sheaves

On a topological space X, a sheaf F assigns to each open set U ⊆ X a set F(U) (often an abelian group, ring, or module) and to each inclusion V ⊆ U a restriction map ρ^U_V: F(U) → F(V). The assignment is compatible with identities and composition of inclusions.

For every open U and every open cover {U_i} of U, two axioms must hold: locality and gluing. Locality says that if s, t ∈ F(U) have s|_{U_i} = t|_{U_i} for all i, then s = t. Gluing says that if sections s_i ∈ F(U_i) agree on overlaps U_i ∩ U_j, there exists a unique s ∈ F(U) with s|_{U_i} = s_i. A presheaf satisfying these axioms is a sheaf.

Examples include the constant sheaf associated to a set A, represented by locally constant maps U → A; the sheaf of continuous real-valued functions C^0(U); the sheaf of holomorphic functions on a complex manifold; and the structure sheaf O_X in algebraic geometry, which assigns to U the ring of regular functions. More generally, F can be a sheaf of abelian groups, rings, or modules, such as the sheaf of sections of a fiber bundle.

The stalk F_x at a point x captures the local data near x and is defined as the colimit of F(U) over neighborhoods U of x. Morphisms of presheaves and the process of sheafification convert presheaves into sheaves. Sheaves provide a flexible language for gluing local data and form the foundation of sheaf cohomology and related constructions.

In practice, sheaves play a central role in algebraic topology and algebraic geometry, where they encode local-to-global principles and support advanced tools like derived functors and cohomology theories.