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NSOP$_1$-like independence in AECats

Mark Kamsma

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.

NSOP$_1$-like independence in AECats

Abstract

The classes stable, simple and NSOP 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 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-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.

Paper Structure

This paper contains 8 sections, 26 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. AECats
  3. Lascar strong Galois types
  4. Independence relations
  5. Independence theorem, 3-amalgamation and stationarity
  6. Long dividing, isi-dividing and long Kim-dividing
  7. Canonicity
  8. More on Lascar strong Galois types

Key Result

Theorem \oldthetheorem

Let $(\mathcal{C}, \mathcal{M})$ be an AECat with the amalgamation property and let $\mathcal{B}$ be some base class. Suppose that $(\mathcal{C}, \mathcal{M})$ satisfies the $\mathcal{B}$-existence axiom and suppose that there is an NSOP$_1$-like independence relation $\mathop{\hbox{$\mid$} \hbox{$\

Theorems & Definitions (103)

  • Theorem \oldthetheorem: Canonicity of NSOP$_1$-like independence
  • Theorem \oldthetheorem: Canonicity of simple independence
  • Theorem \oldthetheorem
  • Definition \oldthetheorem: kamsma_kim-pillay_2020
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • ...and 93 more