Molecular Hamiltonian

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 $\mathbb {C}$ equipped with an involutive, identity-on-object functor $\dagger \colon \mathbb {C} ^{op}\rightarrow \mathbb {C}$.

In detail, this means that it associates to every morphism $f\colon A\to B$ in $\mathbb {C}$ 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 the linear-algebraic sense, not in the category theoretic sense.

Some reputable sources additionally require for a category with involution 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.

Examples

• The category Rel of sets and relations possesses a dagger structure i.e. for a given relation $R:X\rightarrow Y$ in Rel, the relation $R^{\dagger }:Y\rightarrow X$ is the relational converse of $R$.

  A self-adjoint morphism is a symmetric relation.

• The category FdHilb of finite dimensional Hilbert spaces also possesses a dagger structure: Given a linear map $f:A\rightarrow B$, the map $f^{\dagger }:B\rightarrow A$ is just its adjoint in the usual sense.

Remarkable morphisms

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

• unitary if $f^{\dagger }=f^{-1}$;
• self-adjoint if $f=f^{\dagger }$ (this is only possible for an endomorphism $f\colon A\to A$).

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.

See also

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• Dagger symmetric monoidal category
• Dagger compact category

References

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