Is there a classifying topos for schemes?

Question

Is there a topos $\mathcal E$ such that, for any sober topological space $X$, the geometric morphisms $$\mathrm{Sh}\left(\mathcal O\left(X\right)\right)\rightarrow \mathcal E$$ are in correspondance with the schemes whose underlying topological space is $X$?

Answer 1

Interesting question! No, such a topos $\mathcal{E}$ cannot exist.

Fix your favorite topological space $X$ which cannot be made into a scheme, that is, such that for no local sheaf of rings $\mathcal{O}_X$, the pair $(X,\mathcal{O}_X)$ is a scheme. For instance the reals with their Euclidean topology will do (the only open subspace of $\mathbb{R}^1$ which is a spectral space is the empty one).

Then there shouldn't be any geometric morphism $\mathrm{Sh}(X) \to \mathcal{E}$. However, we can easily construct such as a morphism, for instance as the composition of the unique morphism $\mathrm{Sh}(X) \to \mathrm{Sh}(\mathrm{pt})$ and the morphism $\mathrm{Sh}(\mathrm{pt}) \to \mathcal{E}$ corresponding to any of the many scheme structures on $\mathrm{pt}$.