Analogy between topology and algebraic geometry 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. 
 A: I suppose it's true that this is an aspect that deserves to receive more attention. 
One place where algebraic geometry is systematically done via the topos theory of the étale toposes of the given spaces is Jacob Lurie's "Structured Spaces" and generally the 
"E-∞ Geometry" based on that. 
(If you don't want to get into higher geometry you can simply ignore all the $\infty$-prefixes there and still get a discussion of the foundations of algebraic geometry that is conceptually cleaner than most of what one sees offered elsewhere).
For instance one statement of the kind that you are looking for is theorem 1.2.1 in "Quasi-Coherent Sheaves and Tannaka Duality Theorems". This says that a morphism of affine schemes is étale precisely if the induced map on étale toposes is an étale geometric morphism of ringed toposes.
A: 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.
