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

## 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).

• Another note is that something like the "category of sets" is not a set. To talk about it, you need some axioms other than the strict ZFC axioms on which all of Bourbaki's math is based. Thus, to discuss category theory in ways that include its current practice, Bourbaki would need either to augment the existing foundations in one way or another, or to adapt some sort of unconventional and probably very awkward conventions that would make their category theory work in ZFC, probably at the price of making it much harder to use. May 23, 2013 at 22:36
• There are much more qualified people to comment here, but if I recall correctly, it was extensively discussed whether to include categories or not. May 23, 2013 at 22:43
• @Wlodimierz Holsztynski: A related discussion can be read in Armand Borel's Twenty-five years with Nicolas Bourbaki ams.org/notices/199803/borel.pdf on page 378, where a short account of the story of the congrès du foncteur inflexible is given. It discusses Grothendieck's proposal how they should treat sheaf theory and why that route wasn't chosen. May 24, 2013 at 0:23
• @Qfwfq: At least some of the members of Bourbaki did not view what was being done as a pedagogical exercise. For instance, "People just misused [Bourbaki's] books; they were never meant for university teaching..." - JP Serre. (sms.math.nus.edu.sg/smsmedley/Vol-13-1/…) May 24, 2013 at 9:19
• It seems to me that quid's answer, while not bad, was accepted a bit prematurely. In particular, there are people who have made a careful study of the available historical record; among MO users, I think Colin McLarty would be in a position to give a more complete and authoritative answer. I will see whether I can contact him, to alert him to this thread. May 24, 2013 at 11:22

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.

• I find MacLane's statement not convincing. Samuel Eilenberg was a part of Bourbaki, could out-talk any Frenchman (fluency in French was not his problem). By 1956 he & Henri Cartan wrote Homological Algebra. Perhaps Bourbaki invested in its original approach so much that they were not willing to start a gigantic work again. What if there would be still a new revolution? And even the most energetic mathematicians have only so much time and energy, and not more. Bourbaki had no meritoric objections against theory of category, used and developed further theory of categories in their own research. May 24, 2013 at 0:07
• The quote (or rather it seems to be another quote of him) on the page by MacLane also contains "He [Bourbaki] was too conservative to recognize other better descriptions of structure when they arose [some references to category theory]". You say "invested in its original approach so much that they were not willing to start a gigantic work again" yes I would assume there were also practical considerations. This is what I meant with the issue of changing things at the root; eg, by the end of the 50s 'Algebra' (at least large parts thereof) and 'General Topology' already extisted.
– user9072
May 24, 2013 at 0:28
• The article mention by Martin in a comment (on the question) is very interesting and discusses among others some practical considerations in some detail.
– user9072
May 24, 2013 at 0:45

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".

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.

• While I'd agree there is something to Weil having some responsibility here, it is not quite clear to me how to reconcile this strong description in particular "But Weil was a majority of one in the group [...]" with what is generally said regarding the workings of the group (unanimity and alike) and more importantly the fact that Weil (officially) retired from Bourbaki in 1956. (Addded: To stress this point of timeline let me recall that then Grothendieck was aged 28 and Cartier aged 24 [roughly] so they'd had plenty of time.)
– user9072
May 24, 2013 at 19:18
• @quid: regarding "mandatory requirement at age 50", see also the article by Aubin: math.jussieu.fr/~daubin/publis/1997.pdf. In particular, footnote 3 reads: "The historian Liliane Beaulieu, who has worked the most extensively on Bourbaki, told me that she had never come across any written trace of this rule and that in any case it was breached many times." (My emphasis) May 24, 2013 at 20:15
• Weil was the first (while not oldest!) to apply them to himself by letting a letter announcing his retirement be read at the 50th birthday party of Dieudonné (born in 1906, like Weil). [See eg footnote 10 in article mentioned by Martin.] So, at least officially he definitely retired, which is what I claimed. (To what extent his influence continued in a non-official way is not known to me; in case I find time I might reread parts of his correpondence with Cartan for hints.) [There are some other details about the quote I find surprising but I do not want to appear argumentative.]
– user9072
May 24, 2013 at 20:47
• @quid Well, yes Weil was a huge influence on all of modern pure mathematics. But his most important contributions are very hard to grasp even today, and he is utterly underrated by people who only know his low level ideas as in Bourbaki. He was also a brilliant collaborator to people he respected. But I know myself that to call him acerbic is an understatement. He and Grothendieck eventually could not stand each other. May 25, 2013 at 18:41
• It would be worth mentioning (not just in this buried comment) that Krömer (2006; pdf) (subtitled Bourbaki and categories during the 1950s) thoroughly investigates the OP’s question using unpublished internal reports, correspondence and quotes of e.g. Eilenberg (p. 142), Cartier (p. 147), Grothendieck (p. 149), etc. Aug 15, 2019 at 2:51

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.

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.

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.

• I have written in my "Question" above: no fluke [Bourbaki] can overtake Chaos and Jungle. It's fine (almost mandatory :-) to contradict me, but I don't see any logic in your "disagreement" with my "general comments". May 24, 2013 at 21:10