论文标题
完整性和注入性
Completeness and injectivity
论文作者
论文摘要
我们表明，对于任何量子$ \ MATHCAL {q} $，$ \ MATHCAL {q} $ - 类别是骨骼且完整的，并且仅当它对完全忠实的$ \ Mathcal {q} $ - functors offly时。这是由于Hofmann和Stubbe引起的已知定理的一种特殊情况，但是我们使用MacNeille完成$ \ Mathcal {q} $类别的表征作为其注射式信封。对于律师公制空间，我们的结果产生了Kemajou，Künzi和Otafudu的结果。我们指出，它们的Isbell凸度概念可以看作是律师度量空间的分类完整性的几何表述。
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.
