Finite étale topos = étale topos of a field

Question

This implies/suggests that the finite étale topos $\mathrm{Sh}(\mathrm{Spec}(k)_{\mathrm{f, ét}})$ is equivalent to $G\mathbf{Set}$ for $G$ the absolute Galois group of $k$. The quote here, on the other hand, states that étale topos $\mathrm{Sh}(\mathrm{Spec}(k)_{\mathrm{ét}})$ of $k$ is equivalent to $G\mathbf{Set}$ for $G$ the absolute Galois group of $k$.

So is the finite étale topos of $k$ equivalent to the étale topos of $k$?

Answer 1

Let $S$ be a scheme. There are a few (... well, actually, a proper class of ...) things people might mean when they says "the étale topos of $S$". All of them are categories of sheaves on a site where the underlying category is some full subcategory $\mathcal{C} \subseteq \textbf{Sch}_{/ S}$ and the Grothendieck topology is generated by jointly surjective families of étale morphisms.

The first and most important point to check is whether every object in $\mathcal{C}$ is an étale $S$-scheme. If it is, then you are talking about a "petit" topos; otherwise, you are talking about a "gros" topos. My impression is that algebraic geometers are more likely to be talking about petit toposes and topos theorists are more likely to be talking about gros toposes, with the caveat that anyone talking about the functor of points is almost surely talking about gros toposes and not petit toposes.

The next point to check is whether there is a size/complexity condition on the objects of $\mathcal{C}$. In the petit case, it usually does not matter because you get the same topos of sheaves in the end. In the gros case, different conditions can yield inequivalent toposes, or even fail to define a topos at all.

Let me illustrate with some examples of choices of $\mathcal{C}$.

1. The category of finite étale $S$-schemes.

   This is an essentially small category and defines a petit topos.

2. The category of étale $S$-schemes of finite presentation.

   This is an essentially small category and defines a petit topos. If $S$ is the spectrum of a field then it is the same site as 1.

3. The category of étale $S$-schemes where the domain is affine over $\operatorname{Spec} \mathbb{Z}$.

   This is an essentially small category and defines a petit topos equivalent to the petit topos defined by 2.

4. The category of étale $S$-schemes.

   This is usually not an essentially small category, but nonetheless it defines a petit topos. In fact, it is equivalent to the petit topos defined by 2 or 3.

5. The category of $S$-schemes of finite presentation.

   This is an essentially small category and defines a gros topos.

6. The category of $S$-schemes locally of finite presentation over $S$ and affine over $\operatorname{Spec} \mathbb{Z}$.

   This is an essentially small category and defines a gros topos equivalent to the gros topos defined by 5.

7. The category of $S$-schemes locally of finite presentation (over $S$).

   This is usually not an essentially small category but nonetheless defines a gros topos. In fact, it is equivalent to the gros topos defined by 5 or 6.

8. The category of $S$-schemes affine over $\operatorname{Spec} \mathbb{Z}$.

   This is usually not an essentially small category and usually does not define a topos.

9. The category of $S$-schemes.

   This is usually not an essentially small category and usually does not define a topos. However, the category of sheaves obtained is equivalent to the one defined by 8.