I’d recently been wondering whether the archives of the Bourbaki group would be put on-line, and today noticed that there’s a project to do so, with results available here. One can read copies of “La Tribu”, internal reports on the activities of the group, up through 1953. There are a wide variety of interesting mathematical documents, often consisting of attempts to write up one subject or another, efforts that sometimes made it into the published books, often not.
One subject that Bourbaki struggled with over the years was that of how to set out the foundations of differential geometry. My colleague Hervé Jacquet likes to tell about how Chevalley at one point made an effort to do so, with the peculiar starting point of defining things in terms of “cubes”. I wasn’t sure whether to believe him, but here it is. According to Borel, in 1957 Grothendieck presented the group with his own take on the question of manifolds:
Grothendieck lost no time and presented to the next Congress, about three months later, two drafts:
Chap. 0: Preliminaries to the book on manifolds. Categories of manifolds, 98 pages
Chap. I: Differentiable manifolds, The differential formalism, 164 pages
and warned that much more algebra would be needed, e.g., hyperalgebras. As was often the case with Grothendieck’s papers, they were at points discouragingly general, but at others rich in ideas and insights. However, it was rather clear that if we followed that route, we would be bogged down with foundations for many years, with a very uncertain outcome.
I don’t see these documents on the list, perhaps documents from the later years are still to appear.
The documents often start out with some unvarnished comments, here’s an example, from Chevalley’s report on a text about semi-simple Lie algebras:
Au moment d’écrire ces observations, je me demande si ce ramassis des méthodes les plus éculées et les plus pisseuses, ces résultats les moins généraux possibles établis de la manière la plus incompréhensible possible, ne sont pas un canular intrabourbachique monté par le rédacteur. Même s’il en est ainsi, je me laisse prendre au canular et présente les observations suivantes.