What is a superfluous epimorphism? In the definition of projective cover, the term **superfluous epimorphism** is used. > Let $\mathcal{C}$ be a category and $X$ an object in $\mathcal{C}$. A projective cover is a pair $(P,p)$, with $P$ a projective object in $\mathcal{C}$ and $p$ a superfluous epimorphism in $\text{Hom}(P, X)$. There is a note underneath. > If $R$ is a ring, then in the category of $R$-modules, a superfluous epimorphism is then an epimorphism $p : P \to X$ such that the kernel of $p$ is a superfluous submodule of $P$. But this kind of definition doesn't seem to be applicable to the general case. So for an arbitrary category $\mathcal C$, objects $X,Y$ in $\mathcal C$ and $f \in \text{Hom}(X,Y)$, when is $f$ said to be a superfluous epimorphism?

"Superfluous epimorphism" is dual to "essential monomorphism":

An epimorphism $f: X \rightarrow Y$ is superfluous if for any morphism $g: T \rightarrow X$ one has $f \circ g \text{ epi} \Rightarrow g \text{ epi}$.

In the case of modules this is equivalent to $\ker(f)$ being a superfluous submodule of $X$. In one direction, observe that $f\circ g$ epi forces $X = \mathrm{im}(g) + \ker(f)$. In the other direction, for a sum $X = \ker(f) + T$ consider the inclusion $g: T \rightarrow X$.