---- > [!definition] Definition. ([[presheaf]]) > The ability to 'locally restrict data to smaller open subsets'[^1] on a [[topological space]] gives rise to the concept of a **presheaf**. > > Let $X$ be a [[topological space]]. Let $\mathsf{D}$ be some "data [[category]]", such as $\mathsf{Set}$ or $\mathsf{Ab}$. > > A **presheaf $\mathcal{F}$ on $X$** is a [[contravariant functor]] from the [[poset]] of open sets $(\tau_{X}, \subset)$ to $\mathsf{D}$. That is to say, > 1. For each open $U \subset X$, there is an object $\mathcal{F}(U)$ of $\mathsf{D}$. This set is also denoted $\Gamma(U,\mathcal{F})$. Its elements are called **sections of $\mathcal{F}$**. The sections of $\mathcal{F}$ over $X \subset X$ are known as the **global sections of $\mathcal{F}$**. > 2. For each inclusion of open sets $V \subset U$, there is a morphism $\mathcal{F}(U) \xrightarrow{} \mathcal{F}(V)$, variously denoted $\text{res}^{U}_{V}$, $\mathcal{F}_{UV}$, $\mathcal{F}_{U \supset V}$, called a **restriction morphism**. Given a section $s \in \mathcal{F}(V)=\Gamma(V,\mathcal{F})$, the notation $s |_{V}$ is sometimes used to denote $\mathcal{F}_{UV}(s) \in \mathcal{F}(V)$. > 3. The restriction morphisms are (contravariant-) functorial, meaning that $\mathcal{F}_{U \subset U}=1_{\mathcal{F}(U)}$ and given three opens $W \subset V \subset U$ one has $\mathcal{F}_{VW} \circ \mathcal{F}_{UV}=\mathcal{F}_{UW}$. > Presheaves on $X$ and [[natural transformation|natural transformations]] ([[morphism of (pre)sheaves|presheaf morphisms]]) between them form a ([[natural transformation|functor]]) [[category]] $\mathsf{pShv}_{\mathsf{D}}(X)=[\tau_{X}^{\text{op}},\mathsf{D}]$.[^1] Definition taken from Wikipedia (mostly). ---- #### [^1]: $\tau_{X}$ denotes the [[poset]] of open sets in $X$. ---- #### 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 > ```