---- > [!definition] Definition. ([[pullback sheaf]]) > Let $X,Y$ be [[topological space|topological spaces]] and $f:X \to Y$ a [[continuous]] map between them. Let $\mathcal{G}$ be a [[presheaf|(pre)]][[sheaf]] on $Y$. > Define the **pullback presheaf** on $X$ to be the [[presheaf]] > $U \mapsto \{ (V,s) : V \supset f(U), V \text{ open}, s \in \mathcal{G}(V) \} / {\sim}$ [[equivalence relation|where]] $(V,s) \sim (V', s')$ iff there exists $W$ with $f(U) \subset W \subset V \cap V'$ such that $s |_{W}=s' |_{W}$. > Even if $\mathcal{F}$ and $\mathcal{G}$ are earnest [[sheaf|sheaves]], this [[presheaf]] may not be. We define the **pullback sheaf** $f ^{-1}\mathcal{G}$ on $X$ to be the [[sheaf]] [[sheafification|associated to]] it. > $f ^{-1}$ also transfers [[morphism of (pre)sheaves|morphisms]] $\varphi:\mathcal{F} \to \mathcal{G}$ of [[sheaf|sheaves]] over $Y$ to to morphisms $f^{-1}\varphi: f ^{-1} \mathcal{F} \to f ^{-1}\mathcal{G}$ of sheaves over $X$ by taking ($U \subset X$ open) $(f ^{-1}\varphi)_{U}([V, s]):=[V, \varphi_{U}(s)]$.[^1] Thus, $f^{-1}$ is a [[covariant functor|functor]] $\mathsf{Sh}(Y) \to \mathsf{Sh}(X)$, called the **pullback functor for sheaves**.[^2] > [!basicproperties] > (Stalks) For $p \in X$, $(f ^{-1} \mathcal{G})_{p}=\mathcal{G}_{f(p)}$. (Maybe since [[right-adjoint functors commute with limits|left-adjoint functors preserve colimits]] and [[sheaf pullback and pushforward are adjoint functors]]? ) ^properties > [!intuition] > Maybe part of the idea is that, in analogy with the [[pushforward sheaf]], we want to define $(f ^{-1}\mathcal{G} )(U)$ in terms of $f(U)$. But $f(U)$ need not be open, so we are left to look for sections defined on open sets containing $f(U)$, where we focus their on their behavior around $f(U)$ via this (co)limit-esque construction identifying two such sections iff they agree on a neighborhood of the set $f(U)$. ^intuition > [!specialization] > [[(pre)sheaf stalk|Stalks]] are special cases of pullbacks. If $f:\{ p \} \to Y$, then the pullback definition says $(f ^{-1} \mathcal{G})(\{ p \})=\frac{\{ (V,s): V \ni p , V \text{ open}, s \in \mathcal{G}(V)\}}{\big( (V,s) \sim (V', s') \big) \iff \ex p \in W \subset V \cap V' \text{ s.t. } s |_{W}=s |_{W'}}$ which is precisely the definition of $\mathcal{G}_{p}$.[^1] > >Slightly more generally, [[restriction sheaf|restriction sheaves]] arise as pullbacks. > ^specialization ---- #### [^1]: Note that here we have identified [[sheaf]] on the one-point space $X$ with the group $\mathcal{F}(X)$. Note also that $\mathcal{F}(\emptyset)=0$ always. [^2]: Strictly speaking, need to verify functoriality ---- #### 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 > ```