Sheaves

Definition 1.1. (Presheaf)

Let $X$ be a topological space, where we denote $\mathfrak{Top}(X)$ be the category with:

$\text{Ob}(\mathfrak{Top}(X)) = \{\text{open sets of }X\}$.
$\text{Mor}_{\mathfrak{Top}(X)}(U,V) = \begin{cases} \text{inclusion} &,U \subset V\\ \varnothing &, \text{otherwise}\end{cases}$

A presheaf of abelien group on $X$ is a contravariant functor $\mathscr{F} \colon \mathfrak{Top}(X) \to \mathfrak{Ab}$, where $\mathfrak{Ab}$ is the category of abelien groups. Precisely, that is:

$\mathscr{F}$ assigns an abelien group $\mathscr{F}(U)$ to every open set $U \in \mathfrak{Top}(X)$.
$\mathscr{F}(\varnothing) = \text{Trivial group}$.
For each $U \subset V$, $\mathscr{F}$ assigns a group homomorphism $\rho_{VU} \colon \mathscr{F}(V) \to \mathscr{F}(U)$.
$\rho_{UU} = \text{Id}_{U}$.
For each $U \subset V \subset W$, we have $\rho_{WU} = \rho_{WV} \circ \rho_{VU}$

We called $\mathscr{F}(U) = \Gamma(U, \mathscr{F})$ the sections of the presheaf $\mathscr{F}$ over $U$.
We called $\rho_{VU}$ restriction maps, usually for $s \in \mathscr{F}(V)$, we denote $s|_{U} = \rho_{VU}(s)$.

Definition 1.2. (Sheaf)

Let $U \in \mathfrak{Top}(X)$ and $\{V_\alpha\}_{\alpha \in A}$ is an open covering of $U$. If $s \in \mathscr{F}(U)$ is an element such that $s|_{V_{\alpha}} = 0$ for all $\alpha \in A$, then $s = 0$.
Let $U \in \mathfrak{Top}(X)$ and $\{V_\alpha\}_{\alpha \in A}$ is an open covering of $U$. If for all $\alpha \in A$, there exist $s_{\alpha} \in \mathscr{F}(V_\alpha)$ such that for each $\alpha, \beta \in A$, $s_\alpha|_{V_{\alpha} \cap V_{\beta}} = s_\beta|_{V_{\alpha}\cap V_{\beta}}$, then there is an element $s \in \mathscr{F}(U)$ such that $s|_{V_\alpha} = s_{\alpha}$ for each $\alpha \in A$. ($s$ is unique by condition 1.)

Example 1.3.

(Constant Sheaf) Let $A$ be an abelien group with discrete topology, we define the constant sheaf $\mathscr{A}$ on $X$ determined by $A$ be: $$\mathscr{A}(U) = \begin{cases} \{0\} &, U = \varnothing \\ \{\text{continuous maps from } U \text{ to } A\} &, \text{otherwise} \end{cases}$$, in particular, if $U$ is a nonempty connected open set, then $\mathscr{A}(U) \simeq A$ as abelien groups.

Definition 1.4. (Stalk)

Let $p \in X$ a topological space and $\mathscr{F}$ a presheaf on $X$, we define the stalk of $\mathscr{F}$ at $P$ be: \begin{align*}\mathscr{F}_P &= \varinjlim_{U \ni P}\mathscr{F}(U) \\ &= \left\{\langle U, s\rangle \mid U: \text{ ngbh of } P, s \in \mathscr{F}(U)\right\}/\sim\end{align*} $\sim$ is the relation that $\langle U, s \rangle \sim \langle V, t\rangle$ if there exist as open neigbourhood of $P$, $W \subset U \cap V$ such that $s|_W = t|_W$.

Definition 1.5. (Morphism of presheaves)

Let $\mathscr{F}$ and $\mathscr{G}$ be presheaves on $X$. A morphism $\varphi \colon \mathscr{F} \to \mathscr{G}$ consists morphisms of abelien groups $\varphi(U) \colon \mathscr{F}(U) \to \mathscr{G}(U)$ for each $U \in \mathfrak{Top}(X)$. Moreover, if $U \subset V$, then the morphisms should make the following diagram commute: $$\begin{array}{ccc} \mathscr{F}(V) & \overset{\rho_{VU}^{\mathscr{F}}}{\longrightarrow} & \mathscr{F}(U) \\ \bigg\downarrow \varphi(V)& & \bigg\downarrow \varphi(U)\\ \mathscr{G}(V) & \overset{\rho_{VU}^{\mathscr{G}}}{\longrightarrow} & \mathscr{G}(U) \\ \end{array}$$

Proposition 1.6. (Local nature of morphism)

Let $\varphi \colon \mathscr{F} \to \mathscr{G}$ be morphism of presheaves on $X$, then $\varphi$ induces a morphism $\varphi_P \colon \mathscr{F}_P \to \mathscr{G}_P$ on stalks, for every $P \in X$.
Let $\varphi \colon \mathscr{F} \to \mathscr{G}$ be morphism of sheaves on $X$, then $\varphi$ is an isomorphism $\Longleftrightarrow \varphi_P$ is an isomorphism for all $P \in X$.

Proof.

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