Why is surjectivity "harder" than injectivity?

Solution 1:

I’d try to explain this as follows: we are in the habit of understanding a set $S$ in terms of the maps from a singleton $*$ to $S$. This breaks the symmetry dramatically between “epimorphism” and “monomorphism,” the left- and right-cancellations conditions discussed in the comments.

Indeed, if you know $f:S\to T$ is a monomorphism, then you know immediately from the definition that $f$ does not identify any two distinct points $x,y:*\to S$. But if you know $f$ is an epimorphism, it’s much less obvious what this means in terms of maps from the point into $S$ and $T$. You have to know a lot more about the structure of sets to say that, if there were any point of $T$ not in the image of $f$, then one could construct two unequal maps out of $T$ equalized by $f$. And these maps out of $T$ wouldn’t most naturally be into any nice fixed set like $*$! At best, you could use maps $T\to \{0,1\}$.

In fact, if you know enough about $\{0,1\}$, you know you can characterize the epis as precisely the maps $f$ which induce a mono on powersets, $f^*:\{0,1\}^T\to \{0,1\}^S$. So one might measure the difference in difficulty of injective versus surjective maps by considering how much more complex the powerset is than the mere set. For instance, the powerset construction isn’t even available in most other categories, though $\mathbb k$ serves the same role among finite dimensional vector spaces.