A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits
We provide a proof, in ZFC , of Shelah’s eventual categoricity conjecture for abstract elementary classes (AEC’s). Moreover, assuming in addition the Singular Cardinal Hypothesis ( SCH ), we prove a direct generalization to the more general context of accessible categories with directed colimits. If K is such a category, we show that there is a cardinal µ such that if K is λ -categorical for some λ ≥ µ (i.e., it has only one object of internal size λ up to iso-morphism), then K is eventually categorical (i.e., it is λ ′ -categorical for every λ ′ ≥ µ ). When considering cardinalities of models of inﬁnitary theories T of L κ,θ that axiomatize K , the result implies, under SCH , the following inﬁnitary version of Morley’s categoricity theorem: let S be the class of cardinals λ which are of coﬁnality at least θ but are not successors of cardinals of coﬁnality less than θ . Then, if T is a L κ,θ theory whose models have directed colimits and it is λ -categorical for some λ ≥ µ in S , then it is λ ′ -categorical for every λ ′ ≥ µ in S ; moreover, we also exhibit an example that shows that the exceptions in the class S are needed. Along the way we also prove Grossberg conjecture, according to which categoricity in a high enough cardinal implies eventual amalgamation. We establish this result in AEC’s and, assuming in addition SCH , in the more general context of accessible categories whose morphisms are monomorphisms. 1