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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[additive category|B2VP96NJSK]] and let $f \\colon C_\\bullet \\to D_\\bullet$ be a [[chain map|HTK9WQZJ1G]] of [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$, with differentials $d^C$ and $d^D$.\nThe *mapping cone* of $f$ is the chain complex $\\mathrm{cone}(f)$ with\n$$\\mathrm{cone}(f)_n = C_{n-1} \\oplus D_n,$$\nthe [[biproduct|BZEJKTQ76T]], and with differential $\\mathrm{cone}(f)_n \\to \\mathrm{cone}(f)_{n-1}$ given in matrix form by\n$$d(c, y) = \\left(-d^C_{n-1}(c),\\ f_{n-1}(c) + d^D_n(y)\\right).$$\n\n## Remarks\n\nThe condition $d^2 = 0$ holds: $d^2(c, y) = \\left(d^C d^C c,\\ -f d^C c + d^D f c + d^D d^D y\\right) = (0, 0)$, because $f$ is a chain map.\nThe inclusion $D_\\bullet \\to \\mathrm{cone}(f)$ and the projection $\\mathrm{cone}(f) \\to C[-1]_\\bullet$, where $C[-1]_n = C_{n-1}$ with differential $-d^C$, form a degreewise split short exact sequence of complexes.\nIn an abelian category, its long exact homology sequence has connecting maps equal to $H_{n-1}(f)$ up to sign.\nSo $f$ is a [[quasi-isomorphism|C9JEC2WQKV]] if and only if $\\mathrm{cone}(f)$ has zero homology.\nSign conventions differ between texts; they give isomorphic complexes.\n\nExample: let $C_\\bullet = D_\\bullet = \\Z$ in degree $0$ and let $f_0$ be multiplication by $2$.\nThen $\\mathrm{cone}(f)_1 = \\Z$, $\\mathrm{cone}(f)_0 = \\Z$, and the differential is $c \\mapsto 2c$, so $H_0(\\mathrm{cone}(f)) = \\Z/2\\Z \\ne 0$ and $f$ is not a quasi-isomorphism.\n\nNon-example: dropping the sign, $d(c, y) = (d^C c, f c + d^D y)$, does not give a complex in general.\nFor $C_\\bullet = D_\\bullet$ with $C_1 = C_0 = \\Z$, $d_1 = 1_{\\Z}$, and $f$ the identity, the element $(c, 0)$ in degree $2$ goes to $(c, c)$ and then to $(0, 2c) \\ne 0$.\n",
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