Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of fundamental group: one is in de Jong (1995), the other is in the book of Andre on period mappings.
Both of these fundamental groups admit a continuous homomorphism with a dense image to the algebraic fundamental group (comprised of finite etale coverings).
- These two groups generally speaking are not isomorphic, right? For the analytic projective line over $\mathbb{C}_p$, I think de Jong's group surjects on $SL_2(\mathbb{Q}_p)$, while by Prop. 2.1.6 of Andre's book, his group injects into the algebraic group for curves (and the algebraic group is trivial in this case).
- Are there any other definitions? Among all of them, is some definition "morally superior" to others? If not, in which situations which definition would you use?
- As far as I can tell, neither Andre nor de Jong construct a homotopy type. It probably should be technically trivial, we just should verify that the class of coverings in question is reasonable enough to define a topos, and then take the shape of the topos.
- Has this been written down in detail anywhere?
- Is there anything like Artin comparison theorem then (i.e. does the homotopy type determine the algebraic homotopy type and the topological homotopy type of the Berkovich space for normal spaces)?
- Are there any reasons to think that higher homotopy groups of analytic spaces are interesting?