# Wednesday, January 26

:::{.remark}
Recall last time: presheaf vs sheaf properties, images, kernel, cokernel.
We can state the uniqueness sheaf axiom as the following: if $s\in F(U)$ with $\ro{s}{U_i} = 0$ for $\mcu \covers U$, then $s = 0$ in $F(U)$.

- $\mcf \da (\im \phi)^-$ satisfies uniqueness.
- $\mcg \da (\coker \phi)^-$ satisfies existence.
- $\mcf$ fails existence $\iff \mcg$ fails uniqueness
- $\mcf$ fails uniqueness iff $\mcg$ fails existence.

The presheaf image and cokernel can sometimes fail to be a sheaf: use $\Hol(X) \mapsvia{\exp} \Hol(X)\units$.
The kernel presheaf $(\ker \phi)^-$ is already a sheaf.
:::

:::{.exercise title="?"}
Show the following:

- A sheaf $\mcf$ is the zero sheaf iff $\mcf_p =0$ for all $p$.
- $\ker(\phi)_p = \ker(\phi_p)$, which is $\ker(\mcf_p \mapsvia{\phi_p} \mcg_p)$ the kernel of the induced map.
- $\coker(\phi)_p = \coker(\phi_p) \da \coker(\mcf_p \mapsvia{\phi_p} \mcg_p)$.
- $\phi: \mcf\to \mcg$ is injective iff $\phi_p: \mcf_p \to \mcg_p$ is injective for all $p$.
- $\phi: \mcf\to \mcg$ is surjective iff $\phi_p: \mcf_p \to \mcg_p$ is surjective for all $p$.

:::

:::{.remark}
Hints:

- Suppose $s\neq 0$ in $F(U)$, does there exist a $p$ with $s_p = 0$?
- Use that $s_p \in (\ker \phi)_p$ can be regarded as $s\in \ker(F(V) \to G(V))$ mod equivalence.

:::

:::{.definition title="?"}
If there exists an injective morphism $\phi:\mcf\to \mcg$, we regard $\mcf \leq \mcg$ as a **subsheaf** and define the **quotient sheaf** $\mcf/\mcg \da \coker(\mcf \mapsvia{\phi} \mcg)$.
:::

:::{.exercise title="?"}
Show by example that $(\mcf/\mcg)^-$ need not be a sheaf.
:::

:::{.remark}
Note that for $\phi: \mcf\to \mcg$, the image $\im \phi$ is a secondary notion in additive categories, and can instead be defined as either

- $\coker(\ker \phi \to \mcf)$
- $\ker(\mcg \to \coker \phi)$

These need not coincide in general.
:::

:::{.remark}
Defining the direct image: easier using the sheaf axioms.
For $f\in \Top(X, Y)$, define $f_*: \Sh(X) \to \Sh(Y)$ by
\[
f_* \mcf(U) \da \mcf(f\inv(U)), \qquad \in \Sh(Y) \text{ for } \mcf \in \Sh(X)
.\]
:::

:::{.remark}
For the preimage: easier to use the espace étalé.
As a special case, consider $\iota: S \injects Y$ where $S$ is a subspace of $Y$ (with the subspace topology).
Then for $\mcf\in \Sh(Y)$, we can now define sections not only on open subsets $U$ but arbitrary subsets $S$ as
\[
\mcf(S) \da (\iota\inv \mcf)(S)
.\]
:::