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  "body": "\n## Summary\n\nA projective object of a [[category|A2DE66937D]] is an object $P$ such that maps from $P$ lift through every [[epimorphism|RVXDSY04KG]].\nIn a category of modules, the projective objects are exactly the [[projective modules|BFFA03GE3V]], for example the free modules.\nProjective objects are used to build projective resolutions, from which left [[derived functors|NPMC3K6J2F]] are computed.\nThe dual notion is the [[injective object|5QRFTBF4PG]].\n\n## Explanations\n\n### L1 human\n\nThink of an epimorphism $E \\to X$ as a way of covering $X$ by a bigger object $E$.\nAn object $P$ is projective if any map from $P$ into $X$ can be pulled back up to a map into $E$.\nFree objects have this property because one only has to choose where the generators go, and each generator can be sent to any preimage.\nFor example, $\\Z$ is projective among abelian groups, but $\\Z/2\\Z$ is not: the identity of $\\Z/2\\Z$ does not lift along $\\Z \\to \\Z/2\\Z$, since $\\Z$ has no element of order $2$.\n",
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