Dagger category

Examples

• The category Rel of sets and relations possesses a dagger structure: for a given relation ${\displaystyle R:X\to Y}$ in Rel, the relation ${\displaystyle R^{†}:Y\to X}$ is the relational converse of ${\displaystyle R}$. In this example, a self-adjoint morphism is a symmetric relation.
• The category Cob of cobordisms is a dagger compact category, in particular it possesses a dagger structure.
• The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map ${\displaystyle f:A\to B}$, the map ${\displaystyle f^{†}:B\to A}$ is just its adjoint in the usual sense.
• Any monoid with involution is a dagger category with only one object. In fact, every endomorphism hom-set in a dagger category is not simply a monoid, but a monoid with involution, because of the dagger.
• A discrete category is trivially a dagger category.
• A groupoid (and as trivial corollary, a group) also has a dagger structure with the adjoint of a morphism being its inverse. In this case, all morphisms are unitary (definition below).

Remarkable morphisms

In a dagger category ${\displaystyle \mathcal{C}}$, a morphism ${\displaystyle f}$ is called

• unitary if ${\displaystyle f^{†}=f^{-1},}$
• self-adjoint if ${\displaystyle f^{†}=f.}$

The latter is only possible for an endomorphism ${\displaystyle f:A\to A}$. The terms unitary and self-adjoint in the previous definition are taken from the category of Hilbert spaces, where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.