Is the category of monoids cartesian closed? Why?
I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above.
Here I speak of the "category of monoids" as the category with objects monoids and arrows homomorphisms between monoids.
Best Answer
Whenever a non-trivial category is cartesian-closed, the final object $1$ cannot also be an initial object.
Otherwise:
$$\mathrm{Hom}(A,B)\cong \mathrm{Hom}(1\times A,B) \cong \mathrm{Hom}(1,B^A)\cong\{\cdot\}$$
That is, every hom set would have to be a singleton.