---- > [!definition] Definition. ([[sheaf]]) > A **sheaf** is a [[presheaf]] $\mathcal{F}$ whose sections are uniquely determined by their restrictions, in the following sense: > Suppose $U$ is an open set and let $\{ U_{i} \}_{i \in I}$ be an [[cover|open cover]] of $U$[^3] by subsets $U_{i} \subset U$, i.e., $U=\bigcup_{i}U_{i}$. > **1. (Locality)** If $s$, $t \in \mathcal{F}(U)$ satisfy $s |_{U_{i}}=t |_{U_{i}}$[^4] for all $i \in I$, then in fact $s=t$.[^1] > **2. (Gluing)** If a family of sections $\{ s_{i} \in \mathcal{F}(U_{i}) \}_{i}$ has pairwise agreement on all overlap of their domains[^5], then they 'patch together': there is a section $s \in \mathcal{F}(U)$ such that $s |_{U_{i}}=s_{i}$ for all $i \in I$. Such a section is variously called the **gluing**, **concatenation**, or **collation** of the sections $s_{\alpha}$. By **(1)**, it is unique.[^2] > > Note that $\mathcal{F}(\emptyset)$ is always the [[terminal object|final object]] of the data [[category]]. > > $\mathsf{D}$-valued sheaves on $X$ form a full subcategory of $\mathsf{pShv}_\mathsf{D}(X)$, denoted $\mathsf{Shv}_\mathsf{D}(X)$. ^definition [^1]: This is a generalization of the idea that, for functions $U \to \mathbb{R}$, local agreement implies global agreement. [^2]: This is a generalization of the idea for gluing functions in [[the gluing lemma]]. > [!equivalence] > > The locality and gluing conditions may be stated together as asserting that the sequence $0 \to \mathcal{F}(U) \xrightarrow{\alpha} \bigoplus_{i \in I}\mathcal{F}(U_{i}) \overset{\beta_1}{\underset{\beta_2}{\rightrightarrows}} \bigoplus_{i,j \in I}\mathcal{F}(U_{i} \cap U_{j})$ > > > is [[exact sequence|exact]] for all $U \subset X$ open and open covers $\{ U_{i} \}_{i \in I}$ of $U$, where we have defined > - $\alpha(s):=(s |_{U_{i}})_{i \in I}$ > - $\beta_{1}\big( (s_{i})_{i \in I} \big):=( s_{i} |_{U_{i} \cap U_{j}} )_{i,j \in I}$ > - $\beta_{2}\big( (s_{i})_{i \in I} \big):=(s_{j} |_{U_{i} \cap U_{j}})_{i,j \in I}$ > > 'Exactness', in this case, means > 1. $\alpha$ is [[injection|injective]] (this is precisely the **locality** condition) > 2. $\beta_{1} \circ \alpha=\beta_{2} \circ \alpha$ (this automatically holds) > 3. For any $(s_{i})_{i \in I} \in \bigoplus_{i \in I}\mathcal{F}(U_{i})$, with $\beta_{1}\big( (s_{i})_{i \in I} \big)=\beta_{2}\big( (s_{i})_{i \in I} \big)$, there exists $s \in \mathcal{F}(U)$ with $\alpha(s)=(s_{i})_{i \in I}$ (this is the **gluing** condition.) > > [[category|Categorical remark]]: $\alpha$ is the [[equalizer]] of $\beta_{1}, \beta_{2}$. > > (If we have one arrow that is $\beta_{1}-\beta_{2}$ and assuming the data category is something like $R$-$\mathsf{Mod}$, maybe that is the same thing but fits in more with exactness as $R$-$\mathsf{Mod}$.) ---- #### [^3]: Note that each $U_{\alpha} \subset U$ in light of our definition of [[cover|open cover]]. [^4]: Recall that the notation $s |_{V}$ means $\mathcal{F}_{UV}(s)$, the image of $s$ under the restriction map $\mathcal{F}(U) \to \mathcal{F}(V)$. [^5]: That is, if $U_{i} \cap U_{i'} \neq \emptyset$ then $s_{i} |_{U_{i} \cap U_{i'}}=s_{i'} |_{U_{i} \cap U_{i'}}$ for all indices $i,i'$. ---- #### 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 > ```