homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
A $3$-category is any of several concepts that generalize $2$-categories one step in higher category theory. The original notion is that of a globular strict 3-category, but the one most often used here is that of a tricategory. The concept generalizes to $n$-categories.
Fix a meaning of $\infty$-category, however weak or strict you wish. Then a $3$-category is an $\infty$-category such that every 4-morphism is an equivalence, and all parallel pairs of $j$-morphisms are equivalent for $j \geq 4$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-category, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as equality.
3-category