I have come across a certain monoidal category $(C,\otimes,I)$ (let us say it is strictly monoidal to simplify the notations, as usual it does not matter very much) which satisfies the following property: if $f: x^{\otimes n_1}\to y^{\otimes m_1}$ and $g: x^{\otimes n_2}\to y^{\otimes m_2}$ are two morphisms in $C$, then $$ f\otimes g: x^{\otimes n_1+n_2}\to y^{\otimes m_1+m_2}$$ and $$ g\otimes f: x^{\otimes n_1+n_2}\to y^{\otimes m_1+m_2}$$ are equal.

Is there a name for this property ? Obviously if the category is symmetric (or even braided I guess) the two morphisms are isomorphic (in the category of morphisms in $C$), but not equal in general.

I would also be interested in a name for the corresponding property in a monoidal 2-category, namely when for any 1-morphisms $f$ and $g$ as above, $f\otimes g$ and $g\otimes f$ are 2-isomorphic (which amounts to say that the monoidal 1-category we get by identifying the 1-morphisms that are 2-isomorphic satisfies the first version).

  • $\begingroup$ I'm not sure there's a name for this property, but it suggests that something stronger might be true: it might be the case that for any $c \in C$, the symmetry map $c \otimes c \to c \otimes c$ is the identity. $\endgroup$ – tcamps Apr 1 at 1:44
  • $\begingroup$ This is a very good remark. Indeed it might very well be true in my case, and it would give a satisfying explanation for my initial observation. $\endgroup$ – Captain Lama Apr 1 at 4:52

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