a presheaf and sheaf

Imagine you want to keep track of some kind of local data attached to a space — say, the continuous functions defined on each open region of it. To every open set U you assign a collection F(U) of such data, and whenever a smaller open set V sits inside a bigger one U you have a way to restrict the data on U down to V (you just look at the function on the smaller region). That bookkeeping rule, an assignment of data to open sets together with restriction maps that compose correctly, is a presheaf. It is purely formal: nothing forces the data to behave the way local-versus-global geometry actually behaves.

A sheaf is a presheaf that obeys the two rules that genuinely local data must obey. Suppose an open set U is covered by smaller open sets U_i. Then: (1) Locality (separation) — if two sections s, t in F(U) restrict to the same thing on every U_i, they were already equal on U; you cannot have two distinct global objects that look identical everywhere locally. (2) Gluing — if you are given a section s_i on each U_i, and these agree on every overlap U_i intersect U_j, then there is a section s on all of U restricting to each s_i. Together these say: a section over U is exactly the same thing as a compatible family of local sections. Formally a presheaf F on a topological space X is a contravariant functor from the open sets of X to Sets (or Ab, or Rings), and it is a sheaf if for every open cover the sequence F(U) -> product of F(U_i) =>= product of F(U_i intersect U_j) is an equalizer.

The distinction matters because most natural assignments of local data ARE sheaves — continuous functions, smooth functions, holomorphic functions, sections of a bundle — and the sheaf axioms are exactly what let you reconstruct global objects from local ones, which is the whole engine of modern geometry. But not everything is a sheaf: the constant presheaf sending every nonempty U to a fixed group A is famously NOT a sheaf, because on a disconnected open set you can choose different constants on different pieces, and these locally-compatible choices have no single global value. The honest cure for a presheaf that fails the axioms is sheafification, which forces it to become the closest sheaf to it.