nLab
characteristic function

Definition

Of a subset

The characteristic function of a subset $U$ of some set $X$ is a function from $X$ to the set $\mathrm{TV}$ of truth values (which classically is $\mathrm{TV}=\{\perp ,\top \}$) that takes $a$ in $X$ to the truth value of the statement that $a\in U$. That is,

$${\chi }_U(a)\iff a\in U,$$\chi_U(a) \;\Leftrightarrow\; a \in U ,

where ${\chi }_U$ (also often $1_U$) is the characteristic function of $U$.

Of a subobject

More generally, the characteristic morphism of a subobject $U$ of some objects $X$ in a category with a subobject classifier $\Omega$ is the morphism from $X$ to $\Omega$ that classifies $U$; we have that

$$\begin{array}{ccc}U& \hookrightarrow & X\\ \downarrow & & \downarrow & {\chi }_U\\ 1& \underset{\top }{\to }& \Omega \end{array}$$\array { U & \hookrightarrow & X \\ \downarrow & & \downarrow & \chi_U \\ 1 & \underset{\top}\to & \Omega }

is a pullback square.