Hanf numbers via accessible images

TL;DR

This paper explores model-theoretic applications of large cardinal assumptions, demonstrating how they influence the accessibility and properties of models in abstract elementary classes and metric AECs.

Contribution

It generalizes Hanf number computations to accessible categories and shows that under large cardinal assumptions, metric AECs are strongly d-tame.

Findings

Accessible functors have accessible powerful images under large cardinal assumptions.

Joint embedding and amalgamation properties extend to models of all sizes in AECs.

Metric AECs are strongly d-tame under the assumptions.

Abstract

We present several new model-theoretic applications of the fact that, under the assumption that there exists a proper class of almost strongly compact cardinals, the powerful image of any accessible functor is accessible. In particular, we generalize to the context of accessible categories the recent Hanf number computations of Baldwin and Boney, namely that in an abstract elementary class (AEC) if the joint embedding and amalgamation properties hold for models of size up to a sufficiently large cardinal, then they hold for models of arbitrary size. Moreover, we prove that, under the above-mentioned large cardinal assumption, every metric AEC is strongly d-tame, strengthening a result of Boney and Zambrano and pointing the way to further generalizations.