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  "body": "\n## Summary\n\nA Lawvere–Tierney topology on a [[topos|6837RS7N82]] is a morphism $j \\colon \\Omega \\to \\Omega$ from the [[subobject classifier|NKP5TPQ79X]] to itself.\nIt behaves like a modality \"it is locally true that\".\nIt is the elementary counterpart of a [[Grothendieck topology|GDRFSJ8E2B]]: on a presheaf topos, Lawvere–Tierney topologies correspond to Grothendieck topologies on the base category.\nEach such $j$ determines a full subcategory of $j$-sheaves, which is again a topos.\nThe double negation topology $\\neg\\neg$ is a standard example and is used in forcing arguments.\n\n## Explanations\n\n### L1 human\n\nIn a topos, truth values may be partial, like \"true on this part of a space\".\nA Lawvere–Tierney topology is a rule that makes some statements count as true once they hold \"locally enough\".\nApplying the rule twice gives nothing new, true statements stay true, and the rule respects \"and\".\nFor example, take presheaves on the open sets of a space, with the topology given by open covers.\nThen a property counts as true on $U$ as soon as it holds on the pieces of a cover of $U$.\n",
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