We study ideals in, and continuity of, quantaloid-enriched categories (Q-categories for short) as a `many-valued and many-typed' generalization of domain theory. Abstractly, for any (saturated) class Φ of presheaves, we define and study the Φ-continuity of Q-categories. Concretely, we compute three examples of such saturated classes of presheaves - the class of flat ideals, the class of irreducible ideals and the class of conical ideals - which are proper generalizations of ideals in domain theory.
Keywords: Quantaloid, enriched category, domain theory, fuzzy order
2020 MSC: 18B35, 18D20, 06F07
Theory and Applications of Categories, Vol. 39, 2023, No. 24, pp 687-713.