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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[additive category|B2VP96NJSK]].\nThe *homotopy category of chain complexes* $K(\\mathcal{A})$ is the [[category|A2DE66937D]] defined as follows.\n\n1. Its objects are the [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$.\n2. For complexes $C$ and $D$, the morphisms $C \\to D$ are the [[chain maps|HTK9WQZJ1G]] $C \\to D$ modulo [[chain homotopy|4EBHAWJJ8C]]. Here $f \\sim g$ if there are morphisms $s_n \\colon C_n \\to D_{n+1}$ with $f_n - g_n = d_{n+1} s_n + s_{n-1} d_n$ for all $n$.\n3. Composition is induced by the composition of chain maps.\n\n## Remarks\n\nChain homotopy is an equivalence relation compatible with sums and composition, so $K(\\mathcal{A})$ is again additive and composition is well defined.\nExample: in $K$ of abelian groups, the complex $C = (0 \\to \\Z \\xrightarrow{\\id} \\Z \\to 0)$, in degrees $1$ and $0$, is isomorphic to $0$, since $s_0 = \\id \\colon C_0 \\to C_1$ gives $\\id_C = d s + s d$.\nNon-example: the complex $\\cdots \\to \\Z/4\\Z \\xrightarrow{\\cdot 2} \\Z/4\\Z \\xrightarrow{\\cdot 2} \\Z/4\\Z \\to \\cdots$ is exact, but it is not isomorphic to $0$ in $K$: a contraction would split each short exact sequence $0 \\to 2\\Z/4\\Z \\to \\Z/4\\Z \\to 2\\Z/4\\Z \\to 0$, and $\\Z/4\\Z \\not\\cong \\Z/2\\Z \\oplus \\Z/2\\Z$.\nIf $\\mathcal{A}$ is abelian, homology is a functor on $K(\\mathcal{A})$.\nBounded variants $K^b(\\mathcal{A})$, $K^+(\\mathcal{A})$ and $K^-(\\mathcal{A})$ use only complexes that are zero in all but finitely many degrees, in low degrees, or in high degrees.\n",
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