An object in a 2-category is posetal if the category is a preorder (equivalent to a poset) for all objects of . Posetal objects are also called (0,1)-truncated objects since is a (0,1)-category (a poset).
More explicitly, is posetal iff any parallel 2-cells are equal. If has finite limits, this is equivalent to saying that is an equivalence, where is the “walking arrow” and is the “walking parallel pair of arrows.”
We write for the full sub-2-category of posetal objects; it is a (1,2)-category and is closed under limits in .