Dagger category

In 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 Selinger.

Formal definition

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

In detail, this means that it associates to every morphism $f\colon A\to B$ in $\mathcal{C}$ its adjoint $f^\dagger\colon B\to A$ such that for all $f\colon A\to B$ and $g\colon B\to C$ ,

\mathrm{id}_A=\mathrm{id}_A^\dagger\colon A\rightarrow A

(g\circ f)^\dagger=f^\dagger\circ g^\dagger\colon C\rightarrow A

f^{\dagger\dagger}=f\colon A\rightarrow B\,

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 $a\circ c<b\circ c$ for morphisms a, b, c whenever their sources and targets are compatible.