Every presheaf \(\mcf\) admits a morphism \(\theta : \mcf \to \mcf^+\) to a sheaf, universal among morphisms from \(\mcf\) to sheaves. The map \(\theta\) is an isomorphism on every stalk.
The construction is the reason the stalk statement is free: \(\mcf^+(U)\) is the set of functions \(s: U \to \coprod_{p \in U} \mcf_p\) with \(s(p) \in \mcf_p\) that are locally represented by sections of \(\mcf\). Nothing is added to a stalk, because a stalk is already a local object; what is added is the ability to assemble, and what is removed are the sections that were locally zero.
Sheafification is left adjoint to the inclusion of sheaves into presheaves, so it preserves colimits and the inclusion preserves limits. That is the abstract reason kernels of sheaf morphisms need no sheafification and cokernels do.
Dependencies
D-RCCFYPresheaves, sheaves, and the sheaf axiomsuses
Backlinks
D-VJFAPThe espace étalé, and sheafification as continuous sectionsrelated-toPR-IP6ZGDirect limits of sheaves, and the Noetherian hypothesis that removes the sheafificationusesPR-C9ZEKExactness is checked on stalks, and global sections are only left exactusesD-A7LCTKernel, image and cokernel sheaves, and which need sheafifyinguses