In Shelah’s classification of first-order theories we classify theories using combinatorial properties. The most well-known class is that of stable theories, which are very well behaved. Simple theories are more general, and then even more general are the NSOP1 theories. We can characterize those classes by the existence of a certain independence relation. For example, in vector spaces such an independence
relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.
Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP1-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.
This video is part of Masaryk University‘s Algebra seminar.
