Inverse Relation Functor is Contravariant Functor

Theorem

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

Let $C: \mathbf{Rel} \to \mathbf{Rel}$ be the inverse relation functor.

Then $C$ is a contravariant functor.

Proof

For any set $X$. we have:

\(\ds C \operatorname{id}_X\)

\(=\)

\(\ds \operatorname{id}_X^{-1}\)

Definition of Inverse Relation Functor

\(\ds \)

\(=\)

\(\ds \set {\tuple {x, x}: \tuple {x, x} \in \operatorname{id}_X}\)

Definition of Inverse Relation

\(\ds \)

\(=\)

\(\ds \operatorname{id}_X\)

For any two relations $\RR \subseteq X \times Y$ and $\SS \subseteq Y \times Z$, we have:

\(\ds C \paren {\SS \circ \RR}\)

\(=\)

\(\ds \paren {\SS \circ \RR}^{-1}\)

Definition of Inverse Relation Functor

\(\ds \)

\(=\)

\(\ds \RR^{-1} \circ \SS^{-1}\)

Inverse of Composite Relation

\(\ds \)

\(=\)

\(\ds C \RR \circ C \SS\)

Definition of Inverse Relation Functor

Hence $C$ is shown to be a contravariant functor.

$\blacksquare$

Sources

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