Category of Relations is Category

Theorem

Let $\mathbf{Rel}$ be the category of relations.

Then $\mathbf{Rel}$ is a metacategory.

Proof

Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.

We have already defined the composition of morphisms.

For any set $X$, we have the diagonal relation $\operatorname{id}_X$.

By Diagonal Relation is Left Identity and Diagonal Relation is Right Identity it follows that this is the identity morphism for $X$.

Finally by Composition of Relations is Associative, the associative property is satisfied.

Hence $\mathbf{Rel}$ is a metacategory.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.9$: Exercise $1 \ \text{(a)}$