  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.
