Quantale-valued preorders

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory. Two functors, the forward globalization functor and the backward globalization functor, that map a fuzzy set with a fuzzy preorder to a crisp set with a fuzzy preorder are considered. It is shown that both of them preserve cocompleteness with respect to several classes of weights.