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  "body": "\n## Summary\n\nThe mapping cone of a [[chain map|HTK9WQZJ1G]] $f \\colon C_\\bullet \\to D_\\bullet$ is a [[chain complex|WQ82C0M3QB]] $\\mathrm{cone}(f)$.\nIt is built from $D$ and a shifted copy of $C$, with a differential that uses $f$.\nIt measures the failure of $f$ to be a [[quasi-isomorphism|C9JEC2WQKV]]: $f$ is one exactly when the cone has zero homology.\nThe cone fits into a short exact sequence of complexes, and in the [[homotopy category|9Y5Y92R4D5]] and the [[derived category|5ST7NVJFSM]] it gives the [[distinguished triangles|ADKK56VZHW]].\n\n## Explanations\n\n### L1 human\n\nThe cone glues the source of $f$, moved up one degree, onto the target of $f$.\nIn topology, the mapping cone of a map of spaces $X \\to Y$ is formed by attaching the cone on $X$ to $Y$, and its chains are this algebraic cone.\nThe homology of the cone sits in a long exact sequence between the homologies of $C$ and $D$, with the maps $H_n(f)$ as connecting maps.\nSo the cone is acyclic exactly when $f$ induces isomorphisms on homology.\nFor example, the cone of multiplication by $2$ on $\\Z$ in degree $0$ is $0 \\to \\Z \\xrightarrow{2} \\Z \\to 0$, whose homology $\\Z/2\\Z$ is the cokernel of $f$.\n",
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