Dagger category explained

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

l{C} equipped with an involutive contravariant endofunctor \dagger which is the identity on objects.^[1]

In detail, this means that:

• for all morphisms

f: A \to B , there exist its adjoint f ^{\dagger:} B \to A

• for all morphisms

f , (f ^{\dagger)} ^{\dagger} = f

• for all objects

A , id ^{\dagger} _{A} = id _{A}

• for all

f: A \to B and g: B \to C , (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 implies a\circ c<b\circ c for morphisms a , b , c whenever their sources and targets are compatible.

Examples

R:X → Y in Rel, the relation R ^{\dagger: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.

f:A → B , the map f ^{\dagger: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 (definition below).

Remarkable morphisms

In a dagger category

l{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

1. Web site: Dagger category in nLab .

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