Re: Opposites via distributors

Re: Opposites via distributors

[JeanBenabou]

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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Re: Opposites via distributors

[Ronnie Brown]

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



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