Presheaf
The following post are series of definitions in order to build to the concept of schemes which became a central idea in algebraic geometry. The theory of sheaves provides a way to collect local data to deduce global information . Topological space $X$ can be viewed as a category with open sets to be objects with $Hom_X(U,V) = \left \{ \begin{array}{ll} \{ i_{UV} \} & \text{ if } U \subset V \\ \varnothing & \text{ otherwise } \end{array} \right .$ where $i_{UV}$ indicates the inclusion map from $U$ to $V$. Definition : Let $X$ be a topological space, then a presheaf $\mathcal{F}$ (of rings) on $X$ is a contravariant functor from $X$ to the category of rings. We denote the unique restriction map $p_{UV} : \mathcal{F}(U) \rightarrow \mathcal{F}(V)$ when $V \subset U$. An element $s \in \mathcal{F}(U)$ is called a section of $\mathcal{F}$ over $U$, and $s|_V$ be $p_{UV}(s) \in \mathcal{F}(V)$ called the restriction of $s$ to $V$. If, in addition, the following holds,