Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric group $\mathfrak{S}\_n$ on $x^{\otimes n}$ is trivial.)


Is there a paper or book I can cite where basic properties of such categories are derived ?



Symmetric monoidal category with trivial switch morphisms
Coincidentally, terminology for such categories has been introduced very recently:


* John C. Baez and Jade Master. Open Petri Nets. Nov 2018. [arXiv:1808.05415](https://arxiv.org/abs/1808.05415)


More precisely, the authors refer to a strict symmetric monoidal category in which (not only the associator and unitors but also) the symmetry
$$\sigma\_{a,b} : a \otimes b \overset{\sim}\to b \otimes a$$
is the identity morphism as a **commutative monoidal category**, this being the same as a commutative monoid object in $(\mathrm{Cat},\times,1)$.


