General facts on the Scott Adjunction

Abstract

We introduce, comment and develop the Scott adjunction, mostly from the point of view of a category theorist. Besides its technical and conceptual aspects, in a nutshell we provide a categorification of the Scott topology over a posets with directed suprema. From a technical point of view we establish an adjunction between accessible categories with directed colimits and Grothendieck topoi. We show that the bicategory of topoi is enriched over the $2$-category of accessible categories with directed colimits and it has tensors with respect to this enrichment. The Scott adjunction (ri-)emerges naturally from this observation.