Does anyone know in what paper Grothendieck universes were first introduced? I looked at Grothendieck's 1957 paper. I can't read French so I am not entirely sure, but I couldn't find any definitions similar to that of Grothendieck universes in that paper. Did I simply miss it, or are Grothendieck universes first introduced in a different paper? I expect that if Grothendieck was the one to introduce Grothendieck universes, then he defined them under a different name.


Solution 1:

The first use of Grothendieck universes that I'm aware of is in Zermelo's 1930 paper "Über Grenzzahlen und Mengenbereiche" (Fundamenta Mathematicae 16, pp. 29–47). But there might be earlier occurrences of the concept, most likely in work of von Neumann in the 1920's.

Solution 2:

Grothendieck universes were first introduced at the beginning of the Chapter 1 titled "Préfaisceaux" (page 185) of:

Grothendieck, A., & Verdier, J. L. (1972). Volume 1 of Théorie des Topos et Cohomologie Etale des Schémas. Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4). Lecture Notes in Mathematics. doi:10.1007/bfb0081551