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  "body": "\n## Definition\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]] of $A$ and $C$, with inclusion $i_A \\colon A \\to A \\oplus C$ and projection $p_C \\colon A \\oplus C \\to C$.\nThe sequence is *split* if 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\nEquivalently, there is a morphism $t \\colon B \\to A$ with $t \\circ f = 1_A$, or there is a morphism $u \\colon C \\to B$ with $g \\circ u = 1_C$.\nEvery additive functor sends a split exact sequence to a split exact sequence.\n\nExample: $0 \\to \\Z \\xrightarrow{i} \\Z \\oplus \\Z/2\\Z \\xrightarrow{p} \\Z/2\\Z \\to 0$, with $i(n) = (n, 0)$ and $p(n, a) = a$, is split.\nTake $h$ to be the identity of $\\Z \\oplus \\Z/2\\Z$; then $h \\circ i = i_A$ and $p_C \\circ h = p$.\n\nNon-example: $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2\\Z \\to 0$ is short exact but not split.\nA split sequence would give an isomorphism $\\Z \\cong \\Z \\oplus \\Z/2\\Z$, but $\\Z$ has no nonzero element of finite order.\n",
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