Let $\mathcal{E}$ be a non-degenerate topos.

Question: Is it true that the "global sections" functor $\operatorname{Hom}_{\mathcal{E}}(1,-)$ is fully faithful if and only if it is logical?

A non-degenerate topos $\mathcal{E}$ for which $\operatorname{Hom}_{\mathcal{E}}(1,-)$ is faithful is called a well-pointed topos. It is known that in a well-pointed topos, the subobject classifier $\Omega$ can have only two points ($\top$ and $\bot$), so the "global sections" functor preserves the subobject classifier (and of course, (finite) limits) for such a topos.

Hence, to show that $\operatorname{Hom}_{\mathcal{E}}(1,-)$ is logical whenever it is fully faithful, it suffices to show that it preserves exponential objects.

Besides $\mathbf{Set}$ itself, one other example of a topos for which the "global sections" functor is a fully faithful logical functor is $\mathbf{FinSet}$, the topos of finite sets. Also, any (nonempty) Grothendieck universe would work in place of $\mathbf{FinSet}$.

What would not necessarily work include $L$ (the constructible universe) and ultrapowers of $\mathbf{Set}$. In both cases, the "global sections" functor is faithful but not full, nor is it logical (unless $V=L$ or the ultrafilter is principal, respectively).


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