Re: Bourbaki and Categories

[Andre Joyal <joyal.andre@uqam.ca>]

Dear George,

I thank you for your message.

You wrote:

>I insist that Bourbaki group simply did
>not see the importance of category theory.

It is difficult to know. The Boubaki group had shielded itself
in secrecy, like a free mason cell. 
You are surely aware of the interview of Pierre Cartier 
in the Mathematical Intelligencer No1 1998. 
Everyone interested in the history of Bourbaki should read it.

http://ega-math.narod.ru/Bbaki/Cartier.htm

Let me stress a few passages: 

>The fourth generation was more or less a group of students of Grothendieck. But at that time Grothendieck had already left Bourbaki. 
>He belonged to Bourbaki for about ten years but he left in anger. The personalities were very strong at the time. 
>I remember there were clashes very often. There was also, as usual, a fight of generations, like in any family. 
>I think a small group like that repeated more or less the psychological features of a family. 
>So we had clashes between generations, clashes between brothers, and so on. 
>But they did not distract Bourbaki from his main goal, even though they were quite brutal occasionally. 
>At least the goal was clear. There were a few people who could not take the burden of this psychological style, 
>for instance Grothendieck left and also Lang dropped out.


> It is amazing that category theory was more or less the brainchild of Bourbaki. The two founders were Eilenberg and MacLane. 
> MacLane was never a member of Bourbaki, but Eilenberg was, and MacLane was close in spirit. The first textbook on homo-logical 
>algebra was Cartan-Eilenberg, which was published when both were very active in Bourbaki. Let us also mention Grothendieck, who 
>developed categories to a very large extent. I have been using categories in a conscious or unconscious way in much of my work, 
>and so had most of the Bourbaki members. But because the way of thinking was too dogmatic, or at least the presentation in the 
>books was too dogmatic, Bourbaki could not accommodate a change of emphasis, once the publication process was started.

>In accordance with Hilbert's views, set theory was thought by Bourbaki to provide that badly needed general framework. If you need 
> some logical foundations, categories are a more flexible tool than set theory. The point is that categories offer both a general 
> philosophical foundation—that is the encyclopedic, or taxonomic part—and a very efficient mathematical tool, to be used in 
>mathematical situations. That set theory and structures are, by contrast, more rigid can be seen by reading the final chapter in 
>Bourbaki set theory, with a monstrous endeavor to formulate categories without categories.


The interview ends with the following passage.
The bold faces are from me.

>When I began in mathematics the main task of a mathematician was to bring order and make a synthesis of existing material, to create 
>what Thomas Kuhn called normal science. Mathematics, in the forties and fifties, was undergoing what Kuhn calls a solidification period. 
>In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, 
>and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. 
>Now we are again at the beginning of a new revolution. Mathematics is undergoing major changes. We don't know exactly where it will go.
> It is not yet time to make a synthesis of all these things—MAYBE IN TWENTY OR THIRTY CENTURY IT WILL BE TIME FOR A NEW BOURBAKI.
>I consider myself very fortunate to have had two lives, a life of normal science and a life of scientific revolution.

Is it the time for a new Bourbaki?

Best regards, 
André


-------- Message d'origine--------
De: cat-dist@mta.ca de la part de George Janelidze
Date: lun. 15/09/2008 20:03
À: categories@mta.ca
Objet : categories: Re:  Bourbaki and Categories
 
Dear Andree,

Could you please explain this better?:

The only Bourbaki member I new personally was Sammy Eilenberg. As many of
us, I knew him very well and I would say that he was more skeptical about
the Bourbaki Tractate then one can conclude from Andre's message. Having in
mind not just this but the content of Bourbaki's "Homological algebra" and
what we see today from the followers of that Bourbaki group, I protest
against Andre's "two options" and I insist that Bourbaki group simply did
not see the importance of category theory (in spite of being brilliant
mathematicians, as I said in my previous message). I hope Andre will forgive
me and even agree with me.

However, there were three great category-theorists in that group (plus there
is this mysterious story about Chevalley's book of category theory lost in
the train), and "did not see" cannot be said about them of course. On the
other hand I have never heard of any joint work of Charles Ehresmann with
any of the two others, Eilenberg and Grothendieck (and nothing jointly from
them). I think apart from the time issues you describe, the relationship
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).

Is this true, and could you please give details?

Respectfully, and with best regards-

George