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.

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 $R$. 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 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.

• 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.

Remarkable morphisms

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

• unitary if $f^\dagger = f^{-1}$;
• self-adjoint if $f^\dagger = f$

The latter 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

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

References

• Dagger category in nLab