Constructing fundamental group in usual way, one uses $I=[0,1] \in \mathbb R$, so uses real numbers, and gets a group classifying coverings for nice spaces: locally connectible and so on, in fact spaces "with real properties". But one can also go the opposite way and define etalé fundamental group via coverings.
So the questions are
- Whether there is a definition of, for example, etalé fundamental group for p-adic manifolds via "pathes" or something similar?
- It seems that one can correctly define fundamental group verbatim, using an ordered 2-divisible abelian group, such as rational numbers, but does the result make sense?