Editing Preorder (section)

===Category theory===

* A [[Category (mathematics)|category]] with at most one [[morphism]] from any object ''x'' to any other object ''y'' is a preorder. Such categories are called [[thin category|thin]]. Here the [[Object (category theory)|objects]] correspond to the elements of <math>P,</math> and there is one morphism for objects which are related, zero otherwise. In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.
* Alternately, a preordered set can be understood as an [[enriched category]], enriched over the category <math>2 = (0 \to 1).</math>