nLab opposite internal relation

Definition

Let $C$ be a finitely complete category. This means that $C$ is symmetric, and for objects $X$ and $Y$ there exist isomorphisms $B_{X, Y}: X \times Y \cong Y \times X$ and $B_{Y, X}: Y \times X \cong X \times Y$ such that $B_{X, Y} \circ B_{Y, X} = id_{X \times Y}$ and $B_{Y, X} \circ B_{X, Y} = id_{Y \times X}$. Given an internal relation $R\stackrel{(s,t)}\hookrightarrow X \times Y$, the pullback of the internal relation $(s,t)$ along the braiding $B_{Y, X}$ is the opposite internal relation span

$$R \stackrel{\dagger_{R^\op,R}}\leftarrow R^\op\stackrel{(t,s)}\hookrightarrow Y \times X$$

with a morphism $\dagger_{R,R^\op}:R \to R^\op$ such that $\dagger_{R,R^\op} \circ \dagger_{R^\op,R} = id_{R^\op}$ and $\dagger_{R^\op,R} \circ \dagger_{R,R^\op} = id_{R}$.

See also

• internal relation

• opposite relation

• internal antisymmetric relation