Hi, Suppose we define a dcpo of categories, where the ordering relation is a suitably restricted functor. The functor is restricted specificallly so that our collection and ordering reltion form a dcpo. In this dcpo of categories, the familiar categories, those we find in most any textbook, are the compact elements of this dcpo? I feel I also need to add that the dcpo we want is the largest dcpo of categories so defined. I've heard that terminology before.