Isbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions

TL;DR

This paper explores how two-variable adjunctions in quantale-enriched categories underpin Isbell and Kan adjunctions, providing representation theorems for their fixed points.

Contribution

It demonstrates that two-variable adjunctions serve as foundational structures for constructing Isbell and Kan adjunctions in quantale-enriched categories, with new representation theorems.

Findings

Every two-variable adjunction in quantale-enriched categories underpins Isbell adjunctions.

Kan adjunctions are shown to be specific Isbell adjunctions from related two-variable adjunctions.

Representation theorems for fixed points of these adjunctions are established.

Abstract

It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions constructed from suitable associated two-variable adjunctions. Representation theorems are established for fixed points of these adjunctions.