Technical meaning of "profinite circle"

Question

In a private exchange with a professional mathematician, I found the following statement:

the "small etale topos" of a finite field is a "profinite circle", and thus looks like circle.

Could anyone explain me the exact meaning of "profinite circle" in this (or any other) context? And please do not tell me to ask him about it, since I have only very occasional access to him.

Thanks

PS: Apparently, Proposition 8.1 in Chapter 5 of the following text by Grothendiek deals is what is at stake here:

http://arxiv.org/pdf/math/0206203v2.pdf

Answer 1

Just as a group $G$ defines a homotopy type $BG$, its classifying space, a profinite group defines a profinite homotopy type, and "profinite circle" refers to the étale homotopy type of $\text{Spec } \mathbb{F}_q$, which is the profinite homotopy type $B \widehat{\mathbb{Z}}$, where $\widehat{\mathbb{Z}}$ is the profinite group of profinite integers. This is the absolute Galois group of $\mathbb{F}_q$, or equivalently the étale fundamental group of $\text{Spec } \mathbb{F}_q$.

The sense in which this looks like a circle is that the circle, as a homotopy type, is $B \mathbb{Z}$. The simplest version of this relationship is the claim that the category of finite connected covering spaces of the circle is equivalent to the opposite of the category of finite extensions of $\mathbb{F}_q$.