Quantaloidal Completions of Order-enriched Categories and Their Applications

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.

References in corpus (6)

• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Topological categories, quantaloids and Isbell adjunctions
• $\mathcal{Q}$-closure spaces
• 'Hausdorff distance' via conical cocompletion
• Chu connections and back diagonals between $\mathcal{Q}$-distributors
• Fixed points of adjoint functors enriched in a quantaloid