The empty category is the category with no objects (and hence no morphisms). It is a groupoid, so we may call it the empty groupoid. One can similarly speak of the empty n-category, the empty ∞-groupoid, etc etc etc.
The empty category is discrete, hence may be identified with a set: the empty set. This set is a subsingleton, so we may also identify it with a truth value: the false one.
The empty category is initial in Cat (as well as Grpd, ∞Grpd, Set, etc).