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 simply a set (or class).
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 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.
0-category, (0,1)-category