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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
    Commented Jul 13, 2019 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
    Commented Jul 14, 2019 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
    Commented Jul 14, 2019 at 9:31
  • $\begingroup$ They are indeed equivalent in this way. $\endgroup$
    – Simon Henry
    Commented Jul 14, 2019 at 13:48
  • $\begingroup$ @SimonHenry Really? $\endgroup$
    – Martin Brandenburg
    Commented Jan 30, 2020 at 18:04
  • $\begingroup$ @MartinBrandenburg : You know the topic better than I do, but I was under the impression that an appropriate modification of the proof of the strictification theorem was proving this: the condition that the switch map $x \otimes x \rightarrow x \otimes x$ is the identity shows that more general permutation of identical objects in a tensor product all acts as the identity. So one can consider the category MC whose objects are multi-set of objects of C, and morphisms are map between their tensor product (for any ordering) ? $\otimes$ is union and is strictly commutative. $\endgroup$
    – Simon Henry
    Commented Jan 30, 2020 at 18:50
  • $\begingroup$ @MartinBrandenburg Why doesn't the argument presented by Simon Henry work? I had the same intuition: forcing self-symmetries to be identities implies coherence (there is a unique way to permute a given tensor product), and hence should lead to a strictification theorem where permutations are identities. $\endgroup$
    – Léo S.
    Commented Mar 14 at 11:03
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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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  • $\begingroup$ Thank you, nice to see that I was not the first one to have to come up with a name. In the meantime I've been using "strongly symmetric", but "symtrivial" sounds good, if it's the term that has been used up to now I'll try to conform to it. $\endgroup$
    – Captain Lama
    Commented Jan 30, 2020 at 22:27
  • $\begingroup$ I would not say that this is a term which has been used up to now. $\endgroup$
    – Martin Brandenburg
    Commented Feb 2, 2020 at 0:08
  • $\begingroup$ @CaptainLama Did you end up coming up with a terminology for such categories? (and not just for objects satisfying this property) $\endgroup$
    – Léo S.
    Commented Mar 14 at 10:40
  • $\begingroup$ @LéoS. In my article I call both "strongly symmetric". (So the category is strongly symmetric iff all objects are strongly symmetric.) It seems people rediscover this notion once in a while (the sign of a good notion ?), I used Laplaza "Coherence for categories with group structure: an alternative approach" and Ulbrich "Kohärenz in Kategorien mit Gruppenstruktur" as useful earlier references. $\endgroup$
    – Captain Lama
    Commented Mar 14 at 13:27
  • $\begingroup$ Thank you for the quick answer! I also commented Noam's answer with further questions, I would be interested if you have any thoughts on this $\endgroup$
    – Léo S.
    Commented Mar 14 at 18:13

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