Fix a meaning/model 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. This definition may give a concept more general than your preferred definition of -groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of -morphisms as equality.
Specific models
There are various objects that model the abstract notion of -groupoid.
\array{
&& y
\\
& \nearrow &\Downarrow& \searrow
\\
x &&\stackrel{}{\to}&& z
}
in the 2-groupoid, respectively.
Moreover, the 3-simplices in the simplicial set encode the composition operation: given three composable 2-simplex faces of a tetrahedron (a 3-horn)
\array{
y &\to& &\to& z
\\
\uparrow &\seArrow& &\nearrow& \downarrow
\\
\uparrow &\nearrow& &\Downarrow& \downarrow
\\
x &\to&&\to& w
}
\;\;\;
\;\;\;
\array{
y &\to& &\to& z
\\
&\searrow& &\swArrow& \downarrow
\\
&& &\searrow& \downarrow
\\
&&&& w
}
the unique composite of them is is a fourth face and a 3-cell filling the resulting hollow tetrahedron:
\array{
y &\to& &\to& z
\\
\uparrow &\seArrow& &\nearrow& \downarrow
\\
\uparrow &\nearrow& &\Downarrow& \downarrow
\\
x &\to&&\to& w
}
\;\;\;
\stackrel{comp}{\to}
\;\;\;
\array{
y &\to& &\to& z
\\
\uparrow &\searrow& &\swArrow& \downarrow
\\
\uparrow &{}_\kappa\Downarrow& &\searrow& \downarrow
\\
x &\to&&\to& w
}
\,.
The 3-coskeletal-condition says that every boundary of a 4-simplex made up of five such tetrahedra has a unqiue filler. This is the associativitycoherence law on the comoposition operation:
The general notion of -groupoid above is also called weak -groupoid to distinguish from the special case of strict 2-groupoids. A strict -groupoid is a strict 2-category in which all morphisms are strictly invertible. This is equivalently a certain type of Grpd-enriched category.