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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 Abstract Elementary Categories (AECats) 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. [ABSTRACT FROM AUTHOR]

Let ${\cal M}$ be ternary, homogeneous and simple. We prove that if ${\cal M}$ is finitely constrained, then it is supersimple with finite SU-rank and dependence is k -trivial for some k < ω and for finite sets of real elements. Now suppose that, in addition, ${\cal M}$ is supersimple with SU-rank 1. If ${\cal M}$ is finitely constrained then algebraic closure in ${\cal M}$ is trivial. We also find connections between the nature of the constraints of ${\cal M}$ , the nature of the amalgamations allowed by the age of ${\cal M}$ , and the nature of definable equivalence relations. A key method of proof is to "extract" constraints (of ${\cal M}$) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1. [ABSTRACT FROM AUTHOR]

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum--a function of two cardinals κ and λ giving the supremum of the possible number of types over a model of size _ that do not fork over a submodel of size κ. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded κ (ded κ)ω. [ABSTRACT FROM AUTHOR]