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  "body": "\n## Summary\n\nAn injective object of a [[category|A2DE66937D]] is an object $I$ such that maps into $I$ extend along every [[monomorphism|Y2NP1JFR86]].\nIn a category of modules, the injective objects are exactly the [[injective modules|85H7PRHDPZ]].\nAmong abelian groups they are the [[divisible groups|ZXVBEPMTCN]], such as $\\Q$ and $\\Q/\\Z$.\nInjective objects are used to build injective resolutions, from which right [[derived functors|NPMC3K6J2F]] such as sheaf cohomology are computed.\nThe dual notion is the [[projective object|YXMXAGVC1Q]].\n\n## Explanations\n\n### L1 human\n\nSuppose $A$ sits inside a larger object $B$, and a map is given on $A$ with values in $I$.\nThe object $I$ is injective if such a map can always be extended to all of $B$.\nThe rational numbers have this property among abelian groups, because one can always divide in $\\Q$ to find a value for a new element.\nThe integers do not: the map $2\\Z \\to \\Z$ sending $2$ to $1$ cannot be extended to $\\Z$, since its value at $1$ would be an integer whose double is $1$.\n",
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