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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-1-8\">例 1.8</a> · 典范映射与纤维\n>\n> 若 $R$ 是 $X$ 上的等价关系，<u class=\"concept-term\" data-concept-identity=\"urn:uuid:aa2120c3-751f-4f90-944f-26ec8d4c046f\" data-concept-route=\"/concept-identities#concept-aa2120c3-751f-4f90-944f-26ec8d4c046f\" id=\"first-86a675776f33545ca770-zh\">典范映射(canonical mapping)</u> $\\varphi:X\\to X/R$ 定义为 $\\varphi(x)=[x]_R$，它是满射。\n>\n> 反过来，任意映射 $\\varphi:X\\to Y$ 都定义等价关系 $aRb\\Longleftrightarrow\\varphi(a)=\\varphi(b)$。等价类恰好是非空<u class=\"concept-term\" data-concept-identity=\"urn:uuid:850fd218-36a7-4024-b096-39443088c1ec\" data-concept-route=\"/concept-identities#concept-850fd218-36a7-4024-b096-39443088c1ec\" id=\"first-81061e294ac10574c590-zh\">纤维(fiber)</u>：\n>\n> $$\n> X/R=\\{\\varphi^{-1}(\\{y\\})\\mid y\\in\\varphi(X)\\}.\n> $$\n>\n> 映射 $\\psi:\\varphi(X)\\to X/R$，$y\\mapsto\\varphi^{-1}(\\{y\\})$ 是双射。若两个非空纤维相等，从中取 $x$ 就得到 $y=\\varphi(x)=y'$。\n>\n> **编校：**原文把 $\\psi$ 的定义域写为整个 $Y$；必须限制到像 $\\varphi(X)$，否则空纤维不构成等价类，也不能保证单射。",
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