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  "body": "\n## Definition\n\nA *Grothendieck category* is an [[abelian category|Y9CZ3MTNF1]] $\\mathcal{A}$ such that:\n\n1. $\\mathcal{A}$ has all small [[colimits|BJS1CGTKQ5]];\n2. filtered colimits are exact: for every small [[filtered category|975PMY4FG8]] $I$, the functor $\\operatorname{colim} \\colon \\mathcal{A}^I \\to \\mathcal{A}$ is [[exact|S1G61TDW0K]];\n3. $\\mathcal{A}$ has a [[generator|CK08JW32WB]]: an object $G$ such that for any two distinct morphisms $f \\ne g \\colon X \\to Y$ there is a morphism $h \\colon G \\to X$ with $f \\circ h \\ne g \\circ h$.\n\n## Remarks\n\nClauses 1 and 2 together form Grothendieck's axiom AB5.\nEvery Grothendieck category has enough [[injective objects|5QRFTBF4PG]], and it also has all small limits.\n\nExample: for a ring $R$, the category of left [[$R$-modules|0NZ8JA25WK]] is a Grothendieck category.\nIt has all colimits, and a filtered colimit of short exact sequences of modules is exact.\nThe module $R$ is a generator: if $f \\ne g \\colon X \\to Y$, pick $x$ with $f(x) \\ne g(x)$ and let $h \\colon R \\to X$ be $r \\mapsto r x$.\n\nNon-example: the opposite of the category of [[abelian groups|082GHNPJ53]] is abelian, has all small colimits, and has the generator $\\Q/\\Z$, but it fails clause 2.\nFiltered colimits in it are cofiltered limits of abelian groups, and these are not exact.\nFor a prime $p$, take the inverse system of short exact sequences $0 \\to p^n \\Z \\to \\Z \\to \\Z / p^n \\Z \\to 0$ for $n \\ge 1$.\nIts limit is $0 \\to 0 \\to \\Z \\to \\Z_p \\to 0$, where $\\Z_p$ is the group of $p$-adic integers, and the map $\\Z \\to \\Z_p$ is not surjective.\n",
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