General facts on the Scott Adjunction

TL;DR

This paper develops the Scott adjunction, a categorification of the Scott topology, establishing a deep connection between accessible categories with directed colimits and Grothendieck topoi from a category-theoretic perspective.

Contribution

It introduces and explores the Scott adjunction, linking accessible categories and topoi, and demonstrates its natural emergence from enriched bicategory structures.

Findings

Established an adjunction between accessible categories and Grothendieck topoi.

Showed the bicategory of topoi is enriched over accessible categories.

Demonstrated the Scott adjunction naturally arises from this enrichment.

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.