Subsection 2.5.2 (01LC): The Characteristic Function of a Point

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.5: Characteristic Functions
Subsection 2.5.2: The Characteristic Function of a Point (cite)

2.5.2 The Characteristic Function of a Point

Let $X$ be a set and let $x\in X$.

Definition 2.5.2.1.1 ▶ The Characteristic Function of a Point

The characteristic function of $x$ is the function¹

\[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]

defined by

\[ \chi _{x} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{\webleft\{ x\webright\} }, \]

i.e. by

\[ \chi _{x}\webleft (y\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x=y$,}\\ \mathsf{false}& \text{if $x\neq y$} \end{cases} \]

for each $y\in X$.

¹Further Notation: Also written $\chi ^{x}$, $\chi _{X}\webleft (x,-\webright )$, or $\chi _{X}\webleft (-,x\webright )$.

Remark 2.5.2.1.2 ▶ Characteristic Functions of Points as Decategorifications of Representable Presheaves

Expanding upon Remark 2.5.1.1.2, we may think of the characteristic function

\[ \chi _{x}\colon X\to \{ \mathsf{t},\mathsf{f}\} \]

of an element $x$ of $X$ as a decategorification of the representable presheaf and of the representable copresheaf

\begin{align*} h_{X} & \colon \mathcal{C}^{\mathsf{op}} \to \mathsf{Sets},\\ h^{X} & \colon \mathcal{C} \to \mathsf{Sets}\end{align*}

associated of an object $X$ of a category $\mathcal{C}$.

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