TITLE: Symmetric monoidal category with trivial switch morphisms
QUESTION [6 upvotes]: 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 ?

REPLY [3 votes]: 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.