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 $TV$ 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 $U$.

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

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

is a pullback square.

See also

• classifying morphism
• subobject classifier