$\mathcal{Q}$-Sets and Friends: Regarding Singleton and Gluing Completeness

This work is largely focused on extending D. Higgs’ $\mathrm{\Omega }$-sets to the context of quantales, following the broad program of U. Höhle, we explore the rich category of $\mathcal{Q}$-sets for strong, integral and commutative quantales, or other similar axioms. The focus of this work is to study the different notion of ‘completeness’ a $\mathcal{Q}$-set may enjoy and their relations, completion functors, resulting reflective subcategories, their relations to relational morphisms. We establish the general equivalence of singleton complete $\mathcal{Q}$-sets with functional morphisms and the category of $\mathcal{Q}$-sets with relational morphisms; we provide two characterizations of singleton completeness in categorical terms; we show that the singleton complete categorical inclusion creates limits.