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  "body": "\n## Summary\n\nA subobject of an object $X$ in a [[category|A2DE66937D]] is an equivalence class of [[monomorphisms|Y2NP1JFR86]] into $X$.\nTwo monomorphisms are equivalent when each factors through the other.\nIt is the categorical version of a subset, a subgroup or a subspace, defined only through morphisms.\nThe subobjects of $X$ are partially ordered by factorisation, and in a topos they are classified by the subobject classifier.\n\n## Explanations\n\n### L1 human\n\nIn set theory a subset of $X$ is a smaller set sitting inside $X$.\nIn a category there is no \"inside\", so one uses an injective-like arrow $A \\to X$ instead, which shows how $A$ sits in $X$.\nTwo such arrows that differ only by renaming the elements of $A$ give the same subobject.\nFor example, among sets, the inclusion $\\{1, 2\\} \\to \\{1, 2, 3\\}$ and the map $\\{a, b\\} \\to \\{1, 2, 3\\}$ with $a \\mapsto 1$, $b \\mapsto 2$ represent the same subobject, namely the subset $\\{1, 2\\}$.\n",
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