nLab
1-poset

Fix a meaning of $\mathrm{\infty }$-category, however weak or strict you wish. Then a $1$-poset is an $\mathrm{\infty }$-category such that every $2$-morphism is an equivalence and all parallel pairs of $j$-morphisms are equivalent for $j\ge 1$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, not even whether two given parallel $1$-morphisms are equivalent. Up to equivalence, therefore, all that is left in this definition is a poset. Thus one may also say that a $1$-poset is simply a poset.

The point of all this is simply to fill in the general concept of $n$-poset; nobody thinks of $1$-posets as a concept in their own right except simply as posets. Compare $1$-category and $1$-groupoid, which are defined on the same basis.