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Concept
Dagger symmetric monoidal category
Summary
In the mathematical field of , a dagger symmetric monoidal category is a that also possesses a . That is, this category comes equipped not only with a tensor product in the sense but also with a , which is used to describe unitary morphisms and self-adjoint morphisms in : abstract analogues of those found in FdHilb, the . This type of was introduced by Peter Selinger as an intermediate structure between and the that are used in categorical quantum mechanics, an area that now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical concepts.
A dagger symmetric monoidal category is a that also has a such that for all , and all and in ,
and
Here, and are the natural isomorphisms that form the .
The following are examples of dagger symmetric monoidal categories:
The Rel of where the tensor is given by the and where the dagger of a relation is given by its relational converse.
The FdHilb of 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 is a ; both of the above examples are in fact dagger compact.
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