Dagger category

In mathematics, a dagger category (also called involutive category or category with involution [1][2]) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Selinger.[3]

Formal definition

A dagger category is a category ${\displaystyle {\mathcal {C}}}$ equipped with an involutive, identity-on-objects functor ${\displaystyle \dagger \colon {\mathcal {C}}^{op}\rightarrow {\mathcal {C}}}$.

Note that in the previous definition, the term adjoint is used in the linear-algebraic sense, not in the category theoretic sense.

Some reputable sources [4] additionally require for a category with involution that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is a<b implies ${\displaystyle a\circ c for morphisms a, b, c whenever their sources and targets are compatible.

Examples

• 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.

Remarkable morphisms

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.