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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $\\operatorname{Ch}(\\mathcal{A})$ be the category of [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$ and [[chain maps|HTK9WQZJ1G]].\nThe *derived category* of $\\mathcal{A}$ is the [[localization|DNFR3YP0XF]] of $\\operatorname{Ch}(\\mathcal{A})$ at the class of [[quasi-isomorphisms|C9JEC2WQKV]].\nThat is, it is a category $D(\\mathcal{A})$ with a functor $Q \\colon \\operatorname{Ch}(\\mathcal{A}) \\to D(\\mathcal{A})$ such that:\n\n1. $Q(f)$ is an isomorphism for every quasi-isomorphism $f$;\n2. every functor $F \\colon \\operatorname{Ch}(\\mathcal{A}) \\to \\mathcal{C}$ that sends quasi-isomorphisms to isomorphisms factors as $F = G \\circ Q$ for a unique functor $G \\colon D(\\mathcal{A}) \\to \\mathcal{C}$.\n\n## Remarks\n\nThe localization can be computed in two steps: first pass to the [[homotopy category|9Y5Y92R4D5]] $K(\\mathcal{A})$, then invert quasi-isomorphisms there by a calculus of fractions.\nA morphism $X \\to Y$ in $D(\\mathcal{A})$ is then represented by a roof $X \\xleftarrow{s} X' \\to Y$ with $s$ a quasi-isomorphism.\nFor general $\\mathcal{A}$ the morphisms between two objects might not form a set.\nThey do form a set for module categories, and for bounded below complexes when $\\mathcal{A}$ has enough injectives.\nExample: over $\\Z$, let $C$ be $\\Z \\xrightarrow{\\cdot 2} \\Z$ in degrees $1$ and $0$, and let $\\Z/2\\Z$ sit in degree $0$.\nThe quotient map $C_0 = \\Z \\to \\Z/2\\Z$ is a chain map, and it is a quasi-isomorphism since $H_0(C) \\cong \\Z/2\\Z$ and $H_1(C) = 0$; so $C \\cong \\Z/2\\Z$ in $D(\\mathcal{A})$.\nNon-example: $C$ and $\\Z/2\\Z$ are not isomorphic in $K(\\mathcal{A})$, since every chain map $\\Z/2\\Z \\to C$ is zero, because $\\Hom(\\Z/2\\Z, \\Z) = 0$; so $K(\\mathcal{A})$ is not the derived category.\nThe bounded variants $D^b$, $D^+$ and $D^-$ are defined in the same way from bounded complexes.\n",
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