Unexpected properties of locally presentable categories
Let \({\mathcal O}rd\) be the usual category of all ordinals and \({\mathcal G}ra\) the category of all graphs \([=binary\) relations]. The statement that \({\mathcal O}rd\) cannot be fully embedded into \({\mathcal G}ra\) is called Vopěnka's principle. Locally presentable categories are precisely those complete categories which have a small dense subcategory (all subcategories are meant to be full in this paper). The main result is the equivalence of Vopěnka's principle to various properties of locally presentable categories, for example to each of the following three claims: every subcategory of a locally presentable category (i) has a small dense subcategory; (ii) if it is closed under limits, then it is locally presentable; (iii) if it is closed under colimits, then it is coreflective.
scientific article; zbMATH DE number 4148378