Dagger category

In category theory, a branch of mathematics, a dagger category 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 equipped with an involutive contravariant endofunctor which is the identity on objects.
In detail, this means that:

for all morphisms, there exists its adjoint
for all morphisms,
for all objects,
for all and,

Note that in the previous definition, the term "adjoint" is used in a way analogous to 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 implies for morphisms,, whenever their sources and targets are compatible.

Examples

The category Rel of sets and relations possesses a dagger structure: for a given relation in Rel, the relation is the relational converse of. 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, the map 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 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, a morphism is called

unitary if
self-adjoint if

The latter is only possible for an endomorphism. 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.

Dagger category

Formal definition

Examples

Remarkable morphisms