To complement the above answer, let me give a purely abstract explanation.

Apparently, I think the notion you are looking for is... the notion of *monomorphism* --- an object $A$ *is no smaller than* object $B$ if there is no monomorphism from $A$ to $B$. In such a case if $f \colon A \rightarrow B$ is any morphism then there are "points" $a \neq a' \in A$ that are over a single "point" $b \in B$.

Unfortunately, the relation "is no smaller than" does not behave well in case the Cantor–Bernstein–Schroeder property does not hold (i.e. we say that CBS holds if whenever we have monomorphisms $A \rightarrow B$ and $B \rightarrow A$ then $A \approx B$). There is a nice discussion on the subject in the following question:

When does Cantor-Bernstein hold?

Now, let me elaborate, why I think the notion of a monomorphism is the right notion here.

First of all, one may either use a "positive" or a "negative" definition of your property for $f$. The positive definition says that the collection:

$$\{a \in A \colon f(a) = b\}$$

has more than one element for some $b \in B$; whereas, the negative definition says that it is not the case that all elements from the collection are equal for every $b \in B$. The negative definition has a very natural logical meaning --- the following does not hold:

$$f(a) = f(a') \Rightarrow a = a'$$

Such $f$ satisfying the above are called "monomorphisms" (so the negative definition says: "it is not the case that $f$ is a monomorphism").

However, this is not the end of the story. There are also two choices for your term "point" --- let me call them "external" and "internal" respectively.

In the "external" view, one has to understand the above "formula" with respect to the external logic. That is, $a, a' \in A$ have to be understood as generalized elements of $A$ parametrized by some $X$ --- morphisms $a, a' \colon X \rightarrow A$; and applications $f(a), f(a')$, by Yoneda, have to be understood as compositions $f \circ a, f \circ a'$. Thus the above formula, when interpreted in the external logic, becomes the usual definition of a monomorphism:

$$f \circ a = f \circ a' \Rightarrow a = a'$$

In the "internal" view, one has to choose a sufficiently strong internal logic to give a precise meaning for the formula. If $f$ satisfies the formula in such an internal logic, then we say that $f$ is an "internal monomorphism". Fortunately, in many cases, one may show that the notions of "internal monomorphism" and "external monomorphism" coincide. This is, for example, true in the canonical internal logic of any category (more generally, this is obviously true in any logic with external equality). Therefore, if we do not have any natural premises to chose other notions of monomorphism, perhaps the best choice is the usual notion of (external) monomorphism.

actualproblem you're thinking about, and then we can see if there are some finitely generated or finite-dimensional objects lurking in the problem that allow one to employ some kind of size argument $\endgroup$ – Yemon Choi Oct 21 '14 at 19:22