FOM: Grothendieck on Bourbaki

[Colin McLarty]

        It remains a misunderstanding to say Grothendieck attacks Bourbaki
for failing to use toposes.

Simpson replied:

>It wasn't a misunderstanding.  It was based on the quote from
>Grothendieck's unpublished writings attacking Bourbaki, which you
>provided.  That quote mentioned toposes, i.e. Grothendieck toposes,
>presumably, but it didn't mention homological algebra.

        The word "topos" indeed occurs parenthetically in the quote, as an
example of a construct that fits ill in Bourbaki's theory of *structures*.
But Grothendieck complains of actions Bourbaki took in the 1950s before
toposes were created. Leo Corry discusses the dynamics of the debate in
detail in his article "Nicholas Bourbaki and the concept of mathematical
structure" (SYNTHESE 92 (1992) 315-348).

        The remainder of this post quotes Corry's summary of the issue in
MODERN ALGEBRA AND THE RISE OF MATHEMATICAL STRUCTURES page 332:
_________________

        A cursory examination of issues of "La Tribu" [Bourbaki's internal
newsletter] during the fifties uncovers recurring attempts to write chapters
on homological algebra and categories for the ELEMENTS. Eilenberg himself,
who had initiated together with Saunders Mac Lane the study of categories,
was commissioned several times for the preparation of drafts on homology
theories and on categories, while a Fascicule de resultats on categories and
functors was assigned successively to Grothendieck and Cartier. However the
promised chapter on categories never appeared as part of the treatise. The
publication of such a chapter could have proved somewhat problematic when
coupled with Bourbaki's insistence on the centrality of *structures*; the
task of merging both concepts; i.e. categories and *structures* in a
sensible way would have been arduous and not very illuminating, and the
adoption of categorical ideas would probably have necessitated the rewriting
of several chapters of the treatise. In this regard, it is interesting to
notice that when the chapter on homological algebra was finally issued
(1980) the categorical approach was not adopted. Although the conceptual
framework provided by categories had become the standard one for treating
homological concepts since the publication of the above mentioned textbook
of Cartan and Eilenberg, in Bourbaki's own presentation these concepts are
defined in the narrower framework of modules. And naturally the concept of
*structure* is not even mentioned there.