# Projective arrows

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 $f$.

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?

(I'm using "arrow" instead of "morphism" to avoid confilct with the accepted usage of "projective morphism" in Algebraic Geometry).

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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.

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This was very helpful, thanks Fernando! –  Mark Grant Nov 24 '12 at 8:45