Tameness, powerful images, and large cardinals

TL;DR

This paper characterizes large cardinals from weakly compact to strongly compact through closure properties of powerful images of accessible functors, linking them to tameness in abstract elementary classes and extending prior results.

Contribution

It provides a unified framework connecting large cardinal properties with tameness in abstract elementary classes, extending previous research in the area.

Findings

Characterization of large cardinals via powerful images

Equivalence of these properties to tameness in AECs

Extension of prior results in the field

Abstract

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [BU17], [BTR16], [Lie18], and [LR16].