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  "body": "\n## Definition\n\nLet $\\mathbb{T}$ be a geometric theory.\nThis is a first-order theory whose axioms are sequents between formulas built from finite conjunctions, arbitrary disjunctions and existential quantifiers.\nFor a [[Grothendieck topos|S3YVE24AQK]] $\\mathcal{E}$, let $\\mathbb{T}\\text{-}\\mathrm{Mod}(\\mathcal{E})$ be the category of models of $\\mathbb{T}$ in $\\mathcal{E}$.\nFor Grothendieck toposes $\\mathcal{E}$ and $\\mathcal{F}$, let $\\mathbf{Geom}(\\mathcal{E}, \\mathcal{F})$ be the category of [[geometric morphisms|N5R1WAEPFB]] $\\mathcal{E} \\to \\mathcal{F}$, and write $f^*$ for the inverse image part of $f$.\nA *classifying topos* of $\\mathbb{T}$ is a Grothendieck topos $\\mathbf{Set}[\\mathbb{T}]$ together with a model $U_{\\mathbb{T}}$ of $\\mathbb{T}$ in it with the following property.\nFor every Grothendieck topos $\\mathcal{E}$ the functor\n$$\\mathbf{Geom}(\\mathcal{E}, \\mathbf{Set}[\\mathbb{T}]) \\to \\mathbb{T}\\text{-}\\mathrm{Mod}(\\mathcal{E}), \\qquad f \\mapsto f^*(U_{\\mathbb{T}})$$\nis an [[equivalence of categories|K7A04ZBD92]].\nThe model $U_{\\mathbb{T}}$ is called the generic model of $\\mathbb{T}$.\n\n## Remarks\n\nExample: let $\\mathbb{O}$ be the theory with one sort, no symbols and no axioms, so that a model of $\\mathbb{O}$ in $\\mathcal{E}$ is just an object of $\\mathcal{E}$.\nThe functor category $[\\mathbf{FinSet}, \\mathbf{Set}]$, with the inclusion functor as generic model, is a classifying topos for $\\mathbb{O}$; it is called the object classifier.\nNon-example: $\\mathbf{Set}$ is not a classifying topos for $\\mathbb{O}$.\nFor every Grothendieck topos $\\mathcal{E}$ there is, up to isomorphism, only one geometric morphism $\\mathcal{E} \\to \\mathbf{Set}$, while $\\mathcal{E}$ has in general many non-isomorphic objects.\nEvery geometric theory has a classifying topos, and it is unique up to equivalence.\nFor a discrete group $G$, the topos of $G$-sets classifies $G$-torsors.\n",
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