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  "body": "\n## Summary\n\nA chain homotopy between two [[chain maps|HTK9WQZJ1G]] $f, g \\colon C_\\bullet \\to D_\\bullet$ is a family of maps $h_n \\colon C_n \\to D_{n+1}$ with $f - g = dh + hd$.\nChain homotopic maps induce the same maps on [[homology|2J4JJJYP22]].\nThe notion is the algebraic shadow of a [[homotopy|4A7H5AG0NF]] between continuous maps, and it is used to define the [[homotopy category of chain complexes|9Y5Y92R4D5]].\n\n## Explanations\n\n### L1 human\n\nTwo chain maps can look different in each degree yet have the same effect on homology.\nA chain homotopy is a certificate of this: a family of maps one degree up whose \"boundary\" is the difference $f - g$.\nIn topology, a homotopy between two maps of spaces sweeps out a cylinder, and taking chains of that cylinder gives exactly such maps $h_n$.\nFor example, on the complex $0 \\to \\Z \\xrightarrow{1} \\Z \\to 0$ the identity is chain homotopic to zero, which matches the fact that this complex has zero homology.\n",
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