Dagger category

From Wikipedia, the free encyclopedia

In mathematics, a dagger category (also called involutive category or category with involution [3,4]) is a category equipped with a certain structure called dagger or involution.

The dagger structure present in a dagger compact category (introduced in [1] by B. Coecke and S. Abramsky under the name strongly compact closed categories) has been extracted by Peter Selinger in [2]. This structure has its own importance since many categories can possess a dagger structure without being compact closed.

Contents 1 Formal definition 2 Examples 3 Remarkable morphisms 4 See also 5 References

[edit] Formal definition

The following definition is taken from [2].

In mathematics, 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.

[edit] 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.

[edit] 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.

[edit] See also

• Dagger symmetric monoidal category
• Dagger compact category

[edit] References

[1] 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).

[2] 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.

[3] M. Burgin, Categories with involution and correspondences in g-categories, IX All-Union Algebraic Colloquium, Gomel (19680, pp.34–35

[4] J. Lambek, Diagram chasing in ordered categories with involution, Journal of Pure and Applied Algebra 143 (1999), No.1–3, 293–307