Evening discussions FW Lawvere (contd.)

0:00 Lou is just visiting Paris. He's, as I say, he's actually based in Kansas State. Kansas State, yes. Oh, he was at Chicago. Ah, yes, Tom Yetta. No, it's David Yetta. Sorry, he's a different chap. Tom Yetta's somebody else. Yes, he does a lot of stuff in braided categories, doesn't he? Yeah, which are braided... Yeah, braided manoeuvre categories. One arrow tri-categories. Yeah, yeah, yeah. Which these people... These people trying to do so-called category-theoretic versions of quantum logic have really sold on that stuff on symmetric by monoidal categories. They go on and on saying, you know, Bill's a dinosaur, you know, why is everybody hung up, why should anybody be hung up on Cartesian closed categories? I think they've completely missed the point. They haven't thought about it, they've just got a bright shiny new gadget and they, and of course they're sold on the idea that You should quantize everything, you know, including the kitchen sink. They don't have any hazard, you know, they don't have any, there are no open questions in their mind about the foundations of quantum theory itself. They just assume they have to take it as it stands and categorize a protocol version of the formalism. I think that's a completely misconceived approach. The contagion closed categories have been a problem for me. Grab all the fruitful restructure up in the closeness. And so, as I said, they'll have a Cartesian closed category plus another structure, they'll be much better off to have a first order structure, and at the end of it, add on closeness, so they'll lose a lot of the first order structure, and as soon as you get closeness, you get major coherence difficulties, and you're no longer, and you lose your enrichment in cat, not various. I mean, it was second on my hit list after mine. That's what I'm talking about.

2:30 I think it's a very important, very sophisticated and subtle piece of structure that computer science has focused far too much on. Interesting. Well, I think the problem with the symmetric bimanoidal category stuff for quantum systems is almost the exact opposite, that they give up on it much too readily. And I think they're desirable properties for any account of dynamics. I just don't understand what they're saying. Well, again, there's so many different approaches, but... They're unbelievably true. I think that's probably a pretty fair characterisation of their work. But on the other hand, that criticism is typical of almost any work at the boundary between, any work which involves kind of interdisciplinary... That connects two independently quite mature fields and even more in fact when you have one very mathematically mature field which has a great deal of conceptual technical depth when you're trying to transport theorems into another area which is much less well developed and where there are far many more open issues and it may just be premature to try and do that on the other hand in the case of some of these applications of categories in physics.

5:00 It's going to happen, since category theory is clearly going to be the unifying language of 21st century mathematics. Well, we'll see, but it's certainly a strong candidate. Yeah, I think it's a question of redoing the logic, of course, to the point that Bill was discussing yesterday when we were in the restaurant about Grotendieck's program, implicit program. No, sure, there's a separate, as it were, there's the contingent historical and sociological aspects and then there's... Yeah, yeah, yeah, yeah, yeah. This statement is to remember that we say a statement has three poles in it, we say it's category, and we have to follow it precisely. We have to find it. Five years, we don't know precisely. All wrong.

7:30 Well, I'm much less qualified to judge than either of you, but for what it's worth, that's also my very strong impression of Byers, having met him on several occasions. He came to speak at three meetings that I organised. I hadn't corresponded for a time, and there's just so much he just got wrong. One of the things which impressed me, you know, landed me with this impression. And where I think I am qualified to judge is he simply got wrong what other people have been saying in their talks. When I discussed with him afterwards, after Bill's talks, for instance, in Florence, he had clearly just not understood a lot of the things that even I as an outsider, maybe because I was listening to Bill there for the last 20 years, with scrupulous attention. And, you know, he was just... He's always in a hurry to get through. He's a very hasty person. He's got huge amounts of energy. I mean, the fact that he puts out that incredible, I mean, long before the word blog was known, he produced that voluminous thing every month. This week's find in Web Theater. And then there's Ben Caddick at Cafe Blanc that he runs and all that in addition to producing his own papers and giving his own talks, which of course do tend to be rather superficial, very general survey talks, you know, G as in Caddick is very wonderful, but even so, you have to admire the guy's energy, but the problem is that seems to me to be, a lot of it seems to be misdirected, and he does, if you ask him to give His take on somebody else's work, then you often find, I don't say always, but I certainly found on more than one occasion that he had simply not understood what they were saying, or he was taking a just fragment of what they said and trying to force it into a straighter action than he's ever been able to let himself. So I'm inclined to think his own work is probably suspect, although I'm not so qualified to judge that. So I have work to say, that's been my experience, so I've heard it made a big difference. Well the whole end category industry is a perfect example of that because he's its number one chief barker he just completely he's talked it up now for 20 years and so he's got he's got his own tame little house philosophy now the shape of David Caulfield who of course is quite influential because he's the only philosopher his only person in philosophy department who any most people in Britain have heard of.

10:00 Who has actually written anything about the n-categories. And of course who goes around telling philosophers that n-categories are the salvation of the world and are going to be the framework for the whole of 21st mathematics. Gee, aren't they exciting because they show that they've given up in favour of something much more general in some utterly handy wavy way. I mean, no, I think it's real marsh gas, this stuff, intellectual marsh gas. It's potentially very dangerous. My work is really good. I mean, I've done little bits of real, serious, kosher, but it has been hard, serious math. Yeah, yeah. Well, I've read Eugenia Cheng's thesis, and I've read Leinster's thesis, and I would not have been able to see... I certainly couldn't see the wood for the trees of Tom Leinster, because again, all I could talk about from it was that you've got these ten different... Whereas hers, she was clearly trying to prove... She was trying to track the correct notion of category equivalence for at least a part of the spirology. And until you've done that, I mean, I just don't seem to make sense. Your genius thesis is the best thesis I've ever had. I told you that I could sort of take home a car without any trouble. She was obviously trying to do something fairly serious and very worthwhile conceptually, which is, as you say, getting the correct notion of categorical equivalence for at least a part of this construction. I wish I knew more about homotopy theory because then I might be able to take in the... Which seems to me very strange you said that, because I understood from all the talks I've attended that even her construct intended to isolate precisely what the homotopy equivalent is between these maps and their lifting properties between one level of the NKAT tower and another is meant to be. I thought they were tracking the homotopies, not the…

12:30 I don't think that Eugenia is an algebraic topologist. I think she's a categoricalist. So the things that I find valuable from her today are the categorical, theoretic structures. And they're quite often taking someone's definition, you know, taking a few definitions and explaining how to recast that definition in a structurally better way. But I basically ignore any of the things you said about quantum mechanics. Oh, OK. No. On the other hand, it does seem to be quite fundamental to trying to provide a unified notion of impact. Well, that's the impression I got of it from our speakers. The definitions I've seen of high-dimensional category, I mean, there are so many of them that I... Yeah, there's more now than there was when Tom wrote his thesis. There's 13 now, apparently. Well, the last two or three that I've seen, including the one that I presented, did seem to require a lot of homotopy, but that's just my outsider's fate on it. Now, so I haven't seen any homology, but I do remember the strict N category. Yeah, yeah, yeah. Now, that is all about strict homology. I can see obviously how cohomology would be relevant, because it's after all studying sequences of maps and their lifting properties, which is after all what you're looking at here. Yeah, I don't know if he doesn't make words work at all, I'm afraid. I've heard these vague statements about homotopy, but I'm not sure exactly what Bill has in mind with what he was saying there about topological spaces. I mean, I believe what he's saying is... That seems to be a point about algebraic topology rather than about m-category theory, about how one ought to do...

15:00 He was certainly talking about homotopy rather than homology. I heard vague things about homotopy, but it sounds like it's just a book for a reason that makes a lot of sense. But so the Roth was coming out of homology. The Chicago people, I believe, were coming at it from homology. The next main thing I came up with was getting the definition of tri-category. Yeah, right, yeah, which obviously you do need to get right in order to have a satisfactory notion of categories, equivalence of categories for two categories, at least in algebra. And when I talk about Ross and Bob Gordon, and they've been coming out of homophobia. Oh, okay, okay. And then Ross, he's not very well graded among the categories. The notion of gradeable nodal categories is a one object, one arrow tricategory. The notion of gradeable nodal categories resets the notion of tricategory. And the way we got the definitive definition of tricategory was because of the coherence results. Every tricategory is tricomposable for a grade category. So what are the right axioms for a tricategory? Whatever axioms... Make every tricategory triacompatible to a graded category, so that turns the axiom for you and you can have the constructions, basically, you name a category. So that turned the definition of tricategory and it didn't have any homotopy anywhere in it. And the bicategory came along with it by... Yeah, those were strung back, I think. I just saw a bunch of people coming up with a bunch of definitions and the pan sort of came up with something and I couldn't understand his definition and I still can't understand his definition. As you say, there were 10 and counting, as I say, I think 13 at the last count. It was January last year that made me laugh a lot. I've lost all the more now. I found out there was something about Michael McKay earlier in the media, and that was Michael McKay had this stuff by his dog. Yeah. But he's very much in the character too. I know his work. I mean, I've read some of his papers. I've attended a few of his seminars.

17:30 Yeah, he's an absolute nightmare, actually, because he doesn't... In my experience he doesn't give very clear talk. There obviously isn't very serious ideas there indeed, but he's not the world's greatest expositor. Yeah, I've seen some of it. There's also a very fine, two years ago at the place, and I rejected it, and I mean up at McGill when I was a PhD student, and I thought he was far and away the best, most talented category theorist in the place. Yeah. Well, he's clearly got extremely deep ideas, and as I say, I've taken a lot from reading his stuff, but you can see that he might... You've probably talked to him now for the last time. Yeah, he is getting old. I mean, I heard him talk three years ago in Budapest, and he was obviously getting a little old. Right. So he's probably, he probably can't go through now, but then his thinking came from logical stuff and understanding the notion of equality. Yeah, yeah. He came up with a sort of partial definition. But my view is sort of geometric, but in a funny way. We're going to have this question in case things don't happen. The best way to do it is to hear them about it, and on a plane, get them from that to generalising, to two, to anything, because they're thinking of it all. Which made life a lot easier in some respects. So there's sort of a paper sorting out that. And then nothing to do with homotomy. No, no, no. I mean, in the geometry, it was not used to study geometry. It was using a little piece of geometry. Yeah, to study logic. Yeah, to study his...

20:00 To find out how to write things. So those are at least a few different definitions that didn't come from homotomy. But then you've got all this Andre Joyal stuff. Which I don't really understand. I'll go to this question, but I'm not convinced that he was wrong. Yeah, I think he's wasted his talent, basically. And I think he was spent a lot of time on the present. Ah, that's very, very useful. Very helpful. Now to me, do you get... We were talking about Mackay's, some of Mackay's work on generalising equality. It looks like it. Apologies in advance if I have to slip out.