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  "body": "\n## Summary\n\nA chain complex is a sequence of objects, such as abelian groups or modules, joined by morphisms $d_n \\colon C_n \\to C_{n-1}$ with $d_{n-1} \\circ d_n = 0$.\nThe condition says that the image of each map lies in the kernel of the next, so the [[homology|2J4JJJYP22]] $\\ker d_n / \\im d_{n+1}$ is defined.\nChain complexes are the central objects of [[homological algebra|SAHAEZNCVD]].\nThey arise from topological spaces as in [[singular homology|M0Q5CFJ7ZH]], and from modules through resolutions.\n\n## Explanations\n\n### L1 human\n\nA chain complex is a row of objects with maps going down one step at a time, such that two steps in a row always give zero.\nIn topology, $C_n$ is built from $n$-dimensional pieces of a space and $d_n$ takes the boundary of a piece.\nThe boundary of a boundary is empty, which is the rule $d \\circ d = 0$.\nA small algebraic example is $\\cdots \\to \\Z/4\\Z \\xrightarrow{2} \\Z/4\\Z \\xrightarrow{2} \\Z/4\\Z \\to \\cdots$, where doing \"multiply by $2$\" twice multiplies by $4$, which is $0$ in $\\Z/4\\Z$.\n",
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