Dagger category

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

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

• ${\displaystyle \mathrm {id} _{A}=\mathrm {id} _{A}^{\dagger }\colon A\rightarrow A}$
• ${\displaystyle (g\circ \!f)^{\dagger }=f^{\dagger }\!\circ g^{\dagger }\colon C\rightarrow A}$
• ${\displaystyle f^{\dagger \dagger }\!=f\colon A\rightarrow B}$

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[4] 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.

• The category Rel of sets and relations possesses a dagger structure: for a given relation ${\displaystyle R:X\rightarrow Y}$ in Rel, the relation ${\displaystyle R^{\dagger }:Y\rightarrow X}$ is the relational converse of ${\displaystyle R}$. In this example, a self-adjoint morphism is a symmetric relation.
• The category Cob of cobordisms is a dagger compact category, in particular it possesses a dagger structure.
• The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map ${\displaystyle f:A\rightarrow B}$, the map ${\displaystyle f^{\dagger }:B\rightarrow A}$ is just its adjoint in the usual sense.
• Any monoid with involution is a dagger category with only one object. In fact, every endomorphism hom-set in a dagger category is not simply a monoid, but a monoid with involution, because of the dagger.
• A discrete category is trivially a dagger category.
• 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 (definition below).

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

• unitary if ${\displaystyle f^{\dagger }=f^{-1},}$
• self-adjoint if ${\displaystyle f^{\dagger }=f.}$

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.

• *-algebra
• Dagger symmetric monoidal category
• Dagger compact category

• Dagger category in nLab