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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \\colon A \\to B$ and $r \\colon B \\to A$ be [[morphisms|859F0TFVA3]] of $\\mathcal{C}$ with $r \\circ f = 1_A$.\nThen $f$ is a *section* of $r$, and $r$ is a *retraction* of $f$.\nA morphism that has a retraction is a *split monomorphism*.\nA morphism that has a section is a *split epimorphism*.\n\n## Remarks\n\nEvery split monomorphism is a [[monomorphism|Y2NP1JFR86]]: if $f \\circ g_1 = f \\circ g_2$, then $g_1 = r \\circ f \\circ g_1 = r \\circ f \\circ g_2 = g_2$.\nDually, every split epimorphism is an [[epimorphism|RVXDSY04KG]].\nExample: in the category of sets, the inclusion $f \\colon \\set{0} \\to \\set{0, 1}$ and the constant map $r \\colon \\set{0, 1} \\to \\set{0}$ satisfy $r \\circ f = 1_{\\set{0}}$.\nNon-example: in the category of [[abelian groups|082GHNPJ53]], the map $f \\colon \\Z \\to \\Z$, $f(x) = 2x$, is a monomorphism but not a split monomorphism.\nA retraction $r$ would satisfy $r(2) = 1$, so $2 r(1) = 1$ in $\\Z$, which is impossible.\nA morphism is an isomorphism exactly when it is both a split monomorphism and an epimorphism.\nIn the category of sets, the statement that every surjection is a split epimorphism is equivalent to the axiom of choice.\n",
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