---- > [!definition] Definition. ([[presheaf image]]) > Let $X$ be a [[topological space]]. Let $f:\mathcal{F} \to \mathcal{G}$ be a [[morphism of (pre)sheaves]] on $X$. The **presheaf image**, $\text{im }f$, is specified by $(\text{im }f)(U):=\text{im }f_{U}.$ > It is a [[subpresheaf]] of $\mathcal{G}$. ^definition > [!basicnonexample] Warning. > In general, $\im f$ need not be a [[sheaf]] even if $\mathcal{F}$ and $\mathcal{G}$ are. We need to [[sheafification|sheafify]]: see [[sheaf image]]. ^nonexample > [!justification] > We need to show that $\im f$ is a [[subpresheaf]]. The [[covariant functor|functor's]] definition on $U$ has been provided, its definition on morphisms is simply $\begin{align} (\im f)_{U \supset V}: (\im f)(U) &\to (\im f)(V) \\ f_{U}(s) & \mapsto f_{U}(s) |_{V} \end{align}$ which is well-defined (in particular, the codomain is valid) because [[natural transformation|naturality]] of $f$ gives $f_{U}(s) |_{V}=f_{V}(s |_{V}) \in \im f_{V}$. Functoriality is evidently seen. ^justification ---- #### ---- #### 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 > ```