By definition, a monoidal category is a category $\mathbf{C}$ equipped with a bifunctor
$$
\otimes :\mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}
$$
and an object $I$ that is both a left and right identity for $\otimes$, that satisfies some associative conditions.
The question is, for any category $\mathbf{C}$, is it always possible to find a bifunctor $\otimes$ that renders it into a monoidal one?
If not, what is a minimal counterexample to convince a layman? Thanks.
Best Answer
As a consequence of the Eckmann-Hilton argument, the endomorphism monoid of the unit object in a monoidal category is always commutative. (Besides the usual composition monoid structure, it also has a monoid structure coming from the canonical isomorphism $1\otimes 1\to 1$ which you can check is compatible with the composition operation such that the Eckmann-Hilton argument applies.) So, if a category has no object whose endomorphism monoid is commutative, it does not admit a monoidal structure. This of course includes the empty category, but also includes, for instance, the category of sets with more than one element.