---- > [!definition] Definition. ([[restriction sheaf]]) Let $\iota: Z \hookrightarrow X$ be an [[inclusion map|inclusion]] of [[topological space|topological spaces]]. Let $\mathcal{F}$ be a [[sheaf]] on $X$. The [[pullback sheaf]] $\iota ^{-1} \mathcal{F}$ is called the **restriction of $\mathcal{F}$ to $Z$** and denoted $\mathcal{F}|_{Z}$. > If $\iota: U \hookrightarrow X$ is an [[inclusion map|inclusion]] of an *open subset*, then $\mathcal{F}|_{U}$ is just the [[sheaf]] $V \mapsto \mathcal{F}(V)$ for $V \subset U$ open. > Note that, because $\iota ^{-1}$ is a [[covariant functor|functor]] $\mathsf{Sh}(X) \to \mathsf{Sh}(Z)$, we also have the notion of **restriction of a sheaf morphism to a subset $Z \subset X$**. Similar to above, if $Z=U$ is open then this behaves how one would expect. > [!basicnonexample] Warning. A [[morphism of (pre)sheaves|sheaf morphism]] $\mathcal{F} \xrightarrow{f} \mathcal{G}$ can be 'restricted' in two different ways: >1. To a *subset* $Z \subset X$, by pulling back along the [[inclusion map|inclusion of spaces]]: $f |_{Z}:= (\iota ^{-1}f: \mathcal{F} |_{Z} \to \mathcal{G}_{Z}$). In the nicest case where $Z=U$ is open in $X$, we may say *this construction restricts the space, but not the (remnant) sections*. >2. To a *subsheaf* $\mathcal{F}' \subset \mathcal{F}$, by restricting along the [[subsheaf|inclusion of]] *[[sheaf|sheaves]]*: $f |_{\mathcal{F'}}: \mathcal{F}' \hookrightarrow \mathcal{F} \to \mathcal{G}$. That means *this construction restricts the sections, but not the space*. ^warning > [!specialization] > [[(pre)sheaf stalk|Stalks]] arise as the special case $Z=\{ p \}$. Notationally, $\mathcal{F}_{p}= \mathcal{F |_{\{ p \}}}$. ^specialization ---- #### ---- #### 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 > ```