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 dagger structure present in a dagger compact category (introduced in ^[3] by B. Coecke and S. Abramsky under the name strongly compact closed categories) has been extracted by Peter Selinger in ^[4]. This structure has its own importance since many categories can possess a dagger structure without being compact closed.

Formal definition

The following definition is taken from ^[4].

A dagger category is a category equipped with an involutive, identity-on-object 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.

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 R.

  A self-adjoint morphism is a symmetric relation.

• 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.

Remarkable morphisms

In a dagger category , a morphism f 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

• Dagger symmetric monoidal category
• Dagger compact category

References

1. ^ M. Burgin, Categories with involution and correspondences in g-categories, IX All-Union Algebraic Colloquium, Gomel (1968), pp.34–35
2. ^ J. Lambek, Diagram chasing in ordered categories with involution, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307
3. ^ S. Abramsky and B. Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).
4. ^ ^{a b} 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.

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