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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $0 \\to A \\xrightarrow{f} B \\xrightarrow{g} C \\to 0$ be a [[short exact sequence|KJKNJMGKZZ]] in $\\mathcal{A}$.\nLet $A \\oplus C$ be the [[biproduct|BZEJKTQ76T]], with inclusion $i_A$ and projection $p_C$.\nThe following are equivalent:\n\n1. there is a morphism $t \\colon B \\to A$ with $t \\circ f = 1_A$;\n2. there is a morphism $u \\colon C \\to B$ with $g \\circ u = 1_C$;\n3. the sequence is [[split|1ZPMDE4ARK]]: there is an isomorphism $h \\colon B \\to A \\oplus C$ with $h \\circ f = i_A$ and $p_C \\circ h = g$.\n\n## Remarks\n\nFrom (3), the morphisms $t = p_A \\circ h$ and $u = h^{-1} \\circ i_C$ give (1) and (2).\nFrom (1), the morphism $h = (t, g) \\colon B \\to A \\oplus C$ satisfies $h \\circ f = i_A$ and $p_C \\circ h = g$, and it is an isomorphism by the [[five lemma|W6AVEAW2WR]] applied to the map from the given sequence to $0 \\to A \\to A \\oplus C \\to C \\to 0$.\nFrom (2), one argues dually, using $B \\cong A \\oplus C$ through $f$ and $u$.\nExample: in $0 \\to \\Z/2\\Z \\to \\Z/4\\Z \\to \\Z/2\\Z \\to 0$, the quotient map has no right inverse.\nA right inverse would send the generator of $\\Z/2\\Z$ to an element of $\\Z/4\\Z$ of order at most $2$, that is, to $0$ or $2$, and both map to $0$.\nSo the sequence does not split, in agreement with $\\Z/4\\Z \\not\\cong \\Z/2\\Z \\oplus \\Z/2\\Z$.\nThe lemma fails for non-abelian groups: for the symmetric group, $1 \\to A_3 \\to S_3 \\to \\Z/2\\Z \\to 1$ has a right inverse of the quotient map, but $S_3$ is not isomorphic to $A_3 \\times \\Z/2\\Z$.\n\n## Explanations\n\n### L1 human\n\nA short exact sequence builds $B$ from a subobject $A$ and a quotient $C$.\nThe lemma says that $B$ falls apart as $A \\oplus C$ as soon as one can undo either one of the two maps.\nA way to project $B$ back onto $A$, or a way to lift $C$ back into $B$, is already enough.\n\n### L2 human\n\nThe lemma is the standard test for splitting.\nFor example, every short exact sequence ending in a projective object splits, since the identity of $C$ lifts along $g$.\nDually, every short exact sequence starting with an injective object splits.\nFor vector spaces every short exact sequence splits, because a basis of $C$ can be lifted to $B$.\nIn the theory of extensions, the split sequence is the zero element of $\\mathrm{Ext}^1(C, A)$.\nFor groups, condition (2) alone gives a semidirect product, and condition (1) gives a direct product.\n",
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