AMALGAMATION FUNCTORS AND BOUNDARY PROPERTIES IN SIMPLE THEORIES

Abstract. This paper continues the study of generalized amalga-mation properties begun in [1], [2], [3], and [6]. 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 un-stable theories showing, in particular, that there is no straightfor-ward generalization of the groupoid construction in an unstable context