Analytification is the limit of all tropicalizations

Abstract

We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings.

13 pages. v3: expository improvements throughout, especially in section 5 on fields with the trivial valuation. added appendix showing that tropicalization is invariant under extensions of valued fields