Definition:Category of Relations

Definition
The category of (binary) relations, denoted $\mathbf{Rel}$, is the metacategory whose objects are all sets.

A $\mathbf{Rel}$-morphism $\mathcal R: A \to B$ is a binary relation $\mathcal R \subseteq A \times B$.

The composition of $\mathcal R: A \to B$ and $\mathcal S: B \to C$ is their composition as relations, i.e.:


 * $\mathcal S \circ \mathcal R: A \to C \ := \ \left\{{\left({a, c}\right) \in A \times C : \exists b \in B: \left({a, b}\right) \in \mathcal R \land \left({b, c}\right) \in \mathcal S}\right\} \subseteq A \times C$

This forms a metacategory, as shown on Category of Relations is Category.

Note
The reason to call $\mathbf{Rel}$ a metacategory is foundational; allowing it to be a category would bring us to axiomatic troubles.