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 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 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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The first author acknowledges partial support by FCT - Fundação para a Ciência e a Tecnologia under the project SFRH/PROTEC/67529/2010 and by ISCAC - IPC. The second author acknowledges the support of the Centre for Mathematics of the University of Coimbra (funded by the program COMPETE and by the Fundação para a Ciência e a Tecnologia, under the project PEst-C/MAT/UI0324/2013).
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Carvalho, M., Sousa, L. On Kan-injectivity of Locales and Spaces. Appl Categor Struct 25, 83–104 (2017). https://doi.org/10.1007/s10485-015-9413-z
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DOI: https://doi.org/10.1007/s10485-015-9413-z