I was just looking at Grothendieck's statement about schemes in EGA 3: Pour obtenir un langage qui ``colle" sans effort à l'intuition g\'{e}om\'{e}trique, et \'{e}viter des circonlocutions insupportables \`{a} la longue, nous identifions toujours un pr\'{e}sch\'{e}ma $X$ sur un autre $S$ au foncteur \mbox{\clarrow{(\mathrm{Sch}/S)^\mathrm{o}}{\mathrm{Ens}}} qu'il repr\'{e}sente, "To make the language stick to geometric intuition, and to avoid finally unbearable circumlocutions, we will always identify a scheme X over another S, with the functor from Sch/S to Set that it represents." This quote is from the Springer Verlag edition page VI. This edition was printed in 1970. I do not yet know if it is printed in the earlier IHES edition. The IHES edition of EGA chapter 0, printed in 1960, does urge the functorial rather than topological space conception of a sheaf. "We systematically abstain from using espaces etales ... we never consider a sheaf a topological space" (p. 25). best, Colin ___________________________________________________________ t 17:34 29/03/2003 -0500, Lawvere wrote:
Thierry Coquand recently asked me
In your "Comments on the Development of Topos Theory" you refer to a simpler alternative definition of "scheme" due to Grothendieck. Is this definition available at some place?? Otherwise, it it possible to describe shortly the main idea of this alternative definition??
Since several people have asked the same question over the years, I prepared the following summary which, I hope, will be of general interest:
The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme (which had required as a prerequisite the baggage of prime ideals and the spectral space, sheaves of local rings, coverings and patchings, etc.), should be abandoned AS the FUNDAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the "baggage" could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition..