# Wednesday, January 19

:::{.example title="of sheaves"}
Some examples of sheaves:

- For $X \subseteq \CC$ open, consider $\pr_1: X\times \CC\to X$ and consider the space of continuous sections $\OO_X^\cts(U) \da \Hom_\Top(U, U\times \CC)$.

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- Analytic functions $\OO_X^\an$

- $\OO_X^\cts$ where $\CC$ is given the discrete topology instead of the Euclidean topology.
  The opens in $U\times \CC$ are of the form $U\times V$ for $V \subseteq \CC$ any set at all:

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- Constant sheaves $\ul{\CC}(U)$ defined as the locally constant continuous $\CC\dash$valued functions on $U$.


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:::{.remark}
Recall the sheaf properties:

- $U\to F(U)$ and $\iota_{U, V} \mapsto \Res{F(V),F(U)}$.
- $\initial \mapsto F(\initial) = \terminal$.
- Sheaf conditions:
  - Unique gluing: $\mcu \covers X$ with $\Res_{X, U_i} s = \Res_{X, U_i} t \implies s=t\in F(X)$
  - Existence of gluing: $\ts{s_i\in F(U_i)}$ with $\Res_{U_i, U_{ij}} s_i = \Res_{U_j, U_{ij}} s_j$ implies $\exists ! s\in F(X)$ with $\Res_{X, U_i} s = s_i$ for all $i$.

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:::{.example title="?"}
Recall that a **basis** of a topology is a collection $B_i$ where every $U \in \tau_X$ can be written as $\Union_{i\in I} B_i$ for some index set $I = I(X)$.
Some examples:

- For $X\in \Alg\Var\slice k$, the distinguished opens $D(f) = \ts{f\neq 0}$ and $Z(f) = \ts{f=0}$.
- For $X =\spec R \in \Aff\Sch\slice k$, take $D(f) = \ts{\mfp \in \spec R \st f\neq 0 \in R/\mfp} = \ts{\mfp \in \spec R \st f\not\in \mfp}$
  - Note that $\OO_{\spec R}(D(f)) = R\localize{f}$.

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:::{.exercise title="?"}
Formulate the sheaf condition with a basis instead of arbitrary opens.

> Hint: keep all of the same conditions, but since intersections may not be basic opens, write $B_\alpha \intersect B_\beta = \union_k B_k$.

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:::{.remark}
Some upcoming standard notions:

- Stalks $F_x$
- Sheafification $F \mapsto F^+$

A less standard topic:

- The espace etale or "flat space" of $F$.

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:::{.definition title="Stalks"}
Recall that
\[
F_x = \colim_{U\ni x} F(U) = \ts{(U, s\in F(U))} / \sim && (U, s) \sim (V, t) \iff \exists W \contains U, V,\, \Res_{U, W}s = \Res_{V, W} t
.\]

Example: $\OO_{X, p}^\an = \ts{f(z) \da \sum c_k (z-p)^k \st f\text{ has a positive radius of convergence} }$.
Note that $\OO_{X, p}^\cts$ doesn't have such a nice description, since continuous functions can be distinct while agreeing on a small neighborhood.
Similarly, $\ul{\CC}_p = \CC$, since locally constant is actually constant on a small enough neighborhood.
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:::{.remark}
Recall that morphisms of (pre)sheaves are natural transformations of functors.
There is a forgetful functor $\Forget: \Sh(X) \to \Presh(X)$, which has a left adjoint $(\wait)^+: \Presh(X) \to \Sh(X)$.
There is a description of $F^+(U)$ as collections of local compatible sections of $F$ modulo equivalence -- compatibility here means that if $\mcu \covers X$, then writing $U_{ij} = \union V_k$ we have $\Res_{X, V_k}s_i = \Res_{X, V_k}s_j$ for all $i, j$.
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