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  "body": "\n## Summary\n\nAn epimorphism is a [[morphism|859F0TFVA3]] $f$ that can be cancelled on the right: $g_1 \\circ f = g_2 \\circ f$ implies $g_1 = g_2$.\nIt is the categorical version of a [[surjective function|01RVCMSN5N]], and the dual notion of a [[monomorphism|Y2NP1JFR86]].\nIn the categories of sets and of groups, the epimorphisms are exactly the surjective homomorphisms.\nIn other categories an epimorphism need not be surjective: the inclusion of the integers into the rationals is an epimorphism of rings.\n\n## Explanations\n\n### L1 human\n\nA surjective map reaches every point of its target.\nAn epimorphism is defined without points: two maps out of the target that agree after $f$ must already be equal.\nIf $f$ missed a point $b$, two maps that differ only at $b$ would agree after $f$; so for sets an epimorphism must be surjective.\nFor rings the situation is different.\nA ring homomorphism out of $\\Q$ is fixed by its values on $\\Z$, because $1/n$ must go to the inverse of the image of $n$.\nSo $\\Z \\to \\Q$ is an epimorphism of rings although it is far from surjective.\n",
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