Opposite Enriched Category

definition

The opposite of a $\mathcal{V}$-category $\mathcal{X}$ is the $\mathcal{V}$-category ${\mathcal{X}}^{\mathrm{op}}$ with $\mathrm{Ob}({\mathcal{X}}^{\mathrm{op}}):=\mathrm{Ob}(\mathcal{X})$ and ${\mathcal{X}}^{\mathrm{op}}(x,y):=\mathcal{X}(y,x)$. Composition uses the symmetry of $\mathcal{V}$ to swap the two hom-objects (DaoFP Exercise 20.1.1), which is why $\mathcal{V}$ is assumed symmetric.

Sources: 7 Sketches Exercise 2.73; DaoFP §20.1; cf. Opposite Preorder, Opposite Category.

• A dagger $\mathcal{V}$-category is one where the identity function is a $\mathcal{V}$-functor $†:\mathcal{X}\to {\mathcal{X}}^{\mathrm{op}}$, i.e. $\mathcal{X}(x,y)\le \mathcal{X}(y,x)$ — symmetric distances for Cost, equivalence relations for $\mathbf{Bool}$.
• A skeletal $\mathcal{V}$-category is one where $I\le \mathcal{X}(x,y)$ and $I\le \mathcal{X}(y,x)$ imply $x=y$.
• Skeletal dagger $\mathbf{Cost}$-categories are extended metric spaces; skeletal dagger $\mathbf{Bool}$-categories are sets (7S Exercise 2.73, 7S Exercise 1.73).
• $\mathcal{V}$-profunctors are $\mathcal{V}$-functors ${\mathcal{X}}^{\mathrm{op}}\times \mathcal{Y}\to \mathcal{V}$.