In category theory, a branch of 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 Peter Selinger.

Formal definition

A dagger category is a category C {\displaystyle {\mathcal {C}}} equipped with an involutive contravariant endofunctor † {\displaystyle \dagger } which is the identity on objects.

In detail, this means that:

  • for all morphisms f : A → B {\displaystyle f:A\to B}, there exists its adjoint f † : B → A {\displaystyle f^{\dagger }:B\to A}
  • for all morphisms f {\displaystyle f}, ( f † ) † = f {\displaystyle (f^{\dagger })^{\dagger }=f}
  • for all objects A {\displaystyle A}, i d A † = i d A {\displaystyle \mathrm {id} _{A}^{\dagger }=\mathrm {id} _{A}}
  • for all f : A → B {\displaystyle f:A\to B} and g : B → C {\displaystyle g:B\to C}, ( g ∘ f ) † = f † ∘ g † : C → A {\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 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 {\displaystyle a<b} implies a ∘ c < b ∘ c {\displaystyle a\circ c<b\circ c} for morphisms a {\displaystyle a}, b {\displaystyle b}, c {\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 R : X → Y {\displaystyle R:X\rightarrow Y} in Rel, the relation R † : Y → X {\displaystyle R^{\dagger }:Y\rightarrow X} is the relational converse of R {\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 f : A → B {\displaystyle f:A\rightarrow B}, the map f † : B → A {\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 C {\displaystyle {\mathcal {C}}}, a morphism f {\displaystyle f} is called

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

The latter is only possible for an endomorphism f : A → A {\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