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 $†:{\mathbb{ℂ}}^{op}\to \mathbb{ℂ}$.

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

• ${\mathrm{i}\mathrm{d}}_A={\mathrm{i}\mathrm{d}}_A^{†}:A\to A$
• $(g\circ f{)}^{†}=f^{†}\circ g^{†}:C\to A$
• $f^{††}=f:A\to B$

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 $R:X\to Y$ in Rel, the relation $R^{†}:Y\to X$ 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 $f:A\to B$, the map $f^{†}:B\to A$ is just its adjoint in the usual sense.

Remarkable morphisms

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

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

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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