Dagger symmetric monoidal category

A dagger symmetric monoidal category is a monoidal category ${\displaystyle \langle \mathbb {C} ,\otimes ,I\rangle }$ which also possesses a dagger structure; in other words, it means that this category comes equipped not only with a tensor product in the category theoretic sense but also with dagger structure which is used to describe unitary morphism and self-adjoint morphisms in ${\displaystyle \mathbb {C} }$ 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 Selinger[1] as an intermediate structure between dagger categories and the dagger compact categories that are used in categorical quantum mechanics, an area which now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical concepts.

Formal definition

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

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

Here, ${\displaystyle \alpha ,\lambda ,\rho }$ and ${\displaystyle \sigma }$ are the natural isomorphisms that form the symmetric monoidal structure.

Examples

The following categories are examples of dagger symmetric monoidal categories:

A dagger-symmetric category which is also compact closed is a dagger compact category; both of the above examples are in fact dagger compact.