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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.
    1. Has this been written down in detail anywhere?
    2. 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)?
    3. Are there any reasons to think that higher homotopy groups of analytic spaces are interesting?
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