4.5.3 The Characteristic Relation of a Set

Let $X$ be a set.

Definition 4.5.3.1.1 ▶ The Characteristic Relation of a Set

The characteristic relation on $X$1 is the relation2

\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]

on $X$ defined by3

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

for each $x,y\in X$.

  1. 1Further Terminology: Also called the identity relation on $X$.
  2. 2Further Notation: Also written $\chi ^{-_{1}}_{-_{2}}$, or $\mathord {\sim }_{\operatorname {\mathrm{id}}}$ in the context of relations.
  3. 3Under the bijection $\mathsf{Sets}\webleft (X\times X,\{ \mathsf{t},\mathsf{f}\} \webright )\cong \mathcal{P}\webleft (X\times X\webright )$ of Item 2 of Proposition 4.5.1.1.4, the relation $\chi _{X}$ corresponds to the diagonal $\Delta _{X}\subset X\times X$ of $X$.

Remark 4.5.3.1.2 ▶ The Characteristic Relation of a Set as a Decategorification of the Hom Profunctor

Expanding upon Remark 4.5.1.1.2 and Remark 4.5.2.1.2, we may view the characteristic relation

\[ \chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} \]

of $X$ as a decategorification of the $\operatorname {\mathrm{Hom}}$ profunctor

\[ \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets} \]

of a category $\mathcal{C}$.

Proposition 4.5.3.1.3 ▶ Properties of Characteristic Relations

Let $f\colon X\to Y$ be a function.

  1. 1.

    The Inclusion of Characteristic Relations Associated to a Function. Let $f\colon A\to B$ be a function. We have an inclusion1

  1. 1Note: This is the $0$-categorical version of Chapter 11: Categories, Definition 11.5.4.1.1.

Proof of Proposition 4.5.3.1.3.

Item 1: The Inclusion of Characteristic Relations Associated to a Function
The inclusion $\chi _{B}\webleft (f\webleft (a\webright ),f\webleft (b\webright )\webright )\subset \chi _{A}\webleft (a,b\webright )$ is equivalent to the statement “if $a=b$, then $f\webleft (a\webright )=f\webleft (b\webright )$”, which is true.

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