Why did Bourbaki's Élements omit the theory of categories?

Question

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).

Answer 1

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.

Answer 2

As mentioned in a comment, there are some people such as Colin McLarty who I think could give an informed answer. I am not one of those persons, but since this question is likely to be closed soon, I will just mention a few helpful references.

One is McLarty's article The Last Mathematician from Hilbert’s Gottingen: Saunder Mac Lane as Philosopher of Mathematics. Indeed the members of Bourbaki invited Mac Lane to speak to them, but it probably wasn't Mac Lane's French that was the problem in getting them to incorporate category theory into the grand vision. Mac Lane and Weil were of course colleagues at the University of Chicago and presumably had ample opportunity to discuss category theory (in English); as quoted in McLarty's article, Weil writes to fellow Bourbakiste Chevalley in 1951:

As you know, my honourable colleague Mac Lane maintains every notion of structure necessarily brings with it a notion of homomorphism, which consists of indicating, for each of the data that make up the structure, which ones behave covariantly and which contravariantly [...] what do you think we can gain from this kind of consideration?

McLarty explains in his article that Weil didn't understand Mac Lane. If I understand correctly, there were indeed opportunities to incorporate category theory within the Élements, specifically as part of an account of an abstract theory of structures, but (McLarty, page 5):

After the war, Bourbaki hotly debated how to make a working theory. All agreed it must include morphisms. Members Cartier, Chevalley, Eilenberg, and Grothendieck championed categories, as did their visitor Mac Lane. But Weil was a majority of one in the group, so they created a theory with structure preserving functions as morphisms (Bourbaki [1958]). They never used it, and not for lack of trying.

Throughout the discussion of Bourbaki's (or Weil's) attitude toward categories, McLarty mentions the work of Leo Corry, who discusses Bourbaki's structures in his book Modern Algebra and the Rise of Mathematical Structures (reviewed here). Related is a useful online article by Corry, published in Synthese, here. I won't attempt to summarize it, but there is discussion, on the basis of documents, of "the interaction between Bourbaki's work and the first stages of category theory".

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Edit: Although the thread has closed and quid (user9072) has departed, Francois Ziegler recently brought to my attention in a comment below that Ralf Krömer (2006; pdf) (subtitled Bourbaki and categories during the 1950s) has thoroughly investigated the OP’s question, using unpublished internal reports of the meetings of Bourbaki, as well as correspondence and quotes of e.g. Eilenberg (p. 142), Cartier (p. 147), Grothendieck (p. 149), and others. There is quite a rich treasure trove of well-sourced information there, for those who are interested.

Answer 3

These are good answers and I have nothing to add on the particular reasons. To sum it up I would say it was impossible to write a comprehensive Elements of Mathematics based on category theory, because it is impossible to write a comprehensive Elements of Mathematics at all. The brilliant attempt was very good for mathematics, I will maintain, but it could not really be done.

Categories and functors in the 1950s were developed only for immediate applications, and not as a general theory of structure. It was very much easier for Bourbaki to develop a general theory that did not work, than to unify a sprawling mass of working methods into one theory. And nothing was really going to work for their vast project anyway.

So far as I know no one talked about a general theory called "category theory" before biologist Robert Rosen in works like "A relational theory of the structural changes induced in biological systems by alterations in environment" Bull. Math. Biophys. 23 1961 165–171.

Answer 4

It might be interesting to look at the Appendix to Exposé I of SGA 4. In a footnote this is described as follows:

Nous reproduisons ici, avec son accord, des papiers secrets de N. BOURBAKI.

While the appendix treats mainly the theory of universes, it makes use of the language of categories. Moreover, some internal references hint to the existence of some more "papiers secrets" containing a draft of a chapter about categories.

Answer 5

I don't agree with Wlodzimierz's general comments about chaos! In many aspects Bourbaki did a tremendous job in writing clear and clean mathematics, but the aim which I think is professed of writing a complete and so to speak permanent account comes up against the evolving nature of the mathematical project. This is evident in many present and past attitudes to category theory.

A review for the MAA of the 1968 edition of my book "Elements of Modern Topology", now "Topology and Groupoids" (2006), suggested it read like a "book on topology written by a category theorist", and this was presumably not meant as a compliment! Many regarded its use of groupoids as a mistake. Compare a recent review. I would hope most mathematicians now recognise the enormous contribution category theory has made to the unity of mathematics, allowing analogies between constructions in quite disparate parts of the subject, through such terms as limit and colimit, as but one example.

I feel that the progress of mathematics is considerably helped by people trying to write complete, consistent and clear accounts, so that the deficiencies in the attempt, i.e. in current understanding, become apparent.