Dagger symmetric monoidal category

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A dagger symmetric monoidal category is a monoidal category $<\mathbb{C},\otimes, I>$ which also possesses a dagger structure; in other words, it means that this category comes equipped not only with a tensor in the category theoretic sense but also with dagger structure which is used to describe unitary morphism and self-adjoint morphisms in $\mathbb{C}$ that is, a form of abstract analogues of those found in FdHilb, the category of finite dimensional Hilbert spaces. This type of category was introduced by P. Selinger in [2] as an intermediate structure between dagger categories and dagger compact categories introduced in [1] by S. Abramsky and B. Coecke under the name strongly compact closed category .

1 Formal definition
2 Examples
3 See also
4 References

[edit] Formal definition

The following definition is close to the one given in [2].

A dagger symmetric monoidal category is a symmetric monoidal category $\mathbb{C}$ which also has a dagger structure such that for all $f:A\rightarrow B$ , $g:C\rightarrow D$ and all $A, B$ and $C$ in $Ob(\mathbb{C})$ ,

$(f\otimes g)^\dagger=f^\dagger\otimes g^\dagger:B\otimes D\rightarrow A\otimes C$ ;
$\alpha^\dagger_{A,B,C}=\alpha^{-1}_{A,B,C}:(A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)$ ;
$\rho^\dagger_A=\rho^{-1}_A:A\otimes I\rightarrow A$ ;
$\lambda^\dagger_A=\lambda^{-1}_A:I\otimes A\rightarrow A$ and
$\sigma^\dagger_{A,B}=\sigma^{-1}_{A,B}:B\otimes A\rightarrow A\otimes B$ .

Here, $α,λ,ρ$ and $σ$ are the natural isomorphisms from the symmetric monoidal structure.

[edit] Examples

The following categories are examples of dagger symmetric monoidal categories:

The category Rel of sets and relations where the tensor is given by the product and where the dagger of a relation is given by its relational converse.
The category FdHilb of finite dimensional Hilbert spaces is a dagger symmetric monoidal category where the tensor is the usual tensor product of Hilbert spaces and where the dagger of a linear map is given by its hermitian adjoint.

[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.

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Categories: Dagger categories | Monoidal categories