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If $E$ is a topos, is there a nice way to characterize the category of logical morphisms $E\to Set$? Is it complete and/or cocomplete?

The topos $Set$ geometrically represents a point; what does it logically represent?

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I'd be interested if there were any good answers to this. A related question is where even in the case $E = Set$, existence of non-trivial logical endofunctors involve fairly large cardinal hypotheses. – Todd Trimble Sep 12 '10 at 1:20
What is a logical morphism? – Martin Brandenburg Sep 13 '10 at 9:54
@MartinBrandenburg: Roughly, a functor $F:A\to B$ between toposes is called "logical" if it preserves finite limits and power-objects. See – David Spivak May 8 '14 at 18:22

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