This paper continues the study of generalized amalgamation properties begun in , , ,  and . 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 we 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.
Bibliographical noteFunding Information:
∗ The second author was supported by NRF grant 2010-0016044. ∗∗ The third author was partially supported by NSF grant DMS-0901315. Received June 22, 2010 and in revised form May 2, 2011
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