On Kan-injectivity of Locales and Spaces

In the category Top0 of T0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpiński space is Kan-injective, and the Kan-injective hull of the Sierpiński space is the category of continuous lattices and maps preserving directed suprema and arbitrary infima. In the category Loc of locales and localic maps, we give an analogous characterization of flat embeddings; more generally, we characterize n-flat embeddings, for each cardinal n, as those morphisms with respect to which a certain finite subcategory is Kaninjective. Moreover, we show that there is a subcategory of spatial locales whose Kan-injective hull is the entire category Loc. As a consequence, we obtain similar characterizations of the n-flat embeddings in the category Top0, and we show that several well-known subcategories of Loc and Top0 are Kan-injective hulls of finite subcategories.