TITLE: Is the category commutative monoids cartesian closed?
QUESTION [7 upvotes]: Is the category commutative monoids cartesian closed?
I understand it breaks down at the point at which the evaluation map cannot be confirmed, however this is not enough to show it is not cartesian closed.
Thanks!

REPLY [21 votes]: Here's a simple observation: the category of commutative monoids has an object that is both initial and terminal (just like the category of groups or abelian groups), so it cannot be cartesian closed. Indeed, if a cartesian closed category $\mathcal{C}$ has the property that $0 \cong 1$, then
$$\mathcal{C}(X, Y) \cong \mathcal{C}(1, Y^X) \cong \mathcal{C}(0, Y^X) \cong 1$$
and so $\mathcal{C}$ is equivalent to the trivial category with one object and one morphism.