sheaf

A sheaf on a topological space X is a tool for keeping track of local data that can be consistently glued together to form global data. More precisely, for a chosen target category C (such as sets, abelian groups, or rings), a sheaf F assigns to every open subset U ⊆ X an object F(U) in C and to every inclusion V ⊆ U a restriction morphism res^U_V: F(U) → F(V). These assignments are compatible with compositions and identities.

A presheaf satisfies these structures but may fail the gluing and locality conditions. A sheaf satisfies two extra axioms: locality says that if a section s ∈ F(U) restricts to equal sections on every member of an open cover {U_i} of U, then s is determined by those restrictions; gluing says that given sections s_i ∈ F(U_i) that agree on overlaps U_i ∩ U_j, there exists a unique s ∈ F(U) restricting to each s_i. These axioms encode the principle that local data uniquely determine global data.

Stalks capture the behavior of a sheaf at a point x ∈ X. The stalk F_x is the colimit of the system F(U) over all open neighborhoods U of x, intuitively collecting all germs of sections around x.

Examples include the sheaf of real-valued continuous functions, C(U, R), on X, and the locally constant sheaf with value A, whose sections are locally constant functions. The constant presheaf with value A becomes the constant (or constant sheaf) A_X only after sheafification, the process that converts presheaves into actual sheaves. Sheaf theory also studies sheaf morphisms, sheaf cohomology (derived functors of global sections), and the use of sheaves to encode geometric and topological data.