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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[additive category|B2VP96NJSK]] and let $(C_\\bullet, d^C)$ and $(D_\\bullet, d^D)$ be [[chain complexes|WQ82C0M3QB]] in $\\mathcal{A}$.\nA *chain map* $f \\colon C_\\bullet \\to D_\\bullet$ is a family of [[morphisms|859F0TFVA3]] $f_n \\colon C_n \\to D_n$, one for each integer $n$, such that\n$$d^D_n \\circ f_n = f_{n-1} \\circ d^C_n \\quad \\text{for all } n \\in \\Z.$$\n\n## Remarks\n\nChain maps are composed degreewise, and the identity chain map has $f_n = 1_{C_n}$.\nWith these, chain complexes in $\\mathcal{A}$ form a category, often written $\\mathrm{Ch}(\\mathcal{A})$.\nA chain map sends $\\ker d^C_n$ into $\\ker d^D_n$ and $\\im d^C_{n+1}$ into $\\im d^D_{n+1}$.\n\nExample: let $C_\\bullet = D_\\bullet$ be the complex with $C_1 = C_0 = \\Z$, $d_1$ multiplication by $3$, and $C_n = 0$ otherwise.\nThen $f_1 = f_0 = $ multiplication by $2$ is a chain map, since $3 \\cdot 2x = 2 \\cdot 3x$ for $x \\in \\Z$.\n\nNon-example: in the same complex, take $f_1 = 1_{\\Z}$ and $f_0 = 0$.\nThen $d_1 \\circ f_1$ is multiplication by $3$ but $f_0 \\circ d_1 = 0$, so the square in degree $1$ does not commute.\n",
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