The Clowder Project

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

1. We write $\mathrm{Rel}\webleft (A,B\webright )$ for the set of relations from $A$ to $B$.
2. We write $\mathbf{Rel}\webleft (A,B\webright )$ for the sub-poset of $\webleft (\mathcal{P}\webleft (A\times B\webright ),\subset \webright )$ spanned by the relations from $A$ to $B$.
3. Given $a\in A$ and $b\in B$, we write $a\sim _{R}b$ to mean $\webleft (a,b\webright )\in R$.
4. When viewing $R$ as a function
   \[ R\colon A\times B\to \{ \mathsf{t},\mathsf{f}\} \]

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

   ¹

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