A category-theoretic characterization of almost measurable cardinals

TL;DR

This paper provides a category-theoretic characterization of almost measurable cardinals by analyzing the closure properties of powerful images of accessible functors and their implications for Galois types and measurability.

Contribution

It establishes a novel equivalence between almost measurable cardinals and certain categorical properties, connecting set theory with category theory.

Findings

Closure of powerful images under colimits of $ppa$-chains for almost measurable cardinals.

$ppa$-locality of Galois types derived from categorical conditions.

Characterization of almost measurable cardinals through purely category-theoretic properties.

Abstract

Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $\kappa$-chains, $\kappa$ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of Lieberman and Rosicky, implies $\kappa$-locality of Galois types. As this, in turn, implies sufficient measurability of $\kappa$, via a paper of Boney and Unger, we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.