On Kan-injectivity of Locales and Spaces

0101 mathematics 01 natural sciences
DOI: 10.1007/s10485-015-9413-z Publication Date: 2015-10-18T23:03:58Z
ABSTRACT
In the category T o p 0 of T 0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpinski space is Kan-injective, and the Kan-injective hull of the Sierpinski 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 p 0 are Kan-injective hulls of finite subcategories. Moreover, we show that there is a subcategory of spatial locales whose Kan-injective hull is the entire category L o c.
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