Pseudo-element

Remarks

Transitivity of $\sim$ uses that pullbacks of epimorphisms are epimorphisms in an abelian category. A morphism $x$ is equivalent to a zero morphism exactly when $x = 0$. The basic rules are these. A morphism $f \colon A \to B$ is a monomorphism if and only if $f(x) = 0$ implies $x = 0$. It is an epimorphism if and only if every $y \in_p B$ is $f(x)$ for some $x \in_p A$. A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is exact if and only if $g \circ f = 0$ and every $y \in_p B$ with $g(y) = 0$ is $f(x)$ for some $x \in_p A$.

Example: in the category of abelian groups, let $x, y \colon \Z \to \Z$ be $x(n) = n$ and $y(n) = -n$. Take $Z = \Z$, $u = \id$ and $v = y$, both epimorphisms; then $x \circ u = y \circ v$, so $x \sim y$. More generally, two morphisms into an abelian group $A$ are equivalent exactly when they have the same image, so the pseudo-elements of $A$ correspond to the subgroups of $A$.

Non-example: the morphisms $x(n) = n$ and $w(n) = 2n$ from $\Z$ to $\Z$ are not equivalent. If $x \circ u = w \circ v$ with $u, v$ epimorphisms, then $x$ and $w$ have the same image, but these images are $\Z$ and $2\Z$.