Bruce: I'm sitting here with James Dolan, one of the leading lights in higher categories and so on. Um, James, I'd like to get your, kind of, higher bigger picture on the state of mathematics at the moment. At least, you were involved with John Baez in setting up HDA0 - higher dimensional algebra and topological quantum field theory -
James: HDA0... that's the name of a particular paper? I'd really have to try to remember which is which - I'm not very good at remembering which is which.
Bruce: Ok, no, but it sets out a kind of grand program of "categorification" in the mid 90's. To some extent, well this program has developed to a very large degree, and to some extent the very existence of this conference can be traced back to those ideas that you had in the mid 90's.
James: Mmm I do, uh, very much... I think of, you know, practically any paper that has appeared as a joint paper between me and John...from my perspective now, certainly, I always think of those papers as John's papers - that they present more of his point of view than my point of view. It's sort of a very difficult task trying to keep up with John in terms of writing things down. There might have been times when I did try to, uh, make sure that my viewpoint was being expressed, but I quickly found out that it wasn't easy to do that [laughs] and I kind of stopped trying at some point. And, you know, looking back at whatever papers appeared as joint papers between the two of us, I think of them as his papers really, in a way. And, uh, well by now I've probably forgotten the original question, I just wanted to make that point...
James: So, the... have I already forgotten the specific question?
Bruce: No I think you already, almost, partially answered the question. The question was really, to get to the bottom of...
James: Well you did ask something, sort of, about tracing the genesis of the kind of things that are going on at this conference...
Bruce: Yes, yes.
James: ...back to that paper and things like that. Um, I think it probably is true that there was this influence, for good or ill [laughs]. I think it is true. Um, you know, despite, or maybe because of the often ridiculously incomplete nature of things in a lot of those early papers that John wrote, with some involvement from me. Uh... you know, incomplete and vague ideas that were expressed in them many times. Um... I mean the way, the way I see things going, the way I see some of these programs being developed, things being played out, um, it-
Bruce: To what extent does that correspond with your original grand vision? That's what I'd like to know.
James: [laughs] In very weird ways. I mean uh, you know, I feel very much like a spectator and I'm curious about which way it's gonna go. It, sometimes it goes in ways, uh, that, that strike me as along the lines of what I originally thought, and sometimes it goes in pretty different ways. I'm sort of shocked, uh, at the directions it goes sometimes.
Bruce: Can you give examples of that?
James: Well, lets see... it's very tricky to give examples, but often, often uh, you know often when I see some very beautiful development of the subject, I think "Oh yeah that's what I had in mind" [laughs].
Bruce: So what's beautiful and what's ugly at the moment?
James: Well I mean, you know, it's easy to think of some pretty beautiful things. I mean, Joyal's whole theory of quasicategories. I should say, Joyal gave this very interesting talk on quasicategories at this conference. And, uh, I haven't really had a chance to talk with anybody about it yet. In fact, I probably missed some opportunites - I probably could have asked Joyal himself about some of it, although I'm... I think I wasn't really prepared to... I have a feelng I would have had to... Well you know, one thing I really do like about John is talking things over with him at a level of asking really stupid questions and things like that, and I haven't had a chance to talk with John about it. I mean, there were some things that Joyal said in his talk that had me really puzzled. But it could have been based on stupid misunderstandings, I may have misheard what he said or something. Um, this whole business about talking about reduced categories and how it relates to quasicategories and things like that. I've been meaning to talk that over with John. I'm sort of very puzzled about some things that are going on there.
James: Yeah, go ahead.
Bruce: Now, what um, about quasicategories do you like... is it one of your favourite models of ∞-categories?
James: Well, I uh, again, it's sort of just (∞, 1) categories. So in a way it's just a very mild step on the road to, uh, further things that you could hope to develop. But it is very beautifully done. I... I'm not sure I can add too much, I mean when Joyal talks about it [laughs], he always expresses how beautiful it is, and he's right, and I very much agree with what he says. And um, it is interesting that, well I guess that Joyal attributes the definition to Boardman and Vogt, I guess-
James: And I remember, in fact, before the publication of, before the publication of HDA0 - I think it was the publication - it was a long time ago, when John and I were, you know, first starting to think about silly ideas about n-categories, and, you know, not that we were like the first, but that we were doing it in great ignorance of practically what everyone else had done, thinking along those lines. And at one point I remember, we were just floundering around, we had no idea what had already been done, and at one point, and again this was, you know, this was long enough ago that it's hard to remember what the internet was like back then [laughter]. But I guess, there was some sort of forum on which John posted a question, basically sort of asking... some sort of mathematical forum - maybe it was the categories mailing list, it could have been the categories mailing list -
James: -and he asked the question of something to the effect of, "has anybody developed a general theory of weak n-categories", or something like that. I think the question was something along those lines. And there were probably several responses. And one of them was from, uh, whatever his name is - Rainer? Vogt? Boardman and Vogt? I guess his name is, I thought it was Rainer Vogt or something like that, and he answered, he answered John's query, and he answered it with this definition that now, apparently, Joyal has now adopted as the definition of quasicategories.
Bruce: So it goes way back then?
James: Yes. And Joyal attributes it to way back then. And when we say way back then, we mean way back before this, but I mean, this was the first time we heard about it, when he answered this query on the mailing list. And our response at the time was... well you know, first of all we just scratched our heads and tried to figure out what the hell he was talking about. We couldn't tell at first whether he was talking about, you know, we had no idea... I mean, our goal for weak n-categories at that time was that it should be, you know, I guess what in this jargon we would now call it (∞, ∞) categories. And so it took us a while to figure out that what Vogt was saying was just intended to serve as the (∞, 1) categories.
Bruce: I see.
James: And you know, it's sort of amusing, looking back at our attempt to figure out what was going on, and our response when we sort of figured out what was going on, so you know, I'm sure at a certain point we sort of said to ourselves, "Oh, so he's only trying to do the (∞, 1) case" [laughs], you know. We had set some ridiculously hard goal for ourselves [laughs], and uh, I guess that's a typical thing - we tend to, well often we set very ambitious goals for ourselves. We like to set them so ambitious that, you know, there's no shame in failing at them [laughs]. Although, we always seem to discover that, that when we thought that we were being ridiculously ambitious, we usually discover later on that we weren't really being ambitious at all. I mean it often turns out that, you know, things that we thought were ridiculously ambitious goals had actually been done, you know, ten years earlier-
James: -twenty years earlier, in some obscure place, that we-
Bruce: Can you give an example of that?
James: We ran into a case of that recently. I'm trying to remember what it was. Well, this is a little bit imprecise and vague, but, um, there's this concept that John and I have fooled around a lot with, under various names. Some people call it groupoid cardinality, or homotopy cardinality -
James: And at the time that we were thinking about these ideas, we had the feeling that we were doing really strange and bizarre stuff. You know, like, the idea of uh, finding something which when you count it, you get the answer "e", you know, the number "e".
James: - which seems like a really strange thing to do. And we sort of, you know, we really enjoyed the idea that we were, you know, at the edge of the bizarre. And, you know, as we later find out, these ideas are really not bizarre, they're really not even new, they're... we've just found so many ways in which these ideas have been anticipated.
Bruce: Who was the first to...
James: Well, you know, these ideas have been anticipated in so many different ways that it's hard to, uh, to say... um... but what are some examples that we found out, where these ideas have been anticipated... Well... well okay, so one example is, uh, at one point, at one point we uh, we were just looking for examples of spaces that had peculiar cardinalities. And, we started looking at, uh, surfaces, you know, surfaces of genus n, and there's various... there's various nice ways of calculating... Well, right, well one thing that happens of course is that you can simply take the Euler characteristic of a surface, but the funny thing is there's a way to interpret the Euler characteristic of a surface as the ... I forget what the jargon is, but the, you know, the sort of, the sum of the divergent series that you get by counting the number of elements in the fundamental group.
James: And, it's weird that you get the Euler characteristic, which is typically a negative number I guess, for surfaces of high genus. Let's see, is it... actually I'm sorry - I guess we're getting the reciprocal - yeah, I guess we get the reciprocal of the Euler characteristic
Bruce: ... of the Euler characteristic...
James: Is that right? Yeah, that's right. Right, because you took the Euler characteristic as the reciprocal of the size of the fundamental group. And, and the Euler characteristic itself is a negative number, I guess. So we're getting, you know we're adding up, we're summing a divergent series of natural numbers and we're getting as a result, a, uh, a number that's both negative and a fraction! [laughs]
James: And we thought, you know, we had
Bruce: ... something totally weird and new ...
James: Yeah, yeah [laughs]. We thought we had just violated all sorts of rules. And, you know, much later we found out, you know, that this was completely developed really, by other people. And well understood, in a way, these divergent series, and the circumstances under which you can make sense of it.
James: And um... you know, we just had this general feeling that everytime we thought we were doing something on the edge, we were nowhere near the edge! [laughs] They had all been anticipated.
Bruce: Now James, um, what I'd like to know is, you and John, I understand, made a definition of weak n-categories and sent it in a letter to Ross Street. Could you just tell us a little bit more about that, and why you chose to send it in a letter in that way.
James: Ah, okay [laughs]. Well, lets see. How did that happen... So, um, we had been working on this stuff, although a lot of it was, a lot of it was me working by myself on it while John was sort of galavanting around the world, like to Australia [laughs]. And, you know, I'd sometimes tell John, you know, how much progress I thought I was making and things like that. And, he would, uh I mean, this, you know, this is an absolute fact about John, that he overhypes everything [laughs].
Bruce: Oh, okay [laughs].
James: And he, you know, one thing he did was he overhyped, uh... I mean you know, I can see where he would try and blame me for it. He would try and say, he would try and say "Well you said you had this definition!" [laughter]. I said "Yeah, well, I thought I did, but you know, I didn't mean for you to tell everybody about it!" [laughter]. And uh, so one day he told me, you know I guess he had come back from Australia, or he was emailing me from Australia, I think it was after he got back from Australia. He told me all about how he had told all the people in Australia that [laughter] we had a definition of weak n-category. And, you know...
Bruce: So you felt compelled to write a letter -
James: - well I was very non-commital about it. You know, I was, I forget whether I was, I forget whether I said anything to him, like you know, "Why would you tell them that!' [laughs]. But, uh, in the back of my mind certainly I'm thinking, you know, "What the hell is going on here!" I mean, you know, like, I mean I thought... [laughs]. Never mind this particular incident, I definitely had the, I definitely had the general experience a number of times, of situations where...
Bruce: John would hype up and you'd feel uncomfortable...
James: - Well yes, but another phenomenon that would happen is that John and I would be talking about things, and we'd have lots of fun talking, and I'd tell him something as a joke, and later I'd see it written in a paper as a serious statement! [laughs]
Bruce: Any examples of that?
James: Uh, I, I could probably find some. The only one... there's probably some examples I'm thinking of that had to do with, uh... I mean it's probably just some pedagogical example, something like that. Uh... I mean, some of these examples I'm thinking of are things that... I mean some of it, some of it is that, when I would say, things I would say as a joke... I mean there are things I would say as a joke but also seriously use as a joke, in trying to teach people. So you know, sometimes he would steal my joke [Bruce laughs]. That I would, you know, use for teaching purposes and he'd put them into a paper or something like that. And I was sort of ambivalent about that, because, um, I, I don't think of myself as like, a creative researcher or something like that. I think of myself basically as a teacher. And, uh, and uh one of the problems of working with John, is that people really really get the impression that, that like I don't like to communicate with other people. And they think that, you know, that I like, I just like to have John speak for me. I don't like that at all! [laughs] And, um...
Bruce: What would you prefer? I mean, what, what situation would work better for you?
James: Uh just that, you know, I would, I would really like to teach people. For example, there was a recent -
Bruce: Teaching your research? Or teaching, kind of, mathematics at a -
James: Both. I mean, you know, there was a time when I was, you know, when I started out when I sort of didn't have to worry about there being a gap between the kinds of things I was working on, and the kind of things... You know, when I started out I didn't know that much, so I was kind of at the same level where, you know, the things I wanted to teach were the things...
Bruce: Right, right. [laughter]. Have you given any talks at conferences recently? Or...
James: I'd have to think. Uh... uh... only some very informal talks uh, at Minneapolis, that I remember.
James: Um... But like, sometime in the last year or so, I remember, there was uh... a year or two? I don't remember exactly. There was, some sort of uh... John went to Chicago to give some sort of series of talks. And... the way I found out about it was like the day or so before he left. He told me, "By the way, I'm going to Chicago, I'm going to give some lectures". And, the way he phrased it he said something, he said something like "Oh, and by the way, the stuff that I'm going to be talking about is basically a direct steal of all your ideas"... [laughs]
James: That's the way he phrased it.
Bruce: Is this the one, the lectures on cohomology in ...
Aaron Lauda: ... n-categories and cohomology.
James: The what? Is that what it was called?
Aaron: I think it was... I think he says in the beginning that he learnt it all from James Dolan.
James: Well, I, I, if it's the one I'm thinking of he mentions me but he also mentions somebody else... he, he mainly says that none of it is his ideas, but then he mentions several people.
Bruce: Grothendieck, wasn't it? It was you and Grothendieck! [laughs]
James: No-no-no, it was... some, some other people that he was talking about. I forget exactly. But um... Is it really just "n-categories and cohomology"? I remember it as being cohomology and something else, but I can't remember what. I, I don't know. I can't remember. But um...
Bruce: So you weren't.... you don't like that kind of situation where he just pops it on you at the last moment.
James: Well -
Bruce: I mean he has to, he has to... someone's got to say these ideas, I mean... how else are they going to get out of...
James: No no, I mean, no, it's not the last moment thing that bothers me. It's, it's this idea that, I don't like the idea of... I just would really like, you know, the chance to do it myself, rather than, rather than have people... I really dislike the idea that people think I'm deliberately letting him speak for me. I mean, it's not true at all.
Bruce: Okay. Thanks for clearing that up, because I also was wondering about that... okay.
James: Uh-huh. So... I'm trying to answer, I was trying to answer various questions here, and I've already forgotten... I have a feeling that uh...
Bruce: No, you've... I think I've just got one more question to ask -
James: Okay, okay.
Bruce: Um, um, your definition of higher categories, the opetope definition -
James: Yeah, that's probably part of what you started out...
Aaron: The letter and why you sent it to Ross - so you said that the [???] had actually done it, since John... [laughter]
James: Right. Okay, I just want to make one more comment about that - in particular about the letter to, to Ross. So... I guess that when I was writing that letter, I was trying to retribute... you know, I think the letter to Ross was from both John and me.
Bruce: Right. Yes.
James: I think in the letter, I think, I think there was a detectable tone, difference in tone, between John's part of it, and my part of it...
Bruce: Oh. Right.
James: I think, like, I'm trying to be very reticent. I'm trying to emphasize the fact, that, you know, well we were trying to do stuff, but you know, we're not sure how well it worked out! [laughter] And, you know, I...
Bruce: It's your personalities shining through?
James: Well... I mean, that's the way, I think that's the way John thought of it back then. He was always saying things like, he was always saying things like, "Why are you being so overcautious?" He always thought I was being overcautious.
James: I wasn't being overcautious at all. I mean, I think he just didn't realize how uncautious I was being sometimes.
Bruce: So you're not... you weren't completely convinced by your own definition of...
James: Well... well yes, absolutely, right, I mean there were so many things I was worried about. Um... and in particular, well, I mean there are so many different ways I could have been, and was, worried about it. I mean, you could be worried about things being, you know, aspects of it, you could worry about things in the paper being correct, but beyond that, you could also worry about whether it's a good definition.
Bruce: Right, right, right.
James: I, I think in general, in joint work between John and me, um, I think, and again, I think this is an aspect again of how I claim that sort of, you know, I'm not trying to be cautious in framing my discussions of what John is like -
Bruce: Right, right. [laughs]
James: But, um, there's, I mean I've described it before as he overhypes everything. I'm not sure that's the right way of saying it exactly, but something like that is true.
Bruce: Right, right.
James: And in particular, one of the feelings that I've always had, uh, about, uh, joint work between him and me, is that, he never, he never gives a sense of how so much of the stuff that we're working on is, how do I describe it...
Bruce: Still in the air?
Bruce: Still in the air?
James: In the what?
Bruce: In the air, as in...
Aaron: Air. I think is what he's saying
Bruce: It hasn't been completely finalised yet, or... is that what you're trying to say?
James: Yes, yes but I really had trouble understanding the words you were saying.
Bruce: Oh, okay, sorry.
James: Uh, yes, but um how was I going to phrase it. I mean, yes, I think he just doesn't give a sense of when it is that he and I are just, you know, completely pulling stuff out of the sky, and we just, you know, we have no idea of what we're doing. I mean, he makes it sound like we know what we're doing, we've got a plan, and there just isn't. We have no idea what we're doing, we're just completely faking it,
James: - he doesn't, you know, he always papers that over. And I think he does this with everybody. [laughter]. I've been talking to various people about you know, like when they read descriptions in, you know, John's internet columns and things like that, about... You know, John writes descriptions of some work that they've done, where he's heard some talk that they've given or something like that. And I've heard lots of people say, you know, that there's an amusing difference between, you know, what they actually did, how they felt about it, and how they hear John report on it. [laughs]
Bruce: Okay... thanks very much.
Aaron: I was just saying, we were talking about people, using your teaching stories, and then writing them in their papers...
James: Is there one where you do that?
Aaron: Yeah - well I say it's from you, but I mean, maybe you don't like the way I'm attributing it to you [laughter]. "Oh yeah, this is what James Dolan said!"
James: Oh, you know, I guess the thing is that I'm a frustrated teacher. I'm not employed as a teacher. I think I'm a great teacher, and... I'm not employed as a teacher. I think it's kind of tragic, that's all.
Aaron: I had a paper about the "walking ambidextrous adjunction" -
Bruce: - I remember that one! About the hairy eyebrows - did that come from you [to James], originally?
James: Yes - that's probably the original example! You know, I said there were these examples where I say things as a joke, and then I read John writing them in a series of papers. That's probably the original example I had in mind - of something that I thought was, you know, good enough for casual discussion in a classroom... in front of a bunch of students who don't know any better!
Bruce: So Aaron, can you just explain for the listeners, what that joke was about? Aaron: Oh, the walking one?
Bruce: Yes. Aaron: How it became a joke, or the actual...
Bruce: Yes, the actual idea of the joke, I mean -
James: The example of the eyebrows?
Aaron: Wait, I mean, tell the story about the walking one? He [to James] can do it - he did it!
James: Yeah... I mean I probably told it different ways. For some reason I associate this with... I mean probably you [to Aaron] didn't mention Ernie Kovacz,
Aaron: No no...
James: Ernie Kovacz, didn't he have like really bushy eyebrows? I think that was the example I used when originally telling John the... I'm never sure where I got this example [laughter]. Maybe it would be, like, rude if I were to talk about somebody with a big nose, or something. Eyebrows seems very ...
James: Yeah. The example, just, if we're talking to the tape recorder, is just: if somebody has, like, a really bushy pair of eyebrows, you might refer to them as a walking pair of eyebrows. This was supposed to be an illustration of the concept of a universal [laughter].... Let's see, I guess the terminology parses in different ways. A universal property in general, generic thing in general - I'd have to thing seriously about a good way to put it -
Bruce: He's nothing else but his eyebrows, basically?
James: Yeah, yeah.
Bruce: He's just a body that exists to uphold his eyebrows.
James: Yeah, just to uphold the eyebrows. Right.
Aaron: I gave several talks on this walking adjunction -
Aaron: But one time I was giving this talk, the walking pair of eyebrows was in the audience! [Laughter] That was uh... I had to rethink it after that! [laughter].
Bruce: Now, um, James, any final things you'd like to have set straight for the record?
James: No... not that I could leave as final. I'd say that was just a little... tip of the iceberg! I'd have to think a long time before summing it up.
Bruce: Okay, thank you very much.
To set a couple of things straight: in my paper n-Categories and Cohomology, I begin by saying
"nothing new here is due to me: anything not already known by
experts was invented by James Dolan, Toby Bartels or Mike
Shulman". Also, I don't recall ever using the "walking pair
of eyebrows" joke in a published paper, though I explained it in
week173. But, Aaron Lauda explained
this joke on page 20 of his paper
Frobenius algebras and ambidextrous adjunctions - John Baez
© 2007 Bruce Bartlett and James Dolan