Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions

TL;DR

This paper explores how distributors between categories enriched over a small quantaloid induce adjunctions and monads, generalizing Isbell and Kan adjunctions, with functorial properties and implications for cocompletion.

Contribution

It introduces a functorial framework for adjunctions and monads arising from distributors in quantaloid-enriched categories, extending classical category theory concepts.

Findings

Distributors induce two types of adjunctions and monads in enriched categories.

The adjunctions generalize Isbell and Kan extensions.

The free cocompletion functor factors through these adjunctions.

Abstract

Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors.