Sheaf (mathematics) - Turning A Presheaf Into A Sheaf

Turning A Presheaf Into A Sheaf

It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf F and produces a new sheaf aF called the sheaving, sheafification or sheaf associated to the presheaf F. a is called the sheaving functor, sheafification functor, or associated sheaf functor. There is a natural morphism of presheaves i : F → aF that has the universal property that for any sheaf G and any morphism of presheaves f : F → G, there is a unique morphism of sheaves such that . In fact a is the adjoint functor to the inclusion functor from the category of sheaves to the category of presheaves, and i is the unit of the adjunction.