METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

We show that metric abstract elementary classes (mAECs) are, in the sense of [15], coherent accessible categories with directed colimits, with concrete ℵ 1 -directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC —an AEC-like category in which only the κ -directed colimits need be concrete—and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah’s Presentation Theorem and a proof of the existence of an Ehrenfeucht–Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [15] yield a proof that any categorical mAEC is μ- d - stable in many cardinals below the categoricity cardinal.