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In the category T o p ₀ of T ₀-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 L o c 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 Kan-injective. As a consequence, we obtain similar characterizations of the n-flat embeddings in the category T o p ₀, and we show that several well-known subcategories of L o c and T o …