The Scott adjunction

TL;DR

This paper introduces the Scott adjunction, linking accessible categories with directed colimits to topoi, and explores its geometric and logical implications, including categorified Isbell duality and axiomatizations of accessible categories.

Contribution

It presents the Scott adjunction and categorified Isbell duality, providing new insights into the structure of topoi and accessible categories, and connecting these to model theory and categorical frameworks.

Findings

Categorified Isbell duality is idempotent.

The Scott adjunction relates accessible categories to topoi.

Enrichment of the 2-category of topoi over accessible categories.

Abstract

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.