Approximations of superstability in concrete accessible categories

TL;DR

This paper extends the theory of superstability to a broad class of accessible categories, demonstrating that categoricity in a successor implies saturation of directed colimits of saturated objects.

Contribution

It generalizes previous results to coherent accessible categories with concrete directed colimits, establishing new conditions for saturation.

Findings

Directed colimits of saturated objects are saturated in categorical contexts.

Generalization of Baldwin's results to broader accessible categories.

Abstract

We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we prove that if any category of this form is categorical in a successor, directed colimits of saturated objects are themselves saturated.