits of laughter. And I said, look, you don't understand the situation. You are supposed to tell me what theorems you've got. Well, anyway, by that stage the half hour had finished, so we sort of stopped. You never found out if you had any? No, you didn't have any. I mean, there were none in the, there were none in the chapter that never had any, you know, and so, anyway, the half hour had finished, so we just sort of stopped, even though I had part of it. So that was one of the more, I mean, that was, that was the most hilarious example, but I mean, and that was just picking on one bit of it, because a substantial proportion of my students just... If they just get into that. But Martin doesn't, and some of his students don't. And Atiyah, in fact, is never going to be able to do that. So we never found out if Tom Lansford had any feelings? He didn't have any. What about Sestran? Has he had any sense then? Not on that. I mean, he's got feelings in other directions. But not on that. What Martin said later, he said that he thinks Tom's got real talent in combinatorial type things, that he likes his abstract, his kind of abstract ideas. He likes the sort of abstraction that he's not good at. And what he's good at is the combinatorial. And certainly when he went to that. You know, we have the things of misery in Minneapolis, I don't know, five national categories. And anyway, he just gave all four of them. Did you go there? He just gave all four of them. He brought out all. Where is Eugenia? Eugenia? Eugenia. She went on to Chicago. Speaking of difficulties, I mean, Tom Lancer and Eugenia remind me, they both seem to take inspiration. He was a student at Buffalo for about ten years. That's where he learned category theory. But he never ever did anything. He never wrote an exam. After ten years, we decided that he needed to actually pass his qualifying exam.

1:35:00 We considered ourselves to be as generous as possible by just continuing to ask him questions. ...to see if he could finally answer one correctly, and he never did, and you probably think or feel that that was some kind of oppression, but actually we were just hoping, hoping that he would eventually say something right, so at the end of the day we said, well, let's continue tomorrow. So now again, if you look at this from outside, this sounds like some really terrible guy that's insisting that the exam goes on for two days, but he never ever answered anything correctly. You can throw around this jargon, tensor products, algebraic theories, adjuncts, whatever you like, but never answered anything correctly. It was precise and well worked out, and it was kind of thankless because she was bringing together, explaining the relationships between a number of different proposed definitions, and of course, as soon as you do that, like any one individual thinks, well, thank goodness someone's explaining why these other mediocrities should really be doing my thing. As Shannonwell received a request for a recommendation for a book from Baez, and Steve said, well, this is the guy, obviously he has some kind of intelligence somewhere and so forth and so on, but he never managed to do anything concrete, so Baez wrote back, that's exactly the kind of guy I want, so I don't know what kind of a position Goldman had, he doesn't have any degree, but he has some amount of support. John Boyer's got very enthusiastic about it. I mean, I'm not really impressed by John Boyer's. Somehow Peter May's got very impressed by it, but I can't see it, and Martin Ireland can't see it, and then he carries on about Dolan, because Dolan had these kind of basic ideas that he could have picked up from anyone. I mean, he could have picked them up from you.

1:37:30 Probably did. I don't know where they came from, but the ideas are basically fine, but the details of the whole thing. I suspect that's the idea. All of these things come from somewhere else, John Baez, though he doesn't look terribly bright, John Baez kept crediting him with all this stuff. The appeal of John Baez to a lot of these other people is, you know, in theoretical physics, kind of breadth. The alleged connection. The alleged connection, yeah, with civilization, kind of breadth and gravitation and stuff like that. Quantum field theory has something to do with four categories, you know, the big succession of M-categories and somehow related to natural reality. You know, in that meeting in Lawrence that you and O'Hare talked about, you know, Diaz gave a talk about that. And I was completely unconvinced, you know, there wasn't any connection. I think people come... Peter May... Yeah, I think so. And I've got a lot of time, so... I'm not convinced with Ross, but people come at it from different ways. There have been these questions about algebraic topology going down.

1:40:00 Well, the coefficients were more than good. And then some sort of higher dimensional category does seem to have a natural existence, sort of there. And so I can see how from algebraic topology you wind up with a higher dimensional category. But the mathematics in Cambridge was surrounded by some rather impressive mathematicians who knew a few mathematics, and I think they had some influence on him. The way I came out was completely different, because I came from maths, and we had all this work on. You know, categories of structure, and to do that you need some two categories. And if you want to look at, say, categories of essentially algebraic structure, at some point you want to have things like two categories of finite limits that relax above the other, and two functions that deserve finite limits that relax above the other. The only way I could find a period of decency for Hewitt's theorem about it was to evolve a weak version of three categories. So it's really a great category, so you get a great category, it's five categories, that sort of thing. And then by the time I got up to kind of think about three, I realized that sometimes it's just as easy to do it for M as it is to do it for 5. You're a mathematical tool for M. So that's, even when I, at that measuring thing, when I had to give a call, I mean, you know, they probably talk about applications to computer science. So I said, well I can't talk about applications to any categories to computer science, because at the moment... You know, it would take them an hour just to give a definition of weekend category, I mean, no sense could be assigned, they couldn't listen to me for an hour when I could have said the whole thing in terms of common epics, I would have said exactly the same thing, but I can tell a little bit about two categories, you know, and so I wound up doing that, so my angle on it was very much, you know, two categories, I need some three categories for... The three categories are two categories. Yeah, that's right. More than two categories with structures. That's right. So that's just a completely different stuff.

1:42:30 Well, that means that you were coming at it with the applications in mind. Okay, that's true. You were growing from the applications. You were just speculating. I mean, the speculating was sort of tacked on to the end to a genuine foundation. So I buy the argument coming from... I was a very topologist. I was drawing from all this stuff, cosmology, and you see the two characters out there, and then when I take them apart, there's a sense of need. I mean, I know I've got to keep them alive. Well, it's apologies, it's apologies. So, for example, Steve Chandler, my conscience. He doesn't buy the whole idea of n-categories in topology, if they have a relevance to topology. Even outstretched topology? Yeah. I mean, the point being that if you look at spaces and paths and paths between paths and so forth, that does not satisfy the equations of n-categories. It's a different sort of thing, right? The coherence of the... Coherent statements in terms of four cells between three cells. You can paste the things together, but then when you relate them to other things of a similar sort, they don't paste in the right way. Assuming that there's a certain algebraic definition of n categories. On the other hand, the concrete example of spaces, homotopies, homotopies of any type, that it doesn't actually agree, that it doesn't satisfy those equations. It's only after you pass the homotopy that they might hope they agree. So there's somehow the statement that there is this topological application at the beginning of the discussion a little bit. I'll have to jump because it's not that sort of algebra. Presumably it is some other sort of algebra, but the high homotopies do satisfy me. Peter Bailey seems to have something specific in mind. I don't think he's made an elementary mistake.

1:45:00 No, I'm sure what he's doing probably is okay, but there's this conjecture of Grotenberg that homotopy types in the classic life, I mean, infinity group ways, you see, which is one of the possibilities that these people hold up. But again, you see, there's a say. The thing that's derived from an actual topology doesn't agree even with the equations of mathematics. It remains an inigate, not yet achieved conclusion because somehow there's a starting point. Yeah, I mean, yeah, okay, so, yeah, I've seen that. I mean, I've seen a few different, I mean, I've seen people come up with that, which I thought was always, I'm convinced of, right, that, yeah, because they'd say these very naive things about topology, and then say, okay, so we build this drastically complex algebraic spectrum of fieldwork. But there's also these other people, you know, Peter May and that sort of thing, who aren't saying that naive statement. It's something else that they don't fully understand. You know, what's algebraic topology, what's algebraic geometry, which is sort of out of my area of look. Well, I mean, once you linearize, you see a lot of things become fulfilled. There's a gold column theorem that says... The abelian groups in simplecial sets is the same thing as chain complexes. Presumably there's the same thing in the sense of a non-trivial functor which is actually the equivalence. Presumably something similar to the cubical sets if they were properly defined. I never can find out what a cubical set is really. It's nearly zero. But at any rate, once you introduce pluses and minuses, you can do a lot more in the way of translating one thing into another thing. I think that's the part of what's going on there. You were already in the Pennywise situation, looking at chain complex, the complex of chain complex, that you would have... In fact, any given formalism of n-categories is probably applicable there, instead of all becoming equivalent.

1:47:30 In other words, the ten definitions in Tom's book probably become... If you're not known to be equivalent, if you linearize them, they might be more of it. That would have to those people to know how to do these fantastic things like Peter May. Well, I mean, I think he'd be pretty scathing about substantial amounts of the work around there. There is a conceptually hard thing about what does it mean to be a quid one, you know what I mean, because maybe two categories, a pair of categories, a pair of two categories, you know what a quid one is, but what if you got one? Oh my God. What's the definition of equivalence of character? The basic principle that you replace equations with higher cells sort of suggests you can make anything equal to anything else if you just put in enough cells. Yeah, and chemical surgery is including the equivalent. Within what definition of higher dimensional character? Meta-definition. You need a meta-definition. Yeah, in order to state the equivalence. There's quite a lot of care to be made by equivalents and this is where, say, Eugenia sort of cut down to this end of the axiom because the data, I think, was the same. So she could just see what axioms apply to what other axioms. You're not changing the signature. Yeah, I mean, I think she may have had some derived bits of the signature on me, but I mean, it was...

1:50:00 In a form in which you could make a precise statement, you know, a precise, meaningful statement of what equivalence meant. And so, I thought that was worth having. I mean, it took Liz to try to bring some coherence to what was going on. Whereas, what Tom did is, like, he listed these ten things, and that's it. I mean, there's no real analysis. Yeah, we were pretty disappointed in Tom's ability. We're expecting to get some insight into the variety of those and probably further confusion in our own minds. I don't know who, I don't know who will help. The thing is, Eugenia is consistently carrying things tighter and tighter and, you know, narrower and narrower, such as writing paper, because I give her, like, really fast numbers, increasingly little complex. But they're just writing more and more for a kind of a broad audience and so there's lots of explaining some basic ideas and some people just think they're quite nice. A broad audience of would-be physicists? Probably people who only know some basics and carry fairly sophisticated notions of algebra. You've sort of got an adjoint at level n, and you've got to build it up to level n plus 1, and so you need a little bit of reduction and stuff to put it out there. And then there's this guy using Macquarie. When I tried to read his stuff, I couldn't figure out how to make something precise. And then I read Ross's version of his stuff, and I still couldn't figure out how to make things precise. Because Ross, you know, there were choices going on. Ross, he evidently made one choice in his mind, but the thing he thought he wrote to say, he didn't get any choice. He didn't get any choice, he couldn't see how to do it. But he clearly had one thing in his mind, so he couldn't make that decision either.

1:52:30 There's a lot of things that I don't need to say, and probably it's to my very comprehensible purpose that there was that going on, but as you see, it's still increasingly hard to get a precise statement out of it at a level that's sort of appropriate for someone who already understands category theory. And if you ask Martin, you'd be getting these very vague statements, probably not much precision. Some are like André Joël, you'll be getting a very clear vested interest. He's got his own angle on things, which seems to be more appealed to, got more thought into, not as you think it's going to impress the people you're talking to, whereas a Martin would be scathing at that for some precise... There are a bunch of reasons which seem likely to be right to me, and Peter May's got his own version, but it's not the same as some of his... Peter May and that gang have got some clear programming in mind when it comes to mathematics and probability, so they could probably tell you about that program and why they're interested or what specific choices they're making. Martin could probably give some sort of overall picture and you'd probably get frustrated by the lack of detail, by just not understanding that he's saying this is wrong and this is right and you don't understand exactly what that is and you don't understand exactly what that is. You find it believable. You don't regard it. And with others you find they get swallowed up.

1:55:00 We were corresponding for a while about a project that the Columbia had to write a history of algebraic theories, or whatever came of that. Oh, that got published. Really? Yeah, but it's another history of algebraic theories. Well, that's how I understood it. Okay, well, that was... I never... Well, that was... Martin was doing all the corresponding with other people. The thing is that there'd been this massive process of a monastic notion to the computation. Some sort of account, historical account, how the thing came about, what decisions were made, what part, what seemed to be made, what was going to be made, and then because of this very delicacy, Eugenio, Eugenio, where did you go back and see films there at all? I would say on the opposite side of it. Martin communicates with all the people, Martin talks to you in the drama. This is an example of an academic lecture on mathematics, geometry, algebra, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, algebra, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics. This is an example of an academic lecture on mathematics, geometry, mathematics, and physics.

1:57:30 I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I don't want to get into that sort of thing. I mean, I qualify as the most complete outsider in this connection, but I have been going for the last six months in Paris to a series of seminars on applications of the category theory in physics, um, run by, and, um... Have you come across Lou Crane's work? Crane, Crane? He's an American mathematical physicist at Kansas State, actually, but he's published... My visit to Kansas State was for a conference about 15 years ago. Oh, he wasn't at Kansas State that long ago. No, I think he was, he was, he was actually a student. He was a student of Saunders-McLean. He was a student, he did, he got his degree, he got his degree in Chicago, and he was Saunders' research student. Good. So, he's trying to produce a version of what they call categorical quantum gravity, which is an essential attempt to unify quantum field theory and general relativity, which of course both essentially... They naturally live on tens of categories using two category theory and he's very insistent, and this is why I mention it, he's very insistent on this point conceptually that he also strongly rejects this n-category program.

2:00:00 He says that it does, in fact exactly as he was saying, it does have a motivation from algebraic topology. He has quite interesting things to say about how he thinks they ought to be using cubical sets rather than simplicial sets to do what they... And then there would be an actual cut-off at the level of two categories that have a perfectly decent notion of equivalence without having to generalize it all the way up by this kind of group quantification construction. And how good his stuff is from the point of view of the aims of the physics to produce the correct mathematical framework for unifying QMT. Which is obviously a linear theory and GR which is a non-linear theory and what kind of insights it gives you I'm certainly not equipped to judge because I don't have any kind of background in foundations of physics but he's certainly a bright guy from what I've heard he seems to have very much the same take on the conceptual issues as uh as you were presenting just now i.e he thinks that things should There should be a satisfactory equation of semantics for the whole construction at the level of two categories and doesn't want to go beyond that except where you need to introduce there are notions to do with equivalence of categories, but certainly doesn't want to do what these people applying n categories to topological, so-called topological quantum field theory are doing, which is two. To go straight into this whole infinite dimensional construction. So he's clear that's just generalization and there's no satisfactory physical motivation for it. And he's produced a very, he's re-done some of the work which Penrose's people in Oxford did. Some years ago on the so-called theory of spin foams, but showing how one ought to think about the whole notion of a spin foam from the categorical standpoint, which, again, I'm not equipped to judge, but there are category theorists in Paris who seem to be, you know, good mathematicians and who are reasonably impressed with this stuff. So I just thought I'd be interested in just flagging up his work in case you wanted to look at it. Possible. ...possible. He's actually going to be up in Glasgow at the meeting later this month with Byers and these other people, which is on, it's a title called Categorification, Geometry, Geometrization and Representation Theory. But he's very much against the whole Byers-Dolan approach for more or less the reasons that you and Bill have outlined.