The etale fundamental group, as explained in SGA 1 Expose 5 and various other notes I've read, always makes the assumption that the scheme $S$ (for which one intends to construct a fundamental group), is locally noetherian.
How necessary is this assumption?
For example, let $\text{FEt}_S$ be the category of schemes finite etale over a fixed connected scheme $S$. Is $\text{FEt}_S$ a galois category when $S$ isn't locally noetherian?
From the viewpoint of galois categories, it seems that almost all the axioms for a galois category are obvious for $\text{FEt}_S$ (without any noetherian hypotheses), except possibly the conditions:
For any $X\in\text{FEt}_S$ and a finite group $G$ acting on $X$ by automorphisms over $S$, the quotient $X/G$ exists in $\text{FEt}_S$.
For a geometric point $s\in S$, and any $X\in\text{FEt}_S$ acted on by a finite group $G$ of $S$-automorphisms, the fiber functor $F_s$ satisfies $F(X)/G\cong F(X/G)$
The latter appears to follow from SGA 1 Expose 5 beginning of section 2 (as presented here http://arxiv.org/abs/math/0206203)
However, I'm not sure if the first is true. Since $X\rightarrow S$ is finite hence affine, you can reduce to the case where $X,S$ are both affine. In this case if $X = \text{Spec }A$, then $X/G$ is just $\text{Spec }A^G$, but I don't know if $\text{Spec }A^G$ is finite etale over $S$.