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  "body": "\n## Summary\n\nA quasi-isomorphism is a [[chain map|HTK9WQZJ1G]] that induces an isomorphism on [[homology|2J4JJJYP22]] in every degree.\nEvery chain homotopy equivalence is a quasi-isomorphism, but not conversely.\nThe [[derived category|5ST7NVJFSM]] is obtained by formally inverting the quasi-isomorphisms, so that complexes with \"the same homology in a coherent way\" become isomorphic.\n\n## Explanations\n\n### L1 human\n\nHomology is the information one usually wants from a chain complex.\nA quasi-isomorphism is a map between two complexes that matches their homology exactly, even if the complexes themselves look very different.\nFor example, the complex $0 \\to \\Z \\xrightarrow{2} \\Z \\to 0$ and the single group $\\Z/2\\Z$ both have homology $\\Z/2\\Z$ in degree $0$ and nothing else.\nThe reduction map from the first to the second is a quasi-isomorphism.\nThis is how a resolution replaces a module by a nicer complex with the same homology.\n",
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