  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.
