---- > [!definition] Definition. ([[sheaf of sections of a diagram]]) > Let $F$ be a [[diagram]] on a [[poset]] $S$. $F$ induces a [[sheaf]] $\mathcal{F}$ on [[Alexandrov topology|Alexandrov topological space]] $X_{S}$ corresponding to $S$ by taking > $\mathcal{F}(U):=\varprojlim\limits_{s \in U} \ {F(s)}$ > and letting the restriction maps $\mathcal{F}_{V \subset U}$ be naturally determined by the [[universal property]] of the [[categorical limit|limit]]. That is, $\mathcal{F}_{V \subset U}$ is the unique map making the following diagram commute for all incidences $s \leq s'$ in $S$: > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYADEAvgAoAqgEoQY0uky58hFAAZSAJiq1GLNiIlw5CpdjwEiW7bvrNWiEEbgByU4pAYLqomXV2+o4c3LyCwuISAGqmujBQAObwRKAAZgBOEFxImiA4EEjEZiAZWTnU+UhaxaXZiGR5BYg2IAx0AEYwDAAKypZqIOlYCXw4INT2Bk4iAPrAcJwMMACOAARuYvKetUgAzBVNDZ1gUEgAtLvqNZl1+41V1MeniJfXZS8H5XoObJwwAB5YOA4OCrACEq04PH4QgYojEcykkPYcCY7TgMBwqyim2obU6PT6vicWDA2Fg41apOCUDocD48XkFDEQA > \begin{tikzcd} > & \mathcal{F}(U) \arrow[ldd, bend right] \arrow[rdd, bend left] \arrow[d, "\exists ! \mathcal{F}_{U \supset V}" description, dashed] & \\ > & \mathcal{F}(V) \arrow[ld] \arrow[rd] & \\ > F(s) \arrow[rr, "F_{s\leq s'}"'] & & F(s') > \end{tikzcd} > \end{document} > ``` > > > > > Explicitly, in our [[category|categories]] of interest (like $\mathbb{R}\mathsf{Vect}$), $\mathcal{F}(U)$ is given by $\mathcal{F}(U)=\left\{ (x_{s})_{s \in U} \in \prod_{s \in U} F(s) : F_{s \leq s'} x_{s}= x_{s'} \text{ for all } s \leq s' \right\}.$ > and > $\mathcal{F}_{U \supset V}\big( (x_{s})_{s \in U} \big)= (x_{s}) _{s \in V}.$ > That locality and gluing are satisfied is immediate. $\mathcal{F}$ is called the **sheaf of sections** of the [[diagram]] $F$. ---- #### ---- #### 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 > ```