A dagger symmetric monoidal category is a monoidal category 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 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 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.
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