Is the functor from the category of topological spaces to the 2-category of Grothendieck topoi (and geometric morphisms, of course), which sends a space to its topos of sheaves, faithful? (I already know that the functor from sober spaces to Grothendieck topoi is fully faithful.)
Is the functor from schemes to Grothendieck topoi, which sends a scheme to its étale topos, faithful?
Best Answer
Short answer: no.
For topological spaces the obstruction is precisely non-sobriety. The category of sheaves on a non-empty indiscrete space is equivalent to $\textbf{Set}$. Therefore the sheaf topos functor from the 1-category of topological spaces to the 2-category of Grothendieck toposes is not faithful, because geometric morphisms to $\textbf{Set}$ are unique up to unique isomorphism, but there may be distinct continuous maps to an indiscrete space.
For schemes the structure sheaf is crucial, even using the étale topos. For example, étale topos of $\operatorname{Spec} k$ for any separably closed field $k$ is equivalent to $\textbf{Set}$. But there are non-trivial automorphisms $\mathbb{C} \to \mathbb{C}$ – complex conjugation, at least! To say nothing of $\overline{\mathbb{Q}}$ or $\overline{\mathbb{F}_p}$. Therefore the étale topos functor from the 1-category of schemes to the 2-category of Grothendieck toposes is not faithful either.