We introduce the framework of AECats (abstract elementary categories), generalizing 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 characterizes simple theories by the properties dividing independence has. We prove a version of the Kim–Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic.

中文翻译：

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抽象基本范畴的 KIM-PILLAY 定理

我们介绍了AECats（抽象基本范畴）的框架，概括了一些一阶理论的模型范畴和模型子集的范畴。任何 AEC 和任何紧凑的抽象理论（“猫”，由 Ben-Yaacov 介绍）形成一个 AECat。特别是，我们发现在正逻辑和连续逻辑中的应用：正或连续理论的模型（子集）类别是 AECat。一阶逻辑的 Kim-Pillay 定理通过划分独立性所具有的属性来描述简单理论。我们证明了具有合并属性的 AECats 的 Kim-Pillay 定理的一个版本，将一阶版本和现有版本推广为正逻辑。