  We introduce the framework of AECats (abstract elementary categories),
generalising both the category of models of some first-order theory and the
category of subsets of models. Any AEC and any compact abstract theory ("cat",
as introduced by Ben-Yaacov) forms an AECat. In particular, we find
applications in positive logic and continuous logic: the category of (subsets
of) models of a positive or continuous theory is an AECat.
  The Kim-Pillay theorem for first-order logic characterises simple theories by
the properties dividing independence has. We prove a version of the Kim-Pillay
theorem for AECats with the amalgamation property, generalising the first-order
version and existing versions for positive logic.

Logic;Category Theory