[Math] Why did Bourbaki’s Élements omit the theory of categories

bourbakict.category-theoryho.history-overview

QUESTION

They had plenty of time to adopt the theory of categories. They had Eilenberg, then Cartan, then Grothendieck. Did they feel that they have established their approach already, that it's too late to go back and start anew?

I have my very-very general answer: World is Chaos, Mathematics is a Jungle, Bourbaki was a nice fluke, but no fluke can last forever, no fluke can overtake Chaos and Jungle. I'd still like to have a much more complete picture.

Appendix: CHRONOLOGY

  • 1934:   Bourbaki's birth (approximate date);
  • 1942-45:   Samuel Eilenberg & Saunders Mac Lane – functor, natural transformation, $K(\pi,n)$;
  • 1946 & 1952: S.Eilenberg & Norman E. Steenrod publish "Axiomatic…" & "Foundations…";
  • 1956: Henri Cartan & S.Eilenberg publish "Homological Algebra";
  • 1957: Alexander Grothendieck publishes his "Tohoku paper", abelian category.

(Please, feel free to add the relevant most important dates to the list above).

Best Answer

One thing to keep in mind is that Bourbaki started in the 1930s, so in some sense simply too early to include category theory right from the start on, and foundational matters were rather fixed early on and then basically stayed like this. Since (I think) the aim was/is a coherent presentation (as opposed to merely a collection of several books in similar spirit) to change something like this 'at the root' should be a major issue. Some 'add on' seems possible but just does not (yet) exist; and it seems the idea to write something like this was (perhaps is?) entertained (see below).

To support the above here is a quote from MacLane (taken from the French Wikipedia page on Bourbaki which contains a somewhat longer quote and source):

Categorical ideas might well have fitted in with the general program of Nicolas Bourbaki [...]. However, his first volume on the notion of mathematical structure was prepared in 1939 before the advent of categories. It chanced to use instead an elaborate notion of an échelle de structure which has proved too complex to be useful. Apparently as a result, Bourbaki never took to category theory. At one time, in 1954, I was invited to attend one of the private meetings of Bourbaki, perhaps in the expectation that I might advocate such matters. However, my facility in the French language was not sufficient to categorize Bourbaki.

There it is also mentioned that (in the context of the influence of the lack of categories on the discussion of homological algebra, only for modules not for abelian categories):

On peut lire dans une note de bas de page du livre d'Algèbre Commutative: « Voir la partie de ce Traité consacrée aux catégories, et, plus particulièrement, aux catégories abéliennes (en préparation) », mais les propos de MacLane qui précèdent laissent penser que ce livre « en préparation » ne sera jamais publié.

This translates to (my rough translation): One can read in a footnote of the book Commutative Algebra: "See the part of this Treatise dedicated to categories, and, more specificially, to abeliens categories (in preparation)", but the sentiments of Mac Lane expressed above [part of which I reproduced] let one think that this book "in preparation" will never be published.

The precise reference for the footnote according to Wikipedia is N. Bourbaki, Algèbre Commutative, chapitres 1 à 4, Springer, 2006, chap. I, p. 55.

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