This paper is devoted to the proof of the following upward categoricity theorem: Let 𝔎 be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If 𝔎 is categorical in ℵ₁ then 𝔎 is categorical in every uncountable cardinal. More generally, we prove that if 𝔎 is categorical in a successor cardinal λ⁺ then 𝔎 is categorical everywhere above λ⁺.
"Upward categoricity from a successor cardinal for tame abstract classes with amalgamation." J. Symbolic Logic 70 (2) 639 - 660, June 2005. https://doi.org/10.2178/jsl/1120224733