Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset. Following this post it seems that a $n$-poset should be a $(n,n+1)$-category. Now an $(n,r)$-category should be a category such that every $k$-morphism is an equivalence for $k\geq r$ and every pair of parallel $k$-morphisms with $k \geq n$ are equivalent. Now here're my problems: I suppose that this objects should generalize in any some way the notion of poset to higher categorical structure, i.e. it should be a categorification of the notion of poset, but I don't get why this should be the case
could anyone explain to me how $n$-posets generalize the notion of poset?