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  "body": "\n## Definition\n\nAn *elementary topos* is a [[category|A2DE66937D]] $\\mathcal{E}$ such that:\n\n1. $\\mathcal{E}$ has all finite limits (it is [[finitely complete|PYSBNWCHCF]]);\n2. $\\mathcal{E}$ is [[Cartesian closed|BFBNQ36QS8]]: for all objects $A$ and $B$ there is an [[exponential object|RHWD8DY41R]] $B^A$;\n3. $\\mathcal{E}$ has a [[subobject classifier|NKP5TPQ79X]] $\\top \\colon 1 \\to \\Omega$.\n\n## Remarks\n\nAn equivalent definition asks for finite limits and power objects: for each object $A$, an object $PA$ with a natural bijection between subobjects of $B \\times A$ and morphisms $B \\to PA$; then $PA = \\Omega^A$.\nEvery topos also has finite colimits, and its subobjects of any object form a Heyting algebra.\nExample: $\\mathbf{Set}$ is a topos: it has finite limits, the exponential $B^A$ is the set of functions $A \\to B$, and $\\set{0, 1}$ classifies subsets.\nExample: for a group $G$, the category of $G$-sets is a topos with $\\Omega = \\set{0, 1}$ and trivial action.\nNon-example: the category of real vector spaces fails clause 3.\nIts terminal object is $0$, so the pullback of $\\top$ along $\\chi \\colon X \\to \\Omega$ is $\\ker \\chi$.\nIf $\\chi \\colon \\R \\to \\Omega$ classifies the zero subspace, then $\\chi$ is injective, so $2\\chi \\ne \\chi$, yet $\\ker(2\\chi) = 0$ too, and uniqueness fails.\nNon-example: the category $\\mathbf{Top}$ of topological spaces has finite limits but is not Cartesian closed, so it fails clause 2.\n",
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