In category theory, a branch of 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 Peter Selinger.
equipped with an involutive contravariant endofunctor
[4] 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[5] 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
whenever their sources and targets are compatible.
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.