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  "body": "\n## Summary\n\nA monomorphism is a [[morphism|859F0TFVA3]] $f$ that can be cancelled on the left: $f \\circ g_1 = f \\circ g_2$ implies $g_1 = g_2$.\nIt is the categorical version of an [[injective function|09NWETVTNY]].\nIn the categories of sets, groups, rings and vector spaces, the monomorphisms are exactly the injective homomorphisms.\nIts dual notion is the [[epimorphism|RVXDSY04KG]].\n\n## Explanations\n\n### L1 human\n\nAn injective function never sends two different points to the same point.\nA category has no points, so a monomorphism is defined by looking at maps into the domain instead.\nThe arrow $f$ is a monomorphism if two maps $g_1$, $g_2$ that look the same after $f$ were already the same.\nFor sets, test with maps from a one-point set: they pick out points $a_1$, $a_2$, and $f(a_1) = f(a_2)$ must force $a_1 = a_2$.\nSo in the category of sets, monomorphisms are the injective functions.\n",
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