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.



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> **Question:** *Elementary topos can be seen as a generalized space?*
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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).



"Spatial (geometrical)" realization of Elementary topos?
I would like to explain why I think the answer is no, but of course there is no way to prove this, and probably some way to use some geometric insight when talking about elementary toposes.


My main point is that the geometrical aspect of Grothendieck toposes is not related to the fact that they are elementary toposes, but rather to the fact that they are infinitary pretopos with a small generating set.


The fact that infinitary pretopos with a small generating set are also elementary topos is more of an accident.


I have three arguments to defend this claim:


1) Geometric morphism are not the natural notion of morphism of toposes, but the $f^\*$ functors (that preserve arbitrary colimits and finite limits) are the natural notion of morphism between infinitary pre-topos.


2) More general infinitary pre-topos still look a bit like generalized space, which are just "too big" to be toposes, and they are in general not elementary toposes.


3) In predicative mathematics, Grothendieck toposes are still the same as infinitary pre-topos with a small set of generators, they still behave as generalised spaces but they are no longer elementary toposes.


On a slightly related issue, I've heard several times that Grothendieck didn't like the idea of elementary toposes. I have always understood that as related to the fact that elementary topos don't have any of the geometric properties of toposes and maybe should not have been called "toposes" which is a clear reference to geometry and topology, But I have never seen a confirmation on either the initial claim or my interpretation of it, maybe some one can clarify this ? 


