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  "body": "\n## Summary\n\nA classifying topos of a geometric theory $\\mathbb{T}$ is a [[Grothendieck topos|S3YVE24AQK]] that contains a generic model of $\\mathbb{T}$.\nModels of $\\mathbb{T}$ in any Grothendieck topos $\\mathcal{E}$ correspond to [[geometric morphisms|N5R1WAEPFB]] from $\\mathcal{E}$ to the classifying topos.\nEvery geometric theory has a classifying topos, unique up to equivalence.\nThis turns questions about theories into questions about toposes, and lets one read a topos as a theory.\n\n## Explanations\n\n### L1 human\n\nA classifying topos is a universe that contains one generic model of a theory.\nEvery model of the theory in another topos is obtained by pulling back this generic model along a map of toposes.\nIt plays the role that a classifying space plays for vector bundles in topology.\nFor example, the topos of functors from finite sets to sets classifies \"objects\": a map into it from a topos $\\mathcal{E}$ is the same as one object of $\\mathcal{E}$.\n",
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