characteristic function

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

$\chi_U(a) \;\Leftrightarrow\; a \in U ,$

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

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

$\array {
U & \hookrightarrow & X \\
\downarrow & & \downarrow & \chi_U \\
1 & \underset{\top}\to & \Omega
}$

is a pullback square.

Revised on December 5, 2011 18:57:34
by Urs Schreiber
(89.204.139.149)