NSOP$_1$-like independence in AECats

TL;DR

This paper extends the concept of independence relations from first-order theories to AECats, establishing canonicity theorems for stable, simple, and NSOP$_1$-like independence in this broader categorical framework.

Contribution

It generalizes the stability hierarchy and independence concepts to AECats, proving canonicity theorems and defining new independence notions like isi-dividing, isi-forking, and long Kim-dividing.

Findings

Canonicity theorems for stable, simple, and NSOP$_1$-like independence in AECats.

Abstract definitions of independence relations: isi-dividing, isi-forking, long Kim-dividing.

Recovery of parts of the original stability hierarchy in the categorical setting.

Abstract

The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP$_1$ theories it must come from Kim-dividing. We generalise this work to the framework of AECats (Abstract Elementary Categories) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple and NSOP$_1$-like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking and long Kim-dividing.