A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits
Résumé
We provide a proof, in ZF C, 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 isomorphism), 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
Domaines
Mathématiques [math]
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