TITLE: Analogy between topology and algebraic geometry
QUESTION [16 upvotes]: In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in algebraic geometry. 
My question is that do these concepts actually coincide? I mean, for example, a proper morphism of schemes actually induces a proper geometric morphism between some toposes induced by schemes (like etale topos)?
Although I mensioned about only morphisms, I want to know such analogous concepts  in topology and algebraic geometry which are coincide at the level of toposes.

REPLY [3 votes]: I found this question while I was also wondering about proper maps of schemes versus proper geometric morphisms of their small étale toposes. I will add a partial result for future reference.
I'll consider toposes of sheaves of sets (no ringed toposes).
A topos $\mathcal{E}$ is tidy or strongly compact iff the global sections functor $\Gamma : \mathcal{E} \to \mathbf{Sets}$ preserved filtered colimits, and it is proper or compact iff $\Gamma$ preserves directed unions of subterminal objects. All tidy toposes are proper.
It is shown in SGA4, Exposé VI, Exemple 1.22 that the small étale topos $X_\mathrm{\acute{e}t}$ is tidy if and only if $X$ is quasi-compact quasi-separated (and similarly for the small Zariski topos). So if $X$ is a proper scheme, then $X_\mathrm{\acute{e}t}$ is a tidy (in particular, proper) topos. But if $X$ is an affine scheme, then the associated topos $X_\mathrm{\acute{e}t}$ is proper as well.