In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger.

A dagger category is a category ${\displaystyle {\mathcal {C}}}$ equipped with an involutive contravariant endofunctor ${\displaystyle \dagger }$ which is the identity on objects.

In detail, this means that:

Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.

Some sources define a category with involution to be a dagger category with the additional property that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is ${\displaystyle a<b}$ implies ${\displaystyle a\circ c<b\circ c}$ for morphisms ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ whenever their sources and targets are compatible.

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

The latter is only possible for an endomorphism ${\displaystyle f\colon A\to A}$. 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.