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  "body": "\n## Summary\n\nSheafification turns a [[presheaf|AND2WHZC8P]] $P$ on a [[site|NYE0Z94MME]] into the closest [[sheaf|RT8A8FJ566]] $aP$.\nIt comes with a map $P \\to aP$ through which every map from $P$ to a sheaf factors uniquely.\nIt is a functor that is [[left adjoint|TZHF6ZM92S]] to the inclusion of sheaves into presheaves, and it preserves finite limits.\nIt is how one builds sheaves such as cokernels, quotients and constant sheaves, which are not sheaves when computed objectwise.\nOn a small site it can be computed by applying the plus construction twice.\n\n## Explanations\n\n### L1 human\n\nSome natural constructions give data that is defined locally but does not glue, so it is only a presheaf.\nSheafification repairs this: it forces local data that agree on overlaps to glue, and it identifies data that agree locally.\nFor example, on a space with two connected components, the constant presheaf with value $\\Z$ has only constant functions, while its sheafification allows a different integer on each component.\n",
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