TITLE: "Spatial (geometrical)" realization of Elementary topos?
QUESTION [10 upvotes]: It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry.
Note: Grothendieck view of Topos is as an "ultimate" generalization of space.
Also, Elementary topos has many good logical properties. I am interested in elementary topos as a formal geometry.

Question: Elementary topos can be seen as a generalized space?

Note: Can "Elementary higher topos" reflects the geometrical nature of objects in mathematics? This could suggest the physical nature of mathematics (this is vague, only a philosophical note).

REPLY [16 votes]: One way in which elementary toposes can be seen as "fully geometric" is by relativizing them.  An elementary topos is a Grothendieck topos if and only if it admits a bounded geometric morphism to the category of sets, and the 2-category of Grothendieck topoi (and geometric morphisms) is equivalent to the slice category of elementary topoi (and geometric morphisms) equipped with a bounded map to Set.  If we replace Set by any other elementary topos $S$, then the 2-category of elementary topoi equipped with a bounded geometric morphism to $S$ behaves very much like the 2-category of Grothendieck topoi, and in particular is very geometric, especially if $S$ has a NNO.