Categoricity and universal classes

Let ( K , ⊆ ) be a universal class withLS ( K ) = λ categorical in a regular κ > λ +with arbitrarily large models, and let K ∗ be the class of all A ∈ K > λfor which there is B ∈ K ≥ κsuch that A ⊆ B . We prove that K ∗ is totally categorical (i.e., ξ‐categorical for all ξ > LS ( K ) ) andK ≥ ℶ μ +⊆ K ∗for μ = 2 λ +. This result is partially stronger and partially weaker than a related result due to Vasey. In addition to small differences in our categoricity transfer results, we provide a shorter and simpler proof. In the end we prove the main theorem of this paper: the models of K > λ + ∗ are essentially vector spaces (or trivial, i.e., disintegrated).