Dear Thomas, OK, let me see if I understand you correctly. I'll used the language of indexed categories, because it brings out the idea of categories BTop/S of Grothendieck toposes, one for each S, with reindexing along change of base. I'm happy to see it translated into fibred categories (all in a suitable 2- or bi-categorical sense, of course). If we have a geometric morphism g: T -> S (_not_ the inverse image functor), then we have a reindexing functor from BTop/S to BTop/T given by pseudo-pullback along g. In that indexed category I can conjecture geometricity results, such as that the symmetric topos construction is an indexed endofunctor. (There may already be enough in Bunge-Funk to prove this, though I'm bothered by coherence questions.) That's what geometricity says, that the construction commutes (up to ...) with those pullbacks. You seem to want me to use a coindexed category given by composition of geometric morphisms, at least if they are all bounded, and maybe also to think of it as indexed by reversing the base change morphisms (inverse image functors instead of geometric morphisms). Is that because the reindexing and co-reindexing are adjoint, so you can deal with either one of them, and the coindexed one has the advantage of being strict? (Maybe even then the indexed category is cocomplete, with Beck-Chevalley.) All the best, Steve. On 3 Dec 2016, at 19:27, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
Your variances don't seem to match what I had in mind.
I know but what I suggest is the right thing for what you have in mind. BTop/SS is the category of Grothendieck toposes over base topos SS. That you find in my notes on Fibrations and in the Elephant. I understood your question as asking how to conceptualize base change from SS to TT.
I can try to make more explicit what is behind my suggestion. Bounded gm's to SS correspond to Grothendieck toposes over SS. If F -| U : EE -> SS is such a guy the corresponding fibered topos if P_F = F^*P_EE where P_EE is the fundamental ("codomain") fibration. Now if G : TT -> SS is the inverse image part of a bounded gm then G*P_F \cong P_{FG}. But FG is the inverse image part of the composite of the bgm's.
I think that adresses change of base for toposes over a base topos.
What you asked makes sense in itself but doesn't match the aim you described.
Thomas
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