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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] with finite limits, and let $1$ be its [[terminal object|EK85VSSRMR]].\nA *subobject classifier* is an object $\\Omega$ of $\\mathcal{C}$ with a morphism $\\top \\colon 1 \\to \\Omega$ with the following property.\nFor every [[monomorphism|Y2NP1JFR86]] $m \\colon S \\to X$, there is exactly one morphism $\\chi_m \\colon X \\to \\Omega$ for which the square\n$$\\begin{array}{ccc} S \u0026 \\longrightarrow \u0026 1 \\\\ {\\scriptstyle m}\\downarrow \u0026 \u0026 \\downarrow{\\scriptstyle \\top} \\\\ X \u0026 \\xrightarrow{\\chi_m} \u0026 \\Omega \\end{array}$$\nis a [[pullback|Q05K04QJ71]].\nThe morphism $\\chi_m$ is the *characteristic morphism* of $m$.\n\n## Remarks\n\nEquivalently, when $\\mathcal{C}$ is locally small, there is a natural bijection between the [[subobjects|FS1FKAE89H]] of $X$ and $\\Hom(X, \\Omega)$, and $\\Omega$ represents the subobject functor.\nExample: in $\\mathbf{Set}$, take $\\Omega = \\set{0, 1}$ and $\\top(*) = 1$; for $S \\subseteq X$, the unique $\\chi$ is the characteristic function, since the pullback of $\\set{1}$ along $\\chi$ is $\\chi^{-1}(1) = S$.\nNon-example: the category of [[groups|G5S1M9]] has no subobject classifier.\nIts terminal object is the trivial group, so every pullback of $\\top \\colon 1 \\to \\Omega$ is the kernel of a homomorphism, which is a normal subgroup.\nA subgroup of order $2$ in $S_3$ is not normal, so it is not classified.\nIn a presheaf category $\\mathbf{Set}^{\\mathcal{C}^{\\mathrm{op}}}$, the object $\\Omega(c)$ is the set of [[sieves|CNCXNWKC6R]] on $c$, and $\\top$ picks the maximal sieves.\n",
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