sheafification

Sheafification is a construction in sheaf theory that, given a presheaf F on a topological space X, produces a sheaf F+ together with a canonical morphism F → F+. The resulting sheaf is called the associated sheaf or the sheafification of F. The purpose is to enforce the locality and gluing axioms: sections that are locally compatible should glue to global sections.

One standard viewpoint uses the étalé space of a presheaf. Form the étalé space E(F) whose fiber at x ∈ X is the stalk F_x. The topology is chosen so that a local section over an open set U corresponds to an element of F+(U). The sheafification F+(U) can then be described as the set of continuous sections s: U → E(F) with the property that p ∘ s = id_U. This identifies F+ with the sheaf of sections of E(F). Equivalently, F+ is the smallest sheaf that contains the image of F in a suitable sense, obtained by gluing compatible local sections.

The construction satisfies a universal property: for any sheaf G and any morphism of presheaves φ: F → G, there exists a unique morphism of sheaves φ+: F+ → G such that φ = φ+ ∘ η, where η: F → F+ is the canonical map. Consequently, F+ is the initial way to extend F to a sheaf. A basic consequence is that (F+)x ≅ Fx for every x ∈ X, i.e., the stalks of the associated sheaf coincide with the stalks of the original presheaf. If F is already a sheaf, F+ ≅ F.

Example: the constant presheaf with value A assigns to every nonempty open set U the set A. Its sheafification is the constant sheaf of locally constant A-valued functions; on connected X this recovers the constant sheaf with value A.

Sheafification plays a central role in algebraic geometry and topology by turning arbitrary presheaves into sheaves while preserving local data and enabling gluing.