Poset-enrichment of abelian categories

Question

Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another object $y$.

Does there exist a non-trivial partial order on the morphism-sets of $\mathsf{A}$ which behaves well with respect to composition and happens to be monotone with rank, whenever rank is defined?

More specifically, do there exist partial orders $<$ so that for $f,f':x \to y$ and $g, g':y \to z$, we have:

1. $f < f'$ and $g < g'$ implies $gf < g'f'$ in $\mathsf{A}(x,z)$, and
2. if $f < f'$ then there is a monomorphism from the image of $f$ to image of $f'$?

Probably the answer is no, but I'm not sure how to prove that no such thing could possibly exist --- even in the category of vector spaces over $\mathbb{R}$ or $\mathbb{C}$.

Answer 1

Suppose that $V, W$ are vector spaces and $f, g : V \to W$ are two parallel morphisms such that $f \le g$, for some preorder $\le$ satisfying your conditions.

Claim: $f$ is a scalar multiple of $g$.

Proof. The first condition implies that if $v : 1 \to V$ is any vector in $V$ (here $1$ denotes the $1$-dimensional vector space) then $f \circ v \le g \circ v$. The second condition implies that there is a monomorphism from the image of $f \circ v$ to the image of $g \circ v$ (Edit: here I interpret the condition to mean that the monomorphism must be compatible with the inclusion into $W$, but as Eric Wofsey's answer shows, this extra assumption is unnecessary). This is equivalent to the condition that for all $v \in V$ there is a scalar $\lambda(v)$, possibly zero, such that

$$f(v) = \lambda(v) g(v).$$

Observe that

$$f(v + v') = \lambda(v + v') (g(v) + g(v')) = f(v) + f(v') = \lambda(v) g(v) + \lambda(v') g(v').$$

Hence if $g(v)$ and $g(v')$ are linearly independent it follows that $\lambda(v) = \lambda(v') = \lambda(v + v')$. Now we split into cases.

Case: $\text{im}(g)$ has dimension at least $2$. Then the equivalence relation on vectors in $\text{im}(g)$ generated by "linearly independent" identifies every vector, and it follows that $\lambda(v)$ is a constant $\lambda$ and $f = \lambda g$.

Case: $\text{im}(g)$ has dimension $1$. Then $\text{im}(f)$ is contained in $\text{im}(g)$, and hence has dimension at most $1$. Picking $v_0 \in V$ such that $g(v_0) \neq 0$, we then have $f(v_0) = \lambda(v_0) g(v_0)$, and hence it again follows that $f = \lambda g$ (where $\lambda = \lambda(v_0)$).

Case: $\text{im}(g)$ has dimension $0$. Then $f = g = 0$.

This concludes the proof. $\Box$

So possibilities are limited. Over any field, we can take $f \le g$ iff $f = \lambda g$ for some $\lambda$. This is a preorder, but it's pretty boring: linear maps which are nonzero scalar multiples of each other are isomorphic, and the only order relation which isn't an isomorphism is that $0 \le f$ for any nonzero $f$. Over $\mathbb{R}$ we can take $f \le g$ iff $f = \lambda g$ for some $0 \le \lambda \le 1$. This is even a partial order, but it's still not very interesting.

Answer 2

Suppose you have such a preorder $\leq$. Suppose you have $f,f':x\to y$ such that $f\leq f'$ but $\operatorname{im}(f)\not\subseteq\operatorname{im}(f')$. We can then compose with the quotient $q:y\to y/\operatorname{im}(f')$ and conclude that $qf\leq qf'$ by (1). But this contradicts (2), since the image of $qf'$ is trivial but the image of $qf$ is not. Thus if $f\leq f'$, we must have $\operatorname{im}(f)\subseteq\operatorname{im}(f')$. More generally, by precomposing with the inclusions of subobjects of $x$, we must have that for every subobject $i:w\to x$, $\operatorname{im}(fi)\subseteq\operatorname{im}(f'i)$.

Thus the largest that $\leq$ could possibly be is the following: define $f\leq f'$ to hold iff for every subobject $i:w\to x$, $\operatorname{im}(fi)\subseteq\operatorname{im}(f'i)$. In fact, it is easy to see that this definition does satisfy (1) and (2). Note, however, that in many cases this relation may be pretty trivial; Qiaochu's answer shows that for vector spaces, $f\leq f'$ iff $f$ is a scalar multiple of $f'$. However, this is not true for modules over arbitrary rings. For instance, in $Ab$, any two automorphisms of $\mathbb{Q}/\mathbb{Z}$ are $\leq$-equivalent, even though they are usually not multiples of each other.