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> [!definition] Definition. ([[sheaf of sections of a map]])
> Let $\mu:Y \to X$ be a [[continuous|continuous map]]. The "[[section|sections]] of $\mu
quot; form a [[sheaf]] of sets. More precisely, the [[presheaf]] obtained by assigning to each open $U \subset X$ the set of [[continuous|continuous maps]] $s: U \to Y$ satisfying $\mu \circ s=\id |_{U}$ is in fact a [[sheaf]]. This construction is called the **sheaf of sections of $\mu$**.
^definition
> [!justification]
>We have to show this [[presheaf]] (call it $\mathcal{F}$) is indeed a [[sheaf]].
>
**Locality.** This is immediate, since sections are set-functions.
>
**Gluing**. Let $U \subset X$ be an arbitrary open set, and let $\{ U_{i} \}_{i \in I}$ be an [[cover|open cover]] of $U$. Suppose there is a family of sections $s_{i} \in \mathcal{F}(U_{i})$ agreeing on pairwise overlap: $s_{i} |_{U_{i} \cap U_{j}}=s_{j} |_{U_{i} \cap U_{j}}$ for all $i,j \in I$. [[the pasting lemma|The gluing lemma]] gives a [[well-defined]] [[continuous]] map $s:U \to Y$ defined as $s(x):=s_{i}(x)$ if $x \in U_{i} \subset U$. $s$ is indeed a [[section]] of $\mu$ because each $s_{i}$ is.
^justification
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####
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#### 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
> ```