论文标题
Scott的辅助
The Scott adjunction
论文作者
论文摘要
我们介绍并研究了Scott的辅助，将可访问类别与定向的Colimits与Topoi联系起来。我们的重点是双重的，我们研究了其对形式模型理论的应用及其几何解释。从几何学的角度来看，我们引入了分类的ISBEL二元性，将有限的（可能是大的）离子与Topoi相关。分类的ISBEL二元性与Scott的辅助相互作用，在Poset上提供了Scott拓扑的分类（因此名称）。我们表明，分类的ISBELL对偶性是依然的，与其未分类版本类似。从逻辑的角度来看，我们使用此机械为可访问类别的候选（几何）公理提供了定向的colimits。我们讨论了这些辅助设备与对托托伊分类的理论之间的联系。我们将框架与更古典的抽象基础类别联系起来。从更加分类的角度来看，我们表明，Topoi的两类具有定向colimits的可访问类别，我们将此结果与Scott的辅助联系起来。
We introduce and study the Scott adjunction, relating accessible categories with directed colimits to topoi. Our focus is twofold, we study both its applications to formal model theory and its geometric interpretation. From the geometric point of view, we introduce the categorified Isbell duality, relating bounded (possibly large) ionads to topoi. The categorified Isbell duality interacts with the Scott adjunction offering a categorification of the Scott topology over a poset (hence the name). We show that the categorified Isbell duality is idempotent, similarly to its uncategorified version. From the logical point of view, we use this machinery to provide candidate (geometric) axiomatizations of accessible categories with directed colimits. We discuss the connection between these adjunctions and the theory of classifying topoi. We relate our framework to the more classical theory of abstract elementary classes. From a more categorical perspective, we show that the 2-category of topoi is enriched over accessible categories with directed colimits and we relate this result to the Scott adjunction.
