Reply to Jean Benabou

Reply to Jean Benabou

[Marta Bunge]

Dear Jean,

There is an unpublished (untitled and undated) four-pages manuscript which
John Beck gave to me (and I supposed also to many ohers) when he was at
McGill. In it, he states and proves two theorems, the CTT (crude
tripleableness theorem), and the PTT (precise tripleableness theorem). There
is a connection between triples and descent implicit in the PTT. But this is
not the same connection with descent as the Benabou-Roubaud theorem.

The following remarks seem to be relevant to this issue.

In M. Bunge and R. Pare, "Stacks and equivalece of indexed categories",
Cahiers de Topologie et Geometrie Differentielle, vol XX-4 (1979) 373-399,
we state and prove the following version of the Benabou-Roubaud theorem
(which we quote):

Proposition 2.3. (Benabou and Roubaud [10]). let A be an S-indexed category
(S is a topos) for
which Sigma (or Pi) exists and satisfies the Beck condition. Then A is a
stack iff for every regular epi alpha:J--->>I in S, the functor alpha^*: A^I
---> A^J is tripleable (resp. cotripleable).

What Bob Pare and I called "the Beck condition" above is the "Chevalley
property" introduced in your paper. The proof of the Benabou-Roubaud theorem
was not given in your Comptes Rendues note, and this is why we gave an
explicit proof of it, since we needed to apply it to stacks. Indeed, from
the Benabou-Roubaud theorem, using in addition the Beck's tripleableness
theorem, one can obtain applications showing that a certain S-indexed
category A is a stack. In Bunge-Pare, we obtain in this way, using Duskin's
version of the tripleability theorem (J. Duskin, "Variations on Beck's
tripleability criterion", Reports of the Midwest Categories Seminar III, LNM
106, Springer, 1969), the following (actually we give it a more generality):

Corollary 2.5. Any topos S, indexed by itself in the usual way, is a stack.

In turn, this is used in my sequel paper (M. Bunge,Stack completions and
Morita equivalence for categories in a topos", Cahiers de Top. et Geo. Diff.
XX-4 (1979) 401-436), using closure properties of stacks, to
identify/construct stack completions of category objects in S.

It seems then to be an error on the part of Peter Johnstone to have
attributed Proposition 1.5.5 in E1 (page 297) to Beck and not to Benabou and
Roubaud. At the end of this section on "Descent Conditions and Stacks" (page
303), the references given in El 1 are Bourn, Bunge and Pare, Giraud,
Grothendieck, Reiterman and Tholen, but curiously enough, not
Benabou-Roubaud. I am sure that Peter will repair this error should a second
edition of the Elephant ever appear. I would like to add that people who
write such monumental works are bound to make errors of this sort,
particularly in this case, as the manuscript was not (to my knowledge)
distributed around to the topos theorists and other mathematicians for
comments and criticism prior to publication.

With best regards,
Marta



************************************************
Marta Bunge
Professor Emerita
Dept of Mathematics and Statistics
McGill University
805 Sherbrooke St. West
Montreal, QC, Canada H3A 2K6
Office: (514) 398-3810
Home: (514) 935-3618
marta.bunge@mcgill.ca
http://www.math.mcgill.ca/~bunge/
************************************************




>From: JeanBenabou <jean.benabou@wanadoo.fr>
>To: Categories <categories@mta.ca>
>Subject: categories: References
>Date: Fri, 2 Nov 2007 02:55:08 +0100
>
>Dear colleagues,
>
>I hope someone, and in particular Prof. Peter Johnstone, will help me
>with the following information. I thought I had, with Jacques
>Roubaud, proved in our joint note at the "Comptes Rendus" which I
>mentioned in my previous mail proved a theorem on Monads and Descent.
>I must have been mistaken, and also the many persons who quoted this
>note, because in El Proposition 1.5.5 is the same theorem, but
>attributed to J. Beck.
>
>I immediately "rushed" to the monumental bibliography of El to find
>the reference, and there, big surprise, there was no J. Beck at all
>among the 1262 references.
>
>Thus i'd greatly appreciate to have the date and paper of the paper
>where Beck proved this theorem, and the precise statement he made, in
>particular, did he prove his theorem in the general context of
>fibered or indexed categories, or only in some very special case.
>
>Many thanks for your help
>
>
>

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