example. preorder, poset, directed poset [tt-0052]
An example of a preorder category which is not poset is:
An example of a poset category which is not a directed poset is [rosiak2022sheaf, example 3] :
An example that is a directed poset category but not a total order is:
where each pair of nodes has a common upper bound (thus satisfying directedness), but there is no path between the two nodes on the center, thus violating comparability.
A more complicated example of a directed poset category which is not a total order is [spivak2013category, example 3.4.1.3]:
One can see immediately that this is a preorder because length=0 paths give reflexivity and concatenation of paths gives transitivity. To see that it is a partial order we only note that there are no loops.
To see that it is a poset, we note that every pair of nodes from one side or both sides has the central node as an upper bound, thus satisfying directedness.
But this partial order is not a total order because there is no path (in either direction) between some nodes, thus violating comparability.