The Clowder Project

The $2$-category of relations is the locally posetal $2$-category $\boldsymbol {\mathsf{Rel}}$ where

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  Objects. The objects of $\boldsymbol {\mathsf{Rel}}$ are sets.

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  $\mathbf{Hom}$-Objects. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$, we have

  \begin{align*} \operatorname {\mathrm{Hom}}_{\boldsymbol {\mathsf{Rel}}}\webleft (A,B\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}\webleft (A,B\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\mathrm{Rel}\webleft (A,B\webright ),\subset \webright ).\end{align*}
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  Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\boldsymbol {\mathsf{Rel}}\webright )$, the unit map

  \[ \mathbb {1}^{\boldsymbol {\mathsf{Rel}}}_{A} \colon \mathrm{pt}\to \mathbf{Rel}\webleft (A,A\webright ) \]

  of $\boldsymbol {\mathsf{Rel}}$ at $A$ is defined by

  \[ \operatorname {\mathrm{id}}^{\boldsymbol {\mathsf{Rel}}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{A}\webleft (-_{1},-_{2}\webright ), \]

  where $\chi _{A}\webleft (-_{1},-_{2}\webright )$ is the characteristic relation of $A$ of Chapter 4: Constructions With Sets, of .

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  Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\boldsymbol {\mathsf{Rel}}\webright )$, the composition map¹

  \[ \circ ^{\boldsymbol {\mathsf{Rel}}}_{A,B,C}\colon \mathbf{Rel}\webleft (B,C\webright )\times \mathbf{Rel}\webleft (A,B\webright )\to \mathbf{Rel}\webleft (A,C\webright ) \]

  of $\boldsymbol {\mathsf{Rel}}$ at $\webleft (A,B,C\webright )$ is defined by

  \[ S\mathbin {{\circ }^{\boldsymbol {\mathsf{Rel}}}_{A,B,C}}R \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}S\mathbin {\diamond }R \]

  for each $\webleft (S,R\webright )\in \boldsymbol {\mathsf{Rel}}\webleft (B,C\webright )\times \boldsymbol {\mathsf{Rel}}\webleft (A,B\webright )$, where $S\mathbin {\diamond }R$ is the composition of $S$ and $R$ of Chapter 9: Constructions With Relations, .