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  "body": "\n## Definition\n\nA *Grothendieck topos* is a category $\\mathcal{E}$ that is [[equivalent|K7A04ZBD92]] to the category $\\mathrm{Sh}(\\mathcal{C}, J)$ of [[sheaves|RT8A8FJ566]] of sets on some small [[site|NYE0Z94MME]] $(\\mathcal{C}, J)$.\n\n## Remarks\n\nExample: for a topological space $X$, the category $\\mathrm{Sh}(X)$ of sheaves on the site of open sets of $X$ is a Grothendieck topos.\nExample: for a small category $\\mathcal{C}$, the category of [[presheaves|AND2WHZC8P]] on $\\mathcal{C}$ is a Grothendieck topos, for the topology in which only maximal sieves cover.\nNon-example: the category of finite sets is an elementary topos but not a Grothendieck topos, since it lacks infinite coproducts, which every Grothendieck topos has.\nEvery Grothendieck topos is an elementary topos and is complete and cocomplete.\nGiraud's theorem gives an intrinsic characterization.\n",
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