Projective object

Remarks

Equivalently, the hom functor $\Hom(P, -)$ sends epimorphisms to surjections of sets. In an abelian category, $P$ is projective if and only if $\Hom(P, -)$ is an exact functor to abelian groups, and if and only if every short exact sequence $0 \to A \to B \to P \to 0$ splits. In the category of modules over a ring, the projective objects are the projective modules, that is, the direct summands of free modules. In general categories some authors use only regular or strong epimorphisms in the definition; in abelian categories all these versions agree.

Example: $\Z$ is projective in the category of abelian groups. Given an epimorphism $e \colon E \to X$ and $f \colon \Z \to X$, choose $u \in E$ with $e(u) = f(1)$, and let $g(n) = nu$; then $e(g(n)) = n f(1) = f(n)$.

Non-example: $\Z/2\Z$ is not projective in the category of abelian groups. For the reduction $e \colon \Z \to \Z/2\Z$ and $f = 1_{\Z/2\Z}$, a lift $g \colon \Z/2\Z \to \Z$ would have to be $0$, since $\Z$ has no nonzero element of finite order, but then $e \circ g = 0 \ne f$.