Amalgamation functors and boundary properties in simple theories

TL;DR

This paper investigates the structure of amalgamation properties in simple theories, analyzing associated groupoids, their abelian nature, and the relationships between existence and uniqueness properties across different dimensions, with examples in unstable theories.

Contribution

It provides a detailed analysis of groupoids from failure of 3-uniqueness, introduces a family of weaker existence and uniqueness properties, and explores their categorical formulation in simple theories.

Findings

Groupoids from failure of 3-uniqueness are abelian.

The binding group of these groupoids is abelian and linked to automorphism groups.

Examples show no straightforward generalization in unstable theories.

Abstract

This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.