Let $K$ be a number field, and let $X\_K$ be a $K$-variety.


The etale fundamental group of $X\_K$, as defined in SGA1, classifies the automorphisms of finite etale covers of $X\_K$. Some of these etale covers are not geometric. For example, if $L$ is a finite field extension of $K$, then $X\_L\rightarrow X\_K$ is a finite etale cover. A cover $Y$ of $X\_K$ is called regular (think: geometric) if it doesn't have an extension of scalars (to be precise, if $K$ is algebraically closed in the function field of every connected component of $Y$).


I know, however, that there is no "regular etale fundamental group" for $X\_K$, that classifies the automorphisms of just the regular finite etale covers of $X\_K$. I vaguely recall hearing in a conference that this is because the regular etale covers don't form a Galois category. Is that correct? What fails for them to form a Galois category? This question has been in the back of my mind ever since I started working with the etale fundamental group.



Why is there no "regular etale fundamental group"?
$K = \mathbb{Q}$, $X\_K = \mathrm{Spec} \mathbb{Q}[t, t^{-1}]$. Let $Y$ and $Z$ be the covers of $X\_K$ corresponding to adjoining $\sqrt{t}$ and $\sqrt{-t}$ respectively. They are both regular covers, but their composite contains $\sqrt{-1}$ and thus is not.


