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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nThe category $\\mathcal{C}$ *has enough projectives* if for every object $X$ of $\\mathcal{C}$ there are a [[projective object|YXMXAGVC1Q]] $P$ and an [[epimorphism|RVXDSY04KG]] $P \\to X$.\n\n## Remarks\n\nExample: the category of modules over a [[ring|0B1261B5ED]] $R$ has enough projectives.\nFor a module $M$, let $F$ be the [[free module|0DZ9DKAKZR]] with basis $\\set{e_m : m \\in M}$.\nThe map $F \\to M$ with $e_m \\mapsto m$ is surjective, and free modules are [[projective|BFFA03GE3V]].\nNon-example: the category of finite abelian groups has no nonzero projective object.\nFor $n \\ge 2$ the quotient map $\\Z/n^2\\Z \\to \\Z/n\\Z$ does not split, since $\\Z/n^2\\Z$ is cyclic, so $\\Z/n\\Z$ is not projective there.\nEvery nonzero finite abelian group has such a cyclic direct summand, and a direct summand of a projective object is projective.\nSo no epimorphism from a projective object onto $\\Z/2\\Z$ exists.\nThe dual notion is *enough injectives*: every object admits a monomorphism into an [[injective object|5QRFTBF4PG]].\n",
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