---- > [!definition] Definition. ([[subsheaf]]) > A **subsheaf** $\mathcal{F}'$ of a [[sheaf]] $\mathcal{F}$, denoted $\mathcal{F}' \subset \mathcal{F}$, is a [[subpresheaf]] of $\mathcal{F}$ which is also a [[sheaf]] (i.e., satisfies locality and gluing). > Any [[subpresheaf]] can be made into a [[subsheaf]] via [[sheafification]]. > A [[morphism of (pre)sheaves|sheaf morphism]] $\mathcal{F} \xrightarrow{f} \mathcal{G}$ can be **restricted to the subsheaf $\mathcal{F}'$ of $\mathcal{F}$**, by defining $f |_{\mathcal{F}'}:\mathcal{F}' \to \mathcal{G}$ as the composite $\mathcal{F}' \hookrightarrow \mathcal{F} \to \mathcal{G}.$ ![[restriction sheaf#^warning]] ---- #### ---- #### 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 > ```