A quick way to make this precise is to says that a 1-groupoid is a Kan complex which is equivalent (homotopy equivalent) to the nerve of a groupoid: a 2-coskeletal Kan complex. More abstractly this is a 1-truncated ∞-groupoid.
More generally and more vaguely: Fix a meaning of -groupoid, however weak or strict you wish. Then a -groupoid is an -groupoid such that all parallel pairs of -morphisms are equivalent for . Thus, up to equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel -morphisms are equivalent. If you rephrase equivalence of -morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a groupoid. Thus one may also say that a -groupoid is simply a groupoid.
The point of all this is simply to fill in the general concept of -groupoid; nobody thinks of -groupoids as a concept in their own right except simply as groupoids. Compare -category and -poset, which are defined on the same basis.
|homotopy level||n-truncation||homotopy theory||higher category theory||higher topos theory||homotopy type theory|
|h-level 0||(-2)-truncated||contractible space||(-2)-groupoid||true/unit type/contractible type|
|h-level 1||(-1)-truncated||contractible-if-inhabited||(-1)-groupoid/truth value||(0,1)-sheaf/ideal||mere proposition/h-proposition|
|h-level 2||0-truncated||homotopy 0-type||0-groupoid/set||sheaf||h-set|
|h-level 3||1-truncated||homotopy 1-type||1-groupoid/groupoid||(2,1)-sheaf/stack||h-groupoid|
|h-level 4||2-truncated||homotopy 2-type||2-groupoid||(3,1)-sheaf/2-stack||h-2-groupoid|
|h-level 5||3-truncated||homotopy 3-type||3-groupoid||(4,1)-sheaf/3-stack||h-3-groupoid|