Order-preserving reflectors and injectivity
This paper deals with a special type of injectivity, the right-Kan injectivity. In a poset enriched category \(\mathcal X\), an object \(Z\) is right-Kan injective with respect to a morphism \(f: X\to Y\) if, for every \(g: X\to Z\), there is a morphism \(g/f: Y\to Z\) such that \(g/f\cdot f=g\) and \(g/f\) is the join of all morphisms \(t: Y\to Z\) such that \(tf\leq g\). This concept is also formulable for morphisms: a morphism \(k: Z\to W\) is right-Kan injective with respect to \(f: X\to Y\) if \(Z\) and \(W\) are so and, moreover, \(k(g/f)=(kg)/f\) for all morphisms \(g: X\to Z\). Given a subclass \(\mathcal H\) of morphisms of \(\mathcal X\), the objects and morphisms which are right-Kan injective w.r.t. \(\mathcal H\) form a subcategory of \(\mathcal X\). This establishes a Galois connection between classes of morphisms of \(\mathcal X\) and subcategories of \(\mathcal X\) that is investigated in the present paper (specially its relationship with a special kind of subcategories, the KZ-reflective subcategories). It is shown that many examples of injectivity in poset enriched categories encountered in the literature are closely related to this connection. A useful characterization of KZ-reflective subcategories in a poset enriched category is given. Then it is shown that some KZ-reflective subcategories of the category of \(T_0\)-topological spaces and continuous maps (namely, the category of stably compact spaces and stable continuous maps, the category of continuous lattices and maps which preserve directed joins and meets, and the category of continuous Scott domains and maps which preserve directed joins and non empty meets) are right-Kan injective hulls of finite subcategories.
scientific article; zbMATH DE number 5969491
scientific article