A directed type or filtered (0,1)-precategory is a preorder or (0,1)-precategory with
a term
a family of dependent terms
A codirected type or cofiltered (0,1)-precategory is a preorder or (0,1)-precategory with
a term
a family of dependent terms
If the directed type or codirected type is 0-truncated it is called a directed set or codirected set, and if the directed type or codirected type is a poset, then it is called a directed poset or codirected poset, or a filtered (0,1)-category or cofiltered (0,1)-category.