This work is largely focused on extending D. Higgs' Ω-sets to the context of quantales, following the broad program of [7], we explore the rich category of 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 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 Q-sets with functional morphisms and the category of 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.

We extend the classic definition of sheaves on locales introducing an original notion of sheaves on semicartesian quantales. We show that the resulting category and the category of sheaves on locales share similar categorical properties, and discuss the difficulties in concluding whether our sheaves on quantales form a Grothendieck topos. We also prove a base change theorem, which may be useful not only to study the relation between sheaves on locales and sheaves on quantales, but also may be applied in the presence of an isomorphism of commutative and unital rings.