All the usual examples of monoidal categories that one comes across ($\mathbf{Set}$ with $\times$ as product, $R-\mathbf{Mod}$ with $\otimes$ etc.) are symmetric. Does anyone know an example, preferrably but not necessarily one that occurs in a theoretical mathematician's everyday life, of a monoidal category $C=(C,\otimes)$ that is not symmetric (i.e. $\exists A,B\in\operatorname{Obj}(C):A\otimes B\ncong B\otimes A$)?
Monoidal categories that are not symmetric
category-theorymonoidmonoidal-categories
Best Answer
Assuming, by "Abelian", you mean symmetric monoidal, then a good example is the category of endofunctors $(\mathbf C, \circ, \mathrm{Id}_{\mathbf C})$ on a category $\mathbf C$. For instance, pick two distinct objects $X$ and $Y$ in $\mathbf C$, and consider the "constantly $X$" and "constantly $Y$" functors. Clearly, the two composites of these functors are different. Categories of endofunctors are commonly encountered in category theory, since monoids therein are precisely monads.