Dagger category

In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution^[1][2]) is a category equipped with a certain structure called dagger or involution. The name dagger category was coined by Peter Selinger.^[3]

Formal definition

A dagger category is a category ${\displaystyle {\mathcal {C}}}$ equipped with an involutive contravariant endofunctor ${\displaystyle \dagger }$ which is the identity on objects.^[4]

In detail, this means that:

• for all morphisms ${\displaystyle f:A\to B}$, there exist its adjoint ${\displaystyle f^{\dagger }:B\to A}$
• for all morphisms ${\displaystyle f}$, ${\displaystyle (f^{\dagger })^{\dagger }=f}$
• for all objects ${\displaystyle A}$, ${\displaystyle \mathrm {id} _{A}^{\dagger }=\mathrm {id} _{A}}$
• for all ${\displaystyle f:A\to B}$ and ${\displaystyle g:B\to C}$, ${\displaystyle (g\circ f)^{\dagger }=f^{\dagger }\circ g^{\dagger }:C\to A}$

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^[5] 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.

Examples

• 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).

Remarkable morphisms

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.

See also

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

References

• Dagger category at the nLab