Re: Grothendieck's relative point of view

[Colin McLarty <colin.mclarty@case.edu>]

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


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