Notation 5.1.1.1.2 (00HH): Further Notation for Relations

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Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.

Given elements $a\in A$ and $b\in B$ and a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, we write $a\sim _{R}b$ to mean $\webleft (a,b\webright )\in R$.
Viewing $R$ as a function
\[ R\colon A\times B\to \{ \mathsf{t},\mathsf{f}\} \]

via Remark 5.1.1.1.4, we write $R^{b}_{a}$ for the value of $R$ at $\webleft (a,b\webright )$.^[1]

Footnotes

[1] The choice $R^{b}_{a}$ in place of $R^{a}_{b}$ is to keep the notation consistent with the notation we will later employ for profunctors.

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