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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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  • $\begingroup$ 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. $\endgroup$ – Noah Snyder Jul 13 at 18:33
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    $\begingroup$ 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. $\endgroup$ – Captain Lama Jul 14 at 9:27
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Coincidentally, terminology for such categories has been introduced very recently:

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

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  • $\begingroup$ Thanks for the reference. This being said, this seems quite a lot stronger than what I'm thinking about (the example that prompted my question does not satisfy such a strong property). Is it possible that any monoidal category satisfying the property in my question is monoidally equivalent to one as in your answer ? $\endgroup$ – Captain Lama Jul 14 at 9:31
  • $\begingroup$ They are indeed equivalent in this way. $\endgroup$ – Simon Henry Jul 14 at 13:48

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