Presheaf

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

• A set $F(U)$ for every open $U\subseteq X$. It is usually denoted by $\mathrm{\Gamma }(U,F)$. The elements are called sections. And $F(X)$ are called global sections.
• For each inclusion $U\subseteq V$ we have the so called restriction morphism

Often, for $s\in F(V)$ it is denoted $s|U={\text{res}}_{U,V}$(s)

• For every open set $U$, ${\text{res}}_{U,U}=id$
• Given three open sets $U\subseteq V\subseteq W$, ${\text{res}}_{U,V}\circ {\text{res}}_{V,W}={\text{res}}_{U,W}$

They give rise to sheafs.