Yesterday I was wandering for the $n$-lab and I've found the definition of $n$-poset.
Following this [post](http://ncatlab.org/nlab/show/n-poset) 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



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> could anyone explain to me how $n$-posets generalize the notion of poset? 
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What are $n$-poset?
It seems to me that you are effectively asking why a $(0,1)$-category is a poset. Because if that is so, it makes sense to define an $n$-poset to be an $(n-1,n)$-category.


To see why a $(0,1)$-category is a poset, just unwind the definition: it contains possibly non-invertible 1-morphisms, but any two of them that have the same source and target are equivalent (and the space of choices of equivalences between them is contractible). By the characterization of [posets as categories](http://ncatlab.org/nlab/show/partial+order#AsACategoryWithExtraProperties) that means it is a poset.


