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  "body": "\n## Summary\n\nThe derived category $D(\\mathcal{A})$ of an [[abelian category|Y9CZ3MTNF1]] $\\mathcal{A}$ is obtained from the category of [[chain complexes|WQ82C0M3QB]] by declaring every [[quasi-isomorphism|C9JEC2WQKV]] to be an isomorphism.\nIn it, an object and any of its resolutions become isomorphic, so derived functors become ordinary functors between derived categories.\nIt is a [[triangulated category|T2V7GJJ2EN]], and $\\operatorname{Ext}^n(X, Y)$ is the group of morphisms from $X$ to $Y[n]$.\nDerived categories are the standard setting for duality theorems, for sheaf theory and for modern algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nIn homological algebra one replaces an object by a resolution, and many different resolutions carry the same information.\nThe derived category builds this freedom into the category itself: two complexes count as the same object when a map between them induces isomorphisms on homology.\nFor example, the complex $\\Z \\xrightarrow{2} \\Z$ and the group $\\Z/2\\Z$ become isomorphic.\nThis makes the machinery of derived functors much simpler to state and compose.\n",
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