nLab braided monoidal dagger category

Definition

A braided monoidal dagger category is a monoidal dagger category $(C, \otimes, \Iota)$ which is also a braided monoidal category, in that:

• for objects $A \in Ob(C)$ and $B \in Ob(C)$, there is a natural unitary isomorphism? called the braiding of $A$ and $B$

$$\beta_{A,B}: A\otimes B \cong^\dagger B\otimes A$$

• all objects $D \in Ob(C)$, $E \in Ob(C)$, and $F \in Ob(C)$ satisfy the first hexagon identity

$$\alpha_{E,F,D} \circ \beta_{D, E \otimes F} \circ \alpha_{D,E,F} = (id \otimes \beta_{D,F}) \circ \alpha_{E,D,F} \circ (\beta_{D, E} \otimes id)$$

• all objects $D \in Ob(C)$, $E \in Ob(C)$, and $F \in Ob(C)$ satisfy the second hexagon identity

$$\alpha_{F,D,E}^{-1} \circ \beta_{D \otimes E, F} \circ \alpha^{-1}_{D,E,F} = (\beta_{D, F} \otimes id) \circ \alpha^{-1}_{D,F,E} \circ (id \otimes \beta_{E, F})$$

Examples

• symmetric monoidal dagger category, symmetric monoidal groupoid, 2-group

• compact closed dagger category

See also

• dagger category

• monoidal dagger category

• braided monoidal category