Dagger symmetric monoidal category

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In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category ${\displaystyle \langle \mathbf {C} ,\otimes ,I\rangle }$ that also possesses a dagger structure. That is, this category comes equipped not only with a tensor product in the category theoretic sense but also with a dagger structure, which is used to describe unitary morphisms and self-adjoint morphisms in ${\displaystyle \mathbf {C} }$: abstract analogues of those found in FdHilb, the

Formal definition

A dagger symmetric monoidal category is a symmetric monoidal category ${\displaystyle \mathbf {C} }$ that 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 ${\displaystyle Ob(\mathbf {C} )}$,

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

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.

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

See also

• Strongly ribbon category

References