homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
An -groupoid is an ω-category (see there for more details) in which all k-morphisms for all are equivalences.
This is also called an ∞-groupoid. In the literature the term ”-groupoid” is usually reserved for algebraic models instead of geometric models?.
How strict the -category and the inverses must be can vary. Somes authors that use the term ‘-groupoid’ mean strict ω-groupoid by default and speak of weak -groupoid otherwise.