---- $\mathsf{C}$ is some 'nice enough' [[category]], like $\mathsf{Ab}$ or $\mathsf{Ring}$. > [!definition] Definition. ([[(pre)sheaf stalk]]) > > Let $\mathcal{F}$ be a $\mathsf{C}$-[[presheaf]] on a [[topological space]] $X$ and let $p \in X$. The **stalk of $\mathcal{F}$ at $p$** is the object of $\mathsf{C}$[^1] given by $\mathcal{F}_{p}:=\{ (U, s): U \text{ a nbhd of }p , s \in \mathcal{F}(U) \} / \sim$ > where $\sim$ is the [[equivalence relation]] obtained by declaring > $(U, s) \sim (V,t) \iff \exists \text{nbhd }W \subset U \cap V \text{ s.t. } s |_{W}=t |_{W}.$ > The [[equivalence class|class]] $[(U,s)] \in \mathcal{F}_{p}$ is written as $s_{p}$ and called the **germ of the section $s$** at $p$. Thus, the elements of stalks are germs, which is botanically consistent. > > Note that [[morphism of (pre)sheaves|presheaf morphism]] $f:\mathcal{F} \to \mathcal{G}$ induces morphisms of stalks $f_{p}:\mathcal{F}_{p} \to \mathcal{G}_{p}$ by defining $f_{p}\big( [(U,s)] \big):=[\big(U, f_{U}(s)\big)].$ > > Commonly one drops the brackets and just writes $(U,s)$ when they really mean $[(U,s)]$. Or they write $[U,s]$. > > For $p \in U$, is a natural **germ morphism** $\mathcal{F}(U) \to \mathcal{F}_{p}$ given by $s \in \mathcal{F}(U) \mapsto [U, s] \in \mathcal{F}_{p}$, i.e., $s \mapsto s_{p}$. > [!equivalence] > Equivalently, a stalk $\mathcal{F}_{p}$ is a [[categorical colimit|colimit]] of all the $\mathcal{F}(U)$ over all (i.e., indexed by) all open [[neighborhood|neighborhoods]] $U \ni p$. ^equivalence > [!basicproperties] > **Functoriality.** If $f:\mathcal{F} \to \mathcal{G}$ and $g:\mathcal{G} \to \mathcal{H}$, then $(g \circ f)_{p}=g_{p} \circ f_{p}$ for any $p \in X$. ^properties > [!proof] Proof of Basic Properties. > **Functoriality.** $(g \circ f)_{p}$ is defined by assigning a germ $[U,s]$ in $\mathcal{F}_{p}$ to the germ $[U, (g\circ f)_{U}(s)]$ in $\mathcal{H}_{p}$. The composition of two [[natural transformation|natural transformations]] is *defined* (cf. definition of [[natural transformation|functor category]]) to preserve compositions, so this germ equals $[U, g_{U} \circ f_{U}(s)]$. This is precisely $g_{p} \circ f_{p}([U,s])$. ^proof [^1]: For example, if $\mathsf{C}=\mathsf{Ring}$ then germs are added/multiplied via $\begin{align} [U,s]+ [V, t]&:= [U \cap V, s |_{U \cap V}+ t |_{U \cap V}] \\ [U, s] \cdot [V ,t] &= [U \cap V, (s |_{U \cap V})\cdot(t |_{U \cap V})]. \end{align}$ > [!intuition] > The idea is that sheaves are defined on open sets, but the underlying [[topological space]] $X$ consists of points. How can we isolate the behavior of $\mathcal{F}$ around a point? By considering smaller and smaller [[neighborhood|neighborhoods]] — that is, by taking a limit (of some sort)! ^intuition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```