Dear Ross, I have a little more time to answer your mail. I have to recall precisely what I said about you in my long mail about "opposites via distributors" (i) Ross Street's 1980 paper: Fibrations in bicategories, where he .. uses distributors in the enriched case, which he calls "modules", without mentioning my name. I have again consulted your paper, to be absolutely sure. No mention of my name in the text, but that could be a slip, but no mention of my paper on distibutors, which you knew of course, in the bibliography. The only reference you give about the subject is Lawvere' 1974 paper on "Metric spaces". You say you are saddened, an I'm sorry about that. What about me? Isn't there any reason why I should also be "saddened"? I have carefully looked at the Encyclopedia papers you kindly sent me, and I would like to make a few comments, without any "polemic spirit" About distributors you say, I quote you: " There is a bicategory Mod whose objects are (small) categories and whose arrows are modules [St5, St8] (= profunctors = distributors [Bn2] = bimodules [L2]) What does "There is" mean? Was it "god-given", or introduce by your two papers (1981 & 1983) or Lawvere's paper (1974)? Wouldn't it have been "fairer" and more accurate to say "introduced by Benabou, and used, or developed, or whatever you want, by so and so" Now a few more remarks: In your encyclopedia paper you say: "There are several purely categorical motivations for the development of bicategory theory. The first is to study bicategories following the theory of categories but taking into account the 2-dimensionality; .... A given concept of category theory has several generalizations ... " Sorry Ross, much as I respect your work, we don't seem to have the same approach to generalizations. I introduced bicategories because I had a huge amount of mathematically important examples. And I wanted to have a "common denominator" explaining these examples, and others I was sure to find. And these examples were also a guide to indicate what meaningful "abstract notions" were to be investigated. Let me risk a parallel. Formally, a category is "nothing but" a monoid with many objects. Do you think that Eilenberg & Mac Lane's motivation to introduce categories was " following the theory of monoids" but "taking into account" the fact that they had many objects. If they hadn't had so many mathematical examples would the theory of monoids have indicated them that monos, epis, products, equalizers or general limits or colimits were relevant to the study of "monoids with many objects". Would Kan have discovered adjoint functors if he hadn't had in mind many many important mathematical examples? Even such "formal constructions" on categories e.g. categories of fractions, were motivated by mathematics, not abstract formal considerations. You also mention a coherence theorem asserting that every bicategory is equivalent to a 2-category. About this theorem, or the one stating that every fibration is equivalent to a split fibration, I'm tempted, with all due respect, to say so what? They might be interesting if we were concerned by a single bicategory or fibration. But the natural notion of morphism, in both cases does not respect the "stictness properties". Thus, here again, mathematics, not abstract formalism, will tell me what is really important. I apologize for such a long, and probably a bit confuse mail. Very cordially, Jean.
Dear Jean
Thank you for forwarding your message to me although I am quite saddened by it. I have great respect for your work. I have referred to your work in many places: I attach an encyclopedia article published by Kluwer in 2000 as an example. Despite what you might think, I have always tried to use established terminology when it existed. The move from bimodule (as used by Lawvere in his metric space paper --- which paper did refer to your Louvain notes) to module was precisely to make use of the arrow notation and not to interfere with your use of "bi" as in your word "bicategory". Sammy had turned me off "distributor" with some remarks at Oberwolfach.
Best wishes, Ross
<Encyclopedia.pdf>
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There are some interesting general issues which arise out of this correspondence and which could merit further discussion. 1) Having tried and trying to write some background history to a result or theory I am aware of how much work is entailed, and the difficulty of getting it right, so I have sympathy with those complained against. I remember one writer's reaction to a complaint that he did not give enough reference to past work was `it was easier to work it out myself than to read all the relevant literature'. There is historian's ethic here which is important but is somewhat in opposition to the hard job of getting the mathematics correct, in the best format, and publishable. 2) On the other hand appropriate credit is important. Friends will know that I have had my own complaints about `convenient categories for topology' although Eilenberg growled at me once: `Why are you making such a fuss?' Another reaction (G.W. Whitehad) was:`There is no reason why you should not get credit for work you have done.' Credit is important also for the balance and the future of the mathematics and is the more so in these days of research assessment. So my sympathies are also with Jean! 3) In such assessment there is now more importance placed on citations, especially fairly immediate ones. The problems in this are well known, and are illustrated by this discussion. Citations are a bit like the shadows on the wall of Plato's cave, and are only a sparse image of reality. Some forgetful pseudo-functor is clearly involved! There are no agreed rules, certainly no laws; all there is some general feeling that appropriate credit should be given, and often referees or editors have to point out omissions, which can be due to any of: ignorance, forgetfullness, laziness, prejudice, or moral hazard, more likely some of the first three. On forgetfullness, there is the story of Henry Whitehead giving a problem to a research student, who came back 2 weeks later with the comment:`But Henry, you solved this problem in one of your papers!' People have sometimes reacted to me when I came out with a bright idea: `Aah, at last you have seen it!' There is also the question of how far back citations should go? in other words, how long should be the reference list? So the current emphasis on citations in research assessment, and the large business built on it, have a strong element of absurdity. 4) I sometimes wonder if our career structure should not have more in common with music, where the profession and the study involves: performance, composition, musicology, but in all cases involves an assessment of `musicality', as well as technique. 5) The issues Jean raises in his last email on the reasons for the study of a subject are also very important. As Philip Hall was reported to have remarked: `It is important to study the algebra which arises from the geometry, rather than to force the geometry into a standard and known mould.' 6) I do agree it is important for people to speak out on important past trends and influences, and so to show what were the intuitions which inspired an area of work. Not all the aims may have been achieved! I agree that changes of published, cited and accepted terminology has its dangers as have been pointed out by Jean in previous correspondence, with respect to cartesian and cocartesian morphisms. The problem is still that language does and should evolve. Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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JeanBenabou -
Ronnie Brown