The unipotent tropical fundamental group

Abstract We prove that the unipotent completion of the fundamental group of the generic fiber of a tropically smooth family of algebraic varieties can be computed combinatorially: it is canonically isomorphic to the unipotent tropical fundamental group of its tropicalization, a concept that we introduce in this paper. We define the unipotent tropical fundamental group of a polyhedral complex in as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable in that it satisfies a Seifert–Van Kampen theorem and has a description for fans in terms of a bar complex. We then review an intermediate classical object, the unipotent log de Rham fundamental group. We prove our main result by establishing isomorphisms between the unipotent log de Rham fundamental group of the log closed fiber of the family and the unipotent topological and tropical fundamental groups by means of Kato–Nakayama spaces and a descent argument. Our arguments require a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.