We all know that a projective object in a category $\mathcal{C}$ is an object $P$ in $\mathcal{C}$ such that for every epimorphism $f: X\to Y$ in $\mathcal{C}$ and arrow $g\colon P\to Y$ there is a lift $g'\colon P\to X$ of $f$.


Let us define a *projective arrow* in $\mathcal{C}$ to be an arrow $g: P\to Y$ such that for every epimorphism $f: X\to Y$ there is a lift $g': P\to X$ of $f$.



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> 1. What is known about such projective arrows?
> 2. What are the projective arrows in the category of groups?
> 3. Is this just some kind of limit in disguise?
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(I'm using "arrow" instead of "morphism" to avoid confilct with the accepted usage of "projective morphism" in Algebraic Geometry).



Projective arrows
If you assume that your category $\mathcal C$ has enough projectives, e.g. the category of groups, then your projective arrows are the maps which factor through a projective. Let us check this. 


Let $f\colon X\rightarrow Y$ be a projective arrow. Since $\mathcal C$ has enough projectives, we can take an epimorphism $g\colon P\twoheadrightarrow Y$ with projective source, hence $f$ factors through $P$. Conversely, if $f\colon X\rightarrow Y$ factors as


$$f\colon X\stackrel{f''}\rightarrow Q\stackrel{f'}\rightarrow Y$$


with $Q$ projective, then $f'$ can be lifted along any epimorphism $g\colon P\twoheadrightarrow Y$, hence $f$ too.


Maps which factor through a projective are widely studied in homological algebra. The stable category $\underline{\mathcal A}$ of an abelian category $\mathcal A$ is the quotient of $\mathcal A$ by the ideal of morphisms factoring through a projective, i.e. by the ideal of projective maps in your terminology. You can read in wikipedia about the stable category of modules over a ring:


<http://en.wikipedia.org/wiki/Stable_module_category>


if ${\mathcal A}$ is a Frobenius abelian category, i.e. enough projectives and injectives and both classes of objects coincide, then $\underline{\mathcal A}$ is a triangulated category. All algebraic triangulated categories arise in this way, but allowing ${\mathcal A}$ to be an exact category, not just abelian.


