2010/7/7 Eduardo J. Dubuc <edubuc@dm.uba.ar>:
Graham White wrote:
I'm looking for some history of what seems to be called "Grothendieck's relative point of view":
What about "Grothendieck's change of base", so much used in SGA4, and extensively advocated and used by Joyal and others by considering the category of Grothendieck topoi over a given Grothendieck topos (not Sets) ?.
It is an interesting question exactly how Grothendieck came to this. The Grothendieck-Riemann-Roch probably was the first thing that made the idea prominent. But Grothendieck would also have been encouraged to think this way by his interest in sheaves (of sets) and the more-or-less well known fact (in the 1950s) that the idea of a "sheaf X over a sheaf S on space T" has two equivalent definitions: as either an arrow in the category of sheaves on T, or a local homeomorphism to the espace etale of the sheaf S. This is just the special case of what Eduardo remarks, a slice topos over a spatial topos is again a spatial topos. These things arose in the 1950s. But in itself, the relative point of view is an extension to geometry of the usual practice in algebraic number theory which goes back before the 1850s. An algebraic number field is not just a field but a field extension. We constantly view it as an extension of the rational field. But more than that, number theorists have long focused on questions about, say, all cyclic extensions of any fixed algebraic number field, or which groups can occur as Galois groups of one algebraic number field over another. And those are just the natural outgrowths of a much older question, clarified by Abel and Galois but asked long before (e.g. it is the subject of Euclid Book X, stated in terms of constructions rather than polynomials): If a given polynomial has no rational roots, does it have roots expressible in terms of square roots of rationals, or n-th roots? best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]