Metric abstract elementary classes as accessible categories

TL;DR

This paper demonstrates that metric abstract elementary classes can be viewed as accessible categories with specific colimit properties, introduces a broader notion of $-concrete AECs, and explores their theoretical framework and stability results.

Contribution

It establishes a category-theoretic framework for metric AECs, generalizes to $$-concrete AECs, and proves stability properties in many cardinals.

Findings

mAECs are coherent accessible categories with directed colimits.

Develops the theory of $$-concrete AECs including Shelah's Presentation Theorem.

Proves categorical mAECs are $$-d-stable in many cardinals below categoricity.

Abstract

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), 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 [LR] yield a proof that any categorical mAEC is $\mu$-d-stable in many cardinals below the categoricity cardinal.