RE: Partial response to Jean Benabou

RE: Partial response to Jean Benabou

[Marta Bunge]

Dear Peter,

>Incidentally, in reply to a comment in Marta Bunge's posting, I did
>distribute draft copies of the Elephant to a number of colleagues, and
>invite their comments, before it was published. None of them picked up
>this particular point -- though I am not blaming them for that; the
>fault was of course mine.
>

I have now read in the Preface to the Elephant that you did indeed circulate
a draft to "a number of colleagues (including Ieke Moerdijk and Andrew
Pitts, as well as Martin Hyland, Anders Kock and Gavin Wraith) for their
comments and suggestions".

Apologies for my swift comment. I had gotten a different impression from the
reaction of several participants to the Fields Institute Workshop on Galois
Theory etc, where the first two volumes of the Elephant were exhibited,
including comments by Bill Lawvere and several other people whose work was
prominently represented in the book.

Aurelio Carboni and I perused quickly section B4.5 on the symmetric monad,
and were satisfied on the spot with your account of it, but it did not occur
to me to look up section B1.5, even though my paper with Bob Pare on stacks
was cited at the end of the section. Had I done so, I would have pointed out
that Proposition 1.5.5 is due to Benabou and Roubaud as we ourselves had
pointed out in our paper.

I take this opportunity to apologize to Jean Benabou for stating his theorem
with Jacques Roubaud in the context of indexed categories and not on that of
(bi)fibrations without warning the reader, but I also want to point out
that, since the proof  itself only refers to two fibers and a transition map
between them, it is equally meaningful in both contexts. I believe that I
only became aware of the gross difference afterwards, when lecturing on
stack completions at the Benabou Seminar in Paris (whenever that was --
1979?). I still think now that it is easier to work with the indexed
presentations of fibrations than with the fibrations themselves, without
attempting to turn this into a philosophical statement of any kind.

Beck's Tripleability theorem is indeed useful in the applications of the
Benabou-Roubaud theorem, and among them are those we give in the Cahiers
paper on stacks, and in its sequel by myself on stack completions.

Perhaps the relabelling of the Chevalley condition as "the Beck condition"
led to some confusion as to whether the Benabou-Roubaud theorem had been
proved (also) by Jon Beck? I myself never heard him speaking on this, and
never saw any draft written by him of a proof of this theorem. Sadly, we
cannot consult Jon himself on this issue, so we might as well drop it.

Perhaps you would consider, prior to publication of Volume III of the
Elephant, enlarging your list of commentators to include at least those
whose work you include in some form or other.

With best regards,
Marta