Dagger category

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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.

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 $\mathbb{C}$ equipped with an involutive, identity-on-object functor

$\dagger:\mathbb{C}^{op}\rightarrow\mathbb{C}.$

which associates to every morphism $f:A\rightarrow B$ in $\mathbb{C}$ its adjoint $f^\dagger:B\rightarrow A$ such that for all $f:A\rightarrow B$ and $g:B\rightarrow C$ ,

$id_A=id_A^\dagger:A\rightarrow A$
$(g\circ f)^\dagger=f^\dagger\circ g^\dagger:C\rightarrow A$
$f^{\dagger\dagger}=f:A\rightarrow B$

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 $R:X\rightarrow Y$ in Rel, the relation $R^\dagger:Y\rightarrow X$ is the relational converse of $R$ .

The category FdHilb of finite dimensional Hilbert spaces also possesses a dagger structure: Given a linear map $f:A\rightarrow B$ , the map $f^\dagger:B\rightarrow A$ is just its adjoint in the usual sense.

[edit] Remarkable morphisms

In a dagger category $\mathbb{C}$ , a morphism $f$ is called

unitary if $f^\dagger=f^{-1}$ ;
self-adjoint if $f=f^\dagger$ .

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

[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