Sheafification

Summary

Sheafification turns a presheaf $P$ on a site into the closest sheaf $aP$. It comes with a map $P \to aP$ through which every map from $P$ to a sheaf factors uniquely. It is a functor that is left adjoint to the inclusion of sheaves into presheaves, and it preserves finite limits. It is how one builds sheaves such as cokernels, quotients and constant sheaves, which are not sheaves when computed objectwise. On a small site it can be computed by applying the plus construction twice.