nLab
3-poset

A $3$-poset is any of several concepts that generalize $2$-posets one step in higher category theory. One does not usually here about $3$-posets by themselves but instead as special cases of $3$-categories.

$3$-posets can also be called $(2,3)$-categories. The concept generalizes to $n$-posets.

Definition

Fix a meaning of $\mathrm{\infty }$-category, however weak or strict you wish. Then a $3$-poset is an $\mathrm{\infty }$-category such that all parallel pairs of $j$-morphisms are equivalent for $j\ge 3$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, not even whether two given parallel $3$-morphisms are equivalent.