# Monday, January 24 :::{.remark} Recall the definitions of presheaves and sheaves, and sheafification as an adjoint to $\Forget: \Sh(X)\to \Presh(X)$. For $F\in \Presh(X)$ we concretely construct its sheafification $F^+$ using the sheaf space $\pi: Y\da \Disjoint_{x\in X} F_x \to X$. What are the sections of $\pi$? For a basic open $U \subseteq X \ni x$, the fiber is $\pi\inv(x) = F_x \da \colim_{V\ni x} F(V)$, which receives a map $\Res_{U, x}: F(U) \to F_x$. Writing $s\in F(U)$, define $s_x \da \Res_{U, x}(s)$, and set $W_{s, U} \da \ts{s_x \st x\in U}$ to be $\pi\inv(U)$. Then define $F^+$ to be the continuous sections of $Y \mapsvia{\pi} X$. What does such a section look like? For $t:U\to \pi\inv(U)$ and $x\in U$, the vertical fiber is $F_x$. For a basic open $V\ni X$ in the base, there is a basic open $W_{s, V}$ in $Y$ for $s\in F(V)$: ![](figures/2022-01-24_10-39-51.png) There are maps $s_{ij}: U_{ij}\to \pi\inv(U_{ij})$, but note that $\Res(U_i, U_{ij}) s_i$ does not necessarily equal $\Res(U_j,U_{ij}) s_j$ in $F(U_{ij})$ -- instead, there is an open cover $U_{ij} = \Union V_{\alpha}$ with $\Res(U_i, V_\alpha) s_i = \Res(U_j, V_\alpha) s_j$ for each $\alpha$. > Todo ::: :::{.remark} For $f\in \Top(X, Y)$ we have the following constructions: - The direct image $f_*: \Sh(X) \to \Sh(Y)$, which is easy with the sheaf definition, and - The inverse image $f\inv: \Sh(Y)\to \Sh(X)$ which is easier with the sheaf space definition. Recall the definition of a morphism of sheaves as a natural transformation. For sheaves of abelian groups and $\phi: F\to G$ a morphism of sheaves, there are notions of $\ker \phi, \coker \phi, \im \phi$, and extension of a sheaf by zero. To show these exist as presheaves, one only has to show existence of the following blue morphisms of abelian groups: \begin{tikzcd} &&&&& {} \\ 0 && {\ker \phi_U} && {F(U)} && {G(U)} && {\coker \phi_U} && 0 \\ &&&&& {\im \phi_U} \\ 0 && {\ker \phi_V} && {F(V)} && {G(V)} && {\coker \phi_V} && 0 \\ &&&&& {\im \phi_V} \arrow[from=2-5, to=2-7] \arrow[from=2-5, to=4-5] \arrow[from=4-5, to=4-7] \arrow[from=2-7, to=4-7] \arrow[hook, from=2-3, to=2-5] \arrow[two heads, from=2-7, to=2-9] \arrow[hook, from=4-3, to=4-5] \arrow[two heads, from=4-7, to=4-9] \arrow[two heads, from=2-5, to=3-6] \arrow[hook, from=3-6, to=2-7] \arrow[two heads, from=4-5, to=5-6] \arrow[hook, from=5-6, to=4-7] \arrow[from=2-1, to=2-3] \arrow[from=4-1, to=4-3] \arrow[from=2-9, to=2-11] \arrow[from=4-9, to=4-11] \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-3, to=4-3] \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=3-6, to=5-6] \arrow["\exists"{description, pos=0.3}, color={rgb,255:red,92;green,92;blue,214}, dashed, from=2-9, to=4-9] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Write $(\coker \phi)^-$ and $(\im \phi)^-$ for these presheaves. ::: :::{.proposition title="?"} $\ker \phi$ is a sheaf. ::: :::{.proof title="?"} Axiom 1: use that $F$ is a sheaf and $\ker \phi_U \subseteq F(U)$ can be viewed as an inclusion. Axiom 2: write $s_i\in \ker \qty{F(U_i) \mapsvia{\phi_{U_i}} F(U_j) }$, then there exists a unique $s\in F(U)$. Then check that $s\in \ker\qty{F(U) \to G(U)}$ by noting that if $s\mapsto t$ then $\ro{t}{U_i} = 0$ for all $i$, making $t\equiv 0$ by the sheaf property of $G$. ::: :::{.definition title="Cokernel and image sheaves"} Define \[ \coker \phi &\da ( (\coker \phi)^-)^+ \\ \im \phi &\da ( (\im \phi)^-)^+ .\] ::: :::{.example title="of necessity of sheafifying"} Take $X = \CC$ and consider $\exp: \Hol(X) \to G$ the sheaf of nowhere zero holomorphic functions. Then on $U_i \in \CC\smz$, take $z\in G$. Then $z = \exp(f_i)$ in each $U_i$ with $f_i \in \Hol(X)$, so $f_i = \log(z)$ locally and $z = \exp(\log z)$, but there is no global $f\in \Hol(X)$ with $\exp(f) = z$. So $z\in \ker \phi_i(\Hol(U_i) \to G(U_i))$ but $z\not \in \ker \exp$. For the same reason, $z = 0$ in $\coker \phi_i$ since it's locally in the image. but $z\neq 0 \in \coker \exp$ since it's not globally in the image. :::