Sheaves (from Algebraic Geometry by Hartshorne, Section 2.1)

Let's start the study of the second chapter of book Algebraic Geometry by Robin Hartshorne. Knowledge of the first chapter is not required.

Let X be a topological space. A presheaf F of sets on X consists of the following data:

• For each open set U⊆X, there exists a set F(U). This set is also denoted Γ(U,F). The elements in this set are called the sections of F over U. The sections of F over X are called the global sections of F.
• For each inclusion of open sets V⊆U, a function resVU:F(U)→F(V). In view of many of the examples below, the morphisms resVU are called restriction morphisms. If s∈F(U), then its restriction resVU(s) is often denoted s|V by analogy with restriction of functions.

The restriction morphisms are required to satisfy two additional (functorial) properties:

• For every open set U of X, the restriction morphism resUU:F(U)→F(U) is the identity morphism on F(U).
• If we have three open sets W⊆V⊆U, then the composite resWV∘resVU=resWU.

Informally, the second axiom says it does not matter whether we restrict to W in one step or restrict first to V, then to W.