A 0-groupoid or 0-type a set. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, groupoids, 2-groupoids, 3-groupoids, etc. Doing so reveals patterns such as the periodic table. (It also sheds light on the theory of homotopy groups and n-stuff.)
For example, there should be a -groupoid of -groupoids; this is the underlying groupoid of the category of sets. Then a groupoid enriched over this is a -groupoid (more precisely, a locally small groupoid). Furthermore, an enriched category is a category (or -category), so a -groupoid is the same as a 0-category.
h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level | -truncated | homotopy n-type | n-groupoid | | h--groupoid | h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid