Of a subset
The characteristic function of a subset of some set is a function from to the set of truth values (which classically is ) that takes in to the truth value of the statement that . That is,
where (also often ) is the characteristic function of .
Of a subobject
More generally, the characteristic morphism of a subobject of some objects in a category with a subobject classifier is the morphism from to that classifies ; we have that
is a pullback square.