Symmetric monoidal category

Remarks

Under this condition the two hexagon identities of the braiding are equivalent, so one of them suffices. Example: $\mathbf{Vect}_k$ with $\gamma(v \otimes w) = w \otimes v$; applying the swap twice returns $v \otimes w$. Example: $\Z/2$-graded vector spaces with the Koszul sign $\gamma(v \otimes w) = (-1)^{\abs{v}\abs{w}}\, w \otimes v$ on homogeneous elements; the signs cancel when the swap is applied twice. Non-example: $\Z/3$-graded complex vector spaces with $\gamma(v \otimes w) = q^{\abs{v}\abs{w}}\, w \otimes v$, where $q = e^{2\pi i/3}$. This is a braiding because $(a, b) \mapsto q^{ab}$ is a bicharacter of $\Z/3$, but applying it twice multiplies $v \otimes w$ by $q^{2\abs{v}\abs{w}}$, which is not $1$ when $\abs{v} = \abs{w} = 1$. Symmetric monoidal categories form the special case of monoidal categories that model commutative operations.