# Dagger category

In 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 Selinger.^{[3]}

## Formal definition

A **dagger category** is a category equipped with an involutive, identity-on-objects functor .

In detail, this means that it associates to every morphism in its adjoint such that for all and ,

Note that in the previous definition, the term *adjoint* is used in the linear-algebraic sense, not in the category theoretic sense.

Some reputable sources ^{[4]} additionally require for a *category with involution* 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 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 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
**FdHilb**of finite dimensional Hilbert spaces also possesses a dagger structure: Given a 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 (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 , a morphism is called

**unitary**if ;**self-adjoint**if (this 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.

## See also

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## References

- ↑ M. Burgin,
*Categories with involution and correspondences in γ-categories*, IX All-Union Algebraic Colloquium, Gomel (1968), pp.34–35; M. Burgin,*Categories with involution and relations in γ-categories*, Transactions of the Moscow Mathematical Society, 1970, v. 22, pp. 161–228 - ↑ J. Lambek,
*Diagram chasing in ordered categories with involution*, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307 - ↑ P. Selinger,
*Dagger compact closed categories and completely positive maps*, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1, 2005. - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}