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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn object $P$ of $\\mathcal{C}$ is *projective* if, for every [[epimorphism|RVXDSY04KG]] $e \\colon E \\to X$ and every [[morphism|859F0TFVA3]] $f \\colon P \\to X$, there is a morphism $g \\colon P \\to E$ with $e \\circ g = f$.\n\n## Remarks\n\nEquivalently, the [[hom functor|VJXS8AK67E]] $\\Hom(P, -)$ sends epimorphisms to surjections of sets.\nIn an [[abelian category|Y9CZ3MTNF1]], $P$ 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.\nIn the category of modules over a ring, the projective objects are the [[projective modules|BFFA03GE3V]], that is, the direct summands of free modules.\nIn general categories some authors use only regular or strong epimorphisms in the definition; in abelian categories all these versions agree.\n\nExample: $\\Z$ is projective in the category of abelian groups.\nGiven 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 $e(g(n)) = n f(1) = f(n)$.\n\nNon-example: $\\Z/2\\Z$ is not projective in the category of abelian groups.\nFor 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 $e \\circ g = 0 \\ne f$.\n",
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