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Colimit-dense subcategories. (English) Zbl 1474.18012

Recall that that a cocomplete category is locally presentable if and only if it has a strong generator consisting of presentable objects, and that for a subcategory of a category, dense implies colimit-dense and colimit-dense implies strong generator. One of the main results of this paper is that assuming Vopěnka’s Principle, a cocomplete category is locally presentable if and only if it has a colimit-dense subcategory and a presentable generator. Note that Theorem 9 of [J. Rosický et al., Algebra Univers. 27, No. 2, 153–170 (1990; Zbl 0701.18003)] and Theorem 6.35 and Corollary 6.37 of [J. Adámek and J. Rosický, Locally presentable and accessible categories. London Mathematical Society Lecture Note Series 189. Cambridge: Cambridge University Press (1994; Zbl 0795.18007)] are false and the primary goal of this paper was to fix this error. The authors also prove that a \(3\)-element set is colimit-dense in the opposite of the category of sets and that vector spaces of countable dimensions are colimit-dense in the opposite of the category of vector spaces. As a byproduct of the study, the authors present parallel proofs for two classical results: one is that the codensity monad of the embedding of finite sets in the category of sets is the ultrafilter monad. The other result is that the codensity monad of the embedding of finite-dimensional vector spaces into the category of vector spaces is the double-dualization monad.

MSC:

18C35 Accessible and locally presentable categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
03E55 Large cardinals
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