Re: Bourbaki and Categories

[Andree Ehresmann <andree.ehresmann@u-picardie.fr>]

George Janedlize writes

> Could you please explain this better...
> the Bourbaki group simply did not see the importance of category theory...
> However, there were three great category-theorists in that group...
> I have never heard of any joint work of Charles Ehresmann  with any  
> of the two others, Eilenberg and Grothendieck... ...the relation  
> between Bourbaki Tractate and category theory should have been  
> determined by their separate or joint influence and therefore also  
> by their communication with each other (if any).

I'll try to explain why there is no contradiction.
1. Charles only participated actively to the Bourbaki group from 1935  
to the mid forties, at a time he did not know category theory. In 1935  
he had written a first version for the volume "Theorie des ensembles"  
where he introduced the notions of local structures and associated  
pseudogroups of transformations (not so far from groupoids!), but this  
version was not accepted and he did not like the final version  
published much later. After the war, he only participated irregularly  
because he felt that he was no more able to make himself heard, the  
decisions being taken by "those who spoke the more loudly" (as he said  
to me).
2. Around 1950 it was decided that active participation ended at 45  
(the age Charles had then), lessening the influence of those  
(Eilenberg, Cartan, Chevalley and Dieudonne) who  could have stressed  
the importance of categories. I don't know exactly when Grothendieck  
became a member, but it was much later, and I think he did not remain  
for long. Later on, disdain for category theory had developed in  
France...
3. As for the communication between Charles and the other  
category-theorists, he had no contact with Grothendieck who was much  
younger. He was friendly with Eilenberg but did not see him often.  
Before the war he lived in Paris and regularly met Henri Cartan,  
Dieudonne, and more specially, Chevalley (both had regular exchanges  
with the philosophers Cavailles and Lautman). But their communication  
almost ceased after the war when he developed all his activity in  
Strasbourg (up to 1955) and was out of France for a long part of the  
year. Anyway, before our joint work (from the mid sixties up to his  
death), Charles worked essentially alone and published no joint work  
at all, except 6 Notes on Topology or Geometry with some of his  
students. When he began to specialize in category theory in the  
sixties, it was not well understood by other French mathematicians,  
and his influence dwindled up to a real opposition in the seventies.

Andree