Quantaloidal Completions of Order-enriched Categories and Their Applications

TL;DR

This paper introduces quantaloidal completions for order-enriched categories, characterizes them via quotients of power-set completions, and explores their applications such as MacNeille and Down-set completions.

Contribution

It establishes a new framework for quantaloidal completions, linking them to existing completions and providing new insights into their structure and applications.

Findings

Quantaloidal completions are fully characterized as compatible quotients of power-set completions.

The MacNeille completion serves as a special injective hull for order-enriched categories.

The Down-set completion is identified as the free quantaloid over an order-enriched category.

Abstract

By introducing the concept of quantaloidal completions for an order-enriched category, relationships between the category of quantaloids and the category of order-enriched categories are studied. It is proved that quantaloidal completions for an order-enriched category can be fully characterized as compatible quotients of the power-set completion. As applications, we show that a special type of injective hull of an order-enriched category is the MacNeille completion; the free quantaloid over an order-enriched category is the Down-set completion.