---- > [!definition] Definition. ([[sheafification]]) > Let $X$ be a [[topological space]], $\mathcal{F}$ a [[presheaf]] on $X$. The **sheafification of $\mathcal{F}$** is a new [[sheaf]] $\mathcal{F}^{+}$, together with a [[morphism of (pre)sheaves|morphism of (pre)sheaves]] $\theta:\mathcal{F} \to \mathcal{F}^{+}$, satisfying the [[universal property]] that any [[morphism of (pre)sheaves|morphism]] $\mathcal{F} \xrightarrow{\varphi}\mathcal{G}$, $\mathcal{G}$ a [[sheaf]], factors uniquely through this new [[sheaf]] $\mathcal{F}^{+}$ > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYADEAviDGl0mXPkIoAjOSq1GLNpx78hDYAHEJUmdjwEiZRavrNWiDt16Dh4gHoBqSaphQA5vCJQADMAJwguJDIQHAgkZTVbTXZ6ELQ+LElpEFDwyOoYpAAmahsNe05+GBw6EGoGOgAjGAYABVkzBRAQrF8+HEzgsIjEYujYxHiGLDA7ECg6OD4fWoSyhxgADyw4HDgAAgBCPc4IGhgQqZngThS0rAk6xua203k2bt7+sQoxIA > \begin{tikzcd} > \mathcal{F} \arrow[r, "\varphi"] \arrow[d, "\theta"'] & \mathcal{G} \\ > \mathcal{F}^+ \arrow[ru, "\exists ! \overline{\varphi}"', dashed] & > \end{tikzcd} > \end{document} > ``` > > > this defines $\mathcal{F}^{+}$ [[terminal objects are unique up to a unique isomorphism|up to]] [[isomorphism]], if it exists. Indeed, exist it does: $\mathcal{F}^{+}$ may be obtained as $\mathcal{F}^{+}(U)=\left\{ s:U \to \coprod_{p \in U} \mathcal{F}_{p} : \begin{align} > &\textcolor{Thistle}{(1) \ s(p) \in \mathcal{F}_{p} \text{ for all }p} \\ > & \textcolor{Skyblue}{(2) \ \forall p \in U, \exists \text{nbhd } p \in V \subset U \text{ and }} \\ > & \quad \quad \textcolor{Skyblue}{ t \in \mathcal{F}(V) \text{ s.t. } s(q)=[V,t] \ \forall q \in V} > \end{align} \right\}.$ > with the [[natural transformation]] $\theta:\mathcal{F} \to \mathcal{F}^{+}$ specified via $U$-component $\begin{align} > \theta_{U}:\mathcal{F}(U) &\to \mathcal{F}^{+}(U) \\ > s & \mapsto (p \mapsto [U,s]). > \end{align}$ > > [!equivalence] > Another [[sheaf]] satisfying the [[universal property]] is ^equivalence > [!basicproperties] Important Properties. > - If $\mathcal{F}$ is already a [[sheaf]], then $\mathcal{F}^{+}$ is [[isomorphism|isomorphic]] to it. Thus, there is always a way to think of a sheaf's sections as 'functions'. > - Sheafification preserves stalks: $(\mathcal{F}^{+})_{p} \cong_{\theta_{p}} \mathcal{F_{p}}$ for $p \in X$. > - A [[morphism of (pre)sheaves|morphism of presheaves]] $\mathcal{F} \xrightarrow{f}\mathcal{G}$ induces a morphism $\mathcal{F}^{+} \xrightarrow{f^{+}}\mathcal{G}^{+}$ preserving [[(pre)sheaf stalk|stalk maps]]: $(f^{+})_{p}=f_{p}$ > - $f^{+}$ is [[injective sheaf morphism|injective]] when $f$ is. Indeed, [[sheaf morphism injectivity and surjectivity can be tested on stalks|injectivity]] of $f^{+}$ can be tested on [[(pre)sheaf stalk|stalks]] $f_{p}^{+}=f_{p}$, and $f_{p}$ is injective when $f$ is for [[sheaf morphism injectivity and surjectivity can be tested on stalks|any]] morphism of presheaves. Similarly, $f^{+}$ is [[surjective sheaf morphism|surjective]] when $f$ is. ^properties - [ ] as a functor (adjunction with inclusion) ---- #### **Stalk preservation.** **Induced morphism.** Define $\begin{align} f^{+}_{U}: \mathcal{F}^{+}(U) &\to \mathcal{G}^{+}(U) \\ s & \mapsto \big( U \ni p \mapsto f_{p}(s(p)) \big). \end{align}$ ![[CleanShot 2025-02-04 at 22.05.54@2x 1.jpg]] This produces elements in $\mathcal{G}^{+}(U)$ because if $s$ is locally given on $V$ as $[V,t]=t_{p}$ for some $t\in\mathcal{F}(V)$, then $f_{U}^{+}(s)$ is locally given on $V$ by $[V, f_{V}(t)]$ . The map is clearly compatible with restriction, yielding a [[natural transformation]] $f^{+}:\mathcal{F}^{+} \to \mathcal{G}^{+}$ as required. We next show that $\mathcal{F}^{+}_{p} \cong_{\theta_{p}}\mathcal{F_{p}}$, by claiming the map $\begin{align} \psi_{p}: \mathcal{F}_{p}^{+} &\to \mathcal{F}_{p} \\ [U, h:U \to \coprod_{p \in U} \mathcal{F_{p}}] & \mapsto h(p) \end{align}$ to be an inverse to $\theta_{p}$, which recall is given by $\begin{align} \theta_{p}:\mathcal{F}_{p} &\to \mathcal{F}_{p}^{+} \\ [U, s]& \mapsto [U, y \mapsto s_{y}]. \end{align}$ Indeed, $\begin{align} \psi_{p} \circ \theta_{p}([U,s]) &= \psi_{p}([U, y \mapsto s_{y}]) \\ &= s_{p} =[U, s] \end{align}$ while $\begin{align} \theta_{p} \circ \psi_{p} ([U, h]) &= \theta_{p}(h(p)) \\ &= [U, y \mapsto h(p)_{y}] \\ &= [U, h] \\ \end{align}$ (or something very much like that) Note that the above discussion yields a morphism $\mathcal{G} \to \mathcal{G}^{+}$ which is an [[isomorphism]] on stalks, and hence an isomorphism of (pre)sheaves. Next to show the [[universal property]]. We claim that $\varphi^{+}$ gives the unique map $\mathcal{F}^{+} \to \mathcal{G}$. ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYADEAviDGl0mXPkIoyARiq1GLNpx78hDUWIB6AaknSQGbHgJEl5VfWatEHbr0HCA4hKkzL8m6Qq1A4azlpuusBexpKqMFAA5vBEoABmAE4QXEhkIDgQSLZqjprs-DA4dCDUDHQARjAMAAqyVgog6VgJfDimaZnZiLn5SABMwepOLvTpaHxY1SC1Dc2t-s4MMKm9PiAZWUgAzNQjiOPFoS7Y2TX1jS1+1s6d3Ttm+4NFp8cXU5wzcywMVuKwecieHS6PViYiAA \begin{tikzcd} \mathcal{F} \arrow[d, "\theta"'] \arrow[r, "\varphi"] & \mathcal{G} \\ \mathcal{F}^+ \arrow[r, "\varphi^+"'] & \mathcal{G}^+ \arrow[u, "\sim"'] \end{tikzcd} \end{document} ``` It is universal with respect to the property of factoring (pre)sheaf morphisms because, as we have just shown, it is determined on the level of stalks — $\varphi^{+}_{p}=\varphi_{p}$ for all $p$ — and for morphisms of *sheaves* (not merely presheaves) the behavior on stalks uniquely determines the morphism. ---- #### 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 > ```