---- > [!definition] Definition. ([[pushforward sheaf]]) > Let $X,Y$ be [[topological space|topological spaces]] and $f:X\to Y$ a [[continuous]] map between them. Let $\mathcal{F}$ be a [[presheaf|(pre)]][[sheaf]] on $X$. The **pushforward (pre)sheaf** $f_{*}\mathcal{F}$ is the [[presheaf|(pre)]][[sheaf]] on $Y$ given by $(f_{*}\mathcal{F})(U):=\mathcal{F}\big( f ^{-1}(U) \big)$ with restriction maps defined in the obvious way: $(f_{*}\mathcal{F})_{U \supset V}:=\mathcal{F}_{f ^{-1}(U) \supset f^{-1}(V)}$. > If $\mathcal{F}$ is an earnest [[sheaf]], then so is $f_{*}\mathcal{F}$. > $f_{*}$ also transfers [[morphism of (pre)sheaves|morphisms]] $\varphi:\mathcal{F} \to \mathcal{G}$ of [[sheaf|sheaves]] over $X$ to morphisms $f_{*}\varphi:f_{*}\mathcal{F} \to f_{*}\mathcal{G}$ of [[sheaf|sheaves]] over $Y$, by taking ($U \subset Y$ open) $(f_{*}\varphi)_{U}:=\varphi_{f ^{-1}(U)}$. Thus, $f_{*}$ yields a [[covariant functor|functor]] $\mathsf{Sh}(X) \to \mathsf{Sh}(Y)$, called the **pushforward functor for sheaves**.[^1] ---- #### > [!justification] > Need to verify that $f_{*}\mathcal{F}$ is in fact a [[sheaf]] when $\mathcal{F}$ is. This is immediate. > Locality: if $\{ U_{i} \}_{i \in I}$ is an [[cover|open cover]] of $U$, and $s \in \mathcal{f}_{*}\mathcal{F}(U)=\mathcal{F}\big( f ^{-1}(U) \big)$ satisfies $s |_{U_{i}}=0$ for all $i \in I$, then $s=0$ as an element of $\mathcal{F}\big( f ^{-1}(U)\big)$ by the locality condition satisfied by $\mathcal{F}$. > Gluing: If $\{ U_{i} \}_{i \in I}$ is an [[cover|open cover]] of $U$ and we are given $s_{i} \in f_{*}\mathcal{F}(U_{i})$ for each $i$ satisfying $s_{i} |_{U_{i} \cap U_{j}}=s_{j} |_{U_{i} \cap U_{j}}$ for all $i,j\in I$, then the $s_{i}$ patch into some $s \in \mathcal{F}\big( f ^{-1}(U) \big)$ by the gluing condition satisfied by $\mathcal{F}$. ^justification [^1]: 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 > ```