homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
effective epimorphism $\Rightarrow$ regular epimorphism $\Leftrightarrow$ covering
effective monomorphism $\Rightarrow$ regular monomorphism $\Leftrightarrow$ embedding .
See cover, covering space.