Subobject classifier

Summary

A subobject classifier in a category with finite limits is an object $\Omega$ with a morphism $\top \colon 1 \to \Omega$. Every subobject of every object $X$ must be the pullback of $\top$ along exactly one morphism $X \to \Omega$. In the category of sets, $\Omega = \set{0, 1}$ and the morphism $X \to \Omega$ of a subset is its characteristic function. It is one of the defining features of a topos, and $\Omega$ plays the role of the object of truth values. In a sheaf topos, $\Omega$ need not have two elements, and its internal logic is in general intuitionistic.