elementary set theory - Category of Sets w/ 17 Elements: There does not exist a direct product? (Lots of questions here)

I'm having a pretty hard time with this. I'm asked to show that, in the category of sets with exactly 17 elements, no two objects have either a direct product nor direct sum. Part of me doesn't even believe this statement—but whenever I try to come up with a direct product, I get snagged.

Let $(C, \alpha, \beta)$ be a [potential] direct product of $A$ and $B$. Fix some object, $C'$, with mappings, $\alpha'$ and $\beta'$, from $C'$ to $A$ and $B$ respectively. We need a unique $\gamma: C' \rightarrow C$ such that $\alpha \circ \gamma = \alpha'$ and $\beta \circ \gamma = \beta'$.

• $C = A \times B$ just can't work, because $A \times B$ necessarily has more than 17 elements (in fact, no Cartesian-product-like $C$ can work, because the number of elements is fixed). What about some 17-element subset of $A \times B$? (In the category of sets, $\alpha$ and $\beta$ are injective, but not necessarily surjective). But, what if $\alpha'$ and $\beta'$ are both surjective? So, that can't work, because there's no $\gamma$ that could satisfy this (doesn't that mean that, in the general category of sets, $\alpha$ and $\beta$ have to also be surjective? If they don't "touch" every element in both $A$ and $B$, then one can just define a $\alpha'$ or $\beta'$ that touches the elements $\alpha$ or $\beta$ don't—thus making impossible a direct product.)




• Let $\alpha(c_n) = a_n$ and $\beta(c_n) = b_n$. This contains bijective $\alpha$ and $\beta$, but all we have to do is define a $C'$ such that $\alpha'(c_1) = a_1$ and $\beta'(c_1) = b_2$.

Ok. So, $\alpha$ and $\beta$ have to be bijective. Let's try to prove this via negation: Since these mappings are necessarily bijective, they have to have an inverse. Thus, $\gamma$ must be such that $\gamma = \alpha^{-1} \circ \alpha'$ and $\gamma = \beta^{-1} \circ \beta'$. To show that we can choose an object $C'$ where $\gamma$ can't make the graph commute, just choose $\alpha'$ and $\beta'$ such that $\alpha^{-1} \circ \alpha \neq \beta^{-1} \circ \beta$.

1. Can I assume that such an $\alpha'$ and $\beta'$ will always exist?


2. Was there no point to the number of elements being $17$ specifically? This all seems to work for any category of sets with a fixed number of elements.


3. Is there something crucial I'm missing?

Answer

Let $n \geq 2$. If $A,B$ have a product $P$ in the category of sets with $n$ elements, then $\hom(A,P) \cong \hom(A,A) \times \hom(A,B)$ shows $n^n = n^n \cdot n^n$, a contradiction.