Subobject Classifier

In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. Intuitively, as the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements belong to the subobject in question. Because of this role, the subobject classifier is also referred to as the "truth value object". In fact the way in which the subobject classifier classifies subobjects of a given object, is by assigning the values true to elements belonging to the subobject in question, and false to elements not belonging to the subobject. This is why the subobject classifier is widely used in the categorical description of logic.