  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.
