Abstract elementary classes and infinitary logics

AbstractIn 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∈Lω1,ω such that K and Mod(σ) contain precisely the same models of cardinality at least λ