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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nAn object $I$ of $\\mathcal{C}$ is *injective* if, for every [[monomorphism|Y2NP1JFR86]] $m \\colon A \\to B$ and every [[morphism|859F0TFVA3]] $f \\colon A \\to I$, there is a morphism $g \\colon B \\to I$ with $g \\circ m = f$.\n\n## Remarks\n\nEquivalently, the contravariant [[hom functor|VJXS8AK67E]] $\\Hom(-, I)$ sends monomorphisms to surjections of sets.\nIn an [[abelian category|Y9CZ3MTNF1]], $I$ is injective if and only if $\\Hom(-, I)$ is exact, and if and only if every short exact sequence $0 \\to I \\to B \\to C \\to 0$ splits.\nIn the category of modules over a ring, the injective objects are the [[injective modules|85H7PRHDPZ]].\nAn injective object of $\\mathcal{C}$ is the same as a projective object of the opposite category.\n\nExample: $\\Q$ is injective in the category of abelian groups.\nIt is a [[divisible group|ZXVBEPMTCN]], and by Baer's criterion a divisible abelian group is injective: a map $f \\colon n\\Z \\to \\Q$ extends to $\\Z$ by sending $1$ to $f(n)/n$.\n\nNon-example: $\\Z$ is not injective in the category of abelian groups.\nFor $m \\colon \\Z \\to \\Z$, $m(x) = 2x$, and $f = 1_{\\Z}$, an extension $g$ would need $2 g(1) = g(m(1)) = f(1) = 1$, which has no solution in $\\Z$.\n",
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