Authorship of Grothendieck universes

Universes seem to first enter Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is credited to Grothendieck and Verdier. The appendix on them there is credited to N Bourbaki.

Is there any known evidence of who actually wrote the appendix?

• I have no solid evidence, but I believe it was written by Grothendieck himself.
– Joël
Oct 20, 2013 at 6:17
• I believe that too, based the bits of set theory in other things he wrote, as already in the Tohoku paper. I fear there is direct evidence in some long-unread letters or notes somewhere that I will never find. Oct 21, 2013 at 0:57

Pierre Cartier has told me everyone at the time (i.e. everyone in those circles) knew Pierre Samuel wrote the appendix.

Incidentally this makes a third person breaking the general rule that all writings signed N Bourbaki were collective. Weil and Dieudonné wrote historical/philosophic pieces signed Bourbaki, and Samuel wrote this.

• Well, my understanding is that each "rédaction" was written by one member of the group, see here, but then read collectively, corrected and assigned to another member, and so on. In this particular case, it may be that the group decided to abandon the idea of writing on the subject; it may also very well be the case that there were more than one author.
– abx
Mar 1, 2015 at 15:40
• @abx That is how the Elements of Mathematics were written. The Elements are nearly, but not entirely, the only works signed Bourbaki. Oct 12, 2019 at 12:51

This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves.

Namely, universes were introduced and analyzed by Zermelo in his 1930 paper, several decades before Grothendieck:

To be sure, universes are the central focus of this paper, in which Zermelo defines the universe concept, considering them as set-theoretic realms for mathematics. His version of the universe concept allows for (but does not insist upon) a set of urelements. He proves his famous quasi-categoricity result, establishing that the universes are precisely the models of second-order set theory $$\text{ZFC}_2$$, and that they are linearly ordered and connected with the inaccessible cardinals of Hausdorff, and he analysizes their automorphism groups, which are induced by permutations of the urelements, if any. In addition, he considers various philosophical aspects of moving from one universe to the next, in that various proper classes become sets in the next universe, which is a central use case in category theory.

Zermelo also explicitly considers a version of the universe axiom:

...the existence of an unbounded sequence of boundary numbers [heights of universes, or inaccessible cardinals] must be postulated as a new axiom of 'meta-set theory', and in so doing the 'consistency' question must be looked at more closely.

Zermelo's 1930 analysis thus seems in several respects to surpass Grothendieck's later analysis, which to my knowledge does not provide the categoricity result and does not engage with the consistency issue.

In light of their origin in Zermelo's work, the Grothendieck universes are now also known as Zermelo-Grothendieck universes.

• I think that if we are talking about Grothendieck, we mean implicitly categorical language. In that sense, it is fair to say that Grothendieck introduced the notion in categorical foundations. I was of the supposition that it was Tarski who introduced the notion before, but you clarified its introduction in set-theoretical foundations. The question goes in the direction of Grothendieck' seminar and in that way into categorical foundations. Thanks for your clarification, nevertheless. Oct 28 at 15:21
• Yes, evidently Grothendieck was the first to make use of universes in category theory. Meanwhile, the universe concept itself strikes me as fundamentally set-theoretic in nature, even in its use in category theory, which is fundamentally similar to the use in set theory, namely, to delimit a robust realm of mathematical objects and structures, forming a mathematical world of sorts. The universe concept and the universe axiom is the beginning of the large cardinal hierarchy, understood and analyzed most deeply mainly in set theory. Oct 28 at 15:35
• In particular, in my view category theorists should stop saying Grothendieck universe in favor of the Zermelo-Grothendieck universe terminology. Obviously the idea should have Zermelo's name if it has anyone's name. Oct 28 at 15:46
• It seems to me that Zermelo's analysis was a far more significant advance in providing the quasi-categoricity result. Nov 1 at 17:29
• It is fine to call them Zermelo-Grothendieck, or Grothendieck-Zermelo, universes. But Zermelo was also not the first to consider inaccessible cardinals and universes. Hausdorff did that in the Mathematische Annalen in 1908. Tarski and Kuratowski and Baer and Zermelo all followed up on that in the years up to 1930. Steve Givant calls set theory with these universes "Tarski–Grothendieck set theory." Mathematical terminology almost never conveys the full history of concepts. Nov 2 at 0:06

In the D. Monk book "Introduction to Set Theory"

i find that the (first in the bibliography dates order) definition of Universe (in set theory) come from:

Tarski, Alfred (1938). "Über unerreichbare Kardinalzahlen"

Fundamenta Mathematicae 30: 68–89.

And it is exatly what SGA IV.1 reports.

• Yes, I have edited the question to be clear I am talking about Grothendieck's first uses of the term, not about the origin of the term in set theory. Mar 1, 2015 at 19:43
• Zermelo referred to universes of set theory (and these are the same as the Grothendieck universes) in 1930 in his quasi-categoricity paper. Oct 28 at 1:08

Even if there is an accepted answer, I would like to question the hypothesis.

According to

• Ralf Krömer, La « machine de Grothendieck » se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante, Revue d'histoire des mathématiques, Volume 12 (2006) no. 1, pp. 119-162. publisher, HAL, Numdam,

universes (in categorical foundations) were introduced as a Bourbaki "rédaction" ("Sur la formalisation des Catégories et foncteurs" no 307) by Grothendieck in 1958(?) as an alternative of the proposed foundations by (Formalisation des classes et catégories) D.Lacombe inside the group.

It appears that the archives Bourbaki does not have an electronic copy of it. But the cited reference has the appropriate excerpt in pg 149.

• I suspect that even if Grothendieck introduced the concept, it may still have been written up by P. Samuel. But this is a good find. It would be nice to have access to the original documents to get more info. Oct 28 at 0:58