homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
A $0$-category (or $(0,0)$-category) is, up to equivalence, the same as a set (or class).
Although this terminology may seem strange at first, it simply follows the logic of $n$-categories (and $(n,r)$-categories). To understand these, it is very helpful to use negative thinking to see sets as the beginning of a sequence of concepts: sets, categories, 2-categories, 3-categories, 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 $1$-category of $0$-categories; this is the category of sets. Then a category enriched over this is a $1$-category (more precisely, a locally small category). Furthermore, an enriched groupoid is a groupoid (or $1$-groupoid), so a $0$-category is the same as a 0-groupoid.
To some extent, one can continue to define a (−1)-category to be a truth value and a (−2)-category to be a triviality (that is, there is exactly one). These don't fit the pattern perfectly; but the concepts of (−1)-groupoid and (−2)-groupoid for them do work perfectly, as does the concept of 0-poset for a truth value.
Interpreted literally, $0$-category or $(0, 0)$-category would be an $\infty$-category such that every $j$-cell for $j \gt 0$ is an equivalence, and any two such $j$-cells that are parallel are equivalent. The picture that apparently emerges from this description might suggest a set equipped with an equivalence relation, or a what is sometimes called a setoid, or something even more complicated than that. One could thus say that a $0$-category is a “setoid”, when considered just up to isomorphism. But it is more appropriate in higher category theory to consider these things up to equivalence rather than up to isomorphism; when we do this, a $0$-category is equivalent to a set again.
0-category, (0,1)-category