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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[additive category|B2VP96NJSK]], let $(C_\\bullet, d^C)$ and $(D_\\bullet, d^D)$ be [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$, and let $f, g \\colon C_\\bullet \\to D_\\bullet$ be [[chain maps|HTK9WQZJ1G]].\nA *chain homotopy* from $f$ to $g$ is a family of morphisms $h_n \\colon C_n \\to D_{n+1}$, one for each integer $n$, such that\n$$f_n - g_n = d^D_{n+1} \\circ h_n + h_{n-1} \\circ d^C_n \\quad \\text{for all } n \\in \\Z.$$\nThe maps $f$ and $g$ are *chain homotopic*, written $f \\simeq g$, if such a family exists.\n\n## Remarks\n\nChain homotopy is an equivalence relation compatible with composition.\nA chain map $f$ is a *chain homotopy equivalence* if there is a chain map $g \\colon D_\\bullet \\to C_\\bullet$ with $g \\circ f \\simeq 1_C$ and $f \\circ g \\simeq 1_D$.\nA complex is *contractible* if its identity is chain homotopic to the zero map.\nIn an abelian category, chain homotopic maps induce the same morphisms on [[homology|2J4JJJYP22]]: a cycle $z$ is sent by $f - g$ to $d^D h(z)$, which is a boundary.\n\nExample: let $C_\\bullet = D_\\bullet$ have $C_1 = C_0 = \\Z$, $d_1 = 1_{\\Z}$, and $C_n = 0$ otherwise; let $f$ be the identity and $g = 0$.\nTake $h_0 = 1_{\\Z} \\colon C_0 \\to D_1$ and all other $h_n = 0$.\nIn degree $0$, $d_1 h_0 + h_{-1} d_0 = 1 + 0 = 1 = f_0 - g_0$; in degree $1$, $d_2 h_1 + h_0 d_1 = 0 + 1 = 1 = f_1 - g_1$.\n\nNon-example: let $C_\\bullet = D_\\bullet$ be $\\Z$ in degree $0$ and $0$ elsewhere.\nIn degree $0$, the map $h_0$ lands in $D_1 = 0$ and $d^C_0$ lands in $C_{-1} = 0$, so $d_1 h_0 + h_{-1} d_0 = 0$, while $f_0 - g_0 = 1$.\nSo the identity is not chain homotopic to $0$; indeed it induces the identity of $H_0 = \\Z$.\n",
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