# Monoidal categories in which some tensor products of morphisms are equal

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).

• 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. – tcamps Apr 1 at 1:44
• 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. – Captain Lama Apr 1 at 4:52