Categoricity in abstract elementary classes: going up inductive step

TL;DR

This paper explores stability theory in non-elementary classes without compactness, focusing on the inductive step for categoricity transfer in abstract elementary classes.

Contribution

It introduces methods for the inductive step in categoricity transfer within non-elementary classes lacking compactness.

Findings

Develops new techniques for the inductive step in categoricity transfer.

Extends stability theory to broader classes without compactness assumptions.

Provides foundational results for categoricity in abstract elementary classes.

Abstract

We deal with stability theory for ``reasonable'' non-elementary classes without any remanents of compactness (like: above Hanf number or definable by L_{omega_1, omega}).