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. aleph(1)-directed colimits and concrete monomorphisms. More broadly, we define a notion of kappa-concrete AEC-an AEC-like category in which only the kappa-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 mu-d-stable in many cardinals below the categoricity cardinal.