A Note On Continuity Of Ω-categories

Let(Ω,*,I) be a commutative, unitai quantale, which is a monoidal closed category. A category enriched inΩis calledΩ-category for short.Ω-category is the main object in quantitive domain theory. On one hand, as the generalization of domain theory, quantitive domain theory also focuses on approximation and convergence of information, so directed completeness ofΩ-category has received much attention in the literature. On the other hand, anΩ-category can be regarded as aΩ-valued preordered set. In many valued structures there are also definitions for completely distributivity and continuity just like in the classical order structures (the sup functor and ideal sup functor have left adjoints respectively). However, since the complete latticeΩis much more complicated than 2, the implication between these two concepts in many valued situation is not obvious.The notion of way below relation in an ideal completeΩ-category and the notion of well below relation in a completeΩ-category are introduced and it is shown that continuity of ideal completeΩ-categories and completely distributivity of completeΩ-categories can be characterized by the way below relation and well below relation respectively. Then, as a corollary, we show that every completely distributiveΩ-category is continuous when I is 1 inΩ.