Item 2 (02AW) — The Clowder Project

Let $A$ and $B$ be sets.

Let $R\subset A\times B$ be a relation.^1,2

1.
The domain of $R$ is the subset $\mathrm{dom}\webleft (R\webright )$ of $A$ defined by

\[ \mathrm{dom}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in A\ \middle |\ \begin{aligned} & \text{there exists some $b\in B$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \right\} . \]

2.
The range of $R$ is the subset $\mathrm{range}\webleft (R\webright )$ of $B$ defined by

\[ \mathrm{range}\webleft (R\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ b\in B\ \middle |\ \begin{aligned} & \text{there exists some $a\in A$}\\ & \text{such that $a\sim _{R}b$}\\ \end{aligned} \right\} . \]

¹Following , , we may compute the (characteristic functions associated to the) domain and range of a relation using the following colimit formulas:
\begin{align*} \chi _{\mathrm{dom}\webleft (R\webright )}\webleft (a\webright ) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{b\in B}\webleft (R^{b}_{a}\webright )\qquad \webleft (a\in A\webright )\\ & \cong \bigvee _{b\in B}R^{b}_{a},\\ \chi _{\mathrm{range}\webleft (R\webright )}\webleft (b\webright ) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{a\in A}\webleft (R^{b}_{a}\webright )\qquad \webleft (b\in B\webright )\\ & \cong \bigvee _{a\in A}R^{b}_{a}, \end{align*}
where the join $\bigvee $ is taken in the poset $\webleft (\{ \mathsf{true},\mathsf{false}\} ,\preceq \webright )$ of Chapter 4: Constructions With Sets, Definition 3.2.2.1.3.
²Viewing $R$ as a function $R\colon A\to \mathcal{P}\webleft (B\webright )$, we have
\begin{align*} \mathrm{dom}\webleft (R\webright ) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{y\in Y}\webleft (R\webleft (y\webright )\webright )\\ & \cong \bigcup _{y\in Y}R\webleft (y\webright ),\\ \mathrm{range}\webleft (R\webright ) & \cong \operatorname*{\operatorname {\mathrm{colim}}}_{x\in X}\webleft (R\webleft (x\webright )\webright )\\ & \cong \bigcup _{x\in X}R\webleft (x\webright ), \end{align*}

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