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  "body": "\n## Summary\n\nA Grothendieck topos is a category equivalent to the category of [[sheaves|RT8A8FJ566]] of sets on a small [[site|NYE0Z94MME]].\nGrothendieck introduced these categories as generalized spaces.\nA topological space is studied through its category of sheaves, and many sites that are not spaces still give such categories.\nEvery Grothendieck topos is an elementary [[topos|6837RS7N82]], and [[Giraud's theorem|57BXX32WF0]] characterizes Grothendieck toposes by intrinsic properties.\nDifferent sites can give equivalent toposes, so the topos is the invariant object.\n\n## Explanations\n\n### L1 human\n\nA topological space can be studied through the sheaves on it, rather than through its points.\nA Grothendieck topos is any category that looks like \"all sheaves on some space\", where the space may be a generalized one given by a site.\nThe category of sets is the topos of sheaves on a one-point space.\nFor a group $G$, the category of sets with a $G$-action is the topos of presheaves on $G$ viewed as a one-object category, so it is also a Grothendieck topos.\n",
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