Grothendieck toposes

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

For some years now, I have been using the phrase "Grothendieck topos" -
category of sheaves over a site - to allow the site to be in an
arbitrary base elementary topos S (often assumed to have nno). Hence
"Grothendieck topos" means "bounded S-topos". The whole study of
Grothendieck toposes, as of geometric logic, is parametrized by choice of S.

That's presumably not how Grothendieck understood it, and I know some of
his results assumed S = Set, some classical category of sets. Moreover,
the Elephant defines "Grothendieck topos" that way.

On the other hand, if a topos is a generalized space, with a classifying
topos being the space of models of a geometric theory, then that surely
meant Grothendieck topos; and there are various reasons for wanting to
vary S. For example, using Sh(X) as S gives us a generalized topology of
bundles, fibrewise over X.

I'm coming to suspect my usage may confuse.

How do people actually understand the phase "Grothendieck topos"? Do
they hear potential for varying an implicit base S, or do they hear a
firm implication that S is classical?

Steve Vickers.


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