Abstract elementary classes and infinitary logics

In this paper we study abstract elementary classes using infinitary logics and prove a number of results relating them. For example, if ( K , ≺ K ) is an a.e.c. with Löwenheim–Skolem number κ then K is closed under L ∞ , κ + -elementary equivalence. If κ = ω and ( K , ≺ K ) has finite character then K is closed under L ∞ , ω -elementary equivalence. Analogous results are established for ≺ K . Galois types, saturation, and categoricity are also studied. We prove, for example, that if ( K , ≺ K ) is finitary and λ -categorical for some infinite λ then there is some σ ∈ L ω 1 , ω such that K and Mod ( σ ) contain precisely the same models of cardinality at least  λ .