The Unipotent Tropical Fundamental Group

We define the unipotent tropical fundamental group of a polyhedral complex in ${\mathbb{R}}^n$ 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 analogous classical object, the unipotent de Rham fundamental group of a schön subvariety of a toric variety. Our main result is a correspondence theorem between classical and tropical unipotent fundamental groups: there is an isomorphism between the unipotent completion of the fundamental group of a generic fiber of a tropically smooth family over a disc and the tropical unipotent fundamental group of the family’s tropicalization. This theorem is established using Kato–Nakayama spaces and a descent argument. It requires a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.