Completeness and injectivity

Abstract

We show that for any quantale $\mathcal{Q}$, a $\mathcal{Q}$-category is skeletal and complete if and only if it is injective with respect to fully faithful $\mathcal{Q}$-functors. This is a special case of known theorems due to Hofmann and Stubbe, but we provide a different proof, using the characterisation of the MacNeille completion of a $\mathcal{Q}$-category as its injective envelope. For Lawvere metric spaces, our results yield those of Kemajou, Künzi and Otafudu. We point out that their notion of Isbell convexity can be seen as a geometric formulation of categorical completeness for Lawvere metric spaces.

14 pages; v2: final journal version. arXiv admin note: text overlap with arXiv:1909.07620

References in corpus (6)

• Duality and separation theorems in idempotent semimodules
• Categorical structures enriched in a quantaloid: categories, distributors and functors
• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Tight spans, Isbell completions and semi-tropical modules
• Topological categories, quantaloids and Isbell adjunctions
• Fixed points of adjoint functors enriched in a quantaloid