Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts?

This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ but.. is it true in general (esp. in the category of non-commutative algebras)?


The term you want is "distributes over," not "commutes with." (Whatever "$A$ commutes with $B$" means it should be symmetric in $A$ and $B$, which the condition you want is not.) Such monoidal categories are called distributive. Examples include any closed monoidal category with coproducts because in this case $A \otimes (-)$ has a right adjoint and hence preserves all colimits. In particular, this is true in any bicartesian closed category, if $\otimes$ is taken to be categorical product.

But there's no reason for it to be true in general: for example, you can take $\otimes$ to be the coproduct itself.

Incidentally, this is false in both $\text{Grp}$ (for the categorical product) and $\text{CRing}$ (for both the categorical product and the tensor product). In $\text{CRing}$ the coproduct distributes over the product, not the other way around. I think in noncommutative algebras it is false for all of the obvious choices of monoidal structure.


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