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.

Alternative definition

A symmetric monoidal dagger category is a braided monoidal dagger category $C$ which is also a symmetric monoidal category, i.e. such that for all objects $A \in Ob(C)$ and $B \in Ob(C)$, $\beta_{A,B} \circ \beta_{B, A} = \Iota_{A,B}$.

Examples

• compact closed dagger category

• symmetric monoidal groupoid, 2-group

See also

• dagger category

• monoidal dagger category

• braided monoidal dagger category

• symmetric monoidal 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