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Dagger category

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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.

Contents

Formal definition

A dagger category is a category C equipped with an involutive functor † : C o p → C that is the identity on objects, where C o p is the opposite category.

In detail, this means that it associates to every morphism f : A → B in C its adjoint f † : B → A such that for all f : A → B and g : B → C ,

  • i d A = i d A † : A → A
  • ( g ∘ f ) † = f † ∘ g † : C → A
  • f † † = f : A → 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 ∘ c < b ∘ 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 → Y in Rel, the relation R † : Y → 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 → B , the map f † : B → 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 C , a morphism f is called

  • unitary if f † = f − 1 ;
  • self-adjoint if f † = f

    • The latter is only possible for an endomorphism f : A → 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.

    References

    Dagger category Wikipedia
    (Text) CC BY-SA


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