The specialization morphism for the étale fundamental groups of Grothendieck cannot be generalized word-for-word to the more general pro-\’etale fundamental group of Bhatt and Scholze.

It turns out, that one can deal with this problem by applying a rigid-geometric point of view: for a formal scheme X of finite type over a complete rank one valuation ring, we construct a specialization morphism

from the de Jong fundamental group of the rigid generic fiber to the pro-étale fundamental group of the special fiber.  The construction relies on an interplay between admissible blowups of X and normalizations of the irreducible components of X_k, and employs the Berthelot tubes of these irreducible components in an essential way.

I will also mention a generalization of the de Jong’s fundamental group. It is defined using a notion of “geometric arcs” in rigid geometry, enjoys many good properties of the pro-étale fundamental group and allows to answer some old questions of de Jong. This is a joint work with Piotr Achinger and Alex Youcis.