Subsection 7.3.3 (00PK): Representable Relations

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

Let $f\colon A\to B$ and $g\colon B\to A$ be functions.¹

The representable relation associated to $f$ is the relation $\chi _{f}\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ defined as the composition
\[ A\times B\overset {f\times \text{id}_{B}}{\to }B\times B\overset {\chi _{B}}{\to }\{ \mathsf{true},\mathsf{false}\} , \]

i.e. given by declaring $a\sim _{\chi _{f}}b$ iff $f\webleft (a\webright )=b$.
The corepresentable relation associated to $g$ is the relation $\chi ^{g}\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ defined as the composition
\[ B\times A\overset {g\times \text{id}_{A}}{\to }A\times A\overset {\chi _{A}}{\to }\{ \mathsf{true},\mathsf{false}\} , \]

i.e. given by declaring $b\sim _{\chi ^{g}}a$ iff $g\webleft (b\webright )=a$.

¹More generally, given functions

\begin{align*} f & \colon A \to C,\\ g & \colon B \to D \end{align*}

and a relation $B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}D$, we may consider the composite relation

\[ A\times B\overset {f\times g}{\to }C\times D\overset {R}{\to }\{ \mathsf{true},\mathsf{false}\} , \]

for which we have $a\sim _{R\circ \webleft (f\times g\webright )}b$ iff $f\webleft (a\webright )\sim _{R}g\webleft (b\webright )$.

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