6.2.1 The Category of Relations Between Two Sets

Definition 6.2.1.1.1 ▶ The Category of Relations Between Two Sets

The category of relations from $A$ to $B$ is the category $\mathbf{Rel}\webleft (A,B\webright )$ defined by1

\[ \mathbf{Rel}\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}, \]

where $\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}$ is the posetal category associated to the poset $\mathbf{Rel}\webleft (A,B\webright )$ of Item 2 of Notation 6.1.1.1.4 and Chapter 9: Categories, Definition 9.2.7.1.1.

1Here we choose to abuse notation by writing $\mathbf{Rel}\webleft (A,B\webright )$ instead of $\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}$ for the posetal category of relations from $A$ to $B$, even though the same notation is used for the poset of relations from $A$ to $B$.

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