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,
     (1) (Uniqueness) Let $U$ be an open subset of $X$, $s \in \mathcal{F}$, $\{ U_i \}_{i \in I}$ a covering of $U$ by open subsets $U_i$. If $s|_{U_i} = 0$ for every $i$, then $s = 0$.
     (2) (Glueing) If $s_i \in \mathcal{F}(U_i)$, $i \in I$ be sections with $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$, then there exists a section $s \in \mathcal{F}(U)$ with $s|_{U_i} = s_i$.

We can define in the same way sheaves of abelian groups, algebras over a fixed ring, and etc. A subsheaf of $\mathcal{F}'$ of $\mathcal{F}$ is given by: $\mathcal{F}'(U)$ is a subring of $\mathcal{F}(U)$ and $p_{UV}'$ induced by $p_{UV}$.

Suppose $s,t$ be two sections such that $s|_{U_i} = t|_{U_i}$ then $(s-t)|_{U_i} = 0$ for all $i$, hence by uniqueness, $s -t = 0 \Rightarrow s = t$. The element in the glueing property is unique by (1).

Example: Let $X$ be a topological space. For any open subset $U$ of $X$, let $\mathcal{C}(U) = C^0(U, \mathbb{R})$ be the set of continuous functions from $U$ to $\mathbb{R}$. Then we clearly have a restriction map, $p_{UV} : \mathcal{F}(U) \rightarrow \mathcal{F}(V)$ by $f \mapsto f|_V$. The other condition follows easily by looking at evaluation at elements.

A base $\mathcal{B}$ of a topology is collection of open sets such that 
     1. $\mathcal{B}$ covers $X$.
     2. If $B_1, B_2$ are two elements in a base, let $B_3 = B_1 \cap B_2$, then for every element $x \in B_3$, there exists an element in a base called $B_4$ such that $x \in B_4 \subset B_3$. 

$\mathcal{B}$-presheaves  are contravariant functor from the (full) subcategory $\mathcal{B}$ of $X$  to the category of rings and $\mathcal{B}$-sheaves are those defined by replacing "open subset $U$ of $X$" by "open set $U$ belong to $\mathcal{B}$". Then we see that $\mathcal{B}$-sheaf $\mathcal{F}_0$ extends in a unique way to a sheaf $\mathcal{F}$ on $X$. This shows that a sheaf is completely determined by its sections over a base of open sets. 

Definition: Let $\mathcal{F}$ is a presheaf on $X$, and let $x \in X$, then the stalk of of $\mathcal{F}$ at $x$ is the ring

$\mathcal{F}_x = \varinjlim_{x \in U} \mathcal{F}(U)$  

Now consider $\mathcal{F}$ be a presheaf on $X$, then for any open subset $U$ of $X$, we consider $\mathcal{F}^\dagger(U)$, which is set of functions $f: U \rightarrow \coprod_{x \in U} \mathcal{F}_x$ such that for every $x \in U$, there exist an open neighborhood $V$ of $x$ and a section $s \in \mathcal{F}(V)$ verifying $f(y) = s_y$ for every $y \in V$. Then $\mathcal{F}^\dagger$ is a sheaf with an universal property:
     For every morphism $\alpha: \mathcal{F} \rightarrow \mathcal{G}$, where $\mathcal{G}$ is a sheaf, there exists a unique morphism $\widetilde{\alpha} : \mathcal{F}^\dagger \rightarrow \mathcal{G}$ such that $\alpha = \widetilde{\alpha} \circ \theta$.

A morphism of presheaves $\alpha: \mathcal{F} \rightarrow \mathcal{G}$ consists of, for every open subset $U$ of $X$, a group homomorphism $\alpha(U): \mathcal{F}(U) \rightarrow \mathcal{G}(U)$ which is compatible with the restrictions $p_{UV}$.