"Spatial (geometrical)" realization of Elementary topos?

Question

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).

Answer 1

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 ?

Answer 2

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.

Answer 3

It depends on what morphisms you take between topoi. Here is a dumb example: there is a category whose objects are groups and whose morphisms are group homomorphisms. There is another category whose objects are again groups but whose morphisms are arbitrary maps of sets, ignoring the group structure. This is in no sense a category of groups: in fact this category is equivalent to the category of sets.

Here is a less dumb example: there is a category whose objects are rings and whose morphisms are ring homomorphisms. There is also a 2-category, the Morita 2-category, whose objects are rings and whose hom categories are bimodules. The Morita 2-category behaves very differently from the ordinary category of rings; for example, it has biproducts. In fact it behaves in many respects like a categorified version of vector spaces. This is to say that naming only the objects in this 2-category doesn't tell you very much about it; the meat is in the morphisms.

Anyway, here's a path you can take to a very general notion of space, which includes both topoi and various schemes and stacks as a special case, with appropriate morphisms. Namely, consider the following collection of analogies:

• Categories are analogous to sets.
• Cocomplete categories (and cocontinuous functors between them) are analogous to abelian groups (and linear maps between them).
• Monoidal cocomplete categories (this includes the condition that the monoidal product distributes over colimits) are analogous to rings, and symmetric monoidal cocomplete categories are analogous to commutative rings.

Probably we want everything to be presentable too.

Any symmetric monoidal cocomplete category can be thought of as the category of some sort of "sheaves" on some sort of "space," categorifying the standard idea in algebraic geometry to think of any commutative ring as the ring of some sort of "functions" on some sort of "space." One might call this "2-affine" (maybe even "1-affine") or "Tannakian" geometry. See, for example, Chirvasitu and Johnson-Freyd.

For an algebraic geometer the typical example is the symmetric monoidal category $\text{QC}(X)$ of quasicoherent sheaves on a scheme or stack, but any cocomplete cartesian closed category (in particular, any Grothendieck topos) is also an example, where the symmetric monoidal structure is given by product. The first class of examples are "$\text{Ab}$-algebras" while the second class are "$\text{Set}$-algebras."

The point of saying all this is that the natural notion of morphism between such things, following the analogy above, is a symmetric monoidal cocontinuous functor. For Grothendieck topoi this becomes a functor which preserves arbitrary colimits and finite products. This includes geometric morphisms, but from this point of view logical morphisms don't really enter into the story.

To get elementary topoi, as objects, into the game, you can try Ind-completing them.