# Global sections functor is fully faithful if and only if it is logical

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).