Quantaloidal Completions of Order-enriched Categories and Their Applications

Abstract

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