Is this true? About choice toposes.

Is this true? About choice toposes.

[Colin McLarty]

I believe the following theorem is correct.  I would really like it to
be, at least for Grothendieck toposes, and it may be well known.  But
I want to make sure I have not missed something.  The motivation is
Barr covers but I believe it is a result in elementary topos theory.

Theorem:  For any topos A and geometric morphism f^*,f_*:B-->A, where
B satisfies axiom of choice, the direct image f_* preserves module
quotients.

Proof:  In the choice topos B every quotient homomorphism has a right
inverse function (which is generally not a module homomorphism), so
its direct image also does, so the direct image is onto and thus is a
quotient.

Is that good?

Colin


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