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

A category-theoretic characterization of almost measurable cardinals