斯科特附加条款
The Scott adjunction
摘要：
我们介绍和研究的斯科特附加，有关可访问的类别与定向colimits topoi。我们的重点是双重的，我们研究它的应用，正式模型论和它的几何解释。从几何的角度来看，我们介绍了分类Isbell对偶，有界（可能大）ionads拓扑。范畴化的伊斯贝尔对偶与斯科特附加物相互作用，提供了偏序集上的斯科特拓扑的范畴化（因此得名）。我们证明了范畴化的Isbell对偶是幂等的，类似于其未范畴化的版本。从逻辑的角度来看，我们使用这种机器提供候选人（几何）公理化的可访问类别与定向共极限。本文讨论了这些概念与拓扑分类理论之间的联系。我们将我们的框架与更经典的抽象初等类理论联系起来。从一个更范畴的角度来看，我们表明，2类的拓扑丰富的可访问的类别与定向colimits，我们将这一结果的斯科特附加。
英文摘要：
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.
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