A -category is a truth value. Compare the concept of 0-category (a set) and (−2)-category (which is trivial). The point of -categories (a kind of negative thinking) is that they complete some patterns in the periodic table of -categories. (They also shed light on the theory of homotopy groups and n-stuff.)
For example, there should be a -category of -categories; this is the set of truth values, classically
Similarly, -categories form a -category (specifically, the true one).
If we equip the category of -categories with the monoidal structure of conjunction (the logical AND operation), then a category enriched over this is a poset; an enriched groupoid is a set. Notice that this doesn't fit the proper patterns of the periodic table; we see that -categories work better as either -posets or as -groupoids. Nevertheless, there is no better alternative for the term ‘-category’.