Fix a meaning of $\infty$-category, however weak or strict you wish. Then a **$1$-poset** is an $\infty$-category such that every $2$-morphism is an equivalence and all parallel pairs of $j$-morphisms are equivalent for $j \geq 1$. 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.

Last revised on March 17, 2009 at 16:36:57. See the history of this page for a list of all contributions to it.