  Categories enriched over a commutative unital quantale can be studied as
generalized, or many-valued, ordered structures. Because many concepts, such as
complete distributivity, in lattice theory can be characterized by existence of
certain adjunctions, they can be reformulated in the many-valued setting in
terms of categorical postulations. So, it is possible, by aid of categorical
machineries, to establish theories of many-valued complete lattices,
many-valued completely distributive lattices, and so on. This paper presents a
systematical investigation of many-valued complete distributivity, including
the topics: (1) subalgebras and quotient algebras of many-valued completely
distributive lattices; (2) categories of (left adjoint) functors; and (3) the
relationship between many-valued complete distributivity and properties of the
quantale of truth values. The results show that enriched category theory is a
very useful tool in the study of many-valued versions of order-related
mathematical entities.
