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Are all limit-closed subcategories of locally presentable categories reflective? (English) Zbl 0668.18004

Categorical algebra and its applications, Proc. 1st Conf., Louvain-la- Neuve/Belg. 1987, Lect. Notes Math. 1348, 1-18 (1988).
[For the entire collection see Zbl 0644.00009.]
Every reflective subcategory of a locally presentable category is closed under the formation of limits. This interesting paper examines the converse of the above statement. Assuming the non-existence of measurable cardinals, the converse is false. Surprisingly it is true assuming the following Weak Vopěnka Principle: the dual of the well-ordered category Ord of all ordinals does not have a full embedding into the category Graph of all graphs.
More precisely it is shown that, assuming Weak Vopěnka Principle, every locally presentable category K has the following properties: (i) every subcategory of K which is closed under limits is reflective in K; (ii) the reflective subcategories of K form a large complete lattice; (iii) the intersection of two reflective subcategories of K is reflective in K.
Conversely, assuming the negation of Weak Vopěnka Principle, none of the above statements hold in K\(=\underline{Graph}\). Vopěnka Principle (Ord cannot be fully embedded into Graph) implies Weak Vopěnka Principle. It is unknown if the converse is true.
Reviewer: E.Giuli

MSC:

18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18B30 Categories of topological spaces and continuous mappings (MSC2010)

Citations:

Zbl 0644.00009