The Clowder Project

Let $R\subset A\times B$ be a relation.¹²

1. 1. The domain of $R$ is the subset $\mathrm{dom}\left(R\right)$ of $A$ defined by
   $$\mathrm{dom}\left(R\right)\stackrel{\text{def}}{=}\{a\in A|\begin{array}{rl}& \text{there exists some }b\in B\\ & \text{such that }a{\sim }_{R}b\end{array}\}.$$
2. 2. The range of $R$ is the subset $\mathrm{range}\left(R\right)$ of $B$ defined by
   $$\mathrm{range}\left(R\right)\stackrel{\text{def}}{=}\{b\in B|\begin{array}{rl}& \text{there exists some }a\in A\\ & \text{such that }a{\sim }_{R}b\end{array}\}.$$

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¹Following , we may compute the (characteristic functions associated to the) domain and range of a relation using the following colimit formulas:

$$\begin{array}{rl}{\chi }_{\mathrm{dom}\left(R\right)}\left(a\right)& \cong \underset{b\in B}{\text{colim}}\left(R_b^a\right)(a\in A)\\ & \cong \underset{b\in B}{\bigvee }{R}_b^a,\\ {\chi }_{\mathrm{range}\left(R\right)}\left(b\right)& \cong \underset{a\in A}{\text{colim}}\left(R_b^a\right)(b\in B)\\ & \cong \underset{a\in A}{\bigvee }{R}_b^a,\end{array}$$

where the join $\bigvee$ is taken in the poset $(\{\mathsf{true},\mathsf{false}\},⪯)$ of Chapter 2: Constructions With Sets, Definition 1.2.2.1.3.

²Viewing $R$ as a function $R:A\to \mathcal{P}\left(B\right)$, we have

$$\begin{array}{rl}\mathrm{dom}\left(R\right)& \cong \underset{y\in Y}{\text{colim}}\left(R\left(y\right)\right)\\ & \cong \bigcup _{y\in Y}R\left(y\right),\\ \mathrm{range}\left(R\right)& \cong \underset{x\in X}{\text{colim}}\left(R\left(x\right)\right)\\ & \cong \bigcup _{x\in X}R\left(x\right),\end{array}$$