A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits

Abstract

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 $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \geq μ$). When considering cardinalities of models of infinitary theories $\mathbb{T}$ of $\mathcal{L}_{κ, θ}$ that axiomatize $\mathcal{K}$, 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 $\mathbb{T}$ is a $\mathcal{L}_{κ, θ}$ theory whose models have directed colimits and it is $λ$-categorical for some $λ\geq μ$ in $S$, then it is $λ'$-categorical for every $λ' \geq μ$ 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.