Title: Tropical Adic Spaces

Abstract: I will introduce (embedded) tropicalization and give some motivation for it. Embedded tropicalization is not a morphism in any category. Working towards endowing tropical varieties with extra structure, we study the algebra of convergent tropical power series and the topological spaces (of prime congruences) it corresponds to. The construction so far allows us to see a tropicalization map as a natural transformation of functors taking values in the category of topological and uniform spaces