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

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 $(\mathrm{Cat},\times,1)$.


In my thesis I have named objects $x$ whose switch map $x \otimes x \to x \otimes x$ is the identity symtrivial (since I could not find any term in the literature). It was then used by others as well, but right now I can only find this example. Now it is reasonable to call a tensor category symtrivial when every object is symtrivial. The property in Noam's answer is much stronger.

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Symmetric Monoidal Categories

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