nLab symmetric monoidal dagger-category

Definition

A symmetric monoidal $\dagger$-category is a symmetric monoidal category that is also a $\dagger$-category for which:

1. $(f \otimes g)^\dagger = f^\dagger \otimes g^\dagger$ for every pair of morphisms $f,g$

2. the associator, left and right unitors, and braiding are all unitary.

If the category is also a compact closed category in a compatible way, then it is called a dagger-compact category.

References

• 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. web