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.

Formal definition

A dagger category is a category \mathcal{C} equipped with an involutive contravariant endofunctor \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 a < b implies for morphisms a, b, c whenever their sources and targets are compatible.

Examples

Remarkable morphisms

In a dagger category \mathcal{C}, a morphism f is called The latter is only possible for an endomorphism. 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.