nLab unitary morphism

Definition

In a †-category $C$, a morphism $f : x \to y$ is said to be unitary if it is invertible and its inverse $f^{-1}$ is its dagger $f^{\dagger}$:

$$f^{-1} = f^\dagger\colon y \to x \,.$$

For more details, see the entry †-category.

Examples

• The unitary morphisms in $C =$ Hilb are the ordinary unitary operators between Hilbert spaces.

  In particular, the unitary automorphisms of an object in $Hilb$ form its unitary group.

• The unitary morphisms in $C =$ Rel are the ordinary bijections between sets.

  In particular, the unitary automorphisms of an object in $Rel$ form a permutation group.

Related concepts

• self-adjoint morphism

• unitary morphism

• dagger monomorphism

• dagger epimorphism