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A small full subcategory K' of a category K is called dense (or colimit-dense) if every object X of K is a colimit of some diagram in K' (or is the canonical colimit of all maps from objects of K' to X, resp.). The existence of a colimit-dense subcategory is an important property, e.g., the cocompletemness of K then implies that K is complete, see [1]. The dual of the category of sets or vector spaces has a dense subcategory iff all measurable cardinals are bounded. In contrast, the 3-element set is colimit-dense in the dual of Set, and the countably-dimensional space is colimit-dense in the dual of Vec. We also explain why a 'linear analogy' of the concept of an ultrafilter on a set X is a vector of the double dual of a space X. These results stem from [2]. References: [1] J. Adámek, H. Herrlich and J. Reiterman, Cocompleteness almost implies completeness. Proceedings of 'Categorical Topology 1988', World Scientific Publishers, Singapore, 1989, 246-256. [2] J. Adámek, A. Brooke-Raylor, T. Campion, L. Positselski and J. Rosický, Colimit-dense subcategories, submitted, 2019, arXiv:1812.10649.
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