Is the codmain of a surjection smaller?

Question

This might be a bit naive.

When reading about subobjects, it almost seems to me that a monomorphism $m: A \to B$ into a set/object $B$ is identified with (or corresponds to/represents) its subsets/sub-object.

I wonder if a similar correspondence exists between surjections and supersets. That is, if $f : A \to B$ is a surjection, is $B$ smaller than $A$ (in terms of cardinality)?

This seems to be true for (finite) sets: since everything in $B$ is an image of something in $A$, and two elements in $A$ could be mapped to the same element in $B$, so there are more elements in $A$.

In general, is the intuition right that $B$ is smaller, ... and $A$ can be viewed as some kind of superset?

If so, is there a counterpart concept of superojects that generalizes supersets?

Answer 1

This is indeed true - assuming choice. In lieu of the axiom of choice, however, not every surjection splits, and so we may have the following situation: a surjection from $A$ to $B$, an injection from $A$ to $B$, but no injection from $B$ to $A$. In this case, $A$ is smaller than $B$ ($A$ injects into $B$ and not the other way around)! This is really counterintuitive but can happen - e.g. under the Axiom of Determinacy, $\mathbb{R}/\mathbb{Q}$ (the reals up to rational difference) is larger than $\mathbb{R}$.

For this reason, I think what you really want is the dual of a subobject: a monomorphism from $B$ to $A$.

Answer 2

Even in the category of sets, a monomorphism $S\to T$ does not mean $S$ is smaller than $T$ (at least in cardinality) and an epimorphism doesn't mean that $S$ is bigger than $T$. For example, consider the function from the set of rational numbers to the set of integers that takes a rational number to its numerator. That is certainly a surjection, but in fact the set of integers and the set of rational numbers have the same cardinality.

There is a bigger problem. An object in an arbitrary category need not have elements, in which case it is meaningless to say one object has more elements than another.