# Symmetric monoidal category with trivial switch morphisms

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 ?

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• Do you know any examples other than the one object category? For example, I don't think it ever happens for additive categories with an additive tensor product. – Noah Snyder Jul 13 at 18:33
• The example I have in mind is something like the 2-category featured in ncatlab.org/nlab/show/Brauer+group (but I only care about the underlying 1-category), where the objects are Azumaya algebras, and the morphisms are bimodules realizing Morita equivalences. I actually am interested in a hermitian version, where the algebras have an involution. – Captain Lama Jul 14 at 9:27

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