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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?

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I have no solid evidence, but I believe it was written by Grothendieck himself. – Joël Oct 20 '13 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. – Colin McLarty Oct 21 '13 at 0:57
up vote 9 down vote accepted

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.

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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 '15 at 15:40

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.

See also:

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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. – Colin McLarty Mar 1 '15 at 19:43

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