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
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